Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T09:01:43.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Conjugacies for impulsive equations

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  01 November 2011

Luis Barreira  and
Claudia Valls
Luis Barreira
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049001 Lisboa, Portugal (barreira@math.ist.utl.pt; cvalls@math.ist.utl.pt)
Claudia Valls
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049001 Lisboa, Portugal (barreira@math.ist.utl.pt; cvalls@math.ist.utl.pt)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For impulsive differential equations, we construct topological conjugacies between linear and nonlinear perturbations of non-uniform exponential dichotomies. In the case of linear perturbations, the topological conjugacies are constructed in a more or less explicit manner. In the nonlinear case, we obtain an appropriate version of the Grobman–Hartman Theorem for impulsive equations, with a simple and direct proof that involves no discretization of the dynamics.

Keywords

Grobman–Hartman Theoremtopological conjugaciesimpulsive equations

MSC classification

Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 65 - 78
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Bainov, D., Kostadinov, S., Van Minh, N. and Zabreiko, P., Topological equivalence and exponential dichotomy of linear impulsive equations, Int. J. Theor. Phys. 33 (1994), 1581–1597.CrossRefGoogle Scholar
2.Bainov, D., Kostadinov, S., Van Minh, N. and Zabreiko, P., Topological classification of impulsive differential equations, Int. J. Theor. Phys. 34 (1995), 1147–1162.CrossRefGoogle Scholar
3.Barreira, L. and Pesin, Ya., Nonuniform hyperbolicity, Encyclopedia of Mathematics and Its Applications, Volume 115 (Cambridge University Press, 2007).Google Scholar
4.Barreira, L. and Valls, C., Conjugacies for linear and nonlinear perturbations of nonuniform behavior, J. Funct. Analysis 253 (2007), 324–358.CrossRefGoogle Scholar
5.Barreira, L. and Valls, C., Stability of nonautonomous differential equations, Lecture Notes in Mathematics, Volume 1926 (Springer, 2008).Google Scholar
6.Barreira, L. and Valls, C., A simple proof of the Grobman–Hartman theorem for nonuniformly hyperbolic flows, preprint.Google Scholar
7.Belickiĭ, G., Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, Funct. Analysis Applic. 7 (1973), 268–277.Google Scholar
8.Belickiĭ, G., Equivalence and normal forms of germs of smooth mappings, Russ. Math. Surv. 33 (1978), 107–177.Google Scholar
9.Chicone, C. and Swanson, R., Linearization via the Lie derivative, Electron. J. Diff. Eqns, Monograph 02 (2000) (available at http://www.emis.de/journals/EJDE/Monographs/02/chicone.pdf).CrossRefGoogle Scholar
10.Grobman, D., Homeomorphism of systems of differential equations, Dokl. Akad. Nauk SSSR 128 (1959), 880–881.Google Scholar
11.Grobman, D., Topological classification of neighborhoods of a singularity in n-space, Mat. Sb. 56(98) (1962), 77–94.Google Scholar
12.Hartman, P., A lemma in the theory of structural stability of differential equations, Proc. Am. Math. Soc. 11 (1960), 610–620.CrossRefGoogle Scholar
13.Hartman, P., On the local linearization of differential equations, Proc. Am. Math. Soc. 14 (1963), 568–573.Google Scholar
14.Lakshmikantham, V., Bainov, D. and Simeonov, P., Theory of impulsive differential equations, Series in Modern Applied Mathematics, Volume 6 (World Scientific, 1989).Google Scholar
15.McSwiggen, P., A geometric characterization of smooth linearizability, Michigan Math. J. 43 (1996), 321–335.Google Scholar
16.Moser, J., On a theorem of Anosov, J. Diff. Eqns 5 (1969), 411–440.CrossRefGoogle Scholar
17.Palis, J., On the local structure of hyperbolic points in Banach spaces, Annais Acad. Bras. Cienc. 40 (1968), 263–266.Google Scholar
18.Palmer, K., A generalization of Hartman's linearization theorem, J. Math. Analysis Applic. 41 (1973), 753–758.CrossRefGoogle Scholar
19.Palmer, K., The structurally stable linear systems on the half-line are those with exponential dichotomies, J. Diff. Eqns 33 (1979), 16–25.CrossRefGoogle Scholar
20.Perron, O., Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703–728.CrossRefGoogle Scholar
21.Pugh, C., On a theorem of P. Hartman, Am. J. Math. 91 (1969), 363–367.Google Scholar
22.Samoilenko, A. and Perestyuk, N., Impulsive differential equations, Nonlinear Science Series A: Monographs and Treatises, Volume 14 (World Scientific, 1995).Google Scholar
23.Sell, G., Smooth linearization near a fixed point, Am. J. Math. 107 (1985), 1035–1091.CrossRefGoogle Scholar
24.Sternberg, S., Local contractions and a theorem of Poincaré, Am. J. Math. 79 (1957), 809–824.CrossRefGoogle Scholar
25.Sternberg, S., On the structure of local homeomorphisms of Euclidean n-space, II, Am. J. Math. 80 (1958), 623–631.CrossRefGoogle Scholar