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.
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.
We consider $C^r$ ($r\geqslant 1$) diffeomorphisms $f$ defined on manifolds of dimension $\geqslant 3$ with homoclinic tangencies associated to saddles. Under generic properties, we show that if the saddle is homoclinically related to a blender then the diffeomorphism $f$ can be {$C^r$} approximated by diffeomorphisms with {$C^1$} robust heterodimensional cycles. As an application, we show that the classic Simon-Asaoka's examples of diffeomorphisms with $C^1$ robust homoclinic tangencies also display {$C^1$} robust heterodimensional cycles. In a second application, we consider homoclinic tange…
We study bifurcations of non-orientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on non-orientable two-dimensional surfaces. We consider one and two parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.
We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Hénon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Hénon maps.
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 prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms, the ones exhibiting a homoclinic tangency are C^1 dense. This represents a step towards the global understanding of dynamics of surface diffeomorphisms.
We study bifurcations of area-preserving maps, both orientable (symplectic) and non-orientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.
We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.
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.
In this note we show that for any Hamiltonian defined on a symplectic 4-manifold M and any point p in M, there exists a C2-close Hamiltonian whose regular energy surface through p is either Anosov or it contains a homoclinic tangency. Our result is based on a general construction of Hamiltonian suspensions for given symplectomorphisms on Poincaré sections already known to yield similar properties.
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 every $C^1$ generic three-dimensional flow has either infinitely many sinks, or, infinitely many hyperbolic or singular-hyperbolic attractors whose basins form a full Lebesgue measure set. We also prove in the orientable case that the set of accumulation points of the sinks of a $C^1$ generic three-dimensional flow has no dominated splitting with respect to the linear Poincaré flow. As a corollary we obtain that every three-dimensional flow can be $C^1$ approximated by flows with homoclinic tangencies or by singular-Axiom A flows.
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 study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to a map asymptotically close to the 3D Henon map $\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$ which, as known, exhibits wild hyperbolic Lorenz-like attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of Lorenz-like attractors.
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 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 …
Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies
Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Graduate School of Business Administration, Hitotsubashi University（一桥大学经营研究科研究生院）；Department of Mathematics, Imperial College London（帝国理工学院数学系）
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 construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index $c>0$ and homoclinic tangencies of arbitrary codimension $c>0$. These sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite we also construct robust homoclinic tangencies of large codimension which cannot be inside a strong partially hyperbolic set.
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…
We construct partially hyperbolic diffeomorphisms having semi-local robustly transitive sets with $C^1$-robust cycles of any co-index. These constructions also provide a new method to create $C^2$-robust homoclinic, equidimensional and heterodimensional tangencies of large codimension in dimension $d\geq 4$. The method to generate robust homoclinic and equidimensional tangencies also works in the symplectic setting. Thus, these are mechanisms for the robust non-hyperbolicity of symplectomorphisms in higher dimension.
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…
We consider three-dimensional diffeomorphisms having simultaneously heterodimensional cycles and heterodimensional tangencies associated to saddle-foci. These cycles lead to a completely nondominated bifurcation setting. For every $r\geqslant 2$, we exhibit a class of such diffeomorphisms whose heterodimensional cycles can be $C^r$ stabilised and (simultaneously) approximated by diffeomorphisms with $C^r$ robust homoclinic tangencies. The complexity of our nondominated setting with plenty of homoclinic and heteroclinic intersections is used to overcome the difficulty of performing $C^r$ pertur…
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…
We prove that every C1 diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub's entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive and have no symbolic extensions.
Let $f: M \to M$ be a diffeomorphism defined on a compact boundaryless $d$-dimensional manifold $M$, $d\geq 2$. C. Morales has proposed the notion of measure expansiveness. In this note we show that diffeomorphisms in a residual subset far from homoclinic tangencies are measure expansive. We also show that surface diffeomorphisms presenting homoclinic tangencies can be $C^1$-approximated by non-measure expansive diffeomorphisms.
We present here the first part of a program for a classification of the generic dynamics close to homoclinic and heteroclinic points, in the $C^r$ topologies, $r\geq 1$. This paper only contains announcements and a few sketches of proofs; a forthcoming series of papers will present the proofs in details. The two prototypical examples of non-hyperbolic dynamics are homoclinic tangencies and heterodimensional cycles. Palis conjectured that they actually characterize densely non-hyperbolic dynamics. It is therefore important to understand what happens close to those bifurcations. We generalize cl…
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.
The attractors of a dynamical system govern its typical long-term behaviour. The presence of many attractors is significant as it means the behaviour is heavily dependent on the initial conditions. To understand how large numbers of attractors can coexist, in this thesis we study the occurrence of infinitely many stable single-round periodic solutions associated with homoclinic connections in two-dimensional maps. We show this phenomenon has a relatively high codimension requiring a homoclinic tangency and `global resonance', as has been described previously in the area-preserving setting. How…
The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer-van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (whic are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.
We obtain some properties of $C^1$ generic surface diffeomorphisms as finiteness of {\em non-trivial} attractors, approximation by diffeomorphisms with only a finite number of {\em hyperbolic} homoclinic classes, equivalence between essential hyperbolicity and the hyperbolicity of all {\em dissipative} homoclinic classes (and the finiteness of spiral sinks). In particular, we obtain the equivalence between finiteness of sinks and finiteness of spiral sinks, abscence of domination in the set of accumulation points of the sinks, and the equivalence between Axiom A and the hyperbolicity of all ho…
We show that the set of Bernoulli measures of an isolated topologically mixing homoclinic class of a generic diffeomorphism is a dense subset of the set of invariant measures supported on the class. For this, we introduce the large periods property and show that this is a robust property for these classes. We also show that the whole manifold is a homoclinic class for an open and dense subset of the set of robustly transitive diffeomorphisms far away from homoclinic tangencies. In particular, using results from Abdenur and Crovisier, we obtain that every diffeomorphism in this subset is robust…
Newhouse laminations occur in unfoldings of rank-one homoclinic tangencies. Namely, in these unfoldings, there exist codimension $2$ laminations of maps with infinitely many sinks which move simultaneously along the leaves. As consequence, in the space of real polynomial maps, there are examples of: Hénon maps, in any dimension, with infinitely many sinks, quadratic Hénon-like maps with infinitely many sinks and a period doubling attractor, quadratic Hénon-like maps with infinitely many sinks and a strange attractor, non trivial analytic families of polynomial maps with infinitely many sinks.
In the spirit of Kolmogorov typicality, we introduce the notion of germ-typicality: in a space of dynamics, it encompass all these phenomena that occur for a dense and open subset of parameters of any generic parametrized family of systems. For any $2\le r<\infty$, we prove that the Newhouse phenomenon (the coexistence of infinitely many sinks) is locally $C^r$-germ-typical, nearby a dissipative bicycle: a dissipative homoclinic tangency linked to a special heterodimensional cycle. During the proof we show a result of independent interest: the stabilization of some heterodimensional cycles for…
We study the problem of optimizing the performance of a nonlinear spring-mass-damper attached to a class of multiple-degree-of-freedom systems. We aim to maximize the rate of one-way energy transfer from primary system to the attachment, and focus on impulsive excitation of a two-degree-of-freedom primary system with an essentially nonlinear attachment. The nonlinear attachment is shown to be able to perform as a `nonlinear energy sink' (NES) by taking away energy from the primary system irreversibly for some types of impulsive excitations. Using perturbation analysis and exploiting separation…
It has been recently discovered that in smooth unfoldings of maps with a rank-one homoclinic tangency there are codimension two laminations of maps with infinitely many sinks. Indeed, these laminations, called Newhouse laminations, occur also in the holomorphic context. In the space of polynomials of $\mathbb{C}^2$, with bounded degree, there are Newhouse laminations.
We study stability and bifurcations in holomorphic families of polynomial automorphisms of C^2. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of J-stability in one-dimensional dynamics. In the second part of the paper, we prove that under an assumption of moderate dissipativity, the parameters displaying homoclinic tangencies are dense in the bifurcation locus. This confirms one of Palis' Conjectures in the comple…
An arithmetic approach to parabolic multiplicity in complex dynamics
Univ Toulouse, INSA Toulouse, CNRS, IMT（图卢兹大学，图卢兹国立应用科学学院，法国国家科学研究中心，数学研究所）；LAMFA (Laboratoire Amiénois de Mathématique Fondamentale et Appliquée), UMR CNRS 7352, Université de Picardie Jules Verne（亚眠基础与应用数学实验室，CNRS联合研究单位7352，亚眠儒勒·瓦雷斯大学）；Department of Mathematics, The Ohio State University（俄亥俄州立大学数学系）
