121 research outputs found
Isotropy of Angular Frequencies and Weak Chimeras With Broken Symmetry
The notion of a weak chimeras provides a tractable definition for chimera
states in networks of finitely many phase oscillators. Here we generalize the
definition of a weak chimera to a more general class of equivariant dynamical
systems by characterizing solutions in terms of the isotropy of their angular
frequency vector - for coupled phase oscillators the angular frequency vector
is given by the average of the vector field along a trajectory. Symmetries of
solutions automatically imply angular frequency synchronization. We show that
the presence of such symmetries is not necessary by giving a result for the
existence of weak chimeras without instantaneous or setwise symmetries for
coupled phase oscillators. Moreover, we construct a coupling function that
gives rise to chaotic weak chimeras without symmetry in weakly coupled
populations of phase oscillators with generalized coupling
Heteroclinic switching between chimeras
Functional oscillator networks, such as neuronal networks in the brain,
exhibit switching between metastable states involving many oscillators. We give
exact results how such global dynamics can arise in paradigmatic phase
oscillator networks: higher-order network interaction gives rise to metastable
chimeras - localized frequency synchrony patterns - which are joined by
heteroclinic connections. Moreover, we illuminate the mechanisms that underly
the switching dynamics in these experimentally accessible networks
Asynchronous Networks and Event Driven Dynamics
Real-world networks in technology, engineering and biology often exhibit
dynamics that cannot be adequately reproduced using network models given by
smooth dynamical systems and a fixed network topology. Asynchronous networks
give a theoretical and conceptual framework for the study of network dynamics
where nodes can evolve independently of one another, be constrained, stop, and
later restart, and where the interaction between different components of the
network may depend on time, state, and stochastic effects. This framework is
sufficiently general to encompass a wide range of applications ranging from
engineering to neuroscience. Typically, dynamics is piecewise smooth and there
are relationships with Filippov systems. In the first part of the paper, we
give examples of asynchronous networks, and describe the basic formalism and
structure. In the second part, we make the notion of a functional asynchronous
network rigorous, discuss the phenomenon of dynamical locks, and present a
foundational result on the spatiotemporal factorization of the dynamics for a
large class of functional asynchronous networks
Chaotic Weak Chimeras and their Persistence in Coupled Populations of Phase Oscillators
Nontrivial collective behavior may emerge from the interactive dynamics of
many oscillatory units. Chimera states are chaotic patterns of spatially
localized coherent and incoherent oscillations. The recently-introduced notion
of a weak chimera gives a rigorously testable characterization of chimera
states for finite-dimensional phase oscillator networks. In this paper we give
some persistence results for dynamically invariant sets under perturbations and
apply them to coupled populations of phase oscillators with generalized
coupling. In contrast to the weak chimeras with nonpositive maximal Lyapunov
exponents constructed so far, we show that weak chimeras that are chaotic can
exist in the limit of vanishing coupling between coupled populations of phase
oscillators. We present numerical evidence that positive Lyapunov exponents can
persist for a positive measure set of this inter-population coupling strength
Controlling Chimeras
Coupled phase oscillators model a variety of dynamical phenomena in nature
and technological applications. Non-local coupling gives rise to chimera states
which are characterized by a distinct part of phase-synchronized oscillators
while the remaining ones move incoherently. Here, we apply the idea of control
to chimera states: using gradient dynamics to exploit drift of a chimera, it
will attain any desired target position. Through control, chimera states become
functionally relevant; for example, the controlled position of localized
synchrony may encode information and perform computations. Since functional
aspects are crucial in (neuro-)biology and technology, the localized
synchronization of a chimera state becomes accessible to develop novel
applications. Based on gradient dynamics, our control strategy applies to any
suitable observable and can be generalized to arbitrary dimensions. Thus, the
applicability of chimera control goes beyond chimera states in non-locally
coupled systems
Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed
Stabilizing unstable periodic orbits in a chaotic invariant set not only
reveals information about its structure but also leads to various interesting
applications. For the successful application of a chaos control scheme,
convergence speed is of crucial importance. Here we present a predictive
feedback chaos control method that adapts a control parameter online to yield
optimal asymptotic convergence speed. We study the adaptive control map both
analytically and numerically and prove that it converges at least linearly to a
value determined by the spectral radius of the control map at the periodic
orbit to be stabilized. The method is easy to implement algorithmically and may
find applications for adaptive online control of biological and engineering
systems.Comment: 21 pages, 6 figure
Chaos in generically coupled phase oscillator networks with nonpairwise interactions
The Kuramoto-Sakaguchi system of coupled phase oscillators, where interaction
between oscillators is determined by a single harmonic of phase differences of
pairs of oscillators, has very simple emergent dynamics in the case of
identical oscillators that are globally coupled: there is a variational
structure that means the only attractors are full synchrony (in-phase) or splay
phase (rotating wave/full asynchrony) oscillations and the bifurcation between
these states is highly degenerate. Here we show that nonpairwise coupling -
including three and four-way interactions of the oscillator phases - that
appears generically at the next order in normal-form based calculations, can
give rise to complex emergent dynamics in symmetric phase oscillator networks.
In particular, we show that chaos can appear in the smallest possible dimension
of four coupled phase oscillators for a range of parameter values
Controlling Chaos Faster
Predictive Feedback Control is an easy-to-implement method to stabilize
unknown unstable periodic orbits in chaotic dynamical systems. Predictive
Feedback Control is severely limited because asymptotic convergence speed
decreases with stronger instabilities which in turn are typical for larger
target periods, rendering it harder to effectively stabilize periodic orbits of
large period. Here, we study stalled chaos control, where the application of
control is stalled to make use of the chaotic, uncontrolled dynamics, and
introduce an adaptation paradigm to overcome this limitation and speed up
convergence. This modified control scheme is not only capable of stabilizing
more periodic orbits than the original Predictive Feedback Control but also
speeds up convergence for typical chaotic maps, as illustrated in both theory
and application. The proposed adaptation scheme provides a way to tune
parameters online, yielding a broadly applicable, fast chaos control that
converges reliably, even for periodic orbits of large period
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