Some advances on global analysis of nonlinear systems

Abstract

Global analysis in nonlinear dynamics means the study of attractors and their basins of attraction; meanwhile a lot of complex dynamical behaviors and new phenomena are concerned such as fractal basin boundary, Wada basin boundary, infinite unstable periodic orbits embedded in chaotic attractor, chaotic saddle and transient chaos, crises, riddled basin of attractor, stochastic global dynamics, etc.

To analyze the global dynamics analytically is difficult and interesting while the results are few. Then, the numerical analysis for global dynamics is usually the main approach.

Global analysis captures both the interest and imagination of the wider communities in various fields, such as mathematics, physics, meteorology, life science, computational science, engineering, medicine, and others.

Emphasis is put mainly on the development in this global dynamics field in China.

Introduction

Global analysis was regarded as a complex and difficult research topic for nonlinear dynamics systems [1]. Recently, since the chaos and more new phenomena are found, the global analysis in nonlinear dynamics has been given challenging research contents. Generally, in nonlinear dynamics, global analysis means the study of the attractors and their basins of attraction. Sometimes the unstable invariant set study is also involved, and a lot of complex dynamical behaviors and new phenomena relevant to the global region motion are also concerned.

The invariant set of a dynamical system is a more general entity in nonlinear dynamics. One important type of the invariant sets is the attractor which represents an asymptotically stable motion, while the unstable invariant sets possess more complex properties and can affect the global dynamical behaviors and the characteristics of the dynamical systems. There are various kinds of the unstable invariant sets, for examples, a variety of saddles, simple unstable invariant sets, invariant manifolds of saddles and fractal basin boundaries, etc. Chaotic saddle, also known as nonattracting chaotic set, usually leads to chaotic transients, fractal basin boundary and chaotic scattering [2], and the mechanism of a kind of discontinuous bifurcation, crisis, is relevance to the collision of the periodic saddle or the chaotic saddle with a chaotic attractor [3], [4], [5].

Recently, a number of scholars provided more new results in these problems. Sweet et al. [6] provided a general method to locate and visualize chaotic saddle, a common phenomenon in higher dimensional dynamical systems, plane Couette flow. Lai et al. [7] indicated that the symbolic representations of controlled chaotic orbits produced by signal generators can be used for communicating with chaotically behaving signal generators, and can be robust against noise. Jacobs et al. [8] studied on communicating with chaos for being able to determine numerically the topological entropy for chaotic invariant sets. Lai et al. [9] presented that as a system parameter changes, chaotic saddles can evolve via an infinite number of homoclinic or heteroclinic tangencies of their stable and unstable manifolds, and the topological entropy and the fractal dimension of chaotic saddles obey a universal behavior of exhibiting a devil-staircase characteristic. Skufca et al. [10] studied the problem of chaos edge in a parallel shear flow. Rempel and Chain [11] indicated that nonattracting chaotic sets (chaotic saddles) are shown to be responsible for transient and intermittent dynamics in a nonlinear regularized long-wave equation relevant to plasma and fluid studies. Chain et al. [12] revealed and discussed the coexistence of chaos, crisis-induced intermittency, and the chaotic saddles in the business cycles model of the complex economic system. Carr [13] described that the onset of global heteroclinic connections forms the precursor topology of localized homoclinic chaos in driven class-B lasers. Robert [14] examined different types of tangencies of stable and unstable manifolds from orbits of pre-existing invariant sets in the dissipative invertible plane maps, connecting to the explosion of chaotic sets. Banerjee [15] reported the coexisting attractors with fractal basin boundaries in the voltage mode controlled buck converter.

Few results of the global dynamics can be obtained analytically because it is very difficult. Then, the numerical analysis for the global dynamics is usually the main approach.

An effective numerical method for global analysis is the cell mapping method provided and developed systematically by Hsu [1], including simple cell mapping, generalized cell mapping, and partially ordered set digraph generalized cell mapping. Following these ideas a number of scholars improved the cell mapping method. At same times, the bifurcations generated from qualitative changes of the attractors and their basins of attraction may induce many new dynamical problems.

It is known that many new phenomena and complex behaviors of nonlinear dynamical systems have relevance to global orbits and global dynamics. For examples, fractal basin boundary of chaotic attractor, Wada basin boundary, infinite unstable periodic orbits embedded in chaotic attractor, chaotic itineracy of high dimensional system, chaotic saddle and transient chaos, discontinuous bifurcation and crises, and riddled basins of attraction, etc.

The global stochastic analysis of noisy system is also paid attention to recently.

Global analysis captures both the interest and the imagination of the wide communities in various fields, such as in mathematics, physics, meteorology, life science, ecology, computational science, engineering, medicine, and others.

In this summary paper, the emphasis is put mainly on the development of concerning global dynamics problems in China.