Robustness for impulsive equations

For nonautonomous linear impulsive differential equations in Banach spaces, we establish the robustness of exponential contractions and exponential dichotomies, in the sense that the exponential behavior persists under sufficiently small linear perturbations. We also consider the more general case of nonuniform exponential behavior.     

Local analytic first integrals of planar analytic differential systems

We study the existence of local analytic first integrals of a class of analytic differential systems in the plane, obtained from the Chua?s system studied in L.O. Chua (1992, 1995), N.V. Kuznetsov et al. (2011), G.A. Leonov et al. (2012) , ,  and . The method used can be applied to other analytic differential systems.     

Center manifolds and optimal estimates

For sufficiently small perturbations of a nonuniform exponential trichotomy, we establish the existence of $C^k$ invariant center manifolds. We consider the general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. In particular, we obtain optimal estimates for the decay of all derivatives along the trajectories on the center manifolds.     

Exponential behavior in Banach spaces: robustness of trichotomies in discrete time

For difference equations in Banach spaces, we consider a generalization of the notion of exponential dichotomy, usually called trichotomy in the literature, for which the behaviors in \(\mathbb {Z}^+\) and \(\mathbb {Z}^-\) are still exponential but need not agree at the origin. Our main aim is to show that this exponential behavior is robust, in the sense that it persists under sufficiently small linear perturbations.     

Invariant Manifolds for Impulsive Equations and Nonuniform Polynomial Dichotomies

For impulsive differential equations in Banach spaces, we construct stable and unstable invariant manifolds for sufficiently small perturbations of a polynomial dichotomy. We also consider the general case of nonuniform polynomial dichotomies. Moreover, we introduce the notions of polynomial Lyapunov exponent and of regularity coefficient for a linear impulsive differential equation, and we show that when the Lyapunov exponent never vanishes the linear equation admits a nonuniform polynomial dichotomy.     

Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy

We establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that the linear equation v^′=A(t)v admits a nonuniform exponential dichotomy. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of Ck functions with α-Hölder continuous kth derivative. This is a closed subset of the space of continuous functions with the supremum norm, by an apparently not so well-known lemma of Henry (see Proposition 3). The estimates showing that the functions maintain the original bounds when transformed under the fixed-point operator are obtained through a careful application of the Faà di Bruno formula for the higher derivatives of the compositions (see (31) and (35)). As a consequence, we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class Ck.     

Stable manifolds for nonautonomous equations without exponential dichotomy

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.