Chaotic Saddle Research Articles

Dynamics of a Hénon–Lozi-type map

We present and analyze a smooth version of the piecewise linear Lozi map. The principal motivation for this work is to develop a map, which is better amenable for an analytical treatment as compared to the Hénon map and is one that still possesses the characteristics of a Hénon-type dynamics. This paper is a first step. It does the comparison of the Lozi map (which is a piecewise linear version of the Hénon map) with the map that we introduce. This comparison is done for fixed parameters and also through global bifurcation by changing a parameter. If ε measures the degree of smoothness, we prove that, as ε → 0 , the stability and the existence of the fixed points are the same for both maps. We also numerically compare the chaotic dynamics, both in the form of an attractor and of a chaotic saddle.

Partially nearly riddled basins in systems with chaotic saddle

We show that chaotic attractors can have partially nearly riddled basins of attraction, i.e., basins which consist both of large open sets and a set in which small open sets which belong to the basins of different attractors are intermingled. We argue that such basins are robust for systems with the chaotic saddle located between at least two attractors and in the presence of noise cause the uncertainties similar to those implied by riddled basins.

Stagger-and-step method: detecting and computing chaotic saddles in higher dimensions.

Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices. These transients are caused by the presence of chaotic saddles, and they are a common phenomenon in higher dimensional dynamical systems. For many physical systems, chaotic saddles have a big impact on laboratory measurements, but there has been no way to observe these chaotic saddles directly. We present the first general method to locate and visualize chaotic saddles in higher dimensions.

DISCONTINUOUS BIFURCATIONS OF CHAOTIC ATTRACTORS IN FORCED OSCILLATORS BY GENERALIZED CELL MAPPING DIGRAPH (GCMD) METHOD

Discontinuous bifurcations of chaotic attractors are investigated by means of Generalized Cell Mapping Digraph (GCMD) method. The two categories of discontinuous bifurcations of chaotic attractors are found in typical systems described by dissipative ordinary differential equations, namely catastrophic and explosive bifurcations. For the case of a catastrophe, the two different types, namely a regular saddle and a chaotic saddle catastrophe, are studied, in particular, the subtle distinction between them can be explicitly observed in phase portrait. We show that a regular saddle catastrophe results from a collision between a chaotic attractor and a regular saddle on its smooth basin boundary. In such a case the chaotic attractor is suddenly destroyed as the parameter passes through a critical value, leaving behind a nonattracting chaotic saddle in the place of the original chaotic attractor in phase space. We demonstrate that a chaotic saddle catastrophe results from a collision between a chaotic attractor and a chaotic saddle in its fractal basin boundary. In such a case the chaotic attractor is suddenly destroyed as the parameter passes through a critical value, simultaneously the chaotic saddle also undergoes an abrupt enlargement in its size, namely, the chaotic attractor is converted into a new incremental portion of the chaotic saddle after the collision. For the case of an explosion, a chaotic saddle explosion is studied, in which there is a sudden increase in the size of a chaotic attractor as the parameter passes through a critical value. We demonstrate that at the chaotic saddle explosion the chaotic attractor collides with a chaotic saddle already existent when the explosion occurs. This chaotic saddle is an invariant and nonattracting set. We also investigate the origin and evolution of the chaotic saddle. A locally refining process of Generalized Cell Mapping (GCM) method is developed to refine persistent and transient self-cycling sets. The refining procedures of persistent and transient self-cycling sets are respectively given on the basis of their definitions in the cell state space.

Pseudo-riddling in chaotic systems

Previous investigations of riddling have focused in the case where the dynamical invariant set in the symmetric invariant subspace of the system is a chaotic attractor. A situation expected to arise commonly in dynamical systems, however, is that the dynamics in the invariant subspace is in a periodic window. In such a case, there are both a stable periodic attractor and a coexisting non-attracting chaotic saddle in the invariant subspace. We show that riddling can still occur in a generalized sense. In particular, we argue that the basin of the periodic attractor in the invariant subspace contains both an open set and a set of measure zero with riddled holes that belong to the basin of another attractor. We call such basin pseudo-riddled and we argue that pseudo-riddling can be more pervasive than riddling of chaotic attractors because it can occur regardless of whether the chaotic saddle in the invariant subspace is transversely stable or unstable. We construct an analyzable model, derive a scaling law for pseudo-riddling, and provide numerical support. We also show that the chaotic saddle in the invariant subspace can undergo an enlargement into the full phase space when it becomes transversely unstable.

A NEW TYPE OF BOUNDARY CRISES: CHAOTIC BOUNDARY CRISES

Crises of chaotic attractors are typical phenomena in nonlinear dynamical systems. By means of generalized cell mapping digraph (GCMD) method, we show that a chaotic boundary crisis results from a collision between a chaotic attractor and a chaotic saddle in the fractal basin boundary. In such a case the chaotic attractor, together with its basin of attraction, is suddenly destroyed as the parameter passes through a critical value, simultaneously the chaotic saddle also undergoes an abrupt enlargement in its size.

Bi-instability and the global role of unstable resonant orbits in a driven laser

Driven class-B lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior. We describe the onset of global heteroclinic connections which give rise to chaotic saddles. These form the precursor topology which creates both localized homoclinic chaos, as well as global mixed-mode heteroclinic chaos. To locate the relevant periodic orbits creating the precursor topology, approximate maps are derived using matched asymptotic expansions and subharmonic Melnikov theory. Locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser.

Explosions of chaotic sets

Large-scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied. These bifurcations include sudden changes in the size and/or type of the set. An explosion is a bifurcation in which new recurrent points suddenly appear at a non-zero distance from any pre-existing recurrent points. We discuss the following. In a generic one-parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur: (1) a saddle-node bifurcation isolated from other recurrent points, (2) a saddle-node bifurcation embedded in the set of recurrent points, (3) outer homoclinic tangencies, and (4) outer heteroclinic tangencies. (The term “outer tangency” refers to a particular configuration of the stable and unstable manifolds at tangency.) In particular, we examine different types of tangencies of stable and unstable manifolds from orbits of pre-existing invariant sets. This leads to a general theory that unites phenomena such as crises, basin boundary metamorphoses, explosions of chaotic saddles, etc. We illustrate this theory with numerical examples.

Devil-staircase behavior of dynamical invariants in chaotic scattering

A crisis in chaotic scattering is characterized by the merging of two or more nonattracting chaotic saddles. The fractal dimension of the resulting chaotic saddle increases through the crisis. We present a rigorous analysis for the behavior of dynamical invariants associated with chaotic scattering by utilizing a representative model system that captures the essential dynamical features of crisis. Our analysis indicates that the fractal dimension and other dynamical invariants are a devil-staircase type of function of the system parameter. Our results can also provide insight for similar devil-staircase behaviors observed in the parametric evolution of chaotic saddles of general dissipative dynamical systems and in communicating with chaos.

Studying hyperbolicity in chaotic systems

On the basis of method [1] proposed for diagnosing 2-dimensional chaotic saddles we present a numerical procedure to distinguish hyperbolic and nonhyperbolic chaotic attractors in three-dimensional flow systems. This technique is based on calculating the angles between stable and unstable manifolds along a chaotic trajectory in R 3 . We show for three-dimensional flow systems that this serves as an efficient characteristic for exploring chaotic differential systems. We also analyze the effect of noise on the structure of angle distribution for both 2-dimensional invertible maps and a 3-dimensional continuous system.

ENCODING DIGITAL INFORMATION USING TRANSIENT CHAOS

Recent work has demonstrated that symbolic representations of controlled chaotic orbits can be utilized for encoding digital information. So far, this idea has been demonstrated using systems exhibiting sustained chaotic motion on chaotic attractors. The purpose of this work is to explore digital encoding by using transient chaos naturally arising in nonlinear systems. Dynamically, transient chaos is caused by nonattracting chaotic saddles. We argue that there are two major advantages when trajectories on chaotic saddles are exploited as information source: (1) the channel capacity can in general be large, and (2) the influence of channel noise can be reduced. We present a control scheme and also a practical example of encoding a digital message.

TRACKING SUSTAINED CHAOS

Tracking unstable periodic states first introduced in [Schwartz & Triandaf, 1992] is the process of continuing unstable solutions as a systems parameter is varied in experiments. The tracked dynamical objects have been periodic saddles of well-defined finite periods. However, other saddles, such as chaotic saddles, have not been successfully "tracked," or continued. In this paper, we introduce a new yet simple method which can be used to track chaotic saddles in dynamical systems, which allows an experimentalist to sustain chaotic transients far away from crisis parameter values. The method is illustrated on a periodically driven CO 2 laser.

FURTHER STUDY OF INTERIOR CRISES IN SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Crises in systems of ordinary differential equations are investigated by means o f Generalized Cell Mapping Digraph (GCMD) method. We show that a boundary crisis results from a collision between a chaotic attractor and a periodic saddle on i ts basin boundary. In such a case the chaotic attractor, together with its basin of attraction, is suddenly destroyed as the parameter passes through a critical value, leaving behind a nonattracting chaotic saddle in the place of the origin al chaotic attractor in phase space. We focus here on a sudden change in the siz e of a chaotic attractor, namely an interior crisis. We demonstrate that at an i nterior crisis the chaotic attractor collides with a chaotic saddle within its b asin of attraction. This chaotic saddle is an invariant and nonattracting set an d resembles the new portion of the larger chaotic attractor just after the inter ior crisis. We also investigate the origin and evolution of the chaotic saddle. The local refining procedures of persistent and transient self-cycling sets are given.

Abrupt bifurcation to chaotic scattering with discontinuous change in fractal dimension.

One of the major routes to chaotic scattering is through an abrupt bifurcation by which a nonattracting chaotic saddle is created as a system parameter changes through a critical value. In a previously investigated case, however, the fractal dimension of the set of singularities in the scattering function changes continuously through the bifurcation. We describe a type of abrupt bifurcation to chaotic scattering where this physically relevant dimension changes discontinuously at the bifurcation. The bifurcation is illustrated using a class of open Hamiltonian systems consisting of Morse potential hills.

Unstable periodic orbits and the natural measure of nonhyperbolic chaotic saddles.

Chaotic saddles are nonattracting dynamical invariant sets that physically lead to transient chaos. We examine the characterization of the natural measure by unstable periodic orbits for nonhyperbolic chaotic saddles in dissipative dynamical systems. In particular, we compare the natural measure obtained from a long trajectory on the chaotic saddle to that evaluated from unstable periodic orbits embedded in it. Our systematic computations indicate that the periodic-orbit theory of the natural measure, previously shown to be valid only for hyperbolic chaotic sets, is applicable to nonhyperbolic chaotic saddles as well.

Continued Fractions Hierarchy of Rotation Numbers in Planar Dynamics

Global bifurcations such as crises of attractors, explosions of chaotic saddles, and metamorphoses of basin boundaries play a crucial role in understanding the dynamical evolution of physical systems. Global bifurcations in dissipative planar maps are typically caused by collisions of invariant manifolds of periodic orbits, whose dynamical behaviors are described by rotation numbers. We show that the rotation numbers of the periodic orbits created at certain important tangencies are determined by the continued fraction expansion of the rotation number of the orbit involved in the collision.

Metamorphosis of chaotic saddle

Chaotic saddles are nonattracting dynamical invariant sets that can lead to a variety of physical phenomena. We report our finding and analysis of a type of discontinuous global bifurcation (metamorphosis) of chaotic saddle that occurs in high-dimensional chaotic systems with an invariant manifold. A metamorphosis occurs when a chaotic saddle, lying in the manifold, loses stability with respect to perturbations transverse to the invariant manifold. The fractal dimension of the chaotic saddle increases abruptly through the bifurcation. We illustrate our finding by using a system of coupled maps.

The Inverse Scattering Problem for Chaotic Hamiltonian Systems

We propose an analysis of the inverse scattering problem for chaotic Hamiltonian systems. Our main goal will be the reconstruction of the structure of the chaotic saddle from asymptotic data. We will also address the question how to obtain thermodynamic measures and a partition from these data. An essential step in achieving this is the reconstruction of the hierarchical order of the fractal structure of singularities in scattering functions solely from knowledge of asymptotic data. This provides a branching tree which coincides with the branching tree derived from the hyperbolic component of the horseshoe in the Poincaré map taken in the interaction region. We achieve our goal explicitly for two types of systems governed by an external or an internal clock, respectively. Once we have achieved this goal, a discrete arbitrariness remains for the reconstruction of the horseshoe. Here symmetry considerations can help. We discuss the implications for the inverse scattering problem of the effects of finite resolution and the possible use of nonhyperbolic effects. The connection between the formal development parameter of the horseshoe and the topological entropy proves helpful in the systems discussed.

On the entropy devil’s staircase in a family of gap-tent maps

To analyze the trade-off between channel capacity and noise-resistance in designing dynamical systems to pursue the idea of communications with chaos, we perform a measure theoretic analysis the topological entropy function of a ‘gap-tent map’ whose invariant set is an unstable chaotic saddle of the tent map. Our model system, the ‘gap-tent map’ is a family of tent maps with a symmetric gap, which mimics the presence of noise in physical realizations of chaotic systems, and for this model, we can perform many calculations in closed form. We demonstrate that the dependence of the topological entropy on the size of the gap has a structure of the devil’s staircase. By integrating over a fractal measure, we obtain analytical, piece-wise differentiable approximations of this dependence. Applying concepts of the kneading theory we find the position and the values of the entropy for all leading entropy plateaus. Similar properties hold also for the dependence of the fractal dimension of the invariant set and the escape rate.

Bi-instability as a precursor to global mixed-mode chaos.

Bi-instability, in contrast to bistability, is shown to generate unstable chaotic saddles prior to the onset of chaos. The theory and numerics are applied to a CO2 laser model with modulated losses where unstable pairs of saddles coexist, form heteroclinic connections, and allow mixing between local chaotic attractors to produce global mixed-mode chaos.