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Claudia Valls

1049-001 Lisboa, Portugal
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa


Llibre J., Valls  C.
Darboux Polynomials, Balances and Painlevé Property
2017, vol. 22, no. 5, pp.  543-550
For a given polynomial differential system we provide different necessary conditions for the existence of Darboux polynomials using the balances of the system and the Painlevé property. As far as we know, these are the first results which relate the Darboux theory of integrability, first, to the Painlevé property and, second, to the Kovalevskaya exponents. The relation of these last two notions to the general integrability has been intensively studied over these last years.
Keywords: Painlevé property, Darboux polynomial, Kovalevskaya exponents
Citation: Llibre J., Valls  C.,  Darboux Polynomials, Balances and Painlevé Property, Regular and Chaotic Dynamics, 2017, vol. 22, no. 5, pp. 543-550
Barreira L., Dragičević D., Valls  C.
Nonuniform Exponential Dichotomies and Lyapunov Functions
2017, vol. 22, no. 3, pp.  197-209
For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.
Keywords: nonuniform exponential dichotomies, Lyapunov functions
Citation: Barreira L., Dragičević D., Valls  C.,  Nonuniform Exponential Dichotomies and Lyapunov Functions, Regular and Chaotic Dynamics, 2017, vol. 22, no. 3, pp. 197-209
Barreira L., Dragičević D., Valls  C.
Admissibility and Nonuniform Exponential Trichotomies
2015, vol. 20, no. 1, pp.  49-62
For a nonautonomous dynamics defined by a sequence of linear operators acting on a Banach space, we show that the notion of a nonuniform exponential trichotomy can be completely characterized in terms of admissibility properties. This refers to the existence of bounded solutions under any bounded time-dependent perturbation of certain homotheties of the original dynamics. We also consider the more restrictive notion of a strong nonuniform exponential trichotomy and again we give a characterization in terms of admissibility properties. We emphasize that both notions are ubiquitous in the context of ergodic theory. As a nontrivial application, we show in a simple manner that the two notions of trichotomy persist under sufficiently small linear perturbations. Finally, we obtain a corresponding characterization of nonuniformly partially hyperbolic sets.
Keywords: exponential trichotomy, robustness, partially hyperbolic set
Citation: Barreira L., Dragičević D., Valls  C.,  Admissibility and Nonuniform Exponential Trichotomies, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 49-62

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