Chimera patterns and solitary synchronization waves in distributed - PowerPoint PPT Presentation

Chimera patterns and solitary synchronization waves in distributed oscillator populations 1,2 1 Lev Smirnov , Grigory Osipov and 3,1 Arkady Pikovsky 1 Department of Control Theory, Nizhny Novgorod State University, Russia 2 Institute of Applied Physics of the Russian Academy of Sciences, Russia 3 Institute for Physics and Astronomy, University of Potsdam, Germany School and Workshop "Patterns of Synchrony: Chimera States and Beyond" ICTP Trieste, May 6-17, 2019 � 1

Contents of this talk: Chimera patterns in the Kuramoto- Battogtokh model [based on papers L.A. Smirnov, G.V. Osipov, A. Pikovsky J. Phys. A: Math. Theor. 50 , 08LT01 (2017); In: Abcha N., Pelinovsky E., Mutabazi I. (eds) Nonlinear Waves and Pattern Dynamics. Springer, Cham. p 159-180. (2018)] Solitary synchronization waves [based on paper L.A. Smirnov, G.V. Osipov, A. Pikovsky Phys. Rev. E 98 , 062222 (2018)] � 2

Kuramoto-Battogtokh model as a set of partial differential equations Original KB model: Integral equation ∂ t = ω − ∫ L ∂ φ G ( x − ˜ x )sin( φ ( x , t ) − φ (˜ x , t ) − α ) d ˜ x 0 Step 1: introduce a coarse-grained complex order parameter Z ( x , t ) = ⟨ e i φ ( x , t ) ⟩ | x −Δ < x < x + Δ Step 2: Apply the Ott-Antonsen ansatz for the dynamics of the order parameter H = ∫ L ∂ Z ∂ t = i ω Z + 1 2 ( He − i α − H * Z 2 e i α ) G ( x − ˜ x ) Z (˜ x , t ) d ˜ x 0 � 3

Step 3: Exponential kernel corresponds to a di ff erential operator (cf. Lecture of C. Laing) H = ∫ ∞ ∂ 2 H ∂ x 2 − κ 2 H = − κ 2 Z exp[ − κ | x − ˜ ⇔ x | ] Z (˜ x , t ) d ˜ x −∞ Step 4: Apply this to a periodic domain (this slightly modifies the kernel) G ( x ) = κ cosh κ ( | x | − L /2)) H (0) = H ( L ), ∂ x H (0) = ∂ x H ( L ) ⇔ 2 sinh( κ L /2) Result: a system of PDEs with periodic boundary conditions ∂ 2 H ∂ Z ∂ t = i ω Z + 1 2 ( He − i α − H * Z 2 e i α ) ∂ x 2 − κ 2 H = − κ 2 Z We do not solve the consistency equation (nonlinear eigenvalue problem) for fixed length of the domain L, but find periodic in space and time solutions (standing waves) of the system of PDEs � 4

ODE for the chimera patterns ∂ 2 H ∂ Z ∂ t = i ω Z + 1 2 ( He − i α − H * Z 2 e i α ) ∂ x 2 − κ 2 H = − κ 2 Z Z ( x , t ) = z ( x ) e i ( ω + Ω ) t , H ( x , t ) = h ( x ) e i ( ω + Ω ) t Rotating wave ansatz: Quadratic equation for z: e i α h * z 2 + 2 i Ω z − e − i α h = 0 Second-order ordinary di ff erential equation for complex field h Ω 2 − | h | 2 Ω + d 2 dx 2 h − h = β = π /2 − α h * exp[ − i β ] � 5

Fourth-order complex ODE can be reduced, due to phase- rotation invariance, to a three-dimensional real system h ( x ) = r ( x ) e i θ ( x ) , q ( x ) = r 2 ( x ) θ ′ � ( x ) r 2 − Ω 2 r ′ � ′ � = r + q 2 r 3 + Ω r cos β − sin β r <-synchronous domain r 2 − Ω 2 cos β q ′ � = Ω sin β + if | r | > | Ω | Ω 2 − r 2 r ′ � ′ � = r + q 2 r 3 + Ω + cos β r q ′ � = ( Ω + Ω 2 − r 2 ) sin β asynchronous domain -> if | r | < | Ω | � 6

Analytic solutions: one- and two-point chimeras, chimera soliton Case is integrable! α = π /2, β = 0 Dynamics only in the asynchronous domain, but synchrony can be achieved at one or two points d 2 r U ( r ) = − r 2 Ω 2 − r 2 − Ω ln ( dx 2 = − dU ( r ) Ω 2 − r 2 − Ω ) 2 − , dr U − ˆ ˆ U 0 Potential for two-point chimera (a) 0 . 05 Potential for homoclinic one- 0 point chimera − 0 . 05 Potential for one-point chimera − 0 . 1 � | Ω | − 1 . 5 − 1 − 0 . 5 0 0 . 5 1 r � 7

Period-frequency dependencies of singular one- and two-point chimeras solitary (homoclinic) chimera L | z | (b) 0 . 5 20 0 0 3 x 10 0 − 1 − 0 . 75 − 0 . 5 − 0 . 25 Ω One-point chimera Two-point chimera � 8

Perturbation theory close to π 2 − α = β ≪ 1 the integrable case: Synchronous domain now is not a point but has finite length: | Ω | (1 − | Ω | ) ∮ ( R ′ � 2 + R 2 ) dx 8 β L syn ≈ π N SR L syn β = 0 Here is the solution at R Ω = − 0 . 8 1 . 2 and is the number of N SR 0 . 9 synchronous regions (1 or 2) 0 . 6 0 . 3 0 0 0 . 025 0 . 05 0 . 075 β � 9

Chimera patterns as periodic orbits of ODE The system of ODEs for r(x) and q(x) is a reversible third-order system of ODEs with a plethora of solutions, including chaotic ones. Poincare map Examples of chimera patterns q | z | 1 (a) 0.01 0 0.5 − 0.01 (b) − 0.02 0 − − − 2.5 0.83 0.84 7.5 5 0 2.5 5 r x � 10

| z | , | h | 1 A B 0 . 8 Four simple 0 . 6 0 . 4 chimera patterns 0 . 2 0 coexist for a 1 C D particular domain 1 0 . 8 0 . 6 length 0 . 4 0 . 98 0 . 2 0 5 . 6 11 . 2 0 0 2 . 8 5 . 6 8 . 4 0 2 . 8 5 . 6 8 . 4 x Ω Ω − 0 . 2 α = 1.514 α = 1.457 − 0 . 4 B − 0 . 6 C − 0 . 8 A D − 1 α = 0.944 α = 1.229 − 0 . 2 − 0 . 4 − 0 . 6 − 0 . 8 − 1 0 3 6 9 12 0 3 6 9 12 L � 11

Stability properties Essential and discrete spectra [according to O.E. Omel'chenko � � Nonlinearity 26, 2469 (2013); J. Xie et al. PRE 90, 022919 (2014)] Only the “standard” Kuramoto-Battogtokh chimera is stable � 12

Direct numerical simulations � 13

Conclusions to this part • Many chimera patterns can be found as periodic orbits of an ODE (potentially easier than solving a self-consistency problem) • For neutral coupling, one-point and two-point chimera can be found analytically (represented as integrals), for nearly neutral coupling a perturbation theory on top of these solutions is developed • No stable complex chimera patterns found, the only stable one is the KB chimera � 14

Part 2: Solitary synchronyzation waves � 15

Oscillatory medium with Laplacian coupling Start with the KB-type model (1-d medium with non-local coupling) ∂ φ ∫ G ( x − ˜ ∂ t = Im ( He − i φ ) , H ( x , t ) = e − α x ) e i φ (˜ x , t ) d ˜ x With Ott-Antonsen ansatz and coarse-grained order parameter Z H ( x , t ) = ∫ G ( x − ˜ ∂ Z ∂ t = 1 2 ( e − i α H − H * Z 2 e i α ) , x ) Z (˜ x , t ) d ˜ x We assume a kernel with vanishing mean value (Laplacian coupling) ∫ G ( x ) dx = 0 G ( x ) ∼ ( x 2 − σ 2 ) e − x 2 for example 2 σ 2 � 16

Exponential kernel - like mean field coupling, enables synchrony Laplacian coupling - allows for any constant level of synchrony � 17

Lattice with Laplacian coupling / /32 /12 Local dynamics at site n is described by the local order parameter Z n Coupling with nearest neighbours: H n = e − i α ( Z n − 1 + Z n +1 − 2 Z n ) 2 dZ n dt = e − i α ( Z n − 1 + Z n +1 − 2 Z n ) − e i α ( Z * n ) Z 2 n +1 − 2 Z * n − 1 + Z * n A lattice with linear and nonlinear coupling of “complex Ginzburg-Landau” or of “nonlinear Schroedinger” type � 18

Conservative coupling We choose and obtain a conservative lattice α = − π /2 2 dZ n n ) Z 2 dt = i ( Z n − 1 + Z n +1 − 2 Z n ) + i ( Z * n +1 − 2 Z * n − 1 + Z * n Spatially uniform solutions: Z n = ϱ e i θ with any 0 ≤ ϱ ≤ 1 Linear waves on top of this background have dispersion 1 − ϱ 4 (1 − cos k ) ω ( k ) = Phase and group velocities: 1 − ϱ 4 1 − cos k 1 − ϱ 4 sin k λ ph = , λ gr = k � 19

Solitary waves in the limit of full synchrony If all oscillators on a site a synchronised, the problem reduces to a lattice of phase oscillators !"#$%&'( ")*+%%,-".) / /32 /12 !"#$%&'( $"$#%,-+"/)( "0(")*+%%,-".) Z n = e i Θ n The order parameter can be represented as For the phase di ff erence we obtain V n = Θ n − Θ n − 1 dV n dt = cos V n +1 − cos V n − 1 � 20

Compactons and kovatons dV n dt = cos V n +1 − cos V n − 1 Equation has been studied in P . Rosenau, A.P ., PRL (2005); Physica D (2006) V n ( t ) = V ( t − n λ ) Traveling wave ansatz Compactons are solitary waves with compact support 0 < λ < λ c = 4 π Kovatons are glued 0 ↔ π kinks with arbitrary length λ = λ c � 21

Solitary waves close to compactors Z n = ρ n e i θ n v n = θ n − θ n − 1 Full equations for the lattice: Traveling wave ansatz: dt = 1 − ρ 2 d ρ n n ( ρ n − 1 sin v n − ρ n +1 sin v n +1 ) ρ n ( t ) = ρ ( τ ), θ n ( t ) = θ ( τ ) 2 τ = t − n dt = 1 + ρ 2 d θ n n ( ρ n − 1 cos v n + ρ n +1 cos v n +1 − 2 ρ n ) λ 2 ρ n Perturbation approach close to full synchrony (close to true compactons) ϵ = 1 − ϱ ≪ 1 ρ ( τ ) = ϱ + ϵ r 1 ( τ ) + . . . v ( τ ) = V ( τ ) + ϵ v 1 ( τ ) + . . . Analytic expression for the 1st correction: r 1 ( τ ) = 1 − exp [ ∫ τ ] τ ( sin V (˜ τ ) ) d ˜ τ − 1/ λ ) − sin V (˜ −∞ � 22

⇣ ⌘ � Comparison of approximate and exact solitary waves for ϱ = 0.9 Dashed red curves: approximate solution Blue curves: exact solution Exact solitary wave is not compact, but has exponentially decaying, oscillating tails � 23

Examples of compactor-like and kovaton-like solitary waves and the domain on their existence on plane ( ϱ , λ ) � 24

Illustration of solitons in lattice equations 2 dZ n n ) Z 2 dt = i ( Z n − 1 + Z n +1 − 2 Z n ) + i ( Z * n +1 − 2 Z * n − 1 + Z * n � 25