Datenanalyse und Stochastische Modellierung
3. Chaos

### Chaotic systems

• deterministic systems with statistical behavior similar to stochastic processes
• can be discrete or continuous in time
• yield insights to dynamical origins of statistical behavior measured time series

Maps

$x_{t+1} = f(x_{t},t)$

Autonomous if $f=f(x)$

### Bernoulli map

$x_{t+1}=2x_{t} \mbox{ mod } 1$ https://upload.wikimedia.org/wikipedia/commons/6/68/Exampleergodicmap.svg
• stretch-and-fold
• binary code for numbers $z = \sum_{t=0}^\infty \frac{b_t}{2^{t+1}}$
• rational numbers: periodic
• irrational numbers: generally infinite series

### Bernoulli map

$x_{t+1}=2x_{t} \mbox{ mod } 1$
• Pseudo-random numbers: see exercise

### Measure preserving maps F

$\int f(Fx) \rho(x) \mathrm{d}x = \int f(x) \rho(x) \mathrm{d}x$

### Nondecomposable maps F

$FA = A \Rightarrow \int_A \rho(x) dx = 0 \mbox{ or } 1 \; \; \forall A$

### Mixing maps F

$\lim_{n\rightarrow\infty} \rho(F^{n}A\cap B) = \rho(A) \rho(B) \ \ \ \ \forall A,B$
• For measure-preserving maps: $\mbox{Nondecomposable} \Rightarrow \mbox{Mixing}$

### Ergodic maps

A map T is ergodic if

$\lim_{N\rightarrow \infty} \frac{1}{N} \sum_{n=0}^{N-1} f(F^nx) = \int\mathrm{d}\rho f(x)$

for almost all orbits and arbitrary measureable f

• Measure preserving and nondecomposable maps are ergodic

e.g. the Mean Squared Displacement in ergodic systems can be replaced by the time average:

$\langle (x(t)-x(0))^2 \rangle = \langle \frac{1}{T-t} \int_{0}^{T-t} \mathrm{d}t^\prime (x(t^\prime+t)-x(t^\prime))^2 \rangle$

or, in discrete time,

$\langle (x(t)-x(0))^2 \rangle = \langle \frac{1}{T-t} \sum_{t^\prime=1}^{T-t} (x(t^\prime+t)-x(t^\prime))^2 \rangle.$

### Lyapunov Exponents

https://en.wikipedia.org/wiki/Lyapunov_exponent#/media/File:Orbital_instability_(Lyapunov_exponent).png
• Chaotic systems exhibit sensitive dependence on initial conditions
• Exponential approximation of escape of two nearby points $\left|F^t(x_0+\epsilon) - F^t(x_0)\right| \approx \epsilon e^{t\lambda(x_0)}$
• The Lyapunov exponent is defined for each initial point $\lambda (x_0) = \lim_{t\rightarrow \infty} \frac{1}{t} \sum_{n=1}^{t} \log|F^\prime(x_n)|$

### Fixed Points

$x_f=F(x_f)$

Look at environment

$[ x_f-\epsilon, x_f+\epsilon ]$ $\delta_{t+1} = |x_{t+1}-x_f|=|F(x_f\pm \delta_t)-x_f|=\delta_t \left|\frac{F(x_f\pm\delta_t)-F(x_f)}{\delta_t}\right|=\delta_t|F^\prime(x_f)|$ Stable fixed points $1>\left|\frac{\mathrm{d}F}{\mathrm{d}x}(x_f)\right|$ Unstable fixed points $\left|\frac{\mathrm{d}F}{\mathrm{d}x}(x_f)\right|>1$

### Logistic map

$x_{t+1}=rx_{t}(1-x_{t})$

Fixed points?

$x_f = r x_f ( 1 - x_f ) = r x_f - rx_f^2$ $rx_f^2 + (1-r) x_f = 0$ ${x_f}_1 = 0 \; \; \; \; \; {x_f}_2 = \frac{r-1}{r}$

Stability?

$F^\prime (x_f) = r (1-2x_f) = 2-r$
https://fr.wikipedia.org/wiki/Suite_logistique
• Number and types of fixed points depend on r
• $1>r>0 \Rightarrow x \rightarrow 0$ $3>r>1 \Rightarrow x \rightarrow \frac{r-1}{r}$ $1+\sqrt{6} > r > 3 \Rightarrow x \rightarrow \mbox{ 2-periodic limit cycle }$ $3.54 > r > 1+\sqrt{6} \Rightarrow x \rightarrow \mbox{ 4-periodic limit cycle }$ $r>3.54 \Rightarrow x \mbox{ chaotic, with infinite exceptions }$ $r>4 \Rightarrow x \mbox{ diverges }$

### The pendulum

$\ddot{\theta} + g \sin(\theta) =0$

It can be rewritten as

$\dot{\theta}=p$ $\dot{p}=-g\sin(\theta)$

### The damped pendulum

$\ddot{\theta} + \gamma \dot{\theta} + g \sin(\theta) = 0$

In phase space

$\dot \theta = p$ $\dot p = -\gamma p - g \sin(\theta)$
• decays to zero

### The driven pendulum

$\ddot{\theta} + \gamma \dot{\theta} + g \sin(\theta) = A\cos(\omega t)$

In phase space

$\dot \theta = p$ $\dot p = -\gamma p - g \sin(\theta) + A\cos(\omega t)$ simulation

### The kicked rotor

$\dot \theta = p$ $\dot p = g \sin(\theta) \sum_{n=-\infty}^\infty \delta( n-t/T )$ video