Some new chaotic attractors : algebraic, geometric and analytic aspects

若干新混沌吸引子 : 代數, 幾何及其解析本質的研究

Student thesis: Doctoral Thesis

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Author(s)

  • Xiong WANG

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Detail(s)

Awarding Institution
Supervisors/Advisors
Award date15 Jul 2014

Abstract

A great challenge to our scientific understanding of the world is that nature is replete with nonlinear systems. While all linear systems are linear in the same way, nonlinear systems are nonlinear in many different ways. Chaos theory for three-dimensional (3D) autonomous systems has been intensively and extensively studied since the time of Edward N. Lorenz in the 1960s, and the theory has become quite mature today. However, some of the recent findings reveal that the complexity and richness of this subject are far beyond our wildest imagination. In this thesis, some fundamental questions of chaos theory are investigated, with some novel chaotic systems coined and analyzed, and their potential applications are proposed. In particular, the following issues are studied in detail: the algebraic classification of generalized Lorenz systems family; the intrinsic relationships between the algebraic structures and the geometric shapes of chaotic attractors; a gallery of 3D symmetrical autonomous chaotic systems with two quadratic terms; general patterns of Lorenz-like and Chen-like chaotic systems; relationships between the global dynamical behaviors and the number and stability of the equilibria of a chaotic system; non-hyperbolic type of chaotic systems; chaotic systems with no equilibrium or only one and stable equilibrium, or with any number of equilibria having tunable stability. The thesis is organized as follows. Chapter 1 reviews briefly the classical chaos theory and surveys previous works on generalized Lorenz systems family. Chapter 2 discusses the algebraic classification of generalized Lorenz systems family. A simple one-parameter family of 3D quadratic autonomous chaotic systems is discussed. By tuning the only parameter, this system can continuously generate a variety of cascading Lorenz-like attractors, which appears to be richer than the unified chaotic system that contains the Lorenz and the Chen systems as its two extremes. Although this new system belongs to the family of Lorenz-type systems according to some existing classifications such as the generalized Lorenz canonical form, it can generate not only Lorenz-like attractors but also Chen-like attractors. This suggests that there may exist some other unknown yet more essential algebraic characteristics for classifying general 3D quadratic autonomous chaotic systems. Chapter 3 discusses the intrinsic relationship between the algebraic structures and the geometric shapes of attractors of chaotic systems. A gallery of symmetrical Lorenz-like and Chen-like attractors, generated by 3D autonomous systems with two quadratic terms that can maintain the z-axis rotational symmetry, are presented. Some general patterns of the Lorenz-like and Chen-like chaotic systems are found and analyzed, which suggest that the instability of the two saddle-foci of such a system somehow determines the shape of its chaotic attractor. Chapter 4 investigates the intrinsic relationship between the global chaotic dynamical behaviors and the local stability of an equilibrium. This chapter reports the finding of a simple 3D autonomous chaotic system which, very surprisingly, has only one and stable node-focus equilibrium. The discovery of this new system is striking, because with a single stable equilibrium in a 3D autonomous quadratic system, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Yet, unexpectedly, this system is chaotic. The new system is of non-hyperbolic type, therefore the familiar Ši'lnikov homoclinic criterion is not applicable. Although the fundamental chaos theory for autonomous dynamical systems seems to have reached its maturity today, this finding reveals some new mysterious features of chaos. Chapters 5 and 6 further investigate some non-hyperbolic type of chaotic systems. It focuses on the relationship between the number and stability of the equilibria of a chaotic system and the geometrical properties of the attractor that the system generates. Chaotic systems with no equilibrium or with any number of equilibria having tunable stability are constructed. This shows that chaos can appear in a system with any number of stable or unstable equilibria. Chapter 7 summarizes all the discoveries and rethinks about some fundamental questions of chaos. These striking new discoveries help to gain a better understanding of some fundamental questions of chaos theory but still leave more important yet challenging theoretical as well as technical problems for future research.

    Research areas

  • Chaotic behavior in systems, Attractors (Mathematics)