## Stretching and Folding

This post explores the Henon map:
$x_{n+1}=y_n+1-a x_n^2\\\\ y_{n+1}=b x_n$

1000 iterations of a point

Exploring the “fate” of a disc after 10 iterations:

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## Harmony on a Donut

“Harmony” was my sloppy look at the Kuramoto model. Looking at 2 oscillators:
$\dot{\theta_1}=\omega_1+ k\sin (\theta_2-\theta_1)\\\\ \dot{\theta_2}=\omega_2+ k\sin (\theta_1-\theta_2)$

The following starts “knotted” 2:3 with $k=0$, i.e. no coupling and then increasing the coupling…another visualization of harmony.

Visualizing as motion on a circle:

## Knots on a Donut (Doughnut)

Professor Strogatz lecture on quasiperiodicity motivates this post. Just using the simple system:

$\dot{\theta_1}=\omega_1\\\\ \dot{\theta_2}=\omega_2$
when $\omega_1/\omega_2$ is rational the trajectory is closed, when irrational quasiperiodic. In the following the first gif is $\omega_1=2,\omega_2=3$ and the second: $\omega_1=\sqrt{2},\omega_2=2$

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## Relaxation Oscillator

Professor Strogatz Cornell course video lectures on “Nonlinear Dynamics and Chaos” are now online. This post pays homage to lecture 10 on analysis of the  Van der Pol equaltion:

$\ddot{x}+x+\mu \dot{x}(x^2-1)=0$

In the following the case for $\mu =2,\mu =10$ and the examination of the $\mu >>1$. It nicely illustrates the two time scales referred to, the slow then fast parts of the oscillation.

The phase portrait is the $x, y$ plane where $y$ is the the Lienard transformed and rescaled version of the equation.

So the system is:
$\dot{x}=\mu(y-(x^3/3-x))$
$\dot{y}=-x/\mu$

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## Milestone

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