Hausdorff dimension of chaotic attractors in a class of nonsmooth systems

Abstract

Fractal dimension is an important feature of a chaotic attractor. Generally, the rigorous value of Hausdorff dimension of a chaotic attractor is not easy to compute. In this work, we consider a class of discontinuous piecewise linear maps. Initially, we determine a set of parameter values in which the maps have a chaotic attractor with an Sinai-Ruelle-Bowen measure. Then we give a lower bound and an upper bound of the Hausdorff dimension of the attractor. Our rigorous analysis shows that the two bounds are equal, and thus the exact formula of the Hausdorff dimension of the attractor is obtained. Moreover, the relationship between the Hausdorff dimension and the parameter values is discussed in terms of the derived formula.