Graphs of fast-slow systems

Given a dynamical system \(f \colon M \to M\), we consider a skew product \[ \begin{cases} s_{n+1} = \varphi(s_n, x_n, \varepsilon), \\ x_{n+1} = f(x_n). \end{cases} \] with \(x_0\) and \(s_0\) given. Here \(\varphi\) is an \(\mathbb{R}^2\)-valued function. We fix \(n \in \mathbb{N}\) and \(\varepsilon = \varepsilon(n) > 0\), and plot two dimensional càdlàg paths \(S(t) = s_{\lfloor n t \rfloor}\) on \(0 ≤ t ≤ 1\).

Take a look at the preset maps in the form below. Most peculiar, probably, is the collision map for a billiard with a flat cusp.

Use javascript. There are a few preprogrammed functions:

• Liverani-Saussol-Vaienti Pomeau-Manneville intermittent map \[ \text{LSV}(x, \alpha) = \begin{cases} x (1 + 2^\alpha x^\alpha), & x \leq 0.5 \\ 2x - 1, & x \gt 0.5 \end{cases} \]
• Symmetrized version of the intermittent map \[ \text{SLSV}(x, \alpha) = \begin{cases} \text{LSV}(x, \alpha), & x \leq 0.5 \\ 1 - \text{LSV}(1 - x, \alpha), & x \gt 0.5 \end{cases} \]
• Collision map for a billiard with a flat cusp, as in Jung and Zhang arXiv:1611.00879. The billiard table is bounded by the curves \[ x = 1 \qquad \text{and} \qquad y = \pm \alpha x^\beta . \] The code expects that \(\beta \gt 1\) and \(\alpha \gt 0\). Following Jung and Zhang, we suggest \(\beta \gt 2\) and \(\alpha = 1 / \beta\). The javascript code is 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.