2020/10/16 by Simpson, David J. W. · 1 citation
#37G35 #39A28 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.08241
We show how the existence of three objects, Ω\rm trap, \bf W, and C, for a continuous piecewise-linear map f on ℝN, implies that f has a topological attractor with a positive Lyapunov exponent. First, Ω\rm trap ⊂ ℝN is trapping region for f. Second, \bf W is a finite set of words that encodes the forward orbits of all points in Ω\rm trap. Finally, C ⊂ T ℝN is an invariant expanding cone for derivatives of compositions of f formed by the words in \bf W. 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 Ω\rm trap 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 regime 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.