Chimera States

in Networks of Coupled Oscillators

Yuri L. Maistrenko

Institute of Mathematics

and

Centre for Medical and Biotechnical Researches

National Academy of Sciences of Ukraine

Snapshots of chimera state

X(Abram, Strogatz 2004. N=256 oscillators)

(Kuramoto Battogtohk 2002. N=512 oscillators)

Average frequencies

Asymmetric chimera Symmetric chimera

Chimera state =

Partial frequency synchronization!

Phase-locked oscillators co-exist with drifting oscillators

What is chimera, where does it live,

and how can one catch it?

- Chimera is “an animal in a homogeneous world”, it is a hybrid pattern consisting

of two parts: coherent and incoherent.

- One can catch chimeras in non-locally coupled networks, i.g. in the Kuramoto-

Sacaguchi model of N phase oscillators placed at a ring:

e.g. for the the step-like coupling function G:

Come back to symmetric and asymmetric chimeras.

Which of them are robust? How do they behave?

Our finding:

• Symmetric chimera states are transversally unstable with respect

to any small symmetry breaking of initial conditions (e.g. due to

the asymmetry of the Runge-Kutta computational algorithm)

• Asymmetric chimera states are robust and execute chaotic motion

(drift) along the ring

• At long time-scales, the chimera’s drift can be described as a

stochastic Brownian motion

Chaotic drift of the chimera state

Parameters: N=200, = 1.46, r = 0.7

Initial conditions close to the unstable symmetric chimera state

Compare two chimera trajectories

The initial conditions differ from each other by the value 0.001

(in one oscillator only)

How to determine the chimera’s position

Center of the chimera (red curve) moves chaotically!

Time evolution of the chimera’s position

- stochastic brownian motion on large time-scales

- deterministic chaotic motion on short time-scales

Deterministic nature of the chimera’s motion.

Lyapunov spectrum: Hyperchaos

Stochastic nature of the chimera’s motion

We find that decays exponentially!

This is an experimental evidence that on a sufficiently large timescales

chimera state behaves as a stationary stochastic process

Translation motion of chimera vs Brownian motion

Diffusion coefficient D: dependence on parameters

N=100

Collapse of the chimera state

Let decrease the number of oscillators N in the network.

Then “drifting amplitude” D increases resulting apparently in

collapsing the chimera state in short times.

This is clearly observed at N~ 50 or less.

Probability density function for collapse time

N = 40

Average life-time of the chimera states

We conclude: - chimera state is a chaotic transient state,

its average life-time grows exponentially with N

- for large N chimera’s life is so long that one can

count them as an effective attractor

Concluding remarks and open questions

• Location of the chimera states is not fixed in the space, where do they live: they reveal chaotic motion along in the form of stochastic drift.

• Only symmetric chimera states are not moving. But, they are transversally unstable and start drifting at extremely small symmetry breaking.

• “Drifting amplitude” (parameter D) of the chimera states decreases with increasing the network size N and vanishes in the thermodynamic limit N inf.

• Chimeras are chaotic transient states, their average life-time grows exponentially with N.

• The nature of the chimera drift and collapse is not clear yet, it may be related with choosing the model of individual oscillators in the network. Indeed, only the first coupling harmonic is present in the Kuramoto-Sacaguchi model.

• Can chimeras exists for more general phase models? E.g. for Hansel-Mano-Maunier model, where the second harmonic is added in the coupling function? These, and many other questions are still open…

• Study of the low-dim chimera states can help in high-dim case. The smallest chimeras exist for N=5 oscillators (symmetric but transversally unstable). They arise from a Cherry flow on 2Dim torus via homoclinic bifurcation.

Many thanks to everybody

for coming!

проходили тисячоліття, але ніщо

не порушувало спокою у Всесвіті

(з древньо-єгипетської міфології)

aбо

Coherence-incoherence transition

in

networks of chaotic oscillators

Спочатку був Хаос...

Regions of coherence for

non-locally coupled chaotic maps

coupling radius r = P/N

coupling coefficient

Coherence-incoherence bifurcation. Chimera states

incoherence

partial coherence

coherence

ring ring

ring

our chimeras

Therefore, the array of identical oscillators splits into two domains: one coherent,

the other incoherent and chaotic in space. This is a chimera state.

Concluding Remarks

• Strongly asymmetric cluster states are the first to appear in the coherence-incoherence bifurcation as the coupling strength is reduced

• The next steps include riddling and blowout bifurcations for the coherent state

• In the beginning of the transition, asymmetric clusters co-exist with the coherent states but soon after the blowout they are the only network attractors

• After a while, the asymmetric clusters are destroyed and instead, symmetric clusters dominate at the intermediate coupling. Eventually, as coupling decreases more, space-time chaos arise in the network without any evidence of clustering.

• The similar processes are observed in political parties which destruction typically starts with the appearance of small groups of renegades –отщепенцев - and ends, eventually, with two almost equal parties (symmetric two-cluster)