What can we learn from homoclinic orbits in chaotic dynamics?

State diagrams of two model systems involving three variables are constructed. The parameter dependence of different forms of complex nonperiodic behavior, and particularly of homoclinic orbits, is analyzed. It is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system.     

Multi-pulse homoclinic orbits and chaotic dynamics of a parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities

In this paper,the complicated dynamics and multi-pulse homoclinic orbits of a two-degree-of-freedom parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities are studied.The damping,parametrical excitation and the nonlinearities are regarded as weak.The averaged equation depicting the fast and slow dynamics is derived through the method of multiple scales.The dynamics near the resonance band is revealed by doing a singular perturbation analysis and combining the extended Melnikov method.We are able to determine the criterion for the existence of the multi-pulse homoclinic orbits which can form the Shilnikov orbits and give rise to chaos.At last,numerical results are also given to illustrate the nonlinear behaviors and chaotic motions in the nonlinear nano-oscillator.     

Disorder in ballistic systems: A numerical analysis

We present a numerical analysis of the single particle energy spectra of ballistic condensed matter systems and compare with recent theoretical results. We show that the presence even of weak disorder induces full chaoticity on time scales larger than the elastic disorder scattering time. To disentangle the effect of boundary, respectively disorder scattering on the spectral statistics, different types of correlation functions are introduced and discussed.     

Complex wave-interference phenomena: From the atomic nucleus to mesoscopic systems to microwave cavities

Universal statistical aspects of wave scattering by a variety of physical systems ranging from atomic nuclei to mesoscopic systems and microwave cavities are described. A statistical model for the scattering matrix is employed to address the problem of quantum chaotic scattering. The model, introduced in the past in the context of nuclear physics, discusses the problem in terms of a prompt and an equilibrated component: it incorporates the average value of the scattering matrix to account for the prompt processes and satisfies the requirements of flux conservation, causality and ergodicity. The main application of the model is the analysis of electronic transport through ballistic mesoscopic cavities: it describes well the results from the numerical solutions of the Schrödinger equation for two-dimensional cavities.     

A mathematical framework for critical transitions: Bifurcations, fast-slow systems and stochastic dynamics

Bifurcations can cause dynamical systems with slowly varying parameters to transition to far-away attractors. The terms “critical transition” or “tipping point” have been used to describe this situation. Critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance. The main goal of this paper is to give an overview which standard mathematical theories can be applied to critical transitions. We shall focus on early-warning signs that have been suggested to predict critical transitions and point out what mathematical theory can provide in this context. Starting from classical bifurcation theory and incorporating multiple time scale dynamics one can give a detailed analysis of local bifurcations that induce critical transitions. We suggest that the mathematical theory of fast-slow systems provides a natural definition of critical transitions. Since noise often plays a crucial role near critical transitions the next step is to consider stochastic fast-slow systems. The interplay between sample path techniques, partial differential equations and random dynamical systems is highlighted. Each viewpoint provides potential early-warning signs for critical transitions. Since increasing variance has been suggested as an early-warning sign we examine it in the context of normal forms analytically, numerically and geometrically; we also consider autocorrelation numerically. Hence we demonstrate the applicability of early-warning signs for generic models. We end with suggestions for future directions of the theory.     

Critical fluctuation universality in chemically oscillatory systems: A soluble master equation

Multiple time scale arguments are used to show that near a Hopf bifurcation to a chemical oscillation the dynamics of the system reduces to that of a classic soluble limit cycle system. A birth and death master equation is then introduced and the spectrum of the resulting transition operator is shown to be complex. Exact solutions of the master equation are obtained both for the steady and (for a rather general class of systems) excited states. Thus a simple basis of universality of critical properties in chemical oscillations is provided.Research supported in part by a grant from the National Science Foundation.