Related papers: Shadowing for codimension one sectional-Anosov flo…

We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of…

We show that Lorenz flows have neither limit shadowing property nor average shadowing property nor the asymptotic average shadowing property where the reparametrizations related to these concepts relies on the set of increasing…

We study shadowing property for a partially hyperbolic diffeomorphism $f$. It is proved that if $f$ is dynamically coherent then any pseudotrajectory can be shadowed by a pseudotrajectory with "jumps" along the central foliation. The proof…

We consider the dynamics of `nonlinear tent maps': piecewise smooth unimodal maps with nowhere vanishing derivative. We show that if a nonlinear tent map $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently…

In this work we study the existence of singular flows satisfying shadowing-like properties. More precisely, we prove that if C1 -vector field on a closed manifold induces a chain-recurrent flow containing an attached hyperbolic singularity…

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such example as the product metric…

The existence of a pullback attractor is established for the singularly perturbed FitzHugh-Nagumo system defined on the entire space $R^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system…

We study pullback attractors of non-autonomous non-compact dynamical systems generated by differential equations with non-autonomous deterministic as well as stochastic forcing terms. We first introduce the concepts of pullback attractors…

We prove an estimation of the Kolmogorov $\epsilon$-entropy in H of the unitary ball in the space V, where H is a Hilbert space and V is a Sobolev-like subspace of H. Then, by means of Zelik's result [5], an estimate of the fractal…

We identify a class of time-periodic linear symmetric hyperbolic equations that are finite codimension stable, because an associated operator has compact resolvent, sufficiently far to the right in the complex plane. This paper is an…

In this paper, we make several contributions to the theory of asymptotically sectional-hyperbolic (ASH) flows. First, we prove that every star ASH attractor for a $C^2$ vector field is, in fact, sectional-hyperbolic (SH). Second, we…

In this paper, we will show that any geometric Lorenz flow in a definite class satisfies the parameter-shifted shadowing property.

We show that a strong partially hyperbolic diffeomorphism of $\mathbb{T}^3$ isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor.

We prove that every bounded strictly $J$-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We apply this result to the study of the dynamics of pseudo-holomorphic maps.

Bautista and Morales proved the existence of periodic orbits in singular-hyperbolic attracting sets(*). In this paper, we extend their result to singular-hyperbolic Lyapunov stable sets. ((*)"Existence of periodic orbits for…

We extend results on robust exponential mixing for geometric Lorenz attractors, with a dense orbit and a unique singularity, to singular-hyperbolic attracting sets with any number of (either Lorenz- or non-Lorenz-like) singularities and…

We prove that the unique SRB measure for a singular hyperbolic attractor depends continuously on the dynamics in the weak$^\ast$ topology.

We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We…

On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies…

Laminar electrically conducting Couette flows with the hydrodynamically stable quasi-Keplerian rotation profile and non-uniform conductivity are probed for dynamo instability. In spherical geometry the equations for the poloidal and the…