Given a dynamical system \(f \colon M \to M\), an observable \(\varphi \colon M \to \mathbb{R}\), an initial condition \(x_0 \in M\) and a normalizing sequence \(c_n\), we plot càdlàg paths \(S \colon [0,1] \to \mathbb{R}\), \[ S(t) = c_n \sum_{j=0}^{\lfloor nt \rfloor -1} \varphi (f^j(x_0)) . \] If \(x_0\) is random, we expect weak convergence to a Brownian motion in uniform topology, or to a Lévy process in a topology which depends on the system and the observable.
For no specific reason, we compute the p-variation and the quadratic variation of \(S\).
Take a look at the preset maps in the form below. Most peculiar, probably, is the collision map for a billiard with a cusp.
The "subtract mean" option computes the mean as \[ \text{mean} = \frac{1}{2n} \sum_{j=2n}^{4n-1} \varphi (f^j(x_0)) . \]
Use javascript. There are a few preprogrammed functions:
CUSP(x, alpha, beta), where, for example,
x = {
x: 0.5,
y: 0.0,
theta: Math.PI / 2,
phi: 0.0
}
Here (x.x, x.y) is a point on the plane, x.theta is a
velocity angle, measured anti-clockwise with respect to the positive \(x\) axes,
and x.phi is the collision angle, measured anti-clockwise with respect to the normal
vector at collision point.
The initial condition does not have to be on the boundary, and the initial phi is
ignored.
We use hammer.js and Chart.js.