Dynamics of a Hénon-Lozi-type map
The main goal of this paper is to provide a basic analysis of the two-dimensional map \(L_{a,b,\varepsilon}\) along with detailed comparison with the Lozi map when \(\varepsilon\to 0\). Here \(L_{a,b,\varepsilon}\) is defined by \[ L_{a,b,\varepsilon}: \binom {x}{y} \mapsto \binom {1-aS_\varepsilon(x)+y}{bx}, \tag{1} \] where \(a,b\in \mathbb{R}\), \(b\neq 0\) and \(S_\varepsilon(x)\) is defined by \[ S_\varepsilon: \mathbb{R}\to \mathbb{R}, \quad x\mapsto S_\varepsilon(x)= \begin{cases} |x|&\text{if }|x|\geq \varepsilon,\\ \frac{x^2}{2\varepsilon}+ \frac{\varepsilon}{2} &\text{if }|x|\leq \varepsilon, \end{cases} \] where \(\varepsilon\) satisfies \(\varepsilon\in (0,1)\). The authors study existence and local stability of fixed points in different regions of the parameter space of \(a,b\) as the parameter \(\varepsilon\) varies. They also investigate the numerics to compare their chaotic dynamics with that of the Lozi map (period-doubling, fractal structure of the chaotic attractor, etc.) and the influence of \(\varepsilon\) on the Lyapunov exponent.
scientific article; zbMATH DE number 1755332
scientific article