Control of synchronization in complex oscillator networks via - PowerPoint PPT Presentation

Control of synchronization in complex oscillator networks via time-delayed feedback Viktor Novi č enko 2017 Szeged, Hungary

Motivation The synchronous behavior can be desirable or harmful. • Power grids • Parkinson’s disease, essential tremor • Pedestrians on a bridge • Cardiac pacemaker cells • Internal circadian clock The ability to control synchrony in oscillatory networks covers a wide range of real-world applications.

Phase reduction method Phase reduction method allows the approximation of high dimensional dynamics of oscillators with a single-phase variable.     x    has periodic solution  ξ ξ f ( x ) t T t  ϑ  phase gradually increase from 0 to T 1   ϑ   ε Here is a phase response curve – the periodic z  x f ( x ) g ( x , t )   z   T solution of the adjoint equation  z D f ( ξ )     )  ϑ   ε ϑ  ϑ   ξ  T ξ Initial condition for the phase response curve: T 1 z g ( , t z ( 0 ) ( 0 ) 1                      τ τ τ        ε   T T T x t , x t x f g x t , t z t z t A t z t B t       τ where the matrices A t D f ξ ( t ), ξ ( t ) 1              τ ϑ   ε ϑ  ϑ T 1 z g ξ , t B t D f ξ ( t ), ξ ( t ) 2 0           τ  τ   T T Initial condition for the phase response curve: z 0 ξ ( 0 ) z s B s ξ ( s ) ds 1  τ V. Novičenko, K. Pyragas, Physica D 241 , 1090–1098 (2012) K. Kotani et al, Phys. Rev. Lett. 109 , 044101 (2012)

Complex oscillator network – the phase reduction approach Weakly coupled near-identical limit cycle oscillators: without control under the delayed feedback control           ε  ε   τ    ε  ε   x f ( x ) f ( x ) a g ( x , x ) K x t x t x f ( x ) f ( x ) a g ( x , x ) i i i i ij i j i i i i ij i j i i i j j   ϑ                t  τ        τ  x t x t x t T x t x t T T i i i i i i i i i i i π     2 ϕ   ϑ     where t t t By treating a free oscillator as i i i i T       i   ε     x f ( x ) f ( x ) K x t T x t i i i i i i i     ψ  average ϕ t t over the period T i i Applying the phase reduction method for systems with time-delay    ψ ω ε ψ ψ     a h i i ij j i j    ψ  ω  ε ψ  ψ ω      eff eff here the frequencies a h i i ij j i i i j τ        ε T i ω  ω   α  ε  α eff eff i K 1 K Synchronization condition: i i T ψ  ψ   ψ     1 2 N V. Novičenko, Phys. Rev. E 92 , 022919 (2015)

Control of synchronization in a complex oscillator network    K 0 0            ε  ε   τ   x f ( x ) f ( x ) g ( x , x ) K x x a t t  0 0 0 i i i i ij i j i i i    Let’s say K j      0      ψ  ω  ε ψ  ψ  eff eff a h   0 0 0 0 i i ij j i j T τ            1    T i      ε  ξ α  1 where 1 1 ω  ω   α  ε  α C z s s ds eff K eff i 1 K K  i i KC T 0 τ  τ  (i) The delay times are the same T i   ω eff  α K ω  synchronization cannot be controlled i i (ii) The delay times are equal to the natural periods τ   ω  ω eff T i i i i (iii) The delay times are τ  ω eff  T   ω i i i 0       α i T 1 K              ψ  ψ   ψ is a stable solution, under additional assumptions:  h 0 0 , h 0 0 t t t 1 2 N V. Novičenko, Phys. Rev. E 92 , 022919 (2015)

Numerical demonstrations 8 FitzHugh-Nagumo oscillators:    ε   10  ε   eff 4 eff 4 ε   ε  5 3 6 10 1 . 6 10 5 10            F t K x t T x t 1 1 1 1 V. Novičenko, Phys. Rev. E 92 , 022919 (2015)

Odd number limitation              1 x f ( x ) K x t T x t α  1 i i i i i i K  KC According to the odd number limitation theorem,     the periodic solution is unstable, if   ξ ξ t T t i i i   KC 1 E. W. Hooton and A. Amann, Phys. Rev. Lett. 109 , 154101 (2012) What happen for ?   K 1 / C Motion of the Floquet multipliers

Summary The delayed feedback control force applied to a limit cycle oscillator changes its stability properties and, as a consequence, perturbation-induced phase response. The phase model of the oscillator network shows that the coupling strength and the frequencies depend on the parameters of the control. Advantages: • does not require any information about the oscillator model • does not depends on network topology • can be simple realized in experiment • theoretically synchronization can be controlled for the arbitrary small/large coupling strength • the control scheme has only two parameters: control gain and delay time Disadvantages: • the phase model can be derived only for a weak coupling • the control force can disrupt the stability of periodic orbit

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