Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

Abstract

We show how to formally identify chaotic attractors in continuous, piecewise-linear maps on ${\mathbb{R}}^N$. For such a map f, this is achieved by constructing three objects. First, ${\mathrm{\Omega }}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}\subset {\mathbb{R}}^N$ is trapping region for f. Second, $\mathbf{W}$ is a finite set of words that encodes the forward orbits of all points in ${\mathrm{\Omega }}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$. Finally, $C\subset T{\mathbb{R}}^N$ is an invariant expanding cone for derivatives of compositions of f formed by the words in $\mathbf{W}$. The existence of ${\mathrm{\Omega }}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$, $\mathbf{W}$, and C implies f has a topological attractor with a positive Lyapunov exponent. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of ${\mathrm{\Omega }}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p}}$ and C. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large region of parameter space. We also observe how the failure of C to be expanding can coincide with a bifurcation of f. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where f is not differentiable) can be included in the analysis.

Keywords:

• Lyapunov exponent
• trapping region
• symbol sequence
• border-collision
• one-sided directional derivative

2020 Mathematics Subject Classifications:

• 37G35
• 39A28

Acknowledgements

The author thanks Paul Glendinning whose May 2017 visit to Massey University stimulated many of the ideas presented here.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 To the left of ${\tau }_L={\delta }_L+1$, Figure 1 contains 173779 red pixels and 21462 white pixels, giving $89.01\mathrm{\%}$.

2 The Hausdorff distance between sets ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ is defined as $d_H({\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2)=max[\underset{x\in {\mathrm{\Omega }}_1}{sup}\underset{y\in {\mathrm{\Omega }}_2}{inf}\Vert x-y\Vert ,\underset{y\in {\mathrm{\Omega }}_2}{sup}\underset{x\in {\mathrm{\Omega }}_1}{inf}\Vert x-y\Vert ].$

3 Equation (64(64) $\sin \left(\theta \right(x\left)\right)=\frac{S\left(x\right)}{\Vert g\left(x\right)\Vert \Vert g\left(f_L^{-1}\right(x\left)\right)\Vert }.$(64) ) is a version of the well-known formula $\sin \left(\theta \right)=\frac{\Vert u\times v\Vert }{\Vert u\Vert \Vert v\Vert }$ for the sine of the angle between $u,v\in {\mathbb{R}}^3$.

Additional information

Funding

This work was supported by Marsden Fund contract MAU1809, managed by Royal Society Te Apārangi.