Chaotic Invariant Set Research Articles

Stabilizing periodic orbits in a chaotic semiconductor laser

We consider the single mode dynamics of a self-pulsating semiconductor laser model under modulated injection current. The observation of a chaotic regime for a wide range of parameters suggests the usefulness of this model for high-speed secure optical communications. A possible way to achieve symbol encoding by using such a laser as a light source is to control its chaotic dynamics. We study the stabilization of periodic orbits embedded in the chaotic invariant set of the system by applying perturbations on the amplitude of the injection current.

Chaotic Invariant Sets of High-Dimensional Hénon-Like Maps

High-dimensional Hénon-like maps have many applications in the research of spatial chaos and traveling waves of extended systems. Meanwhile, they are of great interest in their own right. The aim of this paper is, by applying the implicit function theorem, to show for high-dimensional Hénon-like maps the existence of chaotic invariant sets and the density of homoclinic points and heteroclinic points in them. Our method is motivated by Aubry's “anti-integrability” concept and is rather different from the traditional techniques such as horseshoes, transversal homoclinic points and heteroclinic cycles, and snap-back repellers.

Wada basins and chaotic invariant sets in the Hénon-Heiles system.

The Hénon-Heiles Hamiltonian is investigated in the context of chaotic scattering, in the range of energies where escaping from the scattering region is possible. Special attention is paid to the analysis of the different nature of the orbits, and the the invariant sets, such as the stable and unstable manifolds and the chaotic saddle. Furthermore, a discussion on the average decay time associated to the typical chaotic transients, which are present in this problem, is presented. The main goal of this paper is to show, by using various computational methods, that the corresponding exit basins of this open Hamiltonian are not only fractal, but they also verify the more restrictive property of Wada. We argue that this property is verified by typical open Hamiltonian systems with three or more escapes.

DRIVING TRAJECTORIES IN CHAOTIC SYSTEMS

This work presents the main ideas and the fundamental procedures for guiding trajectories on chaotic systems and for stabilizing chaotic orbits, all with the use of small perturbations. We consider an extension of the Ott–Grebogi–Yorke method of controlling chaos and an associated procedure for guiding trajectories on chaotic sets of high dimensional systems. We argue that those techniques can be used even if the chaotic invariant set is nonattractive. As nonattractive chaotic invariant sets commonly exist embedded in high dimensional systems, we also argue that we can combine chaotic control techniques with system control strategies to generate a powerful mechanism for manipulating the dynamics. We present an example in which we change and alter system's evolution at will using only small perturbations to some accessible parameter.

Topological aspects of chaotic scattering in higher dimensions.

We investigate the topological properties of the chaotic invariant set associated with the dynamics of scattering systems with three or more degrees of freedom. We show that the separation of one degree of freedom from the rest in the asymptotic regime, a common property in a large class of scattering models, defines a gate which is a dynamical object with phase space separating invariant manifolds. The manifolds form an invariant set causing singularities in the scattering process. The codimension one property of the manifolds ensures that the fractal structure of the invariant set can be studied by scattering functions defined over simple one-dimensional families of initial conditions as usually done in two-degree-of-freedom scattering problems. It is found that the fractal dimension of the invariant set is not due to the gates but to interior hyperbolic periodic orbits.

Characterization of transition to chaos with multiple positive Lyapunov exponents by unstable periodic orbits

We investigate how the transition to chaos with multiple positive Lyapunov exponents can be characterized by the set of infinite number of unstable periodic orbits embedded in the chaotic invariant set. We argue and provide numerical confirmation that the transition is generally accompanied by a nonhyperbolic behavior: unstable dimension variability. As a consequence, the Lyapunov exponents, except for the largest one, pass through zero continuously.

OBSTRUCTION TO DETERMINISTIC MODELING OF CHAOTIC SYSTEMS WITH AN INVARIANT SUBSPACE

A natural process can be deterministically modeled if solutions from its mathematical model stay close to the ones produced by nature. The mathematical model, however, is not exact due to imperfections of the natural system. We describe, in this paper, that there exists a class of models of chaotic processes, for which severe obstruction to deterministic modeling may arise. In particular, such obstruction may occur when unstable periodic orbits embedded in the chaotic invariant set have a distinct number of unstable directions, a type of nonhyperbolicity called unstable-dimension variability. We make these ideas concrete by investigating a class of deterministic models: chaotic systems with an invariant subspace such as systems of coupled chaotic oscillators. We show that unstable-dimension variability can occur in wide parameter regimes of these systems. The implications of our results to scientific modeling are discussed.

Estimating generating partitions of chaotic systems by unstable periodic orbits

An outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1. It has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition. Here we present a general, dimension-independent, and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits. Our algorithm allows us to obtain the approximate location of the generating partition for the Ikeda-Hammel-Jones-Moloney map.

On properties of hyperchaos: Case study

Some properties of hyperchaos are exploited by studying both uncoupled and coupled CML. In addition to usual properties of chaotic strange attractors, there are other interesting properties, such as: the number of unstable periodic points embedded in the strange attractor increases dramatically increasing and a large number of low-dimensional chaotic invariant sets are contained in the strange attractor. These properties may be useful for regarding the edge of chaos as the origin of complexity of dynamical systems.

Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection

We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this ‘cycling chaos’ manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets. Such behaviour is robust to perturbations that preserve the symmetry of the system; we examine bifurcations of this state. We discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace. This differs from the standard scenario for the blowout bifurcation in that in our case, the blowout is neither subcritical nor supercritical. The robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors. The model we consider is a ninth-order truncated ordinary differential equation (ODE) model of three-dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient. Symmetries of the model lead to the existence of invariant subspaces for the dynamics; in particular there are invariant subspaces that correspond to regimes of two-dimensional flows, with variation in the vertical but only one of the two horizontal directions. Stable two-dimensional chaotic flow can go unstable to three-dimensional flow via the cross-roll instability. We show how the bifurcations mentioned above can be located by examination of various transverse Liapunov exponents. We also consider a reduction of the ODE to a map and demonstrate that the same behaviour can be found in the corresponding map. This allows us to describe and predict a number of observed transitions in these models. The dynamics we describe is new but nonetheless robust, and so should occur in other applications.

About Two Mechanisms of Reunion of Chaotic Attractors

Considering a family of two-dimensional piecewise linear maps, we discuss two different mechanisms of reunion of two (or more) pieces of cyclic chaotic attractors into a one-piece attracting set, observed in several models. It is shown that, in the case of so-called contact bifurcation of the 2 nd kind, the reunion occurs immediately due to homoclinic bifurcation of some saddle cycle belonging to the basin boundary of the attractor. In the case of so-called contact bifurcation of the 1 st kind, the reunion is a result of a contact of the attractor with its basin boundary which is fractal, including the stable set of a chaotic invariant hyperbolic set appeared after the homoclinic bifurcation of a saddle cycle on the basin boundary

Calculating topological entropy for transient chaos with an application to communicating with chaos

Recent work on communicating with chaos provides a practical motivation for being able to determine numerically the topological entropy for chaotic invariant sets. In this paper we discuss numerical methods for evaluating topological entropy. To assess the accuracy and convergence of the methods, we test them in situations where the topological entropy is known independently. We also discuss the entropy of invariant chaotic saddles formed by those points in a given attractor that never visit some forbidden {open_quotes}gap{close_quotes} region. Such gaps have been proposed as a means of providing noise immunity in schemes for communication with chaos, and we discuss the dependence of the topological entropy on the size of the gap. {copyright} {ital 1998} {ital The American Physical Society}

Complex Dynamics in a Harmonically Excited Lennard-Jones Oscillator: Microcantilever-Sample Interaction in Scanning Probe Microscopes1

In this paper we model the microcantilever-sample interaction in an atomic force microscope (AFM) via a Lennard-Jones potential and consider the dynamical behavior of a harmonically forced system. Using nonlinear analysis techniques on attracting limit sets, we numerically verify the presence of chaotic invariant sets. The chaotic behavior appears to be generated via a cascade of period doubling, whose occurrence has been studied as a function of the system parameters. As expected, the chaotic attractors are obtained for values of parameters predicted by Melnikov theory. Moreover, the numerical analysis can be fruitfully employed to analyze the region of the parameter space where no theoretical information on the presence of a chaotic invariant set is available. In addition to explaining the experimentally observed chaotic behavior, this analysis can be useful in finding a controller that stabilizes the system on a nonchaotic trajectory. The analysis can also be used to change the AFM operating conditions to a region of the parameter space where regular motion is ensured. [S0022-0434(00)01401-5]

Rotation intervals for chaotic sets

Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction.

Bifurcations and Chaos in a Linear Control System with Saturated Input

An analysis of the dynamical behavior of a saturated input, second-order, linear plant feed back with a simple proportional controller is presented. The synthesized control signal is passed through a saturator before being applied to the plant. Existence and stability of equilibria depending on the controller gain are analyzed. Existence of periodic and homoclinic orbits is also discussed. It is shown that a periodic disturbance and a negative gain can cause the closed-loop system to have a strange (chaotic) invariant set. Numerical and experimental results illustrate the analysis presented.

Cycles homoclinic to chaotic sets; robustness and resonance.

For dynamical systems possessing invariant subspaces one can have a robust homoclinic cycle to a chaotic set. If such a cycle is stable, it manifests itself as long periods of quiescent chaotic behaviour interrupted by sudden transient 'bursts'. The time between the transients increases as the trajectory approaches the cycle. This behavior for a cycle connecting symmetrically related chaotic sets has been called 'cycling chaos' by Dellnitz et al. [IEEE Trans. Circ. Sys. I 42, 821-823 (1995)]. We characterise such cycles and their stability by means of normal Lyapunov exponents. We find persistence of states that are not Lyapunov stable but still attracting, and also states that are approximately periodic. For systems possessing a skew-product structure (such as naturally arises in chaotically forced systems) we show that the asymptotic stability and the attractivity of the cycle depends in a crucial way on what we call the footprint of the cycle. This is the spectrum of Lyapunov exponents of the chaotic invariant set in the expanding and contracting directions of the cycle. Numerical simulations and calculations for an example system of a homoclinic cycle parametrically forced by a Rossler attractor are presented; here we observe the creation of nearby chaotic attractors at resonance of transverse Lyapunov exponents. (c) 1997 American Institute of Physics.

Analysis of the separatrix map in Hamiltonian systems

Abstract A one-degree-of-freedom Hamiltonian system with time periodic perturbation generally display rich dynamics near the separatrix, the understanding of which is important in studying transport and diffusion. We introduce the exact separatrix map which provides an effective way to describe dynamics near the separatrix. We show that the separatrix map and its inverse satisfy twist conditions with logarithmic singularities and, therefore, they have an infinite twist near the transversal intersections of the stable and unstable manifolds of the hyperbolic equilibrium. Using the variational formalism with the generating functions, we show the dynamics near the singularities correspond to the anti-integrable limit. As a consequence, we show the nonexistence of rotational invariant circles with sufficiently large rotation number, first obtained by Lazutkin in standard maps. Moreover, the existence of a uniformly hyperbolic chaotic invariant set is shown in connection with the theory of the anti-integrable limit. The simplest form of the separatrix map is studied in detail to illustrate some of the above results explicitly.

Bifurcation, chaos, and voltage collapse in power systems

A model of a power system with load dynamics is studied by investigating qualitative changes in its behavior as the reactive power demand at a load bus is increased. In addition to the saddle node bilification often associated with voltage collapse, the power system exhibits sub- and supercritical Hopf bifurcations, cyclic fold bifurcation, and period doubling bifurcation. Cascades of period doubling bifurcation terminate in chaotic invariant sets. The presence of these new bifurcations motivates a reexamination of the saddle-node bifurcation as the boundary of the feasible set of power injections.

Crisis and enhancement of chaotic scattering.

Chaotic scattering is characterized by the existence of nonattracting chaotic invariant sets in phase space. There can be several chaotic invariant sets coexisting in phase space when a system parameter value is below some critical value. As the parameter changes through the critical value, stable and unstable foliations of these chaotic invariant sets, which are fractal sets, can become tangent and then cross each other. The first tangency, which provides the linking between chaotic invariant sets, is a crisis in chaotic scattering. Above the crisis, there is an infinite number of such tangencies which keep occurring until the last tangency, above which the stable and unstable foliations cross transversely. As a consequence of this, the fractal dimension of the set of singularities in the scattering function increases in the parameter range determined by the first and the last tangencies. This leads to a proliferation of singularities in the scattering function and, consequently, to an enhancement of chaotic scattering. The phenomenon is investigated by using both simple one-dimensional models and a two-dimensional physical scattering system.

Crisis in chaotic scattering.

We show that in a chaotic scattering system the stable and unstable foliations of isolated chaotic invariant sets can become heteroclinically tangent to each other at an uncountably infinite number of parameter values. The first tangency, which is a crisis in chaotic scattering, provides the link between the chaotic sets. A striking consequence is that the fractal dimension of the set of singularities in the scattering function increases in the parameter range determined by the first and the last tangencies. This leads to a proliferation of singularities in the scattering function and, consequently, to an enhancement of chaotic scattering.