113 research outputs found
Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial
A new computational technique based on the symbolic description utilizing
kneading invariants is proposed and verified for explorations of dynamical and
parametric chaos in a few exemplary systems with the Lorenz attractor. The
technique allows for uncovering the stunning complexity and universality of
bi-parametric structures and detect their organizing centers - codimension-two
T-points and separating saddles in the kneading-based scans of the iconic
Lorenz equation from hydrodynamics, a normal model from mathematics, and a
laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201
Symbolic Toolkit for Chaos Explorations
New computational technique based on the symbolic description utilizing
kneading invariants is used for explorations of parametric chaos in a two
exemplary systems with the Lorenz attractor: a normal model from mathematics,
and a laser model from nonlinear optics. The technique allows for uncovering
the stunning complexity and universality of the patterns discovered in the
bi-parametric scans of the given models and detects their organizing centers --
codimension-two T-points and separating saddles.Comment: International Conference on Theory and Application in Nonlinear
Dynamics (ICAND 2012
A propensity criterion for networking in an array of coupled chaotic systems
We examine the mutual synchronization of a one dimensional chain of chaotic
identical objects in the presence of a stimulus applied to the first site. We
first describe the characteristics of the local elements, and then the process
whereby a global nontrivial behaviour emerges. A propensity criterion for
networking is introduced, consisting in the coexistence within the attractor of
a localized chaotic region, which displays high sensitivity to external
stimuli,and an island of stability, which provides a reliable coupling signal
to the neighbors in the chain. Based on this criterion we compare homoclinic
chaos, recently explored in lasers and conjectured to be typical of a single
neuron, with Lorenz chaos.Comment: 4 pages, 3 figure
Smooth and Non-Smooth Dependence of Lyapunov Vectors upon the Angle Variable on a Torus in the Context of Torus-Doubling Transitions in the Quasiperiodically Forced Henon Map
A transition from a smooth torus to a chaotic attractor in quasiperiodically
forced dissipative systems may occur after a finite number of torus-doubling
bifurcations. In this paper we investigate the underlying bifurcational
mechanism which seems to be responsible for the termination of the
torus-doubling cascades on the routes to chaos in invertible maps under
external quasiperiodic forcing. We consider the structure of a vicinity of a
smooth attracting invariant curve (torus) in the quasiperiodically forced Henon
map and characterize it in terms of Lyapunov vectors, which determine
directions of contraction for an element of phase space in a vicinity of the
torus. When the dependence of the Lyapunov vectors upon the angle variable on
the torus is smooth, regular torus-doubling bifurcation takes place. On the
other hand, the onset of non-smooth dependence leads to a new phenomenon
terminating the torus-doubling bifurcation line in the parameter space with the
torus transforming directly into a strange nonchaotic attractor. We argue that
the new phenomenon plays a key role in mechanisms of transition to chaos in
quasiperiodically forced invertible dynamical systems.Comment: 24 pages, 9 figure
Plykin-like attractor in non-autonomous coupled oscillators
A system of two coupled non-autonomous oscillators is considered. Dynamics of
complex amplitudes is governed by differential equations with periodic
piecewise continuous dependence of the coefficients on time. The Poincar\'{e}
map is derived explicitly. With exclusion of the overall phase, on which the
evolution of other variables does not depend, the Poincar\'{e} map is reduced
to 3D mapping. It possesses an attractor of Plykin type located on an invariant
sphere. Computer verification of the cone criterion confirms the hyperbolic
nature of the attractor in the 3D map. Some results of numerical studies of the
dynamics for the coupled oscillators are presented, including the attractor
portraits, Lyapunov exponents, and the power spectral density.Comment: 11 pages, 9 figure
Self-tuning to the Hopf bifurcation in fluctuating systems
The problem of self-tuning a system to the Hopf bifurcation in the presence
of noise and periodic external forcing is discussed. We find that the response
of the system has a non-monotonic dependence on the noise-strength, and
displays an amplified response which is more pronounced for weaker signals. The
observed effect is to be distinguished from stochastic resonance. For the
feedback we have studied, the unforced self-tuned Hopf oscillator in the
presence of fluctuations exhibits sharp peaks in its spectrum. The implications
of our general results are briefly discussed in the context of sound detection
by the inner ear.Comment: 37 pages, 7 figures (8 figure files
Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps
We study dynamics and bifurcations of two-dimensional reversible maps having
non-transversal heteroclinic cycles containing symmetric saddle periodic
points. We consider one-parameter families of reversible maps unfolding
generally the initial heteroclinic tangency and prove that there are infinitely
sequences (cascades) of bifurcations of birth of asymptotically stable and
unstable as well as elliptic periodic orbits
Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories
Dynamical equations are formulated and a numerical study is provided for
self-oscillatory model systems based on the triple linkage hinge mechanism of
Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic
mechanical constraint of three rotators as well as systems, where three
rotators interact by potential forces. We present and discuss some quantitative
characteristics of the chaotic regimes (Lyapunov exponents, power spectrum).
Chaotic dynamics of the models we consider are associated with hyperbolic
attractors, at least, at relatively small supercriticality of the
self-oscillating modes; that follows from numerical analysis of the
distribution for angles of intersection of stable and unstable manifolds of
phase trajectories on the attractors. In systems based on rotators with
interacting potential the hyperbolicity is violated starting from a certain
level of excitation.Comment: 30 pages, 18 figure
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