Global stability properties of equilibria, periodic, and chaotic solutions
type: Item
instance of: scholarly article
title: Global stability properties of equilibria, periodic, and chaotic solutions
zbMATH Open document ID: 0857.34054
DOI: 10.1016/0307-904X(95)00168-J
author: M. S. Soliman
published in: Applied Mathematical Modelling
publication date: 1996-09-15T00:00:00Z
review text: 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.
reviewed by: Ding Tongren
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C28
Mathematics Subject Classification ID: 34D45
zbMATH DE Number: 927334
zbMATH Keywords: nonlinear dynamical systems
zbMATH Keywords: stability
zbMATH Keywords: transitions across basin boundaries
zbMATH Keywords: basins of attraction
zbMATH Keywords: fractal basin boundaries
zbMATH Keywords: stability of higher order periodic and chaotic solutions
MaRDI profile type: Publication
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Q3751882
cites work: Q3736576
cites work: Q4692840
cites work: Stability by Liapunov's direct method. With applications
cites work: Q4039901
cites work: Q3851672
cites work: BASIN ORGANIZATION PRIOR TO A TANGLED SADDLE-NODE BIFURCATION
cites work: Jumps to Resonance: Long Chaotic Transients, Unpredictable Outcome, and the Probability of Restabilization
cites work: Integrity measures quantifying the erosion of smooth and fractal basins of attraction
cites work: Fractal basin boundaries
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description: scientific article; zbMATH DE number 927334