Estimations of domains with cycles
type: Item
instance of: scholarly article
title: Estimations of domains with cycles
zbMATH Open document ID: 0894.34024
DOI: 10.1016/S0898-1221(97)00130-2
publication date: 1998-09-14T00:00:00Z
review text: The author derives estimates for the locus of possible cycles of \(n\)-dimensional nonlinear autonomous systems (1) \(\dot x=f(x)\). This is accomplished by means of some very useful theorems, such as the following one: For any function \(\psi\in C^\infty(\mathbb{R}^n)\) denote by \(S_\psi\) the set of zeros of the scalar product \(f\cdot\text{grad }\psi\), and let \(\psi|_{S_\psi}\) be the restriction of \(\psi\) to \(S_\psi\). Then any cycle of (1) is contained in the set \(\{x\in\mathbb{R}^n:\inf(\psi|_{S_\psi})\leq \psi(x)\leq \sup(\psi|_{S_\psi})\}\). For applicability of such results which are generalizations of the classical direct method of Lyapunov, it is important to have some knowledge about the boundedness properties of the sets \(S_\psi\) and \(\{x\in\mathbb{R}^n:\psi(x)=\text{const}\}\); corresponding criteria are presented in case \(f\) and \(\psi\) are of polynomial type. The results are then applied to the Lorenz system and the Rössler system, for which suitable quadratic polynomials \(\psi\) are constructed. In this way, effective estimates for the position of possible cycles of the Lorenz system with arbitrary parameter are finally established.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1107127
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: Lorenz systems
zbMATH Keywords: Rössler systems
zbMATH Keywords: cycles
zbMATH Keywords: \(n\)-dimensional nonlinear autonomous systems
author: A. P. Krishchenko
reviewed by: Josef Hainzl
MaRDI profile type: Publication
cites work: Deterministic Nonperiodic Flow
cites work: An equation for continuous chaos
cites work: The Lorenz equations: bifurcations, chaos, and strange attractors
cites work: Analysis and stabilization of nonlinear chaotic systems
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1107127