Position of Periodic Solutions and Invariant Submanifolds in Structurally Stable and Polynomial Dynamical Systems

✍ Scribed by H. G. Bothe

• Publisher: John Wiley and Sons
• Year: 1977
• Tongue: English
• Weight: 616 KB
• Volume: 79
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19770790103

✦ Synopsis

Introduction.

A system of autonomous ordinary differential equations can be regarded as a vector field on Rn or, more generally, on a differentiable manifold M . The integral curves or solutions are maximal differentiable curves in M whose tangent vectors coincide with the vectors of the field. By an orbit we mean the underlying point set of a solution which carries -if it is not a single point -the orientation given by increasing curve parameters. We shall always assume that manifolds and vector fields are smooth i.e. of differentiability class C". If the boundary Bd M of M is not empty we assume moreover that vector fields on M are transverse t o Bd M , i.e. for p p B d M the vector at p is not in the tangent space of Rd M at p . Under these assumptions solutions exist, and, if we consider M floating along these solutions, we get the corresponding flow or dynamical system @. More precisely, @ is defined as the (smooth) mapping @: D + M where