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

SLIDE 1

Control of synchronization in complex oscillator networks via time-delayed feedback

Viktor Novičenko 2017 Szeged, Hungary

SLIDE 2

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.

SLIDE 3

Phase reduction method

Phase reduction method allows the approximation of high dimensional dynamics of oscillators with a single-phase variable. ) (x f x   1  ϑ  has periodic solution

   

t T t ξ ξ   phase gradually increase from 0 to T ) , ( ) ( t x g x f x ε   

    )

, ( 1 t

T

ϑ ϑ ε ϑ ξ g z    

 

ϑ z

Here is a phase response curve – the periodic solution of the adjoint equation

  z

ξ f z

T

D ) (   

1 ) ( ) (  ξ z 

T

Initial condition for the phase response curve:

   

 

   

t t

t t

,

,

x g f x

x x

ε

τ

 





     

t

T

, 1 ϑ ϑ ε ϑ ξ g z    

         

τ τ      t t t t t

T T T

B z A z z 

       

) ( ), ( ) ( ), (

2 1

τ τ     t t D t t t D t ξ ξ f B ξ ξ f A

where the matrices Initial condition for the phase response curve:

     





    1 ) ( ) (

τ

τ τ ds s s s

T T

ξ B z ξ z  

• V. Novičenko, K. Pyragas, Physica D 241, 1090–1098 (2012)
• K. Kotani et al, Phys. Rev. Lett. 109, 044101 (2012)

SLIDE 4

Complex oscillator network – the phase reduction approach

Weakly coupled near-identical limit cycle oscillators:



  

j j i ij i i i i

a ) , ( ) ( ) ( x x g x f x f x ε ε 

 

t

i

ϑ

   

t t t

i i i

    ϑ ϕ

   

T t t

i i

period

• ver the

average ϕ ψ 

 



  

j i j ij i i

h a ψ ψ ε ω ψ 

here the frequencies

   

i i

ω

Synchronization condition:

N

ψ ψ ψ       

2 1

without control under the delayed feedback control

     

t t a

i i i j j i ij i i i i

x x K x x g x f x f x      



τ ε ε ) , ( ) ( ) ( 

              

i i i i i i i i i i

T T t t T t t t τ τ         x x x x x 

By treating a free oscillator as

     

t T t

i i i i i i i

x x K x f x f x      ) ( ) ( ε 

Applying the phase reduction method for systems with time-delay

 



  

j i j ij i i

h a ψ ψ ε ω ψ

eff eff



   

1

eff

     K α τ ω ω T Ti

i i i

 ε

α ε K 

eff

• V. Novičenko, Phys. Rev. E 92, 022919 (2015)

where

i i

T π 2  

SLIDE 5

Control of synchronization in a complex oscillator network

     

t t a

i i i j j i ij i i i i

x x K x x g x f x f x      



τ ε ε ) , ( ) ( ) ( 

• V. Novičenko, Phys. Rev. E 92, 022919 (2015)

                  K K

Let’s say

 



  

j i j ij i i

h a ψ ψ ε ω ψ

eff eff



   

1

eff

     K T Ti

i i i

α τ ω ω

 ε

α ε K 

eff

 

KC K   1 1 α

     





T

ds s s z C

1 1

ξ 

where (i) The delay times are the same

T

i

 τ τ

 

 

i i

K ω α ω eff

(ii) The delay times are equal to the natural periods synchronization cannot be controlled

i i i i

T ω ω τ   

eff

(iii) The delay times are

   

1

eff 

    

i i i i

K T T ω α ω τ

is a stable solution, under additional assumptions:

     

t t t

N

ψ ψ ψ    

2 1

   

,    h h

SLIDE 6

Numerical demonstrations

• V. Novičenko, Phys. Rev. E 92, 022919 (2015)

       

t x T t x K t F

1 1 1 1

  

8 FitzHugh-Nagumo oscillators:

5

10 5



  ε

4 eff

10 6



  ε

3

10  ε

4 eff

10 6 . 1



  ε

SLIDE 7

According to the odd number limitation theorem, the periodic solution is unstable, if

Odd number limitation

What happen for ?

 

KC K   1 1 α

C K / 1  

     

t T t

i i i i i i

x x K x f x     ) ( 

   

t T t

i i i

ξ ξ  

1   KC

• E. W. Hooton and A. Amann, Phys. Rev. Lett. 109,

154101 (2012)

Motion of the Floquet multipliers

SLIDE 8

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

SLIDE 9

The end