Abstract

We consider a homotopic to the identity family of maps, obtained as a discretization of the Lorenz system, such that the dynamics of the last is recovered as a limit dynamics when the discretization parameter tends to zero. We investigate the structure of the discrete Lorenz-like attractors that the map shows for different values of parameters. In particular, we check the pseudohyperbolicity of the observed discrete attractors and show how to use interpolating vector fields to compute kneading diagrams for near-identity maps. For larger discretization parameter values, the map exhibits what appears to be genuinely-discrete Lorenz-like attractors, that is, discrete chaotic pseudohyperbolic attractors with a negative second Lyapunov exponent. The numerical methods used are general enough to be adapted for arbitrary near-identity discrete systems with similar phase space structure.

中文翻译：

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离散类洛伦兹吸引子的数值研究

摘要

我们考虑映射恒等族的同伦，作为洛伦兹系统的离散化获得，使得当离散化参数趋于零时，最后的动力学被恢复为极限动力学。我们研究了图中针对不同参数值显示的离散类洛伦兹吸引子的结构。特别是，我们检查了观察到的离散吸引子的伪双曲性，并展示了如何使用插值向量场来计算近恒等映射的捏合图。对于较大的离散化参数值，该图显示出真正离散的类洛伦兹吸引子，即具有负第二李雅普诺夫指数的离散混沌伪双曲吸引子。所使用的数值方法足够通用，足以适用于具有相似相空间结构的任意近恒等离散系统。