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Phase response curves for systems with time delay Viktor Novičenko and Kestutis Pyragas

ENOC 2011

CENTER

FOR 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 analyze weakly perturbed limit cycle oscillations Most investigations in the field of phase reduction are devoted to the systems described by ODEs The aim of this investigation is to extend the phase reduction 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 cycle is assigned a scalar variable (PHASE) The phase dynamics of the free system satisfies:

1   

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:

?   

Phase reduction

• V. Novičenko and K. Pyragas

Phase response curves for systems with time delay SPI of CPST

Phase reduction of ODE systems

• V. Novičenko and K. Pyragas

Phase response curves for systems with time delay SPI of CPST

Phase reduction of ODE systems

• V. Novičenko and K. Pyragas

Phase response curves for systems with time delay SPI of CPST

Phase reduction of ODE systems

A B

  

Isochron – a set of states with the same phase

Phase dynamics: ,here is periodic vector valued 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

A B

  

Isochron – a set of states with the same phase

) ( ) ( t y G y    

Perturbed system: ) ( ) ( 1 t z

T

     

 

 z

PRC is the periodic solution of an adjoint equation:

 

  z

y DG z

T c

  

With initial condition:

1 ) ( ) ( 

c T

y z 

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

 

) ( ) ( ), ( t t x t x F x      

Perturbed system: Approximation via a delay line:

 

) ( ) , ( , ) , ( ) , ( ) ( ) , ( ), ( t x t s t s t t s t t t x F x                 Discretization of the space variable :

N i ,..,  , / N i si  

) ( ) ( t x t x 

Denote and

) , ( ) ( t s t x

i i

 

We get a final-dimensional system of ODEs:

     

   / ) ( ) ( / ) ( ) ( ) ( ) ( ), (

1 1 1

N t x t x x N t x t x x t t x t x F x

N N N N

     



   

• V. Novičenko and K. Pyragas

Phase response curves for systems with time delay SPI of CPST

Phase reduction of time-delay system: results

) ( ) ( 1 t zT       

Phase dynamic: The adjoint equation for PRC:

) ( ) ( ) ( ) (        t z t B t z t A z

T T



here

   

) ( ), ( ) ( ) ( ), ( ) (

2 1

      t x t x F D t B t x t x F D t A

c c c c

The initial condition: 1 ) ( ) ( ) ( ) ( ) (      



       d x B z x z

c T c T

 

An unstable difference-differential equation of advanced type (backwards integration)

The phase reduced equations for time delay systems have been alternatively derive directly from DDE system without appealing 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 ) ( t x t x t ax dt dx

b

       Unperturbed equation:        ) ( ) ( ) (       

c c

x x

 

) , [     Two different initial conditions: first on the limit cycle and second perturbed by from the first for for Phase response curve:   / ) ( t z  

5

10 7 . 10 2



      b a



• V. Novičenko and K. Pyragas

Phase response curves for systems with time delay SPI of CPST

Example: Mackey-Glass equation

) ( ) ( ) ( 1 ) ( t t x t x t ax dt dx

b

         A perturbation with periodic external signal:

 

    ) 2 sin( ) 2 sin( ) ( t sign t t   

here Frequency mismatch: T / 1      Arnold’s tongues: 7 . 10 2     b a

• V. Novičenko and K. Pyragas

Phase response curves for systems with time delay SPI of CPST

 

 

) ( ) ( ) ( t x t x K t x F x      

System with the stable limit cycle: The adjoint equation for PRC:

) ( ) ( ) ( ) (        t z t B t z t A z

T T



 

K t B K t x DF t A

c

   ) ( ) ( ) (

   

) (t z x DF z

T c

   

(Unstable in both directions)

Phase reduction of chaotic systems subject to a delayed 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

 

 

) ( ) ( ) ( t x t x K t x F x      

System with the stable limit cycle: The adjoint equation for PRC:

) ( ) ( ) ( ) (        t z t B t z t A z

T T



 

K t B K t x DF t A

c

   ) ( ) ( ) (

   

) (t z x DF z

T c

   

The profile of the PRC is invariant with respect to the variation of K

(Unstable in both directions)

Phase reduction of chaotic systems subject to a delayed 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

 

 

) ( ) ( ) ( t x t x K t x F x      

System with the stable limit cycle: The adjoint equation for PRC:

) ( ) ( ) ( ) (        t z t B t z t A z

T T



 

K t B K t x DF t A

c

   ) ( ) ( ) (

   

) (t z x DF z

T c

   

The profile of the PRC is invariant with respect to the variation of K

) ( ) (

) 1 ( ) 2 (

   z z 

The coefficient of the proportionality can be found from the initial condition:





 

  

) 2 ( ) 1 ( ) 1 ( 1

) ( ) ( ) ( ) (



     d x K z x z

c T c T

 

(Unstable in both directions)

Phase reduction of chaotic systems subject to a delayed 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 . ) ( ) ( 2 .

1 3 3 2 2 2 1 2 3 2 1

           x x x t x t x K x x x x x x     K1=0.15 and K2=0.5

558 .  

• 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 class

• f weakly perturbed time-delay systems exhibiting

periodic oscillations An adjoint equation with an appropriate initial condition for the PRC of a time-delay system is derived by two methods The method is demonstrated numerically for the Mackey-Glass system as well as for a chaotic Rossler system subject a DFC The profile of the PRC of a periodic orbit stabilized by the DFC 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 grant

• No. VP1-3.1-ŠMM-07-K-01-025

The end