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What is in this book? The author believes that most of the important ideas and techniques in nonlinear dynamics can be introduced in the setting of the real line or the circle. So, in the first chapter, the focus has been to introduce ideas like structural stability, topological conjugacy, the shift map and homoclinic points in the context of one-dimensional dynamical systems. The second chapter is a study of higher-dimensional dynamical systems while the third chapter describes some works on the complex analytic maps.

Chapter one: One-Dimensional dynamics

Preliminaries from Calculus:

Definition: Function $f$ is of class $C^r$ on $I$ if $f^{(r)}(x)$ exists and is continuous at all $x \in I$ .

Definition: Function $f: X \to Y$ is a homeomorphism if $f$ is a bijection (one-to-one and onto) and both $f$ and $f^{-1}$ are continuous.

Definition: Function $f(x)$ is a $C^r$ -diffeomorphism if both $f(x)$ and $f^{-1}(x)$ are $C^r$ -homeomorphisms.

Mean Value Theorem If $f:[a,b]\to \mathbb{R}$ is $C^1$ , then there exists $c \in [a,b]$ such that $f(b)-f(a) = f'(c)(b-a)$ .

Intermediate Value Theorem If $f:[a,b]\to \mathbb{R}$ is continuous and $f(a) = u$ and $f(b) = v$ , then for any $z \in [u,v]$ , there $c\in [a,b]$ such that $f(c) = z$ .

Implicit Function Theorem Suppose $G: \mathbb{R}^2 \to \mathbb{R}$ is a $C^1$ function, i.e., both the partial derivatives exist and are continuous. Suppose further that $G(x_0,y_0) = 0$ and $\frac{\partial G}{\partial y}(x_0,y_0) \ne 0$ . Then there exist open intervals $I$ about $x_0$ and $J$ about $y_0$ and a $C^1$ function $p$ satisfying $p(x_0) = y_0$ and $G(x, p(x)) = 0$ for all $x \in I$ .

A special case of Brouwer Fixed Point Theorem Let $I = [a,b]$ be and interval and let $f:I \to I$ be continuous. Then $f$ has at least one fixed point in $I$ .

A special case of the Contraction Mapping Theorem Let $f:I \to I$ and assume that $\|f'(x)\| < 1$ for all $x \in I$ . Then there exists a unique fixed point for $f$ in $I$ . Moreover, $\|f(x)-f(y)\| < \|x-y\|$ .

Definition A subset $U$ of $S$ is dense in $S$ if the closure of $U$ (denoted $\bar U$ ) is equal to $S$ .

Bump functions These are an interesting class of $C^\infty$ functions….

Definition The forward orbit of a point $x$ under a function $f$ is denoted $O^+(x)$ and if $f$ is a homeomorphism, we can also define the full orbit $O(x)$ and the backward orbit $O^-(x)$ .

Definition fixed point, periodic point, eventually periodic points.

Definition forward asymptotic points, stable set $W^s(p)$ as set of all points forward asymptotic to $p$ . similarly, backward asymptotic points, unstable set $W^u(p)$ .

Definition critical points: degenerate and non-degenerate.

Goal of dynamical systems; finding periodic points by numerical can be misleading; qualitative and geometrical techniques.

Jacobi’s theorem $T_\lambda(\theta) = \theta + 2\pi \lambda$ , $\lambda \in \mathbb{R}$ are translations on circle. Each orbit of $T_\lambda(\theta)$ is dense in $S^1$ iff $\lambda$ is irrational.

Hyperbolic points