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Mario Bessa

Publications and source records attributed to Mario Bessa.

At least 19 recordsLinked to original sources
arXiv2025

On the periodic orbits of C0-typical impulsive semiflows

Impulsive semiflows modeled by continuous flows and continuous impulsive functions, defined over an impulsive region, are piecewise continuous semiflows with piecewise smooth trajectories. In this paper we contribute to the topological description of typical impulsive semiflows, parameterized by both flows and impulses. We prove that $C^0$-generic continuous flows generate impulsive semiflows with denseness of periodic orbits on the non-wandering set. Additionally, we show that $C^0$-generic impulses generate impulsive semiflows with denseness of periodic orbits on the impulsive non-wandering set.

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arXiv2024

On the stability of $\ddot x(t)+α(t)\dot x(t)+β(t) x(t)=0$

Our main goal is to understand the stability of second order linear homogeneous differential equations $\ddot x(t)+α(t)\dot x(t)+β(t)x(t)=0$ for $C^0$-generic values of the variable parameters $α(t)$ and $β(t)$. For that we embed the problem into the framework of the general theory of continuous-time linear cocycles induced by the random ODE $\ddot x(t)+α(φ^t(ω))\dot x(t)+β(φ^t(ω))x(t)=0$, where the coefficients $α$ and $β$ evolve along the $φ^t$-orbit for $ω\in M$, and $φ^t: M\to M$ is a flow defined on a compact Hausdorff space $M$ preserving a probability measure $μ$. Considering $y=\dot x$, the above random ODE can be rewritten as $\dot X=A(φ^t (ω))X$, with $X=(x,y)^\top$, having a kinetic linear cocycle as fundamental solution. We prove that for a $C^0$-generic choice of parameters $α$ and $β$ and for $μ$-almost all $ω\in M$ either the Lyapunov exponents of the linear cocycle are equal ($λ_1(ω)=λ_2(ω)$), or else the orbit of $ω$ displays a dominated splitting. Applying to dissipative systems ($α<0$) we obtain a dichotomy: either $λ_1(ω)=λ_2(ω)<0$, attesting the stability of the solution of the random ODE above, or else the orbit of $ω$ displays a dominated splitting. Applying to frictionless systems ($α=0$) we obtain a dichotomy: either $λ_1(ω)=λ_2(ω)=0$, attesting the asymptotic neutrality of the solution of the random ODE above, or else the orbit of $ω$ displays a hyperbolic splitting attesting the \emph{uniform} instability of the solution of the ODE above. This last result implies also an analog result for the 1-d continuous aperiodic Schrödinger equation. Furthermore, all results hold for $L^\infty$-generic parameters $α$ and $β$.

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arXiv2023

Genericity of trivial Lyapunov spectrum for Lp-cocycles derived from second order linear homogeneous differential equations

Given an ergodic flow $φ^t\colon M\rightarrow M$ defined on a probability space $M$ we study a family of continuous-time kinetic linear cocycles associated to the solutions of the second order linear homogeneous differential equations $\ddot x +α(φ^t(ω))\dot x+β(φ^t(ω))x=0$, where the parameters $α,β$ evolve along the $φ^t$-orbit of $ω\in M$. Our main result states that for a generic subset of kinetic continuous-time linear cocycles, where generic means a Baire second category with respect to an $L^p$-like topology on the infinitesimal generator, the Lyapunov spectrum is trivial.

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arXiv2023

Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parameters

In the present paper we prove that densely, with respect to an $L^p$-like topology, the Lyapunov exponents associated to linear continuous-time cocycles $Φ:\mathbb{R}\times M\to \text{GL}(2,\mathbb{R})$ induced by second order linear homogeneous differential equations $\ddot x+α(φ^t(ω))\dot x+β(φ^t(ω))x=0$ are almost everywhere distinct. The coefficients $α,β$ evolve along the $φ^t$-orbit for $ω\in M$ and $φ^t: M\to M$ is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation $\ddot x+β(φ^t(ω))x=0$ and for a Schrödinger equation $\ddot x+(E-Q(φ^t(ω)))x=0$, inducing a cocycle $Φ:\mathbb{R}\times M\to \text{SL}(2,\mathbb{R})$.

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arXiv2019

The role of the saddle-foci on the structure of a Bykov attracting set

We consider a one-parameter family $(f_λ)_{λ\, \geqslant \, 0}$ of symmetric vector fields on the three-dimensional sphere $\mathbb{S}^3\subset\mathbb{R}^4$ whose flows exhibit a heteroclinic network between two saddle-foci inside a global attracting set. More precisely, when $λ= 0$, there is an attracting heteroclinic cycle between the two equilibria which is made of two $1$-dimensional connections together with a $2$-dimensional sphere which is both the stable manifold of one saddle-focus and the unstable manifold of the other. After slightly increasing the parameter while keeping the $1$-dimensional connections unaltered, the two-dimensional invariant manifolds of the equilibria become transversal, and thereby create homoclinic and heteroclinic tangles. It is known that these newborn structures are the source of a countable union of topological horseshoes, which prompt the coexistence of infinitely many sinks and saddle-type invariant sets for many values of $λ$. We show that, for every small enough positive parameter $λ$, the stable and unstable manifolds of the equilibria and those infinitely many horseshoes are contained in the global attracting set of $f_λ$. Moreover, we prove that the horseshoes belong to the heteroclinic class of the equilibria. In addition, we verify that the set of chain-accessible points from either of the saddle-foci is chain-stable and contains the closure of the invariant manifolds of the two equilibria.

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arXiv2017

Positive Lyapunov exponents for symplectic cocycles

In the present paper we give a positive answer to a question posed by Viana on the existence of positive Lyapunov exponents for symplectic cocycles. Actually, we prove that for an open and dense set of Holder symplectic cocycles over a non-uniformly hyperbolic diffeomorphism there are non-zero Lyapunov exponents with respect to any invariant ergodic measure with the local product structure.

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arXiv2016

On shadowing and hyperbolicity for geodesic flows on surfaces

We prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the specification properties. Despite the Hamiltonian nature of the geodesic flow, the arguments in the present paper differ completely from those used in [5] for Hamiltonian systems.

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arXiv2016

Uniform hyperbolicity revisited: Index of periodic points and equidimensional cycles

In this paper we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r >= 1). In the case of the C1-topology we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum C1- robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of f are continuous in the weak*-topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The later can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures.

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arXiv2015

Dynamics of conservative Bykov cycles: tangencies, generalized cocoon bifurcations and elliptic solutions

This paper presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a conservative Bykov cycle where trajectories turn in opposite directions near the two saddle-foci. We show that {within the class of divergence-free vector fields that preserve the cycle,} tangencies of the invariant manifolds of two hyperbolic saddle-foci densely occur. The global dynamics is persistently dominated by heteroclinic tangencies and by the existence of infinitely many elliptic points coexisting with suspended hyperbolic horseshoes. A generalized version of the Cocoon bifurcations for conservative systems is obtained.

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arXiv2015

A note on reversibility and Pell equations

We consider hyperbolic toral automorphisms which are reversible with respect to a linear area-preserving involution. We will prove that within this context reversibility is linked to a generalized Pell equation whose solutions we will analyze. Additionally, we will verify to what extent reversibility is a common feature and characterize the generic setting.

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arXiv2015

Generic Hamiltonian Dynamics

In this paper we contribute to the generic theory of Hamiltonians by proving that there is a C2-residual R in the set of C2 Hamiltonians on a closed symplectic manifold M, such that, for any H in R, there is a full measure subset of energies e in H(M) such that the Hamiltonian level (H,e) is topologically mixing; moreover these level sets are homoclinic classes.

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arXiv2014

Explosion of differentiability for equivalencies between Anosov flows on 3-manifolds

For Anosov flows obtained by suspensions of Anosov diffeomorphisms on surfaces, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point, then the conjugacy has a smooth extension to the suspended 3-manifold. These result generalize the similar ones of Sullivan and Ferreira-Pinto for 1-dimensional expanding dynamics and also a result of Ferreira-Pinto for 2-dimensional hyperbolic dynamics.

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arXiv2014

Conservative flows with various types of shadowing

In the present paper we study the C1-robustness of the three properties: average shadowing, asymptotic average shadowing and limit shadowing within two classes of conservative flows: the incompressible and the Hamiltonian ones. We obtain that the first two properties guarantee dominated splitting (or partial hyperbolicity) on the whole manifold, and the third one implies that the flow is Anosov.

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arXiv2014

Positive Lyapunov exponents for Hamiltonian linear differential systems

In the present paper we give a positive answer to some questions posed by Viana on the existence of positive Lyapunov exponents for Hamiltonian linear differential systems. We prove that there exists an open and dense set of Hamiltonian linear differential systems, over a suspension flow with bounded roof function, displaying at least one positive Lyapunov exponent. In consequence, typical cocycles over a uniformly hyperbolic flow are chaotic. Finally, we obtain similar results for cocycles over flows preserving an ergodic, hyperbolic measure with local product structure.

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arXiv2014

A dichotomy in area-preserving reversible maps

In this paper we study R-reversible area-preserving maps f on a two-dimensional Riemannian closed manifold M, i.e. diffeomorphisms f such that Ro f=f^{-1}o R where R is an isometric involution on M. We obtain a C1-residual subset where any map inside it is Anosov or else has a dense set of elliptic periodic orbits. As a consequence we obtain the proof of the stability conjecture for this class of maps. Along the paper we also derive the C1-closing lemma for reversible maps and other perturbation toolboxes.

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arXiv2014

The C0 general density theorem for geodesic flows

Given a closed Riemannian manifold, we prove the C0-general density theorem for continuous geodesic flows. More precisely, that there exists a residual (in the C0-sense) subset of the continuous geodesic flows such that, in that residual subset, the geodesic flow exhibits dense closed orbits.

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arXiv2013

Trivial and simple spectrum for SL(2,R) cocycles with free base and fiber dynamics

Let $AC_D(M,SL(2,\mathbb R))$ denote the pairs $(f,A)$ so that $f\in \mathcal A\subset \text{Diff}^{1}(M)$ is a $C^{1}$-Anosov transitive diffeomorphisms and $A$ is an $SL(2,\mathbb R)$ cocycle dominated with respect to $f$. We prove that open and densely in $AC_D(M,SL(2,\mathbb R))$ (in appropriate topologies) the pair $(f,A)$ has simple spectrum with respect to the unique maximal entropy measure $μ_f$. On the other hand, there exists a residual subset $\mathcal{R}\subset \text{Aut}_{Leb}(M)\times L^\infty(M,SL(2,\mathbb R))$, with respect to the separate topology, such that any element $(f,A)$ in $\mathcal{R}$ has trivial spectrum or it is hyperbolic. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in $\text{Aut}_{Leb}(M)\times L^p(M,SL(2,\mathbb R))$.

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