A numerical renormalization method for quasi–conservative periodic attractors

Abstract

We describe a renormalization method in maps of the plane \begin{document}$ (x, y) $\end{document} , with constant Jacobian \begin{document}$ b $\end{document} and a second parameter \begin{document}$ a $\end{document} acting as a bifurcation parameter. The method enables one to organize high period periodic attractors and thus find hordes of them in quasi-conservative maps (i.e. \begin{document}$ |b| = 1-\varepsilon $\end{document} ), when sharing the same rotation number. Numerical challenges are the high period, and the necessary extreme vicinity of many such different points, which accumulate on a hyperbolic periodic saddle. The periodic points are organized, in the \begin{document}$ (x, y, a) $\end{document} space, in sequences of diverging period, that we call "branches". We define a renormalization approach, by "hopping" among branches to maximize numerical convergence. Our numerical renormalization has met two kinds of numerical instabilities, well localized in certain ranges of the period for the parameter \begin{document}$ a $\end{document} (see [ 3 ]) and in other ranges of the period for the dynamical plane \begin{document}$ (x, y) $\end{document} . For the first time we explain here how specific numerical instabilities depend on geometry displacements in dynamical plane \begin{document}$ (x, y) $\end{document} . We describe how to take advantage of such displacement in the sequence, and of the high period, by moving forward from one branch to its image under dynamics. This, for high period, allows entering the hyperbolicity neighborhood of a saddle, where the dynamics is conjugate to a hyperbolic linear map. The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [ 7 ] for highly dissipative systems.