Adaptive delayed feedback control to stabilize in-phase - PowerPoint PPT Presentation

Adaptive delayed feedback control to stabilize in-phase synchronization in complex oscillator networks Viktor Novičenko September 2019, Rostock

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 network covers a wide range of real-world applications.

Weakly coupled near-identical limit cycle oscillators without control : N � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) j =1

Weakly coupled near-identical limit cycle oscillators without control : N � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) j =1 coupling function G ( x , x ) = 0

Weakly coupled near-identical limit cycle oscillators without control : N � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) j =1 coupling function G ( x , x ) = 0 “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙

Weakly coupled near-identical limit cycle oscillators without control : N � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) j =1 coupling function G ( x , x ) = 0 “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ has a stable periodic solution ξ ( t + T ) = ξ ( t ) and a phase response curve z ( t + T ) = z ( t )

Weakly coupled near-identical limit cycle oscillators without control : N � phase reduction x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) j =1 coupling function G ( x , x ) = 0 “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ has a stable periodic solution ξ ( t + T ) = ξ ( t ) and a phase response curve z ( t + T ) = z ( t )

Weakly coupled near-identical limit cycle oscillators without control : N N � phase reduction ˙ � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) ψ i = ω i + ε a ij h ( ψ j − ψ i ) j =1 j =1 coupling function with frequency G ( x , x ) = 0 ω i = Ω i − Ω “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ and coupling function has a stable periodic solution ξ ( t + T ) = ξ ( t ) T z T � s � s � � � s + χ � ��� h ( χ ) = 1 ˆ � · G d s ξ , ξ and a phase response curve z ( t + T ) = z ( t ) Ω Ω Ω T 0

Weakly coupled near-identical limit cycle oscillators without control : N N � phase reduction ˙ � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) ψ i = ω i + ε a ij h ( ψ j − ψ i ) j =1 j =1 coupling function with frequency G ( x , x ) = 0 ω i = Ω i − Ω “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ and coupling function has a stable periodic solution ξ ( t + T ) = ξ ( t ) T z T � s � s � � � s + χ � ��� h ( χ ) = 1 ˆ � · G d s ξ , ξ and a phase response curve z ( t + T ) = z ( t ) Ω Ω Ω T 0 Weakly coupled near-identical limit cycle oscillators under delayed feedback control : N � x i = f i ( x i , u i ( t )) + ε ˙ a ij G ( x j , x i ) j =1 s i ( t ) = g ( x i ( t )) u i ( t ) = K [ s i ( t − τ i ) − s i ( t )]

Weakly coupled near-identical limit cycle oscillators without control : N N � phase reduction ˙ � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) ψ i = ω i + ε a ij h ( ψ j − ψ i ) j =1 j =1 coupling function with frequency G ( x , x ) = 0 ω i = Ω i − Ω “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ and coupling function has a stable periodic solution ξ ( t + T ) = ξ ( t ) T z T � s � s � � � s + χ � ��� h ( χ ) = 1 ˆ � · G d s ξ , ξ and a phase response curve z ( t + T ) = z ( t ) Ω Ω Ω T 0 Weakly coupled near-identical limit cycle oscillators under delayed feedback control : N � x i = f i ( x i , u i ( t )) + ε ˙ a ij G ( x j , x i ) j =1 s i ( t ) = g ( x i ( t )) u i ( t ) = K [ s i ( t − τ i ) − s i ( t )] with time delays | τ i − T i | ∼ | τ i − T | ∼ ε

Weakly coupled near-identical limit cycle oscillators without control : N N � phase reduction ˙ � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) ψ i = ω i + ε a ij h ( ψ j − ψ i ) j =1 j =1 coupling function with frequency G ( x , x ) = 0 ω i = Ω i − Ω “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ and coupling function has a stable periodic solution ξ ( t + T ) = ξ ( t ) T z T � s � s � � � s + χ � ��� h ( χ ) = 1 ˆ � · G d s ξ , ξ and a phase response curve z ( t + T ) = z ( t ) Ω Ω Ω T 0 Weakly coupled near-identical limit cycle oscillators under delayed feedback control : N � x i = f i ( x i , u i ( t )) + ε ˙ a ij G ( x j , x i ) j =1 phase reduction for s i ( t ) = g ( x i ( t )) system with time delay u i ( t ) = K [ s i ( t − τ i ) − s i ( t )] with time delays | τ i − T i | ∼ | τ i − T | ∼ ε

Weakly coupled near-identical limit cycle oscillators without control : N N � phase reduction ˙ � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) ψ i = ω i + ε a ij h ( ψ j − ψ i ) j =1 j =1 coupling function with frequency G ( x , x ) = 0 ω i = Ω i − Ω “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ and coupling function has a stable periodic solution ξ ( t + T ) = ξ ( t ) T z T � s � s � � � s + χ � ��� h ( χ ) = 1 ˆ � · G d s ξ , ξ and a phase response curve z ( t + T ) = z ( t ) Ω Ω Ω T 0 Weakly coupled near-identical limit cycle oscillators under delayed feedback control : N N ˙ � ψ i = ω eff + ε eff � a ij h ( ψ j − ψ i ) x i = f i ( x i , u i ( t )) + ε ˙ a ij G ( x j , x i ) i j =1 j =1 phase reduction for s i ( t ) = g ( x i ( t )) system with time delay u i ( t ) = K [ s i ( t − τ i ) − s i ( t )] with time delays | τ i − T i | ∼ | τ i − T | ∼ ε V. Novičenko: Delayed feedback control of synchronization in weakly coupled oscillator networks , Phys. Rev. E 92 , 022919 (2015)

Weakly coupled near-identical limit cycle oscillators without control : N N � phase reduction ˙ � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) ψ i = ω i + ε a ij h ( ψ j − ψ i ) j =1 j =1 coupling function with frequency G ( x , x ) = 0 ω i = Ω i − Ω “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ and coupling function has a stable periodic solution ξ ( t + T ) = ξ ( t ) T z T � s � s � � � s + χ � ��� h ( χ ) = 1 ˆ � · G d s ξ , ξ and a phase response curve z ( t + T ) = z ( t ) Ω Ω Ω T 0 Weakly coupled near-identical limit cycle oscillators under delayed feedback control : N N ˙ � ψ i = ω eff + ε eff � a ij h ( ψ j − ψ i ) x i = f i ( x i , u i ( t )) + ε ˙ a ij G ( x j , x i ) i j =1 j =1 phase reduction for ε eff = εα ( KC ) e ff ective coupling strength s i ( t ) = g ( x i ( t )) system with time delay e ff ective frequency u i ( t ) = K [ s i ( t − τ i ) − s i ( t )] = ω i + Ω τ i − T i ω eff [ α ( KC ) − 1] i with time delays | τ i − T i | ∼ | τ i − T | ∼ ε T V . Novičenko: Delayed feedback control of synchronization in weakly coupled oscillator networks , Phys. Rev. E 92 , 022919 (2015)

Weakly coupled near-identical limit cycle oscillators without control : N N � phase reduction ˙ � x i = f i ( x i ) + ε ˙ a ij G ( x j , x i ) ψ i = ω i + ε a ij h ( ψ j − ψ i ) j =1 j =1 coupling function with frequency G ( x , x ) = 0 ω i = Ω i − Ω “central” oscillator with | f ( x ) − f i ( x ) | ∼ ε x = f ( x ) ˙ and coupling function has a stable periodic solution ξ ( t + T ) = ξ ( t ) T z T � s � s � � � s + χ � ��� h ( χ ) = 1 ˆ � · G d s ξ , ξ and a phase response curve z ( t + T ) = z ( t ) Ω Ω Ω T 0 Weakly coupled near-identical limit cycle oscillators under delayed feedback control : N N ˙ � ψ i = ω eff + ε eff � a ij h ( ψ j − ψ i ) x i = f i ( x i , u i ( t )) + ε ˙ a ij G ( x j , x i ) i j =1 j =1 phase reduction for ε eff = εα ( KC ) e ff ective coupling strength s i ( t ) = g ( x i ( t )) system with time delay e ff ective frequency u i ( t ) = K [ s i ( t − τ i ) − s i ( t )] = ω i + Ω τ i − T i ω eff [ α ( KC ) − 1] i with time delays | τ i − T i | ∼ | τ i − T | ∼ ε T T ˆ [ ∇ g ( ξ ( s ))] T · ˙ z T ( s ) · D 2 f ( ξ ( s ) , 0) � � � α ( KC ) = (1 + KC ) − 1 the function and the constant � C = ξ ( s ) d s 0 V . Novičenko: Delayed feedback control of synchronization in weakly coupled oscillator networks , Phys. Rev. E 92 , 022919 (2015)

N N � ˙ � x i = f i ( x i , u i ( t )) + ε ˙ a ij G ( x j , x i ) ψ i = ω eff + ε eff a ij h ( ψ j − ψ i ) (1) i j =1 j =1 s i ( t ) = g ( x i ( t )) ε eff = εα ( KC ) where u i ( t ) = K [ s i ( t − τ i ) − s i ( t )] = ω i + Ω τ i − T i ω eff [ α ( KC ) − 1] i T α ( KC ) = (1 + KC ) − 1