We begin with basic notions that are necessary to describe the dynamical behaviors，and also reach to the most recellt achievements，for example，the chain stability is first proposed in 2008．In this book．we are mainly interested in the geo： metric or topological aspects of the orbits or solutions more than an explicit formula for an orbit．Also，this book is meant to be a graduate textbook and not j ust only a monograph on the subject．
This book contains four chapters．All the definitions．the-orems a

Stability is a classical topic in the study of dynamical sys- terns．Since the time of Lyapunov，mathematicians have pro- posed many different definitions of stability to investigate ba- sic properties of orbits of dynamical systems．This book does not contain all these important results in the literature)but focuses on three special topicsi．e．chain stability,Zhukovskij stability and intertwined basins．
We begin with basic notions that are necessary to describe the dynamical behaviors，and also reach to the most recellt achievements，for example，the chain stability is first proposed in 2008．In this book．we are mainly interested in the geo： metric or topological aspects of the orbits or solutions more than an explicit formula for an orbit．Also，this book is meant to be a graduate textbook and not j ust only a monograph on the subject．
This book contains four chapters．All the definitions．the-orems and formulae are independently numbered by chapter，for example．Theorem 2．8 in Chapter II means the eighth the- orem in Section 2 of the same chapter��

Preface

Chapter I Basic Definitions and Properties
I．1 Some Elementary Concepts
I．2 Recurrent Orbits
I．3 Auslander Recurrence
1．4 Chain Recurrence
I．5 Attractors

Chapter II Chain Stability
II．1 Absolute Stability
II．2 Chain Prolongation and Stability
II．3 Attracting Sets and Quasi-attracting Sets
II．4 Lyapunov Functions
II．5 Chain Stability of Closed Sets

Chapter III Zhukovskij Stability
III．1 Zhukovskij Stability
III：2 Omega Limit Set
III．3 Asymptotical Stability
III．4 Global Structure
III．5 Near Periodicity

Chapter IVIntertwined Basins of Attraction
IV．1 Two-dimensional Systems
IV．2 Intertwining Property
IV．3 Super-intertwining
IV．4 Application
References
……