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Phase response curves for systems with time delay
Viktor Novičenko and Kestutis Pyragas
ENOC 2011
CENTERFOR PHYSICAL SCIENCES
AND TECHNOLOGY
Semiconductor Physics Institute, Vilnius, Lithuania
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Outline
Introduction
Phase reduction of ODE systems
Phase reduction of time-delay systems
Example: Mackey-Glass system
Phase reduction of chaotic systems subjected to a DFC
Example: Rossler system stabilized by a DFC
Conclusions
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Introduction
Phase reduction method is an efficient tool to analyzeweakly perturbed limit cycle oscillations
Most investigations in the field of phase reduction are devotedto the systems described by ODEs
The aim of this investigation is to extend the phasereduction method to time delay systems
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
A dynamical system with a stable limit cycle
For each state on the limit cycle and near the limit cycleis assigned a scalar variable (PHASE)
The phase dynamics of the free system satisfies:
1=ϕ&
Let’s apply an external perturbation to the system.The aim of phase reduction method is to find a dynamical equation the phase of perturbed system:
?=ϕ&
Phase reduction
Phase dynamics: ,here is periodic vectorvalued function - the phase response curve (PRC)
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Phase reduction of ODE systems
AB ϕϕ =
Isochron – a set of states withthe same phase
)()( tyGy εφ+=&Perturbed system:
)()(1 tzT φϕεϕ +=& ( )ϕz
PRC is the periodic solution of an adjoint equation: ( )[ ] zyDGzT
c−=&
With initial condition: 1)0()0( =c
Tyz &
Malkin’s approach:Malkin, I.G.: Some Problems in Nonlinear Oscillation Theory.Gostexizdat,Moscow (1956)
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Phase reduction of time-delay system
( ) )()(),( ttxtxFx εψτ +−=&Perturbed system:
Approximation via a delay line:
( )
)(),0(,),(),(
)(),(),(
txts
ts
t
ts
tttxFx
=∂
∂−=
∂
∂
+=
ξξξ
εψτξ&
Discretization of the space variable : Ni ,..,0=,/ Nisi τ=
)()(0
txtx =Denote and ),()( tstx ii ξ=
We get a final-dimensionalsystem of ODEs:
( )[ ]
[ ] τ
τ
εψ
/)()(
/)()(
)()(),(
1
101
00
Ntxtxx
Ntxtxx
ttxtxFx
NNN
N
−=
−=
+=
−&
K
&
&
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Phase reduction of time-delay system: results
)()(1 tzT ψϕεϕ +=&Phase dynamic:
The adjoint equation for PRC: )()()()( ττ ++−−= tztBtztAzTT
&
here ( )( ))(),()(
)(),()(
2
1
τ
τ
−=
−=
txtxFDtB
txtxFDtA
cc
cc
The initial condition:
1)()()()0()0(
0
=++++ ∫−τ
ϑϑτϑτϑτ dxBzxz c
T
c
T&&
An unstable difference-differentialequation of advanced type(backwards integration)
The phase reduced equations for time delay systems havebeen alternatively derive directly from DDE system withoutappealing to the known theoretical results from ODEs
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Example: Mackey-Glass equation
)()(1
)(tx
tx
tax
dt
dxb
−−+
−=
τ
τUnperturbed equation:
+
+=+
)(
)()(
ϑϕ
εϕϑϕχ
c
c
x
x 0=ϑ
)0,[ τϑ −∈
Two different initial conditions: first on the limit cycle and secondperturbed by from the first
for
for
Phase response curve:
εϕ /)( tz ∆=
510
7.0
10
2
−=
=
=
=
ε
τ
b
a
ε
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Example: Mackey-Glass equation
)()()(1
)(ttx
tx
tax
dt
dxb
εψτ
τ+−
−+
−=
A perturbation with periodic external signal:
[ ]
=)2sin(
)2sin()(
tsign
tt
πυ
πυψhere
Frequency mismatch:
T/1−=∆ υυ
Arnold’s tongues:
7.0
10
2
=
=
=
τ
b
a
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
( ) [ ])()()( txtxKtxFx −−+= τ&System with the stable limit cycle:
The adjoint equation for PRC: )()()()( ττ ++−−= tztBtztAzTT
&
( )KtB
KtxDFtA c
=
−=
)(
)()(
( )[ ] )(tzxDFzT
c−=&
τ
The profile of the PRC is invariant with respectto the variation of K
)()()1()2( ϕαϕ zz = The coefficient of the proportionality can
be found from the initial condition:α
∫−
− ++=0
)2()1()1(1)()()0()0(
τ
ϑϑϑτα dxKzxz c
T
c
T&&
(Unstable in both directions)
Phase reduction of chaotic systems subject to adelayed feedback control
The delay time is equal to PRC period,so the adjoint equation can be simplified to:
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Example: Rossler system stabilized by DFC
[ ])7.5(2.0
)()(2.0
133
22212
321
−+=
−−++=
−−=
xxx
txtxKxxx
xxx
&
&
&
τ
K1=0.15 and K2=0.5
558.0≈α
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Conclusions
A phase reduction method is applied to a general classof weakly perturbed time-delay systems exhibitingperiodic oscillations
An adjoint equation with an appropriate initial conditionfor the PRC of a time-delay system is derived by twomethods
The method is demonstrated numerically for theMackey-Glass system as well as for a chaoticRossler system subject a DFC
The profile of the PRC of a periodic orbit stabilized by theDFC algorithm is independent of the control matrix
V. Novičenko and K. Pyragas
Phase response curves for systems with time delay
SPI of CPST
Acknowledgements
This work was supported by the Global grantNo. VP1-3.1-ŠMM-07-K-01-025
