Chaotic dynamics can be quite heterogeneous in the sense that some regions are unstable more directions than other regions. When trajectories wander between these regions, is complicated. We say a chaotic invariant set when arbitrarily close to each point of there different periodic points with numbers dimensions. call such chaos (or hetero-chaos), While we believe it common for physical system...

It is shown how functions that are defined by evaluation programs involving the absolute value function abs() (besides smooth elementals), can be approximated locally by piecewise-linear models in the style of algorithmic, or automatic, differentiation (AD). The model can be generated by a minor modification of standard AD tools [GJU96b] and it is Lipschitz continuous with respect to the base p...