Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Jacobi matrices and transversality

Giorgio Fuscoa1 and Waldyr Muniz Olivaa2

a1 Dipartimento di Metodi e Modelli, Matematici per le Scienze Applicate, Via Antonio Scarpa 10, 00161 Rome, Italy

a2 Departamento de Matemática Aplicada, Universidade de Sāo Paulo, Caixa Postal 20.570 (Ag. Iguatemi), Sāo Paulo, SP, Brasil

Synopsis

The paper deals with smooth nonlinear ODE systems in ℝn, = 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 = 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 ℝn is then explicitly constructed.

(Received November 14 1986)

(Revised October 28 1987)