Related papers: A trichotomy for generic sectional-hyperbolic chai…
We prove for a generic star vector field $X$ that, if for every chain recurrent class $C$ of $X$ all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular hyperbolic. We also prove that every Lyapunov…
This thesis attempts to contribute to the study of differentiable dynamics both from a semi-local and global point of view. The center of study is differentiable dynamics in manifolds of dimension 3 where we are interested in the…
In a smooth dynamical system, a homoclinic connection is a closed orbit returning to a saddle equilibrium. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and with chaos in higher dimensions. Homoclinic…
This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary…
Works of Liao, Ma\~n\'e, Franks, Aoki and Hayashi characterized lack of hyperbolicity for diffeomorphisms by the existence of weak periodic orbits. In this note we announce a result which can be seen as a local version of these works: for…
The renormalization paradigm for low-dimensional dynamical systems is that of hyperbolic horseshoe dynamics. Does this paradigm survive a transition to more physically relevant systems in higher dimensions? This article addresses this…
We show that for a $C^1$ residual subset of diffeomorphisms far away from homoclinic tangency, the stable manifolds of periodic points cover a dense subset of the ambient manifold. This gives a partial proof to a conjecture of C. Bonatti.
In this work a homoclinic-like loop of a piecewise smooth vector field passing through a typical singularity is analyzed. We have shown that such a loop is robust in one-parameter families of Filippov systems. The basin of attraction of…
A homoclinic class of a vector field is the closure of the transverse homoclinic orbits associated to a hyperbolic periodic orbit. An attractor (a repeller) is a transitive set to which converges every positive (negative) nearby orbit. We…
The classification problem is solved for some type of nonlinear lattices. These lattices are closely related to the lattices of Ruijsenaars-Toda type and define the Backlund auto-transformations for the class of two-component hyperbolic…
We review the theory of strange attractors and their bifurcations. All known strange attractors may be subdivided into the following three groups: hyperbolic, pseudo-hyperbolic ones and quasi-attractors. For the first ones the description…
The aim of this paper is twofold. First, we introduce standard blenders (special hyperbolic sets) and their variations, and prove their fundamental properties on the generation of $C^1$-robust tangencies. In particular, these blenders…
The well known stability conjecture of Palis and Smale states that if a diffeomorphism is structurally stable then the chain recurrent set is hyperbolic. It is natural to ask if this type of results is true for an individual chain class,…
We study various types of shadowing properties and their implication for C1 generic vector fields. We show that, generically, any of the following three hypotheses implies that an isolated set is topologically transitive and hyperbolic: (i)…
A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least…
We develop a "local theory" of multidimensional quasiperiodic $\SL(2,\R)$ cocycles which are not homotopic to a constant. It describes a $C^1$-open neighborhood of cocycles of rotations and applies irrespective of arithmetic conditions on…
This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with…
The paper deals with topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finitedimensional smooth systems can exist in three different forms.…
In this paper we introduce a class of polygonal complexes for which we can define a notion of sectional combinatorial curvature. These complexes can be viewed as generalizations of 2-dimensional Euclidean and hyperbolic buildings. We focus…
Some of the guiding problems in partially hyperbolic systems are the following: (1) Examples, (2) Properties of invariant foliations, (3) Accessibility, (4) Ergodicity, (5) Lyapunov exponents, (6) Integrability of central foliations, (7)…