Time-delay feedback control of nonlinear oscillators Viktor Urumov, PMF, Skopje 30 juni 2010, Niš

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PMF - Skopje Primeri nelinearnih oscilatora Fazni prelaz kod modela Kuramoto Nestabilne fiksne ta~ke i wihova

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Current programme - part 2(semesters 5-8, physics teachers branch)

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Nonlinear oscillator

sin sin

sin sin

x b x x A t

x y

y x by A t

The Lorenz system

Chaotic attractor of theunperturbed system (F(t)=0)

E. N. Lorenz, “Deterministic nonperiodic flow,”J. Atmos. Sci. 20 (1963) 130.

Fixed points: C0 (0,0,0)C± (±8.485, ±8.485,27)

Eigenvalues:(C0) = {-22.83, 11.83, -2.67}(C±) = {-13.85, 0.09+10.19i, 0.09-10.19i}

van der Pol oscillator

2

2

( 1) 0

(1 )

x x x x

x y

y x y x

Limit cycle

- 2 - 1 0 1 2displacement xHtL- 2

- 1

0

1

2yticolev

xvHtL

b=0.5

Rössler oscillator with harmonic forcing

sin( )

( )

extx y z E t

y x ay

z f z x c

Historical example from Biology

The glowworms ... Represent another shew, which settle on some Trees, like a fiery cloud, with this surprising circumstance, that a whole swarm of these insects, having taken possession of one Tree, and spread themselves over its branches, sometimes hide their Light all at once, and a moment after make it appear again with the utmost regularity and exactness …

Engelbert Kaempfer description from his trip in Siam (1680)

Further examples

• The Moon facing the Earth; Gallilean satelites; Kirkwood gaps

• Cyclotron and other accelerators

• Stroboscope; Fax-machine

• Biological clocks; Jet lag

• Pacemakers

• Farmacological actions of steroids

Further examples 2

• Cardiorespiratory system

• Entrainment of cardial and locomotor rhythms

• Cardiovascular coupling during anesthesia

• Synchronization between parts of the brain

• Magnetoencephalographic fields and muscle activity of Parkinsonian patients

Modelot na Kuramoto

Parametar na poredok i sinhronizacija

1r 0r

Re{enie na modelot na Kuramoto (1975)

2/

2/

2 )sin(cos

dKrgKrr

re{enija

0r i 0r

)0(/2 gK c

KKrg c /1/

)(22

INTRODUCTION - THE PYRAGAS CONTROL METHOD

- Time-delayed feedback control (TDFC)- Time-delayed autosynchronization (TDAS)

K. Pyragas, Phys. Lett. A 170 (1992) 421

Applications

Delays are natural in many systems

• Coupled oscillators

• Electronic circuits

• Lasers, electrochemistry

• Networks of oscillators

• Brain and cardiac dynamics

Pyragas control force:

VARIABLE DELAY FEEDBACK CONTROL OF USS

VDFC force:

- saw tooth wave:

- sine wave:

- random wave:

- noninvasive for USS and periodic orbits

- piezoelements, noise

A. Gjurchinovski and V. Urumov – Europhys. Lett. 84, 40013 (2008)

VARIABLE DELAY FEEDBACK CONTROL OF USS

THE MECHANISM OF VDFC

DELAY MODULATIONS



2D UNSTABLE FOCUS WITH A DIAGONAL COUPLING

original system : comparison system :

– sufficiently large

Characteristic equation of the comparison system (2D focus):


TDAS VDFC VDFC VDFC


The effect of including variable delay into TDAS for small

• condition for the roots lying on the imaginary axis for =0 to move to the left half-plane as increases from zero

CONCLUSION: the stability domain will expand in all directions within the half-space K>K0, as soon as is increased from zero, independent of the precise way in which the delay is varied


2D unstable focus withand

Pyragas

Increase of the stability domain for small

(brown)

(green)

(yellow)


diagrams for a saw tooth wave modulation (T0=1)



Stability analysis for the Lorenz system (saw tooth wave)

C+ (8.485, 8.485,27)

C0 (0,0,0)

C- (-8.485, -8.485,27)

10, r 28, b 8/3



The Rössler system (sawtooth wave)

O.E. Rössler, Phys. Lett. A 57, 397 (1976).

Fixed points: C1 (0.007,-0.035,0.035)C2 (5.693, -28.465,28.465)

Eigenvalues:(C1) = {-5.687,0.097+0.995i,0.097-0.995i}(C2) = {0.192,-0.00000459+5.428i, -0.00000459-5.428i}

0 0.5

1 2

STABILIZATION OF UPO BY VDFC

SQUARE WAVE MODULATION

• periodic change of the delay, e. g. between T0 and 2T0, K fixed (VDFC)

• periodic change of the delay, K varied (VDFC + SCHUSTER, STEMMLER)

T(t)

T0

2T0

t

- half-period of the wave (optimal choice: T0)

T(t)

T0

2T0

t

K(t)

K/2

K

t

+


•PYRAGASRössler T0=5.88

•VDFC (square wave)

•SCHUSTER, STEMMLER

•VDFC (square wave) + SCH-ST

F(t)=K [y(t-T0)-y(t)]

F(t)=K [y(t-T(t))-y(t)]

F(t)=K(t) [y(t-T0)-y(t)]

F(t)=K(t) [y(t-T(t))-y(t)]


Rössler T0=11.75 Rössler T0=17.5


•VDFC + SCHUSTER

K periodically varied between K and K/4 (Rössler, T0=17.5)

•Restricted VDFC + SCHUSTER F(t)=K(t) Sin [y(t-T(t))-y(t)]


Rössler T0=5.88VDFC (square wave)

= T0

= 2T0

= T0/2

STABILITY ANALYSIS - RDDE

Retarded delay-differential equations

• GOAL: stabilization of unstable steady states by a variable-delay feedback control in a nonlinear dynamical systems described by a scalar autonomous retarded delay-differential equation (RDDE)

• MOTIVATION: extension of the delay method to infinite dimensional systems

• INTEREST: frequent occurrence of scalar RDDE in numerous physical, biological and engineering models, where the time-delays are natural manifestation of the system’s dynamics

T. Erneux, Applied Delay Differential Equations (Springer, New York, 2009)

Retarded delay-differential equationsGeneral scalar RDDE

system:

T1 ≥ 0 – constant delay time

F – arbitrary nonlinear function of the state variable x

Linearized system around the fixed point x*:

DELAY-DIFFERENTIAL EQUATIONS

Characteristic equation for the stability of steady state x* of the free-running system:

A. Gjurchinovski and V. Urumov – Phys. Rev. E 81, 016209 (2010)


Retarded delay-differential equations

Controlled RDDE system:

u(t) – Pyragas-type feedback force with a variable time delay

K – feedback gain (strength of the feedback) T2 – nominal delay value f – periodic function with zero mean – amplitude of the modulation – frequency of the modulation


Stability of the unperturbed system


Stability under variable-delay feedback control

Limitation of the VDFC for RDDE systems:

• A kind of analogue to the odd-number limitation in the case of delayed feedback control of systems described by ordinary differential equations:

W. Just et al., Phys. Rev. Lett. 78, 203(1997)H. Nakajima, Phys. Lett. A 232, 207 (1997)

• … refuted recently:

B. Fiedler et al., Phys. Rev. Lett. 98, 114101 (2007).B. Fiedler et al., Phys. Rev. E 77, 066207 (2008).


Representation of the control boundaries parametrized by = Im()

(K,T2) plane:

EXAMPLES AND SIMULATIONS

Mackey-Glass system

• A model for regeneration of blood cells in patients with leukemia

M. C. Mackey and L. Glass, Science 197, 28 (1977).

• M-G system under variable-delay feedback control:

• For the typical values a = 0.2, b = 0.1 and c = 10, the fixed points of the free-running system are:

• x1 = 0 – unstable for any T1, cannot be stabilized by VDFC• x2 = +1 – stable for T1 [0,4.7082)• x3 = -1 – stable for T1 [0,4.7082)


Mackey-Glass system (without control)

(a) T1 = 4

(b) T1 = 8

(c) T1 = 15

(d) T1 = 23


Mackey-Glass system (VDFC)

(a) = 0 (TDFC)

(b) = 0.5 (saw)

(c) = 1 (saw)

(d) = 2 (saw)

T1 = 23



(a) = 1 (sin)

(b) = 2 (sin)

(c) = 1 (sqr)

(d) = 2 (sqr)

T1 = 23



(a) = 0 (TDFC)

(b) = 2 (saw)

(c) = 2 (sin)

(d) = 2 (sqr)

K = 0.5



T1 = 23, T2 = 18, K = 2, = 2, = 5

saw

sin

sqr






Ikeda system

• Introduced to describe the dynamics of an optical bistable resonator, incorporating the round-trip time of light in an optical cavity via the time delay T1

K. Ikeda, Opt. Commun. 39, 257 (1979)K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987).

• Ikeda system under variable-delay feedback control:

• For = 4 and x0 = /4, the fixed points of the free-running system are:

• x1 = 3.05708 – stable for T1 [0, 0.82801)• x2 = 1.05136 – unstable for any T1, cannot be stabilized by

VDFC• x3 = -1.86979 – stable for T1 [0, 0.54767)


Sprott system

• The simplest one-parameter RDDE system with a sinusoidal nonlinearity

J. C. Sprott, Phys. Lett. A 366, 397 (2007)

• Sprott system under variable-delay feedback control:

• The fixed points of the free-running system are:

• x2n = 2n – unstable for any T1, cannot be stabilized by VDFC

• x2n+1 = (2n+1) – stable for T1 [0, /2)

FRACTIONAL DIFFERENTIAL EQUATIONS

Fractional Rössler system

Caputo fractional-order derivative:


Fractional Rössler system


Fractional Rössler system - stability diagrams

Time-delayed feedback control

Variable delay feedback control

(sine-wave, =1, =10)








Desynchronisation in systems of coupled oscillators

Hindmarsh - Rose oscillators

Mean field

Global coupling

Delayed feedback control

M. Rosenblum and A. Pikovsky, Phys. Rev. Lett. 92, 114102; Phys. Rev. E 70, 041904 (2004)


Feedback switched on at t=5000

System of 1000 H-R oscillators

=const=72.5

K=0.0036

Kmf=0.08




(sine-wave, =40, =10)

Suppression coefficient

X – Mean field in the absence of feedback

Xf – Mean field in the presence of feedback

T=145 – average period of the mean field in the absence of feedback

CONCLUSIONS AND FUTURE PROSPECTS

• Enlarged domain for stabilization of unstable steady states in systems of ordinary/delay/fractional differential equations in comparison with Pyragas method and its generalizations

• Agreement between theory and simulations for large frequencies in the delay variability

• The enlargement of the control domain may undergo a complex rearrangement depending on the type of the delay modulation

• Extended area of stabilization of periodic orbits by noninvasive variable-delay feedback control

• Variable delay feedback control provides increased robustness in achieving desynchronization in wider domain of parameter space in system of coupled Hindmarsh-Rose oscillators interacting through their mean field

• The influence of variable-delay feedback in other systems (neutral DDE, PDE, networks, …)

• Experimental verification

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Time-delay feedback control of nonlinear oscillators Viktor Urumov, PMF, Skopje 30 juni 2010, Niš