Chaos

We finally will define what it means for a system to be chaotic.

Typical behavior:

• Fixed Points
  • Sink
  • Source
  • Saddle
• Periodic Orbits
  • also have those 3
• Unbounded

Unstable behavior may be transcient, gives way to stability in the long run.

Initial conditions that are near Sources don’t necessarily end up in the basin of a sink. A chaotic orbit will be one that experiences the unstable behavior associated with being near a source, Sensitive Dependence on Initial Conditions, but that is not itself fixed or periodic.

Lyapunov Number

Let $f$ be a continuous map on $\mathbb{R}$. The Lyapunov number $L(x_1)$ of the orbit $\{x_1,x_2,\dots \}$ is defined as:

$$L(x_1)=\underset{n\to \infty }{\lim }(|f^{{}^{\prime }}(x_1)|\cdot |f^{{}^{\prime }}(x_2)|\dots |f^{{}^{\prime }}(x_N)|{)}^{\frac{1}{n}}ifitexists$$

Example

If $x_1$ is in the basin of a fixed point sink $p$, then:

\begin{align*} L(x_{1}) &= \lim_{ n \to \infty } \left( \begin{array} f \text{product of n} \\ \text{slopes approaching } \\ f^{'}(p) \end{array} \right) \\ &= |f^{'}(p)| \end{align*}

If $x_1$ is in the basin of a period-2 sink $\{p_1,p_2\}$, then:

$$\begin{array}{rl}L(x_1)& =\sqrt{|f^{{}^{\prime }}(p_1)|\cdot |f^{{}^{\prime }}(p_2)|}\\ & =(|f^{{}^{\prime }}(p_1)|\cdot |f^{}(p_2)|{)}^{\frac{1}{2}}\end{array}$$ $$\begin{array}{rl}L<1& \Rightarrow sink\\ L>1& \Rightarrow source\text{(or something else)}\end{array}$$

Example

$$f(x)=3x(\mathrm{mod}1)$$

imagine $x_1\in \text{Irrational}$ $L(x_1)=3$

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Lyapunov Exponents

$h(x_1)$ is defined as:

$$h(x_1)=\underset{n\to \infty }{\lim }\frac{1}{n}\left(\ln |f^{{}^{\prime }}(x_1)|+\ln |f^{{}^{\prime }}(x_2)|+\cdots +\ln |f^{{}^{\prime }}{(}_n)|\right)$$

if it exists, and $\ln (L)=h$,

If $x_1$ is a period-K point, then:

$$h(x_1)=\underset{\text{limit is now gone}}{\underbrace{\frac{1}{K}\left(\ln |f^{{}^{\prime }}(x_1)|+\ln |f^{{}^{\prime }}(x_2)|+\dots \ln |f^{{}^{\prime }}(x_n)|\right)}}$$

We are describing the average local stretching or shrinking along an orbit.

$$\begin{array}{rl}h>0,L>1& \Rightarrow \text{stretching}\\ h<0,0<L<1& \Rightarrow \text{shrinking}\end{array}$$

Let $f$ be a smooth map on $\mathbb{R}$. An orbit $\{x_1,x_2,\dots ,x_n\}$ is called asymptotically periodic(A.P.) if it converges to a periodic orbit as $n\to \infty$.

In other words, there exists a periodic orbit given by $\{y_1,y_2,\dots ,y_k,y_1,y_2,\dots y_k\dots \}$ such that:

$$\underset{n\to \infty }{\lim }|x_n-y_n|=0$$

orbits that are eventually periodic (E.P.) will also be asymptotically periodic.

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Chaotic Orbits

Let $f$ be a smooth map on $\mathbb{R}$ and let $\{x_1,x_2,\dots \}$ be an orbit of $f$. The orbit is chaotic if:

1. the orbit must be bounded ( not in basin of $\infty$ )
2. the orbit must not asymptotically periodic
3. the Lyapunov Exponent $h(x_1)>0$

Example

Consider the map

$$f(x)=(x+q)(\mathrm{mod}1)\text{where }q\in (0,1)\text{ is irrational}$$

Desmos

Transcription

this ia test

Chaotic Orbits?

1. $f$ has no periodic orbits. - YES
2. no periodic orbits to BE approaching - YES
3. $h>0$? - NO
   • because $h=0$ everywhere

If $q$ were rational, then rational Initial Conditions would be Eventually Periodic, and irrationals would bounce around forever, never repeating, but also never separating from the neighbors.

A bounded orbit that is not Asymptotically Periodic and does not exhibit SDIC is Quasi Periodic