In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) a hyperbolic basic set which exhibits $C^2$-robust homoclinic tangency of the largest codimension for any higher dimensional manifold, using blenders. The former implies that an analog of Moreira's theorem on Cantor sets in the real line does not hold in higher dimension. The latter solves a question posed by Barrientos and A.Raibekas.
We give a detailed proof for the estimates of products of norms for flows away from homoclinic tangency. The estimates we can prove is weaker than the expectation of experts.
In this paper we study the effect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of entropy variation in the $C^{\infty}$-topology. We also prove results about variation of entropy in other topologies and when the tangency does not correspond to a basic set with maximal entropy. We also show an example of discontinuity of the entropy among $C^{\infty}$ diffeomorphisms of three dimensional manifolds.
We study main bifurcations of multidimensional diffeomorphisms having a non-transversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighbourhood of the homoclinic orbit. Also a relation of our results to well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.
We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinksnear the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.
It was established in 2006 that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is non-simple.
We show that any diffeomorphism of a compact manifold can be C1 approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.
Adding small random parametric noise to an arc of diffeomophisms of a manifold of dimension 3, generically unfolding a codimension one quadratic homoclinic tangency q associated to a sectionally dissipative saddle fixed point p, we obtain not more than a finite number of physical probability measures, whose ergodic basins cover the orbits which are recurrent to a neighborhood of the tangency point $q$. This result is in contrast to the extension of Newhouse's phenomenon of coexistence of infinitely many sinks obtained by Palis and Viana in this setting. There is a similar result for the simple…
We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387.
We study generic two- and three-parameter unfoldings of a pair of orbits of quadratic homoclinic tangency in strongly dissipative systems. We prove that the corresponding stability windows for periodic orbits have various universal forms: the so-called "shrimps" (cross-road areas), as well as spring and saddle areas, and (in three-parameter unfoldings) "pregnant shrimps" (specific types of transitions between the shrimps and spring or saddle areas).
We prove that any vector field on a three-dimensional compact manifold can be approximated in the C1-topology by one which is singular hyperbolic or by one which exhibits a homoclinic tangency associated to a regular hyperbolic periodic orbit. This answers a conjecture by Palis. During the proof we obtain several other results with independent interest: a compactification of the rescaled sectional Poincaré flow and a generalization of Mañé-Pujals-Sambarino theorem for three-dimensional C2 vector fields with singularities.
In this paper, we show that the Hénon map $φ_{a,b}$ has a generically unfolding cubic tangency for some $(a,b)$ arbitrarily close to $(-2,0)$ by applying results of Gonchenko-Shilnikov-Turaev [12]-[16]. Combining this fact with theorems in Kiriki-Soma [20], one can observe the new phenomena in the Hénon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.
The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its stable manifold intersects a non-differentiable point of its unstable manifold (or vice-versa). This is a codimension-one bifurcation analogous to a homoclinic tangency of a smooth map, referred to here as a homoclinic corner. This paper presents an unfolding of generic homoclinic corners for saddle fixed points of planar piecewise-smooth continuous maps. I…
A diffeomorphism $f$ has a $C^1$-robust homoclinic tangency if there is a $C^1$-neighbourhood $\cU$ of $f$ such that every diffeomorphism in $g\in \cU$ has a hyperbolic set $\La_g$, depending continuously on $g$, such that the stable and unstable manifolds of $\La_g$ have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with $C^1$-robust homoclinic tangencies. Using blender-horseshoes, we prove that homoclinic classes of $C^1$-generic diffeomorphisms containing saddles …
We show that for a $C^1$ residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it's the Hausdorff limit of a family of sources.
We show that there exists a polynomial automorphism $f$ of $\mathbb{C}^{3}$ of degree 2 such that for every automorphism $g$ sufficiently close to $f$, $g$ admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each $d \ge 2$, there exists an open set of polynomial automorphisms of degree at most $d$ in which the automorphisms having infinitely many sinks are dense. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Diaz.
Homoclinic tangencies and singular hyperbolicity are involved in the Palis conjecture for vector fields. Typical three dimensional vector fields are well understood by recent works. We study the dynamics of higher dimensional vector fields that are away from homoclinic tangencies. More precisely, we prove that for \emph{any} dimensional vector field that is away from homoclinic tangencies, all singularities contained in its robustly transitive singular set are all hyperbolic and have the same index. Moreover, the robustly transitive set is {$C^1$-generically }partially hyperbolic if the vector…
A discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, an explosion is a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. Newhouse and Palis conjectured in 1976 that planar explosions are generically the result of either tangency or saddle node bifurcations. In this paper, we prove this conjecture for one-dimensional maps. Furthermore, we give a full classification for all possible tangency bifurcations and whether they lead to explosions.
Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tangency, in this paper we study one-parameter perturbations of typical globally resonant homoclinic tangencies. We assume the tangencies are formed by the stable and unstable manifolds of saddle fixed points of two-dimensional maps. We show the perturbations display two infinite sequences of bifurcations, one saddle-node the other period-doubling, between which…
In this paper, we study a two-parameter family of two-dimensional diffeomorphisms such that it has a cubic homoclinic tangency unfolding generically which is associated with a dissipative saddle point. Our first theorem presents an open set in the parameter-plane such that, for any parameter value in the open set, there exists a one-parameter subfamily through this value exhibiting cubically related persistent contact-making and contact-breaking quadratic tangencies. Moreover, the second theorem shows that any such two-parameter family satisfies Wang-Young's conditions which guarantee that it …
We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a $\mathbb{Z}_2$ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.
We consider a homoclinic orbit to a saddle fixed point of an arbitrary $C^\infty$ map $f$ on $\mathbb{R}^2$ and study the phenomenon that $f$ has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory, this requires $f$ to have a homoclinic tangency. We show it also necessary for $f$ to satisfy a `global resonance' condition and for the eigenvalues associated with the fixed point, $λ$ and $σ$, to satisfy $|λσ| = 1$. The phenomenon is codimension-three in the case $λσ= -1$, but codimension-four in the case $λσ= 1$ because here the coefficients of the…
In this paper we consider horseshoes containing an orbit of homoclinic tangency accumulated by periodic points. We prove a version of the Invariant Manifolds Theorem, construct finite Markov partitions and use them to prove the existence and uniqueness of equilibrium states associated to Hölder continuous potentials.
In this paper, we explore the three-dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions. We extend previous results on the field and we show that the first return map to a given cross section may be approximated by a map exhibiting heteroclinic tangencies associated to two periodic orbits. When the cycle is broken, under an additi…
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.
An interesting problem in solid state physics is to compute discrete breather solutions in $\mathcal{N}$ coupled 1--dimensional Hamiltonian particle chains and investigate the richness of their interactions. One way to do this is to compute the homoclinic intersections of invariant manifolds of a saddle point located at the origin of a class of $2\mathcal{N}$--dimensional invertible maps. In this paper we apply the parametrization method to express these manifolds analytically as series expansions and compute their intersections numerically to high precision. We first carry out this procedure …
Let $\{f_{a,b}\}$ be the (original) Hénon family. In this paper, we show that, for any $b$ near $0$, there exists a closed interval $J_b$ which contains a dense subset $J'$ such that, for any $a\in J'$, $f_{a,b}$ has a quadratic homoclinic tangency associated with a saddle fixed point of $f_{a,b}$ which unfolds generically with respect to the one-parameter family $\{f_{a,b}\}_{a\in J_b}$. By applying this result, we prove that $J_b$ contains a residual subset $A_b^{(2)}$ such that, for any $a\in A_b^{(2)}$, $f_{a,b}$ admits the Newhouse phenomenon. Moreover, the interval $J_b$ contains a dense…
