Abstract : The emerging discipline known as \"chaos theory\" is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for \"chaos\" as it applies to general dynamical systems. Various approaches range from topological methods of a qualitative description; to physical notions of randomness, information, and entropy in ergodic theory; to the development of computational definitions and algorithms designed to obtain quantitative information. This thesis develops some of the current definitions and discusses several quantitative measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems, it serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.

Introduction of chaotic dynamical systems

On shadowing: Ordinary and ergodic

https://moothathu.files.wordpress.com/2013/03/11tksm-implicationsofpotp.pdf

Implications of pseudo-orbit tracing property for continuous maps on compacta

This paper studies distributional chaos in non-autonomous discrete systems generated by given sequences of maps in metric spaces. In the case that the metric space is compact, it is shown that a system is Li–Yorke δ-chaotic if and only if it is distributionally δ′-chaotic in a sequence; and three criteria of distributional δ-chaos are established, which are caused by topologically weak mixing, asymptotic average shadowing property, and some expanding condition, respectively, where δ and δ′ are positive constants. In a general case, a criterion of distributional chaos in a sequence induced by a Xiong chaotic set is established.

On distributional chaos in non-autonomous discrete systems

Shift spaces, ω-chaos and specification property

P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy

We provide a necessary and sufficient condition for the Higson compactification to be perfect for the noncompact, locally connected, proper metric spaces. We also discuss perfectness of the Smirnov compactification.

/pdf/perfectness-of-the-higson-and-smirnov-compactifications-1u3e2hzqxz.pdf

Naotsugu Chinen

Papers

P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy

Perfectness of the Higson and Smirnov compactifications