Center manifolds depending on a parameter

For differential equations u^′=A(t)u+f(t,u,λ) obtained from sufficiently small C¹ perturbations of a nonuniform exponential trichotomy, we establish the C¹ dependence of the center manifolds on the parameter λ. Our proof uses the fiber contraction principle to establish the regularity property. We note that our argument also applies to linear perturbations, without further changes.     

Stable manifolds for delay equations and parameter dependence

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x^′=L(t)_xt+f(t,_xt,λ)$x^{\prime }=L\left(t\right)x_t+f(t,x_t,\lambda )$, assuming that the linear equation x^′=L(t)_xt$x^{\prime }=L\left(t\right)x_t$ admits a nonuniform exponential dichotomy and that f$f$ is a sufficiently small Lipschitz perturbation. We also show that the stable invariant manifolds are Lipschitz in the parameter λ$\lambda$.     

Center manifolds for infinite delay

We obtain a C¹ center manifold theorem for perturbations of delay difference equations in Banach spaces with infinite delay. Our results extend in several directions the existing center manifold theorems. Besides considering infinite delay equations, we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable, unstable and central behaviors with respect to arbitrary asymptotic rates ecρ(n) for some diverging sequence ρ(n). This includes as a very special case the usual exponential behavior with ρ(n)=n.     

Analytic invariant manifolds for sequences of diffeomorphisms

We obtain real analytic invariant manifolds for trajectories of maps assuming only the existence of a nonuniform exponential behavior. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We emphasize that the maps that we consider are defined in a real Euclidean space, and thus, one is not able to obtain the invariant manifolds from a corresponding procedure to that in the nonuniform hyperbolicity theory in the context of holomorphic dynamics. We establish the existence both of stable (and unstable) manifolds and of center manifolds. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the invariant manifolds, but also for all their derivatives.     

Smooth center manifolds for nonuniformly partially hyperbolic trajectories

We establish the existence of unique smooth center manifolds for ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that v^′=A(t)v admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting.     

Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs

We obtain global and local theorems on the existence of invariant manifolds for perturbations of nonautonomous linear differential equations assuming a very general form of dichotomic behavior for the linear equation. Besides some new situations that are far from the hyperbolic setting, our results include, and sometimes improve, some known stable manifold theorems.     

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.     

Stability in delay difference equations with nonuniform exponential behavior

We study the stability under perturbations for delay difference equations in Banach spaces. Namely, we establish the (nonuniform) stability of linear nonuniform exponential contractions under sufficiently small perturbations. We also obtain a stable manifold theorem for perturbations of linear delay difference equations admitting a nonuniform exponential dichotomy, and show that the stable manifolds are Lipschitz in the perturbation.     

Invariant manifolds around equilibria of Newtonian equations: Some pathological examples

Let the equation be periodic in time, and let the equilibrium x_∗≡0 be a periodic minimizer. If it is hyperbolic, then the set of asymptotic solutions is a smooth curve in the plane ; this is stated by the Stable Manifold Theorem. The result can be extended to nonhyperbolic minimizers provided only that they are isolated and the equation is analytic (Ureña, 2007 [6]). In this paper we provide an example showing that one cannot say the same for C² equations. Our example is pathological both in a global sense (the global stable manifold is not arcwise connected), and in a local sense (the local stable manifolds are not locally connected and have points which are not accessible from the exterior).     

Stability of dichotomies in difference equations with infinite delay

For delay difference equations with infinite delay we consider the notion of nonuniform exponential dichotomy. This includes the notion of uniform exponential dichotomy as a very special case. Our main aim is to establish a stable manifold theorem under sufficiently small nonlinear perturbations. We also establish the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. Finally, we characterize the nonuniform exponential dichotomies in terms of strict Lyapunov sequences. In particular, we construct explicitly a strict Lyapunov sequence for each exponential dichotomy.     

A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics

We establish a version of the Grobman-Hartman theorem in Banach spaces for nonuniformly hyperbolic dynamics. We also consider the case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. More precisely, we consider sequences of Lipschitz maps Am+fm such that the linear parts Am admit a nonuniform exponential dichotomy, and we establish the existence of a unique sequence of topological conjugacies between the maps Am+fm and Am. Furthermore, we show that the conjugacies are Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. To the best of our knowledge this statement appeared nowhere before in the published literature, even in the particular case of uniform exponential dichotomies, although some experts claim that it is well known in this case. We are also interested in the dependence of the conjugacies on the perturbations fm: we show that it is Hölder continuous, with the same Hölder exponent as the one for the conjugacies. We emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial. In particular, we need to consider several additional Lyapunov norms.     

Conjugacies and invariant manifolds via evolution semigroups

For a linear evolution family with a nonuniformly hyperbolic behavior, we give simple proofs of the existence of topological conjugacies and stable invariant manifolds under sufficiently small perturbations. The proofs are obtained via evolution semigroups, which allows us to pass from the original nonautonomous nonuni-formly hyperbolic dynamics to one that is autonomous and uniformly hyperbolic.     

Nonexistence of codimension one Anosov flows on compact manifolds with boundary

A flow is Anosov if it exhibits contracting and expanding directions forming with the flow a continuous tangent bundle decomposition. An Anosov flow is codimension one if its contracting or expanding direction is one-dimensional. Examples of codimension one Anosov flows on compact boundaryless manifolds can be exhibited in any dimension ?3. In this paper, we prove that there are no codimension one Anosov flows on compact manifolds with boundary. The proof uses an extension to flows of some results in Hirsch [On Invariant Subsets of Hyperbolic Sets, Essays on Topology and Related Topics, Memoires dédiés à Georges de Rham, 1970, pp. 126-135] related to Question 10(b) in Palis and Pugh [Fifty problems in dynamical systems, in: J. Palis, C.C. Pugh (Eds.), Dynamical Systems-Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E.C. Zeeman on his fiftieth birthday), Lecture Notes in Mathematics, vol. 468, Springer, Berlin, 1975, pp. 345-353].     

Center manifolds for impulsive equations under nonuniform hyperbolicity

We establish the existence of smooth center manifolds under sufficiently small perturbations of an impulsive linear equation. In particular, we obtain the C¹ smoothness of the manifolds outside the jumping times. We emphasize that we consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential trichotomy.     

Torsion and conformally Anosov flows in contact Riemannian geometry

We prove that on a compact (non Sasakian) contact metric 3-manifold with critical metric for the Chern-Hamilton functional, the characteristic vector field ξ is conformally Anosov and there exists a smooth curve in the contact distribution of conformally Anosov flows. As a consequence, we show that negativity of the ξ-sectional curvature is not a necessary condition for conformal Anosovicity of ξ (this completes a result of [4]). Moreover, we study contact metric 3-manifolds with constant ξ-sectional curvature and, in particular, correct a result of [13].     

Center manifolds for smooth invariant manifolds

We study dynamics of flows generated by smooth vector fields in in the vicinity of an invariant and closed smooth manifold . By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of ) based on the information of the linearization along , which contains every locally bounded solution and is persistent under small perturbations.

     

Minimal Morse flows on compact manifolds

In this paper we prove, using the Poincaré-Hopf inequalities, that a minimal number of non-degenerate singularities can be computed in terms only of abstract homological boundary information. Furthermore, this minimal number can be realized on some manifold with non-empty boundary satisfying the abstract homological boundary information. In fact, we present all possible indices and types (connecting or disconnecting) of singularities realizing this minimal number. The Euler characteristics of all manifolds realizing this minimal number are obtained and the associated Lyapunov graphs of Morse type are described and shown to have the lowest topological complexity.     

Frobenius manifolds and central invariants for the Drinfeld-Sokolov bihamiltonian structures

The Drinfeld-Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of Miura-type transformations.     

Several sufficient conditions for sensitive dependence on initial conditions

This paper is concerned with sensitive dependence on initial conditions for maps and semi-flows. Several sufficient conditions for sensitive dependence are given, where it is not required that maps and semi-flows are continuous and spaces are compact. In particular, it is shown that if a measure-preserving map (resp. a measure-preserving semi-flow) on a metric probability space with a fully supported measure is topologically strongly ergodic, then it has sensitive dependence. These results relax and extend the conditions of some existing results.