Stable manifolds for nonautonomous equations without exponential dichotomy

By Luís Barreira, Cláudia Valls. 2006

58 citations recorded by the source corpus.

Abstract We consider nonautonomous ordinary differential equations v ′ = A ( t ) v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v ′ = A ( t ) v + f ( t , v ) , which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v ′ = A ( t ) v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/44bc235d2d7eee5f3f481cd98e05307763ee98e7

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In Applied Mathematics

1. César M. Silva
2. A. J. G. Bento
3. Luís Barreira
4. Cláudia Valls
5. Daniel Thompson

In Computational Theory and Mathematics

1. César M. Silva
2. A. J. G. Bento
3. Luís Barreira
4. Cláudia Valls
5. Daniel J. Thompson

In Control and Systems Engineering

1. César M. Silva
2. A. J. G. Bento
3. Cláudia Valls
4. Luís Barreira
5. Adam Czornik
6. Michal Niezabitowski

Citations in the corpus

• Generalized Nonuniform Dichotomies and Local Stable Manifolds (2010, methodology): “If μ(t) = ν(t) = e, we get the local stable manifold theorem obtained by Barreira and Valls in [1].”
• Robustness of nonuniform exponential dichotomies in Banach spaces (2008, result): “In particular, in [2,5] we established the existence and smoothness of stable and unstable invariant manifolds in R, assuming a sufficiently small nonuniformity of the dichotomy when compared to the Lyapunov exponents.”