Global stability properties of equilibria, periodic, and chaotic solutions
This article gives an instructive and interesting exposition for understanding nonlinear dynamical systems. Simply speaking, using some basic results of the differential equation \(\ddot x+2\beta\dot x+f(x)=F\sin \omega t\) \((x\in\mathbb{R}^1)\) as a concrete model, the author introduces and organizes, in a natural way, the basic concepts of stability, transitions across basin boundaries, highly intertwined basins of attraction (or fractal basin boundaries) and stability of higher order periodic and chaotic solutions. In the conclusion of the paper, the author stresses the following topics. When analysing the global stability characteristics of higher periodic and chaotic solutions, it is found that qualitative changes in long-term behavior depends not only on the structure of coexisting basins but also on the phase at which a disturbance was applied. Such global considerations are further highlighted by the fact that systems may exhibit highly intertwined basins of attraction. In such cases not only may a disturbance cause a loss of stability, but there may also be the added complexity of a loss of predictability.
stability of higher order periodic and chaotic solutions
scientific article; zbMATH DE number 927334
scientific article