AACIMP 2010 Summer School lecture by Yuri Maistrenko. "Applied Mathematics" stream. "Nonlinear Dynamics of Coupled Oscillators: Theory and Application" course.
More info at http://summerschool.ssa.org.ua
in Networks of Coupled Oscillators
Yuri L. Maistrenko
Institute of Mathematics
Centre for Medical and Biotechnical Researches
National Academy of Sciences of Ukraine
Snapshots of chimera state
Phase-locked oscillators co-exist with drifting oscillators
Asymmetric chimera Symmetric chimera
(Abram, Strogatz 2004. N=256 oscillators)
Chimera state =
Partial frequency synchronization!
(Kuramoto Battogtohk 2002. N=512 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?
• 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
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.
Спочатку був Хаос...
проходили тисячоліття, але ніщо
не порушувало спокою у Всесвіті
(з древньо-єгипетської міфології)
networks of chaotic oscillators
Regions of coherence for
non-locally coupled chaotic maps
coupling radius r = P/N
Coherence-incoherence bifurcation. Chimera states
Therefore, the array of identical oscillators splits into two domains: one coherent,
the other incoherent and chaotic in space. This is a chimera state.
• 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
• In the beginning of the transition, asymmetric clusters co-exist with the
coherent states but soon after the blowout they are the only network
• 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