Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-20T06:01:52.597Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Angenent, S. B.. The Morse–Smale Property for a Semi-Linear Parabolic Equation. J. Differential Equations 62 (1986), 427–442.CrossRefGoogle Scholar
2Henry, D. B.. Some Infinite-Dimensional Morse–Smale Systems defined by Parabolic Partial Differential Equations. J. Differential Equations 59 (1985), 165–205.CrossRefGoogle Scholar
3Gantmacher, F. R.. The Theory of Matrices Vol. 2 (New York: Chelsea, 1959).Google Scholar
4Hale, J. K.. Infinite Dimensional Dynamical Systems. LCDS Report 81–22 (Providence, R.I.: Division of Applied Math., Brown University, 1981).CrossRefGoogle Scholar
5Hale, J. K., Magalhães, L. T. and Oliva, W. M.. An Introduction to Infinite Dynamical Systems—Geometric Theory, Springer Appl. Math. Sciences 47 (Berlin: Springer, 1984).Google Scholar
6Oliva, W. M.. Stability of Morse–Smale Maps. Rel. Tec. MAP 8301 (Sāo Paulo: Inst. Mat. Est., Univ. of Sāo Paulo, 1983).Google Scholar
7Palis, J. and Mello, W. De. Geometric Theory of Dynamical Systems—An Introduction (New York: Springer, 1982).CrossRefGoogle Scholar
8Stone, M. H.. Linear Transformations in Hilbert Space and their Applications to Analysis, AMS Coll. Publ., Vol. XV (Providence, R.I.: American Mathematical Society, 1932).Google Scholar