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CENTER FOR PHYSICAL SCIENCES AND TECHNOLOGY Semiconductor Physics Institute, Vilnius, Lithuania Phase response curves for systems with time delay Viktor Novi čenko and Kestutis Pyragas ENOC 2011

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. Novi č enko 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. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction 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:    ? V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of ODE systems V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of ODE systems V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of ODE systems    B A Isochron – a set of states with the same phase V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of ODE systems    B A Isochron – a set of states with the same phase Malkin’s approach: Malkin, I.G.: Some Problems in Nonlinear Oscillation Theory.Gostexizdat, Moscow (1956)     ( ) ( ) y G y t Perturbed system:         T   Phase dynamics: ,here is periodic vector 1 z ( ) ( t ) z valued function - the phase response curve (PRC)   z     T  PRC is the periodic solution of an adjoint equation: z DG y c  T  z ( 0 ) y ( 0 ) 1 With initial condition: c V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of time-delay system         x F x ( t ), x ( t ) ( t ) Perturbed system: Approximation via a delay line:         x F x ( t ), ( , t ) ( t )     ( s , t ) ( s , t )     , ( 0 , t ) x ( t )   t s    / N , i 0 ,.., N Discretization of the space variable : s i i    Denote and x ( t ) x ( t ) x ( t ) ( s , t ) 0 i i       ( ), ( ) ( ) x F x t x t t 0 0 N       We get a final-dimensional x x ( t ) x ( t ) N / 1 0 1 system of ODEs:        x x ( t ) x ( t ) N /  N N 1 N V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of time-delay system: results        z T Phase dynamic: 1 ( ) ( t )         T T The adjoint equation for PRC: ( ) ( ) ( ) ( ) z A t z t B t z t   An unstable difference-differential    here A ( t ) D F x ( t ), x ( t ) equation of advanced type 1 c c      (backwards integration) B ( t ) D F x ( t ), x ( t ) 2 c c The initial condition: 0              T  T  z ( 0 ) x ( 0 ) z ( ) B ( ) x ( ) d 1 c c   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. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Example: Mackey-Glass equation   dx ax ( t )   x ( t ) Unperturbed equation:    b dt 1 x ( t ) Two different initial conditions: first on the limit cycle and second  perturbed by from the first      for  0 x ( )      c  ( )   [      for , 0 )  x ( ) c  Phase response curve: a 2      z ( ) t / 10 b   0 . 7    5 10 V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Example: Mackey-Glass equation A perturbation with periodic external signal:     dx ax ( t ) sin( 2 ) t       x ( t ) ( t ) here  ( t )      b  dt 1 x ( t )  sign sin( 2 t ) Frequency mismatch:      1 / T Arnold’s tongues:  a 2  b 10   0 . 7 V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of chaotic systems subject to a delayed feedback control           System with the stable limit cycle: x F x ( t ) K x ( t ) x ( t )        T T  The adjoint equation for PRC: z A ( t ) z ( t ) B ( t ) z ( t )     A ( t ) DF x ( t ) K  c The delay time is equal to PRC period,  B ( t ) K so the adjoint equation can be simplified to:       T  z DF x z ( t ) (Unstable in both directions) c V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of chaotic systems subject to a delayed feedback control           System with the stable limit cycle: x F x ( t ) K x ( t ) x ( t )        T T  The adjoint equation for PRC: z A ( t ) z ( t ) B ( t ) z ( t )     A ( t ) DF x ( t ) K  c The delay time is equal to PRC period,  B ( t ) K so the adjoint equation can be simplified to:       T  z DF x z ( t ) (Unstable in both directions) c The profile of the PRC is invariant with respect to the variation of K V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

Phase reduction of chaotic systems subject to a delayed feedback control           System with the stable limit cycle: x F x ( t ) K x ( t ) x ( t )        T T  The adjoint equation for PRC: z A ( t ) z ( t ) B ( t ) z ( t )     A ( t ) DF x ( t ) K  c The delay time is equal to PRC period,  B ( t ) K so the adjoint equation can be simplified to:       T  z DF x z ( t ) (Unstable in both directions) c The profile of the PRC is invariant with respect to the variation of K      The coefficient of the proportionality can ( 2 ) ( 1 ) z ( ) z ( ) be found from the initial condition: 0   T T         1 ( 1 )  ( 1 ) ( 2 )  z ( 0 ) x ( 0 ) z ( ) K x ( ) d c c V. Novi č enko and K. Pyragas   Phase response curves for systems with time delay SPI of CPST

Example: Rossler system stabilized by DFC     x x x 1 2 3          x x 0 . 2 x K x ( t ) x ( t ) 2 1 2 2 2     x 0 . 2 x ( x 5 . 7 ) 3 3 1 K 1 =0.15 and K 2 =0.5   0 . 558 V. Novi č enko 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 of 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. Novi č enko 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 V. Novi č enko and K. Pyragas Phase response curves for systems with time delay SPI of CPST

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