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

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