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In some maps the existence of an attractor with a positive Lyapunov exponent can be proved by constructing a trapping region in phase space and an invariant expanding cone in tangent space. If this approach fails it may be possible to adapt the strategy by considering an induced map (a first return map for a well-chosen subset of phase space). In this paper we show that such a construction can be applied to the two-dimensional border-collision normal form (a continuous piecewise-linear map) if a certain set of conditions are satisfied and develop an algorithm for checking these conditions. The algorithm requires relatively few computations, so it is a more efficient method than, for example, estimating the Lyapunov exponent from a single orbit in terms of speed, numerical accuracy, and rigor. The algorithm is used to prove the existence of an attractor with a positive Lyapunov exponent numerically in an area of parameter space where the map has strong rotational characteristics and the consideration of an induced map is critical for the proof of robust chaos.
Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide forma
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 $f$ has a topological attractor with a positive Lyapunov exponent. First, $Omega_{rm trap} subs
The collision of a fixed point with a switching manifold (or border) in a piecewise-smooth map can create many different types of invariant sets. This paper explores two techniques that, combined, establish a chaotic attractor is created in a border-
The goal of this paper is to construct invariant dynamical objects for a (not necessarily invertible) smooth self map of a compact manifold. We prove a result that takes advantage of differences in rates of expansion in the terms of a sheaf cohomolog
We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs-Markov-Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invaria