Computation, continuation, and bifurcation of torus solutions for dissipative maps and ordinary differential equations
The smooth and dissipative map \(s:X\to {\mathbb{R}}^ n\), \(X\subset {\mathbb{R}}^ n\) is called stroboscopic map, and \(\Sigma (x)=\{x,S(x),S^ 2(x),...\}\) strobed orbit. The author considers the stroboscopic maps derived from the underlying dynamical system governed by an iterative map or by ordinary differential equation (autonomous, periodically forced or bi-periodically forced). If a strobed orbit is periodic then S can be considered as a fixed point of the Poincaré map. This equivalence is generalized to be bi-periodic case and the fixed point is studied in dependence of any control parameter. Moreover, the author investigates the bifurcation of a torus solution from a periodic solution. He gives a method of localization of bifurcation points. The approach is algorithmic and illustrated by suitable numerical results.
scientific article; zbMATH DE number 4000431
scientific article