10 research outputs found
Regular and irregular patterns of self-localized excitation in arrays of coupled phase oscillators
We study a system of phase oscillators with nonlocal coupling in a ring that
supports self-organized patterns of coherence and incoherence, called chimera
states. Introducing a global feedback loop, connecting the phase lag to the
order parameter, we can observe chimera states also for systems with a small
number of oscillators. Numerical simulations show a huge variety of regular and
irregular patterns composed of localized phase slipping events of single
oscillators. Using methods of classical finite dimensional chaos and
bifurcation theory, we can identify the emergence of chaotic chimera states as
a result of transitions to chaos via period doubling cascades, torus breakup,
and intermittency. We can explain the observed phenomena by a mechanism of
self-modulated excitability in a discrete excitable medium.Comment: postprint, as accepted in Chaos, 10 pages, 7 figure
Periodic solutions in next generation neural field models
We consider a next generation neural field model which describes the dynamics
of a network of theta neurons on a ring. For some parameters the network
supports stable time-periodic solutions. Using the fact that the dynamics at
each spatial location are described by a complex-valued Riccati equation we
derive a self-consistency equation that such periodic solutions must satisfy.
We determine the stability of these solutions, and present numerical results to
illustrate the usefulness of this technique. The generality of this approach is
demonstrated through its application to several other systems involving delays,
two-population architecture and networks of Winfree oscillators.Comment: 15 pages, 11 figure
Bumps, chimera states, and Turing patterns in systems of coupled active rotators
Self-organized coherence-incoherence patterns, called chimera states, have
first been reported in systems of Kuramoto oscillators. For coupled excitable
units similar patterns, where coherent units are at rest, are called bump
states. Here, we study bumps in an array of active rotators coupled by
non-local attraction and global repulsion. We demonstrate how they can emerge
in a supercritical scenario from completely coherent Turing patterns: single
incoherent units appear in a homoclinic bifurcation with a subsequent
transition via quasiperiodic and chaotic behavior, eventually transforming into
extensive chaos with many incoherent units. We present different types of
transitions and explain the formation of coherence-incoherence patterns
according to the classical paradigm of short-range activation and long-range
inhibition
Hopf Bifurcations of Twisted States in Phase Oscillators Rings with Nonpairwise Higher-Order Interactions
Synchronization is an essential collective phenomenon in networks of
interacting oscillators. Twisted states are rotating wave solutions in ring
networks where the oscillator phases wrap around the circle in a linear
fashion. Here, we analyze Hopf bifurcations of twisted states in ring networks
of phase oscillators with nonpairwise higher-order interactions. Hopf
bifurcations give rise to quasiperiodic solutions that move along the
oscillator ring at nontrivial speed. Because of the higher-order interactions,
these emerging solutions may be stable. Using the Ott--Antonsen approach, we
continue the emergent solution branches which approach anti-phase type
solutions (where oscillators form two clusters whose phase is apart) as
well as twisted states with a different winding number.Comment: 24 pages, 8 figure
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Bumps, chimera states, and Turing patterns in systems of coupled active rotators
Self-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest, are called bump states. Here, we study bumps in an array of active rotators coupled by non-local attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition
A Tweezer for Chimeras in Small Networks
We propose a control scheme which can stabilize and fix the position of
chimera states in small networks. Chimeras consist of coexisting domains of
spatially coherent and incoherent dynamics in systems of nonlocally coupled
identical oscillators. Chimera states are generally difficult to observe in
small networks due to their short lifetime and erratic drifting of the spatial
position of the incoherent domain. The control scheme, like a tweezer, might be
useful in experiments, where usually only small networks can be realized
Bumps, chimera states, and Turing patterns in systems of coupled active rotators
Self-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest, are called bump states. Here, we study bumps in an array of active rotators coupled by non-local attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition
Optimal design of tweezer control for chimera states
Chimera states are complex spatio-temporal patterns which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyze the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHugh-Nagumo oscillators, and determine the ranges of optimal control parameters. We demonstrate that the tweezer control scheme allows for stabilization of chimera states with different shapes, and can be used as an instrument for controlling the coherent domains size, as well as the maximum average frequency difference of the oscillators