Jacobi matrices and transversality

Abstract

Synopsis The paper deals with smooth nonlinear ODE systems in ℝ n , ẋ = f ( x ), such that the derivative f ′( x ) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f ′( x ( t )) y . Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝ n is then explicitly constructed.