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 . For such a map f, this is achieved by constructing three objects. First, is trapping region for f. Second, is a finite set of words that encodes the forward orbits of all points in . Finally, is an invariant expanding cone for derivatives of compositions of f formed by the words in . The existence of , , 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 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.