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Jacobi matrices and transversality

Published online by Cambridge University Press:  14 November 2011

Giorgio Fusco  and
Waldyr Muniz Oliva
Giorgio Fusco
Affiliation:
Dipartimento di Metodi e Modelli, Matematici per le Scienze Applicate, Via Antonio Scarpa 10, 00161 Rome, Italy
Waldyr Muniz Oliva
Affiliation:
Departamento de Matemática Aplicada, Universidade de Sāo Paulo, Caixa Postal 20.570 (Ag. Iguatemi), Sāo Paulo, SP, Brasil
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Synopsis

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

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 3-4 , 1988 , pp. 231 - 243
Copyright
Copyright © Royal Society of Edinburgh 1988

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