Bifurcation of a stable invariant torus from an equilibrium
type: Item
instance of: scholarly article
title: Bifurcation of a stable invariant torus from an equilibrium
zbMATH Open document ID: 0723.34029
DOI: 10.1007/BF01164258
published in: Mathematical Notes
publication date: 1990-01-01T00:00:00Z
review text: The system of differential equations \((1)\quad \dot x=Ax+X(x),\quad x\in R^ n,\) is considered where \(Y(0)=0\), \(\partial X(0)/\partial x=0\), X is sufficiently smooth in a neighbourhood of \(x=0\), det A\(=0\). The implicit function theorem yields that a small smoothly perturbed system \((2)\quad \dot x=Ax+X(x)+\epsilon F(x,\epsilon)\) has an equilibrium state for each sufficiently small \(\epsilon >0\). If for such \(\epsilon\) system (1) has an invariant manifold homeomorphic to a torus, then it is said that a bifurcation of invariant tori from the equilibrium state arises. If the matrix A has no eigenvalues on the imaginary axix then such a bifurcation does not occur. In the paper a sufficient condition (of the type that some matrix is stable) is obtained for system (1) to possess such a bifurcation, provided the matrix A has exactly m pairs of purely imaginary eigenvalues.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34D35
zbMATH DE Number: 4192162
zbMATH Keywords: invariant manifold homeomorphic to a torus
zbMATH Keywords: bifurcation of invariant tori
author: Yu. N. Bibikov
MaRDI profile type: Publication
cites work: Q3913687
cites work: INVARIANT ASYMPTOTICALLY STABLE TORI OF THE PERTURBED KORTEWEG-DE VRIES EQUATION
cites work: Integral manifolds of perturbed differential systems
cites work: Local theory of nonlinear analytic ordinary differential equations
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description: scientific article; zbMATH DE number 4192162