Bifurcation of a stable invariant torus from an equilibrium
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.
invariant manifold homeomorphic to a torus
scientific article; zbMATH DE number 4192162
scientific article