217 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
Bifurcation Phenomena in Two-Dimensional Piecewise Smooth Discontinuous Maps
In recent years the theory of border collision bifurcations has been
developed for piecewise smooth maps that are continuous across the border, and
has been successfully applied to explain nonsmooth bifurcation phenomena in
physical systems. However, many switching dynamical systems have been found to
yield two-dimensional piecewise smooth maps that are discontinuous across the
border. The theory for understanding the bifurcation phenomena in such systems
is not available yet. In this paper we present the first approach to the
problem of analysing and classifying the bifurcation phenomena in
two-dimensional discontinuous maps, based on a piecewise linear approximation
in the neighborhood of the border. We explain the bifurcations occurring in the
static VAR compensator used in electrical power systems, using the theory
developed in this paper. This theory may be applied similarly to other systems
that yield two-dimensional discontinuous maps
Delay-induced multistability near a global bifurcation
We study the effect of a time-delayed feedback within a generic model for a
saddle-node bifurcation on a limit cycle. Without delay the only attractor
below this global bifurcation is a stable node. Delay renders the phase space
infinite-dimensional and creates multistability of periodic orbits and the
fixed point. Homoclinic bifurcations, period-doubling and saddle-node
bifurcations of limit cycles are found in accordance with Shilnikov's theorems.Comment: Int. J. Bif. Chaos (2007), in prin
Chaotic scattering in solitary wave interactions: A singular iterated-map description
We derive a family of singular iterated maps--closely related to Poincare
maps--that describe chaotic interactions between colliding solitary waves. The
chaotic behavior of such solitary wave collisions depends on the transfer of
energy to a secondary mode of oscillation, often an internal mode of the pulse.
Unlike previous analyses, this map allows one to understand the interactions in
the case when this mode is excited prior to the first collision. The map is
derived using Melnikov integrals and matched asymptotic expansions and
generalizes a ``multi-pulse'' Melnikov integral and allows one to find not only
multipulse heteroclinic orbits, but exotic periodic orbits. The family of maps
derived exhibits singular behavior, including regions of infinite winding. This
problem is shown to be a singular version of the conservative Ikeda map from
laser physics and connections are made with problems from celestial mechanics
and fluid mechanics.Comment: 29 pages, 17 figures, submitted to Chaos, higher-resolution figures
available at author's website: http://m.njit.edu/goodman/publication
Homoclinic organization in the Hindmarsh-Rose model: A three parameter study
Bursting phenomena are found in a wide variety of fast-slow systems. In this article, we consider the Hindmarsh-Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast-slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast-slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh-Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book, " gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme
Computational Study in Chaotic Dynamical Systems and Mechanisms for Pattern Generation in Three-Cell Networks
A computational technique is introduced to reveal the complex intrinsic structure of homoclinic and heteroclinic bifurcations in a chaotic dynamical system. This technique is applied to several Lorenz-like systems with a saddle at the center, including the Lorenz system, the Shimizu-Morioka model, the homoclinic garden model, and the laser model. A multi-fractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities is explored on a bi-parametric plane of those dynamical systems. Also a great detail is explored in the Shimizu-Morioka model as an example. The technique is also applied to a re exion symmetric dynamical system with a saddle-focus at the center (Chua\u27s circuits). The layout of the homoclinic bifurcations near the primary one in such a system is studied theoretically, and a scalability ratio is proved. Another part of the dissertation explores the intrinsic mechanisms of escape in a reciprocally inhibitory FitzHugh-Nagumo type threecell network, using the phase-lag technique. The escape network can produce phase-locked states such as pace-makers, traveling-waves, and peristaltic patterns with recurrently phaselag varying
Homoclinic puzzles and chaos in a nonlinear laser model
We present a case study elaborating on the multiplicity and self-similarity
of homoclinic and heteroclinic bifurcation structures in the 2D and 3D
parameter spaces of a nonlinear laser model with a Lorenz-like chaotic
attractor. In a symbiotic approach combining the traditional parameter
continuation methods using MatCont and a newly developed technique called the
Deterministic Chaos Prospector (DCP) utilizing symbolic dynamics on fast
parallel computing hardware with graphics processing units (GPUs), we exhibit
how specific codimension-two bifurcations originate and pattern regions of
chaotic and simple dynamics in this classical model. We show detailed
computational reconstructions of key bifurcation structures such as Bykov
T-point spirals and inclination flips in 2D parameter space, as well as the
spatial organization and 3D embedding of bifurcation surfaces, parametric
saddles, and isolated closed curves (isolas).Comment: 28 pages, 23 figure
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