Spiral Wave Chimeras

Spiral Wave Chimeras Carlo R. Laing IIMS, Massey University, Auckland, New Zealand Erik Martens MPI of Dynamics and Self-Organization, Göttingen, Germany Steve Strogatz Dept of Maths, Cornell University, Ithaca, NY, USA

Networks of Coupled Oscillators Oscillator: anything varying periodically in time. Neurons (and anything they control: breathing, walking...), heartbeat, fireflies, clocks... Coupled: oscillators can feel one another (e.g. neurons are chemically and electrically coupled) form a network. Simplification: oscillator i described by single angular variable, θ i. Example: Kuramoto model (1984): for i = 1... N. Movie dθ i dt = ω i + K N N j=1 sin (θ j θ i )

For that system, oscillators with small ω i locked, remainder incoherent. Similar behaviour occurs in networks of identical oscillators (unusual). Discovered by Kuramoto c. 22; named chimeras by Strogatz.

For that system, oscillators with small ω i locked, remainder incoherent. Similar behaviour occurs in networks of identical oscillators (unusual). Discovered by Kuramoto c. 22; named chimeras by Strogatz.

One example: Two groups, all-to-all coupling within and between groups. dθ 1 i dt dθ 2 i dt = ω + µ N = ω + µ N N ) sin (θ j 1 θ1 i α j=1 N j=1 ) sin (θ j 2 θ2 i α + ν N + ν N N ) sin (θ j 2 θ1 i α j=1 N j=1 ) sin (θ j 1 θ2 i α Movie

Another example: Oscillators on a ring, coupled with strength that depends on distance. dθ i dt = ω 1 N where G is 2π-periodic. N j=1 ( ) 2π i j G cos (θ i θ j β) N.4.3 G(x).2.1 1 2 3 4 5 6 x Movie.

System we study: planar network, Gaussian coupling in space. Spatially continuous version: θ(x, t) t = ω G( x y ) sin [θ(x, t) θ(y, t) + α]dy R2 where θ(x, t) is phase of oscillator at position x R 2 at time t, ω is natural frequency, α is a phase shift. G is normalised Gaussian: G( x y ) = e x y 2 π (Induces characteristic length-scale of 1.)

Simulation of spatially-discretised version shows spiral wave, with incoherent core. (Colour is sin θ). Movie 5 1.5.5 5 5 5 1 1.5.5 1 Oscillators in spiral arms synchronised, core desynchronised chimera.

Analysis Take continuum limit. System described by probability density function f(x, θ, t) (Probability of oscillator at position x R 2 having state θ, at time t). Conservation of oscillators gives continuity equation where v is velocity: v = ω G( x y ) R2 and θ = θ(x) and θ = θ(y). f t + θ (fv) = π π sin [θ θ + α]f(y, θ, t)dθ dy

Defining an order parameter R(x, t) G( x y ) R2 we can write π π e iθ f(y, θ, t)dθdy [Re i(θ+α) Re i(θ+α)] v = ω 1 2i where overbar indicates complex conjugate. Now f(x, θ, t) is periodic in θ, so write as Fourier series: f(x, θ, t) = 1 2π [ 1 + ] h n (x, t)e inθ + c.c. n=1

Substitute into continuity equation, f t + (fv) =, θ get infinite number of evolution equations: h 1 t h 2 t = =. Ott and Antonsen (28) tried h n (x, t) = [a(x, t)] n, found all of these equations are the same!

Nontrivial, removes one dimension (θ) from the problem. a(x, t) t = iωa + (1/2) [Re iα Re iα a 2] where R(x, t) = G( x y )a(y, t)dy R2 i.e. a nonlocal differential equation.

Spiral can be frozen by moving to coordinate frame rotating in θ direction with uniform speed: Movie If this speed is Ω, then z = ae iωt should be stationary. Now z satisfies z(x, t) t where = i(ω Ω)z + (1/2) R(x, t) = [ Re iα Re iα z 2] (1) G( x y )z(y, t)dy (2) R2 Stationary states of (1)-(2) describe spiral wave chimeras. Linearisation about them gives stability.

Physical interpretation of z At steady state and position x, density in θ is F (θ) n= [z(x)] n e inθ If z = re iφ and r = 1, then F (θ) = δ(θ φ) i.e. all oscillators are locked at angle φ. If < r < 1 then F (θ) = 1 r 2 2π[1 2r cos (θ φ) + r 2 ] i.e. oscillators are asynchronous, but most likely to be found at θ = φ.

Example densities F (θ):.2 r=.5 r=.5,φ=π 3 r=.99,φ=π.15.1.5.4.3.2.1 2 1.5 1 θ/(2π).5 1 θ/(2π).5 1 θ/(2π) Remember, r and φ are functions of x.

Typical solutions 5 r 5 sin(θ).8.6.4.2 5 5 5 5 5 5

Steady states satisfy i(ω Ω)z + (1/2) [ Re iα Re iα z 2] = where R(x) = G( x y )z(y)dy R2 Eliminate z: R(x) = ie R iα G( x y ) 2 2 R(y) 2 dy R(y) where Ω ω. Non-local eigenvalue problem.

Typical R: 5 R 5 sin(arg(r)).8.6.4.2 5 5 5 5 5 5 Seems reasonable to use spiral wave ansatz: where (r, θ) are polar coordinates. R(x) = A(r)e i(θ+ψ(r))

Substitute into functional equation, obtain A(r)e iψ(r) = ie iα K(r, s)e iψ(s) 2 A 2 (s) ds A(s) where K(r, s) = 2s π G = 2se (r2 +s 2) I 1 (2rs) ( ) r 2 + s 2 2rs cos θ cos θ dθ and I 1 is a modifed Bessel function. Now have two unknown functions of a single variable, A(r) and Ψ(r), and unknown scalar.

Expand about α =. When α =, original system is gradient system in coordinate frame rotating at speed ω. All attractors in this frame are fixed points. In original frame these are phase-locked states rotating with Ω = ω. So when α =, ( Ω ω) is also zero. Thus series expansion: = 1 α + O(α 2 ) A(r) = A (r) + A 1 (r)α + O(α 2 ) Ψ(r) = Ψ (r) + Ψ 1 (r)α + O(α 2 )

Results to O(α): 1 = 1, A (r) = ( ) π 2 [ ] re r2 /2 I (r 2 /2) + I 1 (r 2 /2) 1.8 A (r).6.4.2 2 4 6 8 1 r and A 1 (r) = Ψ (r) =.

Results in two predictions as α : 1. The spiral arms will rotate at angular velocity = α, if ω =. 2. The radius of the incoherent core is ρ = (2/ π)α.

Results in two predictions as α : 1. The spiral arms will rotate at angular velocity = α, if ω =. 2. The radius of the incoherent core is ρ = (2/ π)α..5.4 (a).6 (b).3.4.2.1 ρ.2.1.2.3.4 α.1.2.3.4 α

What about other kernels? Questions that arise Using Gaussian for G meant we could explicitly do the integral π ( ) K(r, s) = 2s G r 2 + s 2 2rs cos θ cos θ dθ Try three others, do this integral numerically. To facilitate comparison, all kernels have been scaled such that and 2πG(r)r dr = 1 2πG(r)r 2 dr = π/2 ( same integral over R 2 ) (same standard deviation)

.3.2 (a) A.1 B C D.1.2.3 α ρ.4.3.2.1 (b).1.2.3 α Kernels 2πG(r) given by: (A) πk ( πr) (B) 2e r2 (C) (16/π)e 4r/ π (D) 32H(r)H(3 π/4 r)/(9π), where H is the Heaviside step.

What about non-identical oscillators? If natural frequencies ω i are chosen from a distribution g(ω), certain quantities are replaced by their integral over ω, weighted by g(ω). If g(ω) is a Lorentzian with mean zero and HWHM D, i.e. g(ω) = 1 2πi ( ) 1 ω id 1 ω + id and b(ω) is well-behaved, by contour integration and residue theory b(ω)g(ω) dω = b( id) i.e. these integrals can be done explicitly; D is a new parameter.

Non-phase oscillators? Spiral wave chimeras first observed in non-locally coupled FitzHugh- Nagumo system (2 ODEs). Also seen in non-locally coupled complex Ginzburg-Landau equation.

Validity of Ott/Antonsen ansatz (h n = a n ) Proven to describe attractors for infinite network of non-identical oscillators with g(ω) as above (Ott and Antonsen, 29). For infinite network of identical oscillators, probably need Watanabe/Strogatz ansatz (1994). For finite network of identical oscillators, finite-size fluctuations seem to stabilise marginal modes. Attractors are described by O/A ansatz with D =, i.e. analysis presented here.

Summary Chimera: mixed synchronous/incoherent state in network of coupled identical oscillators. In 2D, appear as spiral with incoherent core. Described using Ott/Antonsen ansatz to derive PDE for z(x, t). Steady states analysed using spiral wave ansatz and expansion about α =. Several testable predictions, verified by simulation. [More: paper appeared in this week s issue of Physical Review Letters.]