HOMOTOPY OF DYNAMICAL SYSTEMS ON MANIFOLDS AND MORSE THEORY ON COVERING SPACE

摘要

We consider a dynamical system X on a compact differentiable manifold M and the induced dynamical system X(ρ), on the universal covering space M of M. We develop algebraic topology methods for estimating the lower bounds on the number of codimension one surfaces (i.e. on the number of index one equilibria) on the boundary of regions of stability on M. We also develop a method of constructively verifying that the number of index one equilibria on the boundary of any region of stability in M is preserved during a homotopy of vector fields, avoiding a verification of the transversality condition. This approach allows us to get lower bounds for the index one equilibria on the boundary of stability regions of dynamical systems on noncompact manifolds and get stronger estimates than the ones afforded by the classical Morse -Smale theory.