Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

Methods for control of deterministicMethods for control of deterministic chaoschaos

bull a continuous external perturbation R Lima et al Phys Rev A 41 (1990) 726

bull a time-discrete conditioned perturbation (OGY) E Ott et al Phys Rev Lett 64 (1990) 1196

bull delayed feedback loop K Pyragas Phys Lett A 170 (1992) 421

bull a continuous external perturbation R Lima et al Phys Rev A 41 (1990) 726

bull a time-discrete conditioned perturbation (OGY) E Ott et al Phys Rev Lett 64 (1990) 1196

bull delayed feedback loop K Pyragas Phys Lett A 170 (1992) 421

Three main methodsThree main methods

Delayed feedback control of chaosDelayed feedback control of chaos

)()( txtxK minusτminus

ττ==ττ=T=T

How does it workHow does it work

Chaos control Example IChaos control Example I

S Bielawski et al Phys Rev E 49 (1994) R971S Bielawski et al Phys Rev E 49 (1994) R971

CO2 laserCO2 laser

without controlwithout control

with controlwith control

EXPERIMENTEXPERIMENT

Electrochemical oscillatorsElectrochemical oscillators

P Paramanade et al Phys Rev E 59 (1999) 5266P Paramanade et al Phys Rev E 59 (1999) 5266

control

no control no control

Chaos control Example IIChaos control Example II

EXPERIMENTEXPERIMENT

Chaos control Example IIIChaos control Example III

NUMERICAL RESULTSNUMERICAL RESULTS

G FranceschiniG Franceschini et al Phys Rev E 60 (1999) 5426 et al Phys Rev E 60 (1999) 5426

Semiconductor structureSemiconductor structure

no control

control

Chaos control Example IVChaos control Example IV

EXPERIMENTEXPERIMENT

C Beta et al Phys Rev E 67 (2003) 046224C Beta et al Phys Rev E 67 (2003) 046224

Catalytic CO oxidation on platinumCatalytic CO oxidation on platinum

controlno control

Delayed feedback control of chaos main known factsDelayed feedback control of chaos main known facts

(more than 700 citations)(more than 700 citations)K Pyragas Phys Lett A 170K Pyragas Phys Lett A 170 421 421 (1992) (1992)

1) At fixed τ=T there is a range of K for which the orbit is stabilized2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized 3) Feedback can induce multistability and death of oscillations

ME Bleich et al Phys Lett A 210 87 (1996) ME Bleich et al Phys Lett A 210 87 (1996) W Just et al Phys Rev Lett 203 78 (1997) W Just et al Phys Rev Lett 203 78 (1997) H Nakajima Phys Lett A 232 H Nakajima Phys Lett A 232 207 (1997) 207 (1997) W Just et al Phys Lett A 254 158 (1999)W Just et al Phys Lett A 254 158 (1999)

limitations of delayed feedbacklimitations of delayed feedback

W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998)What if τ is not exactly equal to T but close to it

1) The orbit changes its shape and period 2) A method to estimate the orbit period from the given two values of τ

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

Methods for control of deterministicMethods for control of deterministic chaoschaos

bull a continuous external perturbation R Lima et al Phys Rev A 41 (1990) 726

bull a time-discrete conditioned perturbation (OGY) E Ott et al Phys Rev Lett 64 (1990) 1196

bull delayed feedback loop K Pyragas Phys Lett A 170 (1992) 421

bull a continuous external perturbation R Lima et al Phys Rev A 41 (1990) 726

bull a time-discrete conditioned perturbation (OGY) E Ott et al Phys Rev Lett 64 (1990) 1196

bull delayed feedback loop K Pyragas Phys Lett A 170 (1992) 421

Three main methodsThree main methods

Delayed feedback control of chaosDelayed feedback control of chaos

)()( txtxK minusτminus

ττ==ττ=T=T

How does it workHow does it work

Chaos control Example IChaos control Example I

S Bielawski et al Phys Rev E 49 (1994) R971S Bielawski et al Phys Rev E 49 (1994) R971

CO2 laserCO2 laser

without controlwithout control

with controlwith control

EXPERIMENTEXPERIMENT

Electrochemical oscillatorsElectrochemical oscillators

P Paramanade et al Phys Rev E 59 (1999) 5266P Paramanade et al Phys Rev E 59 (1999) 5266

control

no control no control

Chaos control Example IIChaos control Example II

EXPERIMENTEXPERIMENT

Chaos control Example IIIChaos control Example III

NUMERICAL RESULTSNUMERICAL RESULTS

G FranceschiniG Franceschini et al Phys Rev E 60 (1999) 5426 et al Phys Rev E 60 (1999) 5426

Semiconductor structureSemiconductor structure

no control

control

Chaos control Example IVChaos control Example IV

EXPERIMENTEXPERIMENT

C Beta et al Phys Rev E 67 (2003) 046224C Beta et al Phys Rev E 67 (2003) 046224

Catalytic CO oxidation on platinumCatalytic CO oxidation on platinum

controlno control

Delayed feedback control of chaos main known factsDelayed feedback control of chaos main known facts

(more than 700 citations)(more than 700 citations)K Pyragas Phys Lett A 170K Pyragas Phys Lett A 170 421 421 (1992) (1992)

1) At fixed τ=T there is a range of K for which the orbit is stabilized2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized 3) Feedback can induce multistability and death of oscillations

ME Bleich et al Phys Lett A 210 87 (1996) ME Bleich et al Phys Lett A 210 87 (1996) W Just et al Phys Rev Lett 203 78 (1997) W Just et al Phys Rev Lett 203 78 (1997) H Nakajima Phys Lett A 232 H Nakajima Phys Lett A 232 207 (1997) 207 (1997) W Just et al Phys Lett A 254 158 (1999)W Just et al Phys Lett A 254 158 (1999)

limitations of delayed feedbacklimitations of delayed feedback

W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998)What if τ is not exactly equal to T but close to it

1) The orbit changes its shape and period 2) A method to estimate the orbit period from the given two values of τ

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

Delayed feedback control of chaosDelayed feedback control of chaos

)()( txtxK minusτminus

ττ==ττ=T=T

How does it workHow does it work

Chaos control Example IChaos control Example I

S Bielawski et al Phys Rev E 49 (1994) R971S Bielawski et al Phys Rev E 49 (1994) R971

CO2 laserCO2 laser

without controlwithout control

with controlwith control

EXPERIMENTEXPERIMENT

Electrochemical oscillatorsElectrochemical oscillators

P Paramanade et al Phys Rev E 59 (1999) 5266P Paramanade et al Phys Rev E 59 (1999) 5266

control

no control no control

Chaos control Example IIChaos control Example II

EXPERIMENTEXPERIMENT

Chaos control Example IIIChaos control Example III

NUMERICAL RESULTSNUMERICAL RESULTS

G FranceschiniG Franceschini et al Phys Rev E 60 (1999) 5426 et al Phys Rev E 60 (1999) 5426

Semiconductor structureSemiconductor structure

no control

control

Chaos control Example IVChaos control Example IV

EXPERIMENTEXPERIMENT

C Beta et al Phys Rev E 67 (2003) 046224C Beta et al Phys Rev E 67 (2003) 046224

Catalytic CO oxidation on platinumCatalytic CO oxidation on platinum

controlno control

Delayed feedback control of chaos main known factsDelayed feedback control of chaos main known facts

(more than 700 citations)(more than 700 citations)K Pyragas Phys Lett A 170K Pyragas Phys Lett A 170 421 421 (1992) (1992)

1) At fixed τ=T there is a range of K for which the orbit is stabilized2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized 3) Feedback can induce multistability and death of oscillations

ME Bleich et al Phys Lett A 210 87 (1996) ME Bleich et al Phys Lett A 210 87 (1996) W Just et al Phys Rev Lett 203 78 (1997) W Just et al Phys Rev Lett 203 78 (1997) H Nakajima Phys Lett A 232 H Nakajima Phys Lett A 232 207 (1997) 207 (1997) W Just et al Phys Lett A 254 158 (1999)W Just et al Phys Lett A 254 158 (1999)

limitations of delayed feedbacklimitations of delayed feedback

W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998)What if τ is not exactly equal to T but close to it

1) The orbit changes its shape and period 2) A method to estimate the orbit period from the given two values of τ

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

Chaos control Example IChaos control Example I

S Bielawski et al Phys Rev E 49 (1994) R971S Bielawski et al Phys Rev E 49 (1994) R971

CO2 laserCO2 laser

without controlwithout control

with controlwith control

EXPERIMENTEXPERIMENT

Electrochemical oscillatorsElectrochemical oscillators

P Paramanade et al Phys Rev E 59 (1999) 5266P Paramanade et al Phys Rev E 59 (1999) 5266

control

no control no control

Chaos control Example IIChaos control Example II

EXPERIMENTEXPERIMENT

Chaos control Example IIIChaos control Example III

NUMERICAL RESULTSNUMERICAL RESULTS

G FranceschiniG Franceschini et al Phys Rev E 60 (1999) 5426 et al Phys Rev E 60 (1999) 5426

Semiconductor structureSemiconductor structure

no control

control

Chaos control Example IVChaos control Example IV

EXPERIMENTEXPERIMENT

C Beta et al Phys Rev E 67 (2003) 046224C Beta et al Phys Rev E 67 (2003) 046224

Catalytic CO oxidation on platinumCatalytic CO oxidation on platinum

controlno control

Delayed feedback control of chaos main known factsDelayed feedback control of chaos main known facts

(more than 700 citations)(more than 700 citations)K Pyragas Phys Lett A 170K Pyragas Phys Lett A 170 421 421 (1992) (1992)

1) At fixed τ=T there is a range of K for which the orbit is stabilized2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized 3) Feedback can induce multistability and death of oscillations

ME Bleich et al Phys Lett A 210 87 (1996) ME Bleich et al Phys Lett A 210 87 (1996) W Just et al Phys Rev Lett 203 78 (1997) W Just et al Phys Rev Lett 203 78 (1997) H Nakajima Phys Lett A 232 H Nakajima Phys Lett A 232 207 (1997) 207 (1997) W Just et al Phys Lett A 254 158 (1999)W Just et al Phys Lett A 254 158 (1999)

limitations of delayed feedbacklimitations of delayed feedback

W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998)What if τ is not exactly equal to T but close to it

1) The orbit changes its shape and period 2) A method to estimate the orbit period from the given two values of τ

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

Electrochemical oscillatorsElectrochemical oscillators

P Paramanade et al Phys Rev E 59 (1999) 5266P Paramanade et al Phys Rev E 59 (1999) 5266

control

no control no control

Chaos control Example IIChaos control Example II

EXPERIMENTEXPERIMENT

Chaos control Example IIIChaos control Example III

NUMERICAL RESULTSNUMERICAL RESULTS

G FranceschiniG Franceschini et al Phys Rev E 60 (1999) 5426 et al Phys Rev E 60 (1999) 5426

Semiconductor structureSemiconductor structure

no control

control

Chaos control Example IVChaos control Example IV

EXPERIMENTEXPERIMENT

C Beta et al Phys Rev E 67 (2003) 046224C Beta et al Phys Rev E 67 (2003) 046224

Catalytic CO oxidation on platinumCatalytic CO oxidation on platinum

controlno control

Delayed feedback control of chaos main known factsDelayed feedback control of chaos main known facts

(more than 700 citations)(more than 700 citations)K Pyragas Phys Lett A 170K Pyragas Phys Lett A 170 421 421 (1992) (1992)

1) At fixed τ=T there is a range of K for which the orbit is stabilized2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized 3) Feedback can induce multistability and death of oscillations

ME Bleich et al Phys Lett A 210 87 (1996) ME Bleich et al Phys Lett A 210 87 (1996) W Just et al Phys Rev Lett 203 78 (1997) W Just et al Phys Rev Lett 203 78 (1997) H Nakajima Phys Lett A 232 H Nakajima Phys Lett A 232 207 (1997) 207 (1997) W Just et al Phys Lett A 254 158 (1999)W Just et al Phys Lett A 254 158 (1999)

limitations of delayed feedbacklimitations of delayed feedback

W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998)What if τ is not exactly equal to T but close to it

1) The orbit changes its shape and period 2) A method to estimate the orbit period from the given two values of τ

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

Chaos control Example IIIChaos control Example III

NUMERICAL RESULTSNUMERICAL RESULTS

G FranceschiniG Franceschini et al Phys Rev E 60 (1999) 5426 et al Phys Rev E 60 (1999) 5426

Semiconductor structureSemiconductor structure

no control

control

Chaos control Example IVChaos control Example IV

EXPERIMENTEXPERIMENT

C Beta et al Phys Rev E 67 (2003) 046224C Beta et al Phys Rev E 67 (2003) 046224

Catalytic CO oxidation on platinumCatalytic CO oxidation on platinum

controlno control

Delayed feedback control of chaos main known factsDelayed feedback control of chaos main known facts

(more than 700 citations)(more than 700 citations)K Pyragas Phys Lett A 170K Pyragas Phys Lett A 170 421 421 (1992) (1992)

1) At fixed τ=T there is a range of K for which the orbit is stabilized2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized 3) Feedback can induce multistability and death of oscillations

ME Bleich et al Phys Lett A 210 87 (1996) ME Bleich et al Phys Lett A 210 87 (1996) W Just et al Phys Rev Lett 203 78 (1997) W Just et al Phys Rev Lett 203 78 (1997) H Nakajima Phys Lett A 232 H Nakajima Phys Lett A 232 207 (1997) 207 (1997) W Just et al Phys Lett A 254 158 (1999)W Just et al Phys Lett A 254 158 (1999)

limitations of delayed feedbacklimitations of delayed feedback

W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998)What if τ is not exactly equal to T but close to it

1) The orbit changes its shape and period 2) A method to estimate the orbit period from the given two values of τ

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

Chaos control Example IVChaos control Example IV

EXPERIMENTEXPERIMENT

C Beta et al Phys Rev E 67 (2003) 046224C Beta et al Phys Rev E 67 (2003) 046224

Catalytic CO oxidation on platinumCatalytic CO oxidation on platinum

controlno control

Delayed feedback control of chaos main known factsDelayed feedback control of chaos main known facts

(more than 700 citations)(more than 700 citations)K Pyragas Phys Lett A 170K Pyragas Phys Lett A 170 421 421 (1992) (1992)

1) At fixed τ=T there is a range of K for which the orbit is stabilized2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized 3) Feedback can induce multistability and death of oscillations

ME Bleich et al Phys Lett A 210 87 (1996) ME Bleich et al Phys Lett A 210 87 (1996) W Just et al Phys Rev Lett 203 78 (1997) W Just et al Phys Rev Lett 203 78 (1997) H Nakajima Phys Lett A 232 H Nakajima Phys Lett A 232 207 (1997) 207 (1997) W Just et al Phys Lett A 254 158 (1999)W Just et al Phys Lett A 254 158 (1999)

limitations of delayed feedbacklimitations of delayed feedback

W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998)What if τ is not exactly equal to T but close to it

1) The orbit changes its shape and period 2) A method to estimate the orbit period from the given two values of τ

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

Delayed feedback control of chaos main known factsDelayed feedback control of chaos main known facts

(more than 700 citations)(more than 700 citations)K Pyragas Phys Lett A 170K Pyragas Phys Lett A 170 421 421 (1992) (1992)

1) At fixed τ=T there is a range of K for which the orbit is stabilized2) Orbits with an odd number of Floquet multipliers greater than 1 cannot be stabilized 3) Feedback can induce multistability and death of oscillations

ME Bleich et al Phys Lett A 210 87 (1996) ME Bleich et al Phys Lett A 210 87 (1996) W Just et al Phys Rev Lett 203 78 (1997) W Just et al Phys Rev Lett 203 78 (1997) H Nakajima Phys Lett A 232 H Nakajima Phys Lett A 232 207 (1997) 207 (1997) W Just et al Phys Lett A 254 158 (1999)W Just et al Phys Lett A 254 158 (1999)

limitations of delayed feedbacklimitations of delayed feedback

W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998)What if τ is not exactly equal to T but close to it

1) The orbit changes its shape and period 2) A method to estimate the orbit period from the given two values of τ

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

General bifurcation picture in the General bifurcation picture in the plane plane

of parameters of delayed feedbackof parameters of delayed feedbackK - K - ττ

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

)( mxzbz

ayxy

zyx

minus+=+=minusminus=

amp

amp

amp

System for the studySystem for the study

RRoumloumlssler systemssler system

a=02 b=02 m=65a=02 b=02 m=65

period-one limit cycleperiod-one limit cycleperiod-two limit cycleperiod-two limit cycle

Unstable orbitsUnstable orbits

euro

+K(x(t minus τ) minus x)

Bifurcation diagram (main result)Bifurcation diagram (main result)

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Multistability and hysteretic transitions Multistability and hysteretic transitions

1D bifurcation diagrams1D bifurcation diagrams

Multistability and hysteretic transitions Multistability and hysteretic transitions

Control forceControl force ττ increasesincreases

ττ decreasesdecreases

General structure of bifurcation diagramGeneral structure of bifurcation diagram

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Hopf bifurcation

stable fixed point

Multi-leaf structureMulti-leaf structureSize of period-1 orbit

Orbit is stable

Orbit is unstable

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 0Bifurcation diagram Leaf 0

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 1Bifurcation diagram Leaf 1

Bifurcation diagram Leaf 2Bifurcation diagram Leaf 2Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Multistability and hysteretic transitions Multistability and hysteretic transitions

1D bifurcation diagrams1D bifurcation diagrams

Multistability and hysteretic transitions Multistability and hysteretic transitions

Control forceControl force ττ increasesincreases

ττ decreasesdecreases

General structure of bifurcation diagramGeneral structure of bifurcation diagram

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Hopf bifurcation

stable fixed point

Multi-leaf structureMulti-leaf structureSize of period-1 orbit

Orbit is stable

Orbit is unstable

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 0Bifurcation diagram Leaf 0

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 1Bifurcation diagram Leaf 1

Bifurcation diagram Leaf 2Bifurcation diagram Leaf 2Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Multistability and hysteretic transitions Multistability and hysteretic transitions

Control forceControl force ττ increasesincreases

ττ decreasesdecreases

General structure of bifurcation diagramGeneral structure of bifurcation diagram

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Hopf bifurcation

stable fixed point

Multi-leaf structureMulti-leaf structureSize of period-1 orbit

Orbit is stable

Orbit is unstable

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 0Bifurcation diagram Leaf 0

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 1Bifurcation diagram Leaf 1

Bifurcation diagram Leaf 2Bifurcation diagram Leaf 2Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

General structure of bifurcation diagramGeneral structure of bifurcation diagram

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Hopf bifurcation

stable fixed point

Multi-leaf structureMulti-leaf structureSize of period-1 orbit

Orbit is stable

Orbit is unstable

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 0Bifurcation diagram Leaf 0

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 1Bifurcation diagram Leaf 1

Bifurcation diagram Leaf 2Bifurcation diagram Leaf 2Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Hopf bifurcation

stable fixed point

Multi-leaf structureMulti-leaf structureSize of period-1 orbit

Orbit is stable

Orbit is unstable

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 0Bifurcation diagram Leaf 0

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 1Bifurcation diagram Leaf 1

Bifurcation diagram Leaf 2Bifurcation diagram Leaf 2Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 0Bifurcation diagram Leaf 0

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 1Bifurcation diagram Leaf 1

Bifurcation diagram Leaf 2Bifurcation diagram Leaf 2Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 1Bifurcation diagram Leaf 1

Bifurcation diagram Leaf 2Bifurcation diagram Leaf 2Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Bifurcation diagram Leaf 2Bifurcation diagram Leaf 2Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Leaf 2 transition to chaosLeaf 2 transition to chaos

ττ=135=135

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Bifurcation diagram Leaf 3Bifurcation diagram Leaf 3

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Bifurcation diagram Leaf 4Bifurcation diagram Leaf 4

Hopf bifurcation

period-doubling bifurcation

crisis of attractors

subcritical Neimark-Sacker bifurcation

supercritical Neimark-Sacker bifurcation

stable fixed point chaos

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (I)I)In In W Just et al Phys Rev Lett 8 562 (1998)W Just et al Phys Rev Lett 8 562 (1998) it is shownit is shown

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

ΘΘ((KKττ)) ndash period of the resulting cycle in the system under controlndash period of the resulting cycle in the system under control

TT ndash period of the unperturbed orbitndash period of the unperturbed orbit κ κ ndash ndash a parameter characterizing the effect of the control force on the system a parameter characterizing the effect of the control force on the system

dynamics for thedynamics for the particular orbit Does not depend on KK or or ττ - - derivative of derivative of ΘΘ((KKττ)) over over ττ

Facts

11 ΘΘ(KT)=T(KT)=T22 Control force Control force F(t)=x(t-F(t)=x(t-ττ)-x(t)=)-x(t)=00 not only for not only for ττ=T=T but alsobut also for for

any any ττ=nT=nT nn=12hellip=12hellip

33 ThusThus Θ Θ(KnT)=T(KnT)=T

euro

partτΘ

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (II)II)

euro

partτΘ τ=nT =K

nK minus κ n = 012

euro

κpartτΘ τ=T +TKpartτ(τ Θ) τ=T = 0

210

2101

2101

210

2

2

==Θpartminus+Θpartκ

==Θpartminus+Θpartκ

==ΘpartΘτ

minusΘ

+Θpartκ

==⎟⎠

⎞⎜⎝

⎛Θτ

part+Θpartκ

=τ

=τ

=τ=τ=τ

=τ

τ=ττ

τ=ττ

τ=ττ

τ=ττ

nnKK

nTnT

TKT

TK

nTKTK

nTK

nT

nT

nTnTnT

nT

nT

nT

nT

nT

Substitute ττ=nT=nT into

Take into account the facts from the previous slide

And obtain an expressionfor the derivative of Θ with respect to τ

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Expand Expand ΘΘ into a Taylor series around into a Taylor series around ττ=nT=nT

Period of period-1 periodic orbit for ldquoanyrdquo Period of period-1 periodic orbit for ldquoanyrdquo ττ ( (III)III)

euro

Θ(K τ) = T + partτΘ τ= nT (τ minus nT) + Ο((τ minus nT)2)

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT) n = 012

euro

partτΘ τ=nT =K

nK minus κ n = 012Substitute this into ()

()

and obtain an approximate expression for the and obtain an approximate expression for the period-1 orbit of the system under control period-1 orbit of the system under control

for the case of for the case of ττ close to close to nTnT

()

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Period of period-m periodic orbit for ldquoanyrdquo Period of period-m periodic orbit for ldquoanyrdquo τ τ

21210)(

)( ==minusτκminus

+asympτΘ mnnmTmnK

KmTK

Thus for the cycle of the period Thus for the cycle of the period mTmT m=12hellipm=12hellip

Facts

1 It is not likely that a period-m orbit has a period exactly mT

2 However it is likely that the period of period-m orbit is close to mT

33 ΘΘ(KmT)=T(KmT)=T

44 For a different orbit For a different orbit κκ will be different will be different

55 Observation for a period-Observation for a period-mm orbit orbit κκmm becomes becomes κκmm

66 Eq () should hold for period-m orbits approximatelyEq () should hold for period-m orbits approximately

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

K=013K=013

K=08K=08

K=20K=20

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-1 orbits numerics vs analytics of period-1 orbits numerics vs analytics

euro

Θ(K τ) asymp T +K

nK minus κ(τ minus nT)

n = 01234

n - the number of the leaf of the bifurcation diagram

Numerical fit for κ is 035

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

K=K=0808

by formulaby formula

numericallynumerically

Period Period ΘΘ of period-2 and period-4 orbits of period-2 and period-4 orbits numerics vs analytics numerics vs analytics

m=m=22 n= n=012012period-2 orbitperiod-2 orbit

m=m=44 n= n=0101period-4 orbitperiod-4 orbit

)(

)(

nmTmnK

KmT

K

minusτκminus

+asymp

asympτΘ

n - the number of the leaf of the bifurcation diagramm ndash the integer ldquoperiodrdquo of the periodic orbit

n=0

n=1 n=2

n=0n=1K=K=2020

Numerical fit for κ is 035

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Limit cycles classification Limit cycles classification

nn

euro

Θ(K τ) asymp mT +K

nK minus κ m(τ minus nmT) n = 012m =12

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Main conclusions Main conclusions

The bifurcation diagram in The bifurcation diagram in ((ττKK)) parameter space has essentially parameter space has essentially multi-leaf structuremulti-leaf structure

bull The lager the period of the orbit the smaller the domain of its The lager the period of the orbit the smaller the domain of its stabilization isstabilization is

bull The same limit cycle can have several domains of stabilityThe same limit cycle can have several domains of stability

bull Increase of both Increase of both ττ and and KK leads to severe multistability leads to severe multistability

AG Balanov NB Janson E Scholl ldquoDelayed feedback control of chaos Bifurcationanalysisrdquo Phys Rev E 71 016222 (2005)

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30

Delayed feedback control of chaos Delayed feedback control of chaos bifurcation analysisbifurcation analysis

N Janson (Loughborough)N Janson (Loughborough)

Collaborators A Balanov and E SchCollaborators A Balanov and E Schoumlll (TUB)oumlll (TUB)

Technische UniversitTechnische Universitaumlt aumlt BerlinBerlin

Loughborough UniversityLoughborough University

• PowerPoint Presentation
• Slide 2
• Slide 3
• Slide 4
• Slide 5
• Slide 6
• Slide 7
• Slide 8
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Slide 20
• Slide 21
• Slide 22
• Period of period-1 periodic orbit for ldquoanyrdquo t (II)
• Slide 24
• Period of period-m periodic orbit for ldquoanyrdquo t
• Slide 26
• Slide 27
• Slide 28
• Slide 29
• Slide 30