Dynamics of a Hénon–Lozi-type map

We present and analyze a smooth version of the piecewise linear Lozi map. The principal motivation for this work is to develop a map, which is better amenable for an analytical treatment as compared to the Hénon map and is one that still possesses the characteristics of a Hénon-type dynamics. This paper is a first step. It does the comparison of the Lozi map (which is a piecewise linear version of the Hénon map) with the map that we introduce. This comparison is done for fixed parameters and also through global bifurcation by changing a parameter. If ε measures the degree of smoothness, we prove that, as ε → 0, the stability and the existence of the fixed points are the same for both maps. We also numerically compare the chaotic dynamics, both in the form of an attractor and of a chaotic saddle.