BIFURCAÇÕES GENÉRICAS DE EQUAÇÕES DIFERENCIAIS ORDINÁRIAS

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Publications

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

Publication . Bessa, Mário; Rodrigues, Alexandre A. P.

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 non-uniformly hyperbolic suspended horseshoes. A generalized version of the Cocoon bifurcations for conservative systems is obtained.

2016Journal article

Open access

Generic area-preserving reversible diffeomorphisms

Publication . Bessa, Mário; Carvalho, Maria; Rodrigues, Alexandre A. P.

Let M be a surface and R : M → M an area-preserving C∞ diffeomorphism which is an involution and whose set of fixed points is a submanifold with dimension one. We will prove that C1 -generically either an area-preserving R-reversible diffeomorphism, is Anosov, or, for μ-almost every x ∈ M, the Lyapunov exponents at x vanish or else the orbit of x belongs to a compact hyperbolic set with an empty interior. We will also describe a nonempty C1- open subset of area-preserving R-reversible diffeomorphisms where for C1-generically each map is either Anosov or its Lyapunov exponents vanish from almost everywhere.

2015Journal article

Restricted access

A dichotomy in area-preserving reversible maps

Publication . Bessa, Mário; Rodrigues, Alexandre A. P.

In this paper we study R-reversible area-preserving maps f : M → M on a two-dimensional Riemannian closed manifold M, i.e. diffeomorphisms f such that R ◦ f = f−1 ◦ R where R: M → M is an isometric involution. We obtain a C1-residual subset where any map inside it is Anosov or else has a dense set of elliptic periodic orbits, thus establishing the stability conjecture in this setting. Along the paper we derive the C1-Closing Lemma for reversible maps and other perturbation toolboxes.

2015Journal article

Open access

A note on reversibility and Pell equations

Publication . Bessa, Mário; Carvalho, Maria; Rodrigues, Alexandre A. P.

This note concerns hyperbolic toral automorphisms which are reversible with respect to a linear area-preserving involution. Due to the low dimension, we will be able to associate the reversibility with a generalized Pell equation from whose set of solutions we will infer further information. Additionally, we will show that reversibility is a rare feature and will characterize the generic setting.

2017Journal article

Open access