434 research outputs found
Multistable behavior above synchronization in a locally coupled Kuramoto model
A system of nearest neighbors Kuramoto-like coupled oscillators placed in a
ring is studied above the critical synchronization transition. We find a
richness of solutions when the coupling increases, which exists only within a
solvability region (SR). We also find that they posses different
characteristics, depending on the section of the boundary of the SR where the
solutions appear. We study the birth of these solutions and how they evolve
when {K} increases, and determine the diagram of solutions in phase space.Comment: 8 pages, 10 figure
Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators
We show that a wide class of uncoupled limit cycle oscillators can be
in-phase synchronized by common weak additive noise. An expression of the
Lyapunov exponent is analytically derived to study the stability of the
noise-driven synchronizing state. The result shows that such a synchronization
can be achieved in a broad class of oscillators with little constraint on their
intrinsic property. On the other hand, the leaky integrate-and-fire neuron
oscillators do not belong to this class, generating intermittent phase slips
according to a power low distribution of their intervals.Comment: 10 pages, 3 figure
Helicoidal instability of a scroll vortex in three-dimensional reaction-diffusion systems
We study the dynamics of scroll vortices in excitable reaction-diffusion
systems analytically and numerically. We demonstrate that intrinsic
three-dimensional instability of a straight scroll leads to the formation of
helicoidal structures. This behavior originates from the competition between
the scroll curvature and unstable core dynamics. We show that the obtained
instability persists even beyond the meander core instability of
two-dimensional spiral wave.Comment: 4 pages, 5 figures, revte
Forced Symmetry Breaking from SO(3) to SO(2) for Rotating Waves on the Sphere
We consider a small SO(2)-equivariant perturbation of a reaction-diffusion
system on the sphere, which is equivariant with respect to the group SO(3) of
all rigid rotations. We consider a normally hyperbolic SO(3)-group orbit of a
rotating wave on the sphere that persists to a normally hyperbolic
SO(2)-invariant manifold . We investigate the effects of this
forced symmetry breaking by studying the perturbed dynamics induced on
by the above reaction-diffusion system. We prove that depending
on the frequency vectors of the rotating waves that form the relative
equilibrium SO(3)u_{0}, these rotating waves will give SO(2)-orbits of rotating
waves or SO(2)-orbits of modulated rotating waves (if some transversality
conditions hold). The orbital stability of these solutions is established as
well. Our main tools are the orbit space reduction, Poincare map and implicit
function theorem
Scroll waves in isotropic excitable media : linear instabilities, bifurcations and restabilized states
Scroll waves are three-dimensional analogs of spiral waves. The linear
stability spectrum of untwisted and twisted scroll waves is computed for a
two-variable reaction-diffusion model of an excitable medium. Different bands
of modes are seen to be unstable in different regions of parameter space. The
corresponding bifurcations and bifurcated states are characterized by
performing direct numerical simulations. In addition, computations of the
adjoint linear stability operator eigenmodes are also performed and serve to
obtain a number of matrix elements characterizing the long-wavelength
deformations of scroll waves.Comment: 30 pages 16 figures, submitted to Phys. Rev.
Theory of spiral wave dynamics in weakly excitable media: asymptotic reduction to a kinematic model and applications
In a weakly excitable medium, characterized by a large threshold stimulus,
the free end of an isolated broken plane wave (wave tip) can either rotate
(steadily or unsteadily) around a large excitable core, thereby producing a
spiral pattern, or retract causing the wave to vanish at boundaries. An
asymptotic analysis of spiral motion and retraction is carried out in this
weakly excitable large core regime starting from the free-boundary limit of the
reaction-diffusion models, valid when the excited region is delimited by a thin
interface. The wave description is shown to naturally split between the tip
region and a far region that are smoothly matched on an intermediate scale.
This separation allows us to rigorously derive an equation of motion for the
wave tip, with the large scale motion of the spiral wavefront slaved to the
tip. This kinematic description provides both a physical picture and exact
predictions for a wide range of wave behavior, including: (i) steady rotation
(frequency and core radius), (ii) exact treatment of the meandering instability
in the free-boundary limit with the prediction that the frequency of unstable
motion is half the primary steady frequency (iii) drift under external actions
(external field with application to axisymmetric scroll ring motion in
three-dimensions, and spatial or/and time-dependent variation of excitability),
and (iv) the dynamics of multi-armed spiral waves with the new prediction that
steadily rotating waves with two or more arms are linearly unstable. Numerical
simulations of FitzHug-Nagumo kinetics are used to test several aspects of our
results. In addition, we discuss the semi-quantitative extension of this theory
to finite cores and pinpoint mathematical subtleties related to the thin
interface limit of singly diffusive reaction-diffusion models
Dynamics of lattice spins as a model of arrhythmia
We consider evolution of initial disturbances in spatially extended systems
with autonomous rhythmic activity, such as the heart. We consider the case when
the activity is stable with respect to very smooth (changing little across the
medium) disturbances and construct lattice models for description of
not-so-smooth disturbances, in particular, topological defects; these models
are modifications of the diffusive XY model. We find that when the activity on
each lattice site is very rigid in maintaining its form, the topological
defects - vortices or spirals - nucleate a transition to a disordered,
turbulent state.Comment: 17 pages, revtex, 3 figure
Optimal phase response curves for stochastic synchronization of limit-cycle oscillators by common Poisson noise
We consider optimization of phase response curves for stochastic
synchronization of non-interacting limit-cycle oscillators by common Poisson
impulsive signals. The optimal functional shape for sufficiently weak signals
is sinusoidal, but can differ for stronger signals. By solving the
Euler-Lagrange equation associated with the minimization of the Lyapunov
exponent characterizing synchronization efficiency, the optimal phase response
curve is obtained. We show that the optimal shape mutates from a sinusoid to a
sawtooth as the constraint on its squared amplitude is varied.Comment: 15 pages, 7 figure
The branch processes of vortex filaments and Hopf Invariant Constraint on Scroll Wave
In this paper, by making use of Duan's topological current theory, the
evolution of the vortex filaments in excitable media is discussed in detail.
The vortex filaments are found generating or annihilating at the limit points
and encountering, splitting, or merging at the bifurcation points of a complex
function . It is also shown that the Hopf invariant of knotted
scroll wave filaments is preserved in the branch processes (splitting, merging,
or encountering) during the evolution of these knotted scroll wave filaments.
Furthermore, it also revealed that the "exclusion principle" in some chemical
media is just the special case of the Hopf invariant constraint, and during the
branch processes the "exclusion principle" is also protected by topology.Comment: 9 pages, 5 figure
Theory of Spike Spiral Waves in a Reaction-Diffusion System
We discovered a new type of spiral wave solutions in reaction-diffusion
systems --- spike spiral wave, which significantly differs from spiral waves
observed in FitzHugh-Nagumo-type models. We present an asymptotic theory of
these waves in Gray-Scott model. We derive the kinematic relations describing
the shape of this spiral and find the dependence of its main parameters on the
control parameters. The theory does not rely on the specific features of
Gray-Scott model and thus is expected to be applicable to a broad range of
reaction-diffusion systems.Comment: 4 pages (REVTeX), 2 figures (postscript), submitted to Phys. Rev.
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