Chaotic Invariant Set Research Articles

A nonintegrable model in general relativity

The geodesic flow of a perturbation of the Schwarzschild metric is shown to possess a chaotic invariant set. The perturbed meric is a relativistic analogue of Hill's problem in classical celestial mechanics in that is models the effects of a distant third body.

Chaos in a model of forced quasi-geostrophic flow over topography: an application of Melnikov's method

We demonstrate the existence of a chaotic invariant set of solutions of an idealized model for wind-forced quasi-geostrophic flow over a continental margin with variable topography. The model (originally formulated to investigate mean flow generation by topographic wave drag) has bottom topography that slopes linearly offshore and varies sinusoidally alongshore. The alongshore topographic scales are taken to be short compared to the cross-shelf scale, allowing Hart's (1979) quasi-two-dimensional approximation, and the governing equations reduce to a non-autonomous system of three coupled nonlinear ordinary differential equations. For weak (constant plus time-periodic) forcing and weak friction, we apply a recent extension (Wiggins & Holmes 1987) of the method of Melnikov (1963) to test for the existence of transverse homoclinic orbits in the model. The inviscid unforced equations have two constants of motion, corresponding to energy E and enstrophy M, and reduce to a one-degree-of-freedom Hamiltonian system which, for a range of values of the constant G = E − M, has a pair of homoclinic orbits to a hyperbolic saddle point. Weak forcing and friction cause slow variations in G, but for a range of parameter values one saddle point is shown to persist as a hyperbolic periodic orbit and Melnikov's method may be applied to study the perturbations of the associated homoclinic orbits. In the absence of time-periodic forcing, the hyperbolic periodic orbit reduces to the unstable fixed point that occurs with steady forcing and friction. The method yields analytical expressions for the parameter values for which sets of chaotic solutions exist for sufficiently weak time-dependent forcing and friction. The predictions of the perturbation analysis are verified numerically with computations of Poincaré sections for solutions in the stable and unstable manifolds of the hyperbolic periodic orbit and with computations of solutions for general initial-value problems. In the presence of constant positive wind stress τ0 (equatorward on eastern ocean boundaries), chaotic solutions exist when the ratio of the oscillatory wind stress τ1 to the bottom friction parameter r is above a critical value that depends on τ0/r and the bottom topographic height. The analysis complements a previous study of this model (Samelson & Allen 1987), in which chaotic solutions were observed numerically for weak near-resonant forcing and weak friction.

Transition to chaotic scattering.

This paper addresses the question of how chaotic scattering arises and evolves as a system parameter is continuously varied starting from a value for which the scattering is regular (i.e., not chaotic). Our results show that the transition from regular to chaotic scattering can occur via a saddle-center bifurcation, with further qualitative changes in the chaotic set resulting from a sequence of homoclinic and heteroclinic intersections. We also show that a state of ``fully developed'' chaotic scattering can be reached in our system through a process analogous to the formation of a Smale horse- shoe. By fully developed chaotic scattering, we mean that the chaotic-invariant set is hyperbolic, and we find for our problem that all bounded orbits can be coded by a full shift on three symbols. Observable consequences related to qualitative changes in the chaotic set are also discussed.

Bifurcation to chaotic scattering

We investigate a novel type of bifurcation to chaos which occurs in the context of chaotic scattering. In chaotic scattering the deflection angle versus impact parameter is singular on a set of impact parameters which is fractal. This behavior is caused by the presence of a chaotic invariant set of unstable bounded orbits. In the bifurcation considered here the chaotic set arises abruptly (in a sense to be discussed) as the particle energy E decreases from above a critical value E m, to below. We call this transition an abrupt bifurcation to fully developed chaotic scattering. Numerical computation of the dimension, d, of the chaotic set shows that, in agreement with theoretical prediction, d ≈ 1/In[(E m - E) −1] near the abrupt bifurcatio

Chaotic scattering in several dimensions

For chaotic scattering in two-degree-of-freedom ( N=2), time-independent, Hamiltonian systems, scattering functions (i.e., plots of the dependence of a phase space variable after scattering versus a phase space variable before scattering) typically display singularities on a fractal set. For N#62;2, however, scattering functions typically do not have fractal properties (even when the chaotic invariant set is fractal), unless the fractal dimension of the chaotic set is large enough. A numerical investigation of this phenomenon is presented for a scatterer consisting of four reflecting spheres at the vertices of a regular tetrahedron.

Nonlinear oscillators, iterated maps, symbolic dynamics, and knotted orbits

We illustrate some recent developments in the theory of dynamical systems. Concentrating on periodically forced second-order ordinary differential equations, and using the Josephson (pendulum) and Duffing equations as examples, we show how the method of symbolic dynamics allows one to study complicated (chaotic) invariant sets in the Poincaré maps of such systems. Under certain conditions, these two-dimensional invertible maps can be reduced to one-dimensional (noninvertible) maps on the interval or circle. In this situation, a fairly complete kneading theory is available and one can use it to study bifurcation sequences occurring as parameters vary. We apply these ideas to the Josephson equation. In contrast to reducing the flow to a lower dimensional Poincaré map, Birman and Williams have exploited reduction to a semiflow on a branched two-dimensional manifold by collapsing orbits along stable contracting directions. Using their fundamental result that the knot types of periodic orbits (closed curves in 3-space) are preserved under this collapse--we study some of the knots arising in the Duffing equation. Since the knot type of a periodic orbit is invariant under continuous deformations, one can use it to characterize families of such orbits in parameterized equations. This notion permits us to follow bifurcating sequences of orbits for Poincaré maps of heavily damped (almost one-dimensional) and lightly damped (almost area-preserving) systems and to show that certain "universal," one-dimensional bifurcation sequences are completely reversed for area-preserving maps.

Study of noise-induced transitions in the Lorenz system using the minimum action method

We investigate noise-induced transitions in non-gradient systems when complex invariant sets emerge. Our example is the Lorenz system in three representative Rayleigh number regimes. It is found that before the homoclinic explosion bifurcation, the only transition state is the saddle point, and the transition is similar to that in gradient systems. However, when the chaotic invariant set emerges, an unstable limit cycle continues from the homoclinic trajectory. This orbit, which is embedded in a local tube-like manifold around the initial stable stationary point as a relative attractor, plays the role of the most probable exit set in the transition process. This example demonstrates how limit cycles, the next simplest invariant set beyond fixed points, can be involved in the transition process in smooth dynamical systems.