Dynamical Systems
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 37 (Dynamical systems)

Prerequisites:

# Getting Oriented

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Dynamical systems…

# The Basics

## Dynamical Systems

Discrete-Time Dynamical System
a pair $(X,f)$ with $f:X\to X$, such that iterates $f^n$ satisfy $f^{t+s}=f^t \circ f^s$.
Continuous Time Dynamical System
a one-parameter family of maps $\{f^t:X\to X\}$ with $f^{t+s}=f^t \circ f^s$ and $f^0=\id$. It is a flow if $t\in\mathbb{R}$ and a semiflow if $t\in\mathbb{R}_0^+$.
Conjugacy
the notion of equivalence for dynamical systems. Given systems $(X,f)$ and $(Y,g)$, a semiconjugacy is a surjective map $\pi: Y \to X$ such that $f\circ\pi=\pi\circ g$. A conjugacy is an invertible semiconjugacy.
Orbits
a point has positive (forward) and negative (backward) semiorbits, whose union is the point's orbit. A periodic point $x$ of period $T$ satisfies $f^T(x)=x$. If $f(x)=x$, $x$ is a fixed point.

A fixed point $x_0$ can be either attracting (when a neighborhood of $x_0$ converges to $x_0$ under $f$), or repelling (when the inverse $f^{-1}$ is attracting at $x_0$). Points can also be eventually periodic.

A subset $A\subset X$ for which $f(A)\subset A$ is $f$-invariant. If $X$ is the only $f$-invariant subset, $(X,f)$ is called minimal. Closely related is the ergodic dynamical system, whose only measurable invariant subsets have either measure zero or full measure.

## Canonical Examples

Systems on $S^1$
Consider the circle $S^1$ as the unit interval $I=[0,1]$ with endpoints identified. The rotation map $R_\alpha$ is defined by $R_\alpha(x)=x+\alpha \mod 1$. If $\alpha$ is rational, every orbit is periodic, while if $\alpha$ is irrational, every (positive) semiorbit is dense.

The map defined by $E_m(x)=mx \mod 1, m \in \mathbb{Z}$ is the expanding endomorphism of $S^1$, and is an $|m|$-to-$1$ function.

Sequences and Shifts
Let $\Sigma^+_m=\{0,\ldots,m-1\}^\mathbb{N}$, \ie the set of (one-sided) sequences of $m$ letters. The shift is the $m$-to-$1$ map $\sigma: (x_1,x_2,\ldots) \mapsto (x_2,x_3,\ldots)$. Replacing $\mathbb{N}$ with $\mathbb{Z}$, we obtain the set of two-sided sequences $\Sigma_m$, for which the shift $\sigma$ is invertible.

Given a directed graph with $m$ nodes, we have a subshift (of $\Sigma_m$ or $\Sigma_m^+$) whose elements are sequences corresponding to paths in the graph.

Many systems are conjugate or semiconjugate to shifts. For example, $\phi: \Sigma_m^+ \to S^1$ is a semiconjugacy between $(\Sigma_m^+,\sigma)$ and $(S^1,E_m)$ given by the base $m$ expansion of a sequence in $\Sigma$.

the canonical quadratic map is $q_\mu(x)=\mu x (1-x), \mu>0$. The behavior of this system is very different for $\mu\leq 4$ and $\mu>4$. As an example, for $\mu>4$, there are periodic points of any period $T\in\mathbb{N}$.
The Gauss Transformation
the map on $[0,1]$ given by
(1)
\begin{align} \phi(x)=\begin{cases}\tfrac{1}{x}-[\tfrac{1}{x}]& x\neq 0,\\ 0& x=0,\end{cases} \end{align}

where $[x]$ is the greatest integer less than or equal to $x$, for $x\in\mathbb{R}$. The map $\phi$ is closely related to continued fractions: if $x=[a_1,a_2,\ldots,a_n]$ is a continued fraction, then $\phi(x)=[a_2,\ldots,a_n]$. Thus, $\phi$ corresponds (via a conjugacy) to shifts on sequences of integers. This property can be used to show that any rational number is uniquely represented by a finite continued fraction.

Hyperbolic Total Automorphisms
consider the torus $T^2$ as the unit square with opposite sides identified. Then, a $2 \times 2$ integer matrix $A$ induces an automorphism of the torus with $A: \binom{x_1}{x_2} \mapsto A\binom{x_1}{x_2} \mod 1.$ If $A$ is invertible ($|\det A|=1$) and the eigenvalues of $A$ do not lie on the unit circle, then $A: T^n \to T^n$ is called a hyperbolic toral automorphism because it expands in contracts in complementary directions.
The Horseshoe
a system on the unit square with map $f$ defined by ‘bending’ the square into a horseshoe and placing it atop the square. Thus, $f$ maps the square to two horizontal strips, and $f^{-1}$ maps the square to two vertical strips.

The horseshoe set $H$ is the $f$-invariant subset of the square, and is the product of two Cantor sets. The system $(H,f)$ is conjugate with the shift system $\Sigma_2$. It is hyperbolic because it expands and contracts in different directions, but its behavior survives under small perturbations.

The Solenoid
the subset $S$ of the solid torus $T=S^1\times D^2$ forward invariant under
(2)
\begin{align} F(\phi,x,y)=(2\phi,\lambda x+\tfrac{1}{2}\cos 2\pi\phi, \lambda y + \tfrac{1}{2}\sin 2\pi\phi), \quad \lambda\in(0,\tfrac{1}{2}) \text{ fixed}. \end{align}

$S$ is a Cantor set, and $(S,F)$ is a hyperbolic system. It is conjugate to the symbolic system $\Phi$ consisting of sequences $(\phi_1,\phi_2,\ldots)$ in $(S^1)^\mathbb{N}$ for which $\phi_i=2\phi_{i+1} \mod 1$, with map $\alpha:(\phi_1,\phi_2,\ldots)\mapsto(2\phi_1,\phi_1,\phi_2,\ldots)$.

## Basic Concepts

Flows, Suspension, and Cross-Section
Given an autonomous differential equation $\dot x=F(x)$, we can consider the flow $f^t(x)$. Given any flow and a subset $A\subset X$ for which any point $x\in A$ implies $f^t(x)\in A$ a ‘discrete’ time $t$ later, one can define a map $g: A \to A$ by $g(x)=f^{\tau(x)}(x)$, where $\tau(x)$ is the ‘return time’ of $x$, i.e., the first intersection of the orbit of $x$ and $A$. This gives the cross-section dynamical system $(A,g)$.

Conversely, given a map $f: X \to X$ we have a quotient space $X^*=X\times\mathbb{R}^+/~$, where $(f(x),0)~(x,1)$. Then the suspension of $f$ is the semiflow $f^t: X^* \to X^*$ given by flowing from $(x,0)$ to $(x,1)$, then from $(f(x),0)$ to $(f(x),1)$, and so on.

Chaos and Lyapunov Exponents
Chaos is essentially sensitive dependence on initial conditions. A simple example is the circle endomorphism $(S^1,E_m)$, because the orbits of nearby points diverge. The sensitivity of initial conditions strongly correlates with a positive Lyapunov Exponent, which is defined by
(3)
\begin{align} \mathbb{C}hi(x,v)=\limsup_{n\to\infty}\frac{1}{n}\log\|df^n(x)v\|. \end{align}

This measures the exponential growth rate of tangent vectors along orbits.

Attractors
Given a map $f:X \to X$, a compact subset $C\subset X$ is an attractor if there is an open set $U \supset C$ with $f(\overline{U}) \subset U$ and $C=\cap_{n\geq 0} f^n(U)$. Thus, nearby points converge to $C$. The set of points which converge to $C$ is called the basin of attraction $BA(C)$. An open set $U$ with compact closure and $f(\overline{U})\subset U$ is called a trapping region because $\cap_{n\geq 0} f^n(U)$ is an attractor. The existence of an attractor is often proven by finding a trapping region. Attractors can come in many forms: fixed points, periodic orbits, hyperbolic attractors, and chaotic attractors. Examples of chaotic attractors include the Lorenz attractor and the H\'enon Attractor.