We present new examples of generic diffeomorphisms without attractors. Also, we study how these wild classes are accumulated by infinitely many other classes (obtaining that the chain recurrence classes different from the only quasi-attractor are contained in center stable manifolds). The construction relies on some derived from Anosov (DA) constructions and uses strongly the semiconjugacy obtained by these diffeomorphisms. An interesting feature of this examples is that we can show that robustly, they present a unique attractor in the sense of Milnor.
We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be $C^1$ approximated by vector fields with orbit-flip homoclinic orbits.
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 show that a generic C1 vector field on a closed n-manifold has either infinitely many homoclinic classes or a finite collection of attractors (repellers) whose basins form an open-dense set. This result gives an approach to a conjecture by Palis. We also prove the existence of a locally residual subset of C1 vector fields on a 5-manifold having finitely many …
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.
Given a closed smooth four-dimensional manifold, we construct a diffeomorphism that has a homoclinic class whose continuation locally generically satisfies the following condition: it does not admit any kind of dominated splittings whereas any periodic points belonging to it never have unstable index one.
We study the dynamics of homoclinic classes on three dimensional manifolds under the robust absence of dominated splittings. We prove that if such a homoclinic class contains a volume-expanding periodic point, then, $C^1$-generically, it contains a hyperbolic periodic point whose index (dimension of the unstable manifold) is equal to two.
In this article we construct the parameter region where the existence of a homoclinic orbit to a zero equilibrium state of saddle type in the Lorenz-like system will be analytically proved in the case of a nonnegative saddle value. Then, for a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the detected parameter region is carried out to discover several new interesting bifurcation scenarios.
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 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.
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 establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for $N$-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solut…
We study small perturbations of a sectional hyperbolic set of a vector field on a compact manifold. Indeed, we obtain robustly finiteness of homoclinic classes on this scenary. Moreover, since attractor and repeller sets are particular cases of homoclinic classes, this result improve (A. M. López B, Finiteness and existence of attractors and repellers on sectional hyperbolic sets, Discrete and Continuous Dynamical Systems-A 37).
Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a peri…
The asymmetrically forced, damped Duffing oscillator is introduced as a prototype model for analyzing the homoclinic tangle of symmetric dissipative systems with \textit{symmetry breaking} disturbances. Even a slight fixed asymmetry in the perturbation may cause a substantial change in the asymptotic behavior of the system, e.g. transitions from two sided to one sided strange attractors as the other parameters are varied. Moreover, slight asymmetries may cause substantial asymmetries in the relative size of the basins of attraction of the unforced nearly symmetric attracting regions. These cha…
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 prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially hyperbolic (it has a finite number of transitive hyperbolic attractors with open and dense basin of attraction).
We present an example of a new strange attractor which, as we show, belongs to a class of wild pseudohyperbolic spiral attractors. We find this attractor in a four-dimensional system of differential equations which can be represented as an extension of the Lorenz system.
We prove that for every smooth compact manifold $M$ and any $r \ge 1$, whenever there is an open domain in $\mathrm{Diff}^r(M)$ exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic saddle, topologically generic diffeomorphisms in this domain have Lyapunov unstable Milnor attractors. This implies, in particular, that the instability of Milnor attractors is locally topologically generic in $C^1$ if $\mathrm{dim}\,M \ge 3$ and in $C^2$ if $\mathrm{dim}\,M = 2$. Moreover, it follows from the results of C. Bonatti, L. J. Díaz and E. R. Pujals th…
Lorenz attractors are important objects in the modern theory of chaos. The reason from one side is that they are met in various natural applications (fluid dynamics, mechanics, laser dynamics, etc.). At the same time, Lorenz attractors are robust, in the sense that they are generally not destroyed by small perturbations (autonomous, non-autonomous, stochastic). This allows us to be sure that the observed in the experiment object is exactly the chaotic attractor, rather than a long-time periodic orbit. Discrete-time analogs of the Lorenz attractor possess even more complicated structure -- they…
We give the first examples of flows which exhibit robust singular attractors containing a wild hyperbolic set (in the sense of Newhouse). A hyperbolic set is said to be wild, if it has tangencies between its stable and unstable manifolds, in a robust way. The only restriction on the ambient manifold is that its dimension should be at least 5.
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 paper we present the scenario of the occurrence of strongly dissipative mixed dynamics in two-dimensional reversible diffeomorphisms, using as an example the system describing a motion of two point vortices under the influence of wave perturbation and shear flow. For mixed dynamics of this type the chaotic attractor intersects with the chaotic repeller, but their intersection forms a "thin" set. The main stage of this scenario is the appearance of homoclinic structures for a symmetric saddle orbit which arise after crisis of a homoclinic attractor and repeller.
Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviours. Such systems can have chaotic dynamics for $N>4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies showed that some of chaotic attractors in this system are organized by heteroclinic networks. In present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also …
We present new examples of open sets of diffeomorphisms such that a generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that except from one particular class, the other classes are contained in periodic surfaces. This study allows us to obtain existence of Milnor attractors as well as studying ergodic properties of the diffeomorphisms in those open sets by using the ideas and results from [BV] and [BF].
Let $D$ be the set of $β\in (1, 2]$ such that $f_β$ is a symmetric tent map with finite critical orbit. For $β\in D$, by operating Denjoy like surgery on $f_β$, we constructed a $C^1$ unimodal map $\tilde{g}_β$ admitting a thick hyperbolic repelling invariant Cantor set which contains a wild Cantor attractor. The smoothness of $\tilde{g}_β$ is ensured by the effective estimation of the preimages of the critical point as well as the prescribed lengths of the inserted intervals. Furthermore, $D$ is dense in $(1, 2]$, and $\tilde{g}_β$ can not be $C^{1+α}$ because the hyperbolic repelling invaria…
Fibonacci unimodal maps can have a wild Cantor attractor, and hence be Lebesgue dissipative, depending on the order of the critical point. We present a one-parameter family $f_λ$ of countably piecewise linear unimodal Fibonacci maps in order to study the thermodynamic formalism of dynamics where dissipativity of Lebesgue (and conformal) measure is responsible for phase transitions. We show that for the potential $ϕ_t = -t\log|f'_λ|$, there is a unique phase transition at some $t_1 \le 1$, and the pressure $P(ϕ_t)$ is analytic (with unique equilibrium state) elsewhere. The pressure is majorised…
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…
We prove the results in [1] using Theorem 1 of the recent paper [2] by Crovisier and Yang. References: [1] Arbieto, A., Rojas, C., Santiago, B., Existence of attractors, homoclinic tangencies and singular-hyperbolicity for flows, arXiv:1308.1734v1 [math.DS] 8 Aug 2013. [2] Crovisier, S., Yang, D., On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis, arXiv:1404.5130v1 [math.DS] 21 Apr 2014.
In the space of polynomial maps of $\mathbb R^2$ of degree at least two, there are codimension $3$ laminations of maps with at least $3$ period doubling Cantor attractors. The leafs of the laminations are real-analytic and they have uniform diameter. The closure of each lamination contains the codimension one tangency locus of a saddle point. Asymptotically, the leafs of each lamination align with the leafs of the eigenvalue foliation. This is an example of general coexistence theorems valid for higher dimensional real-analytic unfoldings of two dimensional homoclinic tangencies.
We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust geometric property of partially hyperbolic laminations that we show to hold after perturbations of the dynamics. This technique also allows to prove that $C^1$-generic diffeomorphisms far from homoclinic tangencies in dimension $3$ either have at most finitely many attractors, or satisfy Newhouse phenomenon.
We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich-Morioka-Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for the birth of the Lorenz attractor; we also supply a proof for this criterion. The results are applied in order to give an analytic proof of the existence of a robust, pseudohyperbolic strange attractor (the so-called discrete Lorenz attractor) for an o…
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…
Suppose that a closed surface $S \subseteq \mathbb{R}^3$ is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points $W$ is totally disconnected, we prove that (up to an ambient homeomorphism) it has to be contained in a straight line. Using this result and a modification of the classical construction of a wild sphere due to Antoine we show that there exist uncountably many different $2$--spheres in $\mathbb{R}^3$ none of which can be realized as an attractor for a homeomorphism.
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.
Lorenz attractors play an important role in the modern theory of dynamical systems. The reason is that they are robust, i.e. preserve their chaotic properties under various kinds of perturbations. This means that such attractors can exist in applied models and be observed in experiments. It is known that discrete Lorenz attractors can appear in local and global bifurcations of multidimensional diffeomorphisms. However, to date, only partial cases were investigated. In this paper bifurcations of homoclinic and heteroclinic cycles with quadratic tangencies of invariant manifolds are studied. A f…
In this paper we study homoclinic tangles formed by transversal intersections of the stable and the unstable manifold of a {\it non-resonant, dissipative} homoclinic saddle point in periodically perturbed second order equations. We prove that the dynamics of these homoclinic tangles are that of {\it infinitely wrapped horseshoe maps}. Using $μ$ as a parameter representing the magnitude of the perturbations, we prove that (a) there exist infinitely many disjoint open intervals of $μ$, accumulating at $μ= 0$, such that the entire homoclinic tangle of the perturbed equation consists of one single…
In this paper, we study heterodimensional cycles of two-parameter families of 3-dimensional diffeomorphisms some element of which contains nondegenerate heterodimensional tangencies of the stable and unstable manifolds of two saddle points with different indexes, and prove that such diffeomorphisms can be well approximated by another element which has a quadratic homoclinic tangency associated to one of these saddle points. Moreover, it is shown that the tangency unfolds generically with respect to the family. This result together with some theorem in Viana, we detect strange attractors appear…
We consider a scenario when a stable and unstable manifolds of compact center manifold of a saddle-center coincide. The normal form of the ODE governing the system near the center manifold is derived and so is the normal form of the return map to the neighbourhood of the center manifold. The limit dynamics of the return map is investigated by showing that it might take the form of a Henon-like map possessing a Lorenz-like attractor or satisfy 'cone-field condition' resulting in partial hyperbolicity. We consider also motivating example from game theory.
In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic a…
In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full support. These interesting systems including singular hyperbolic attractors, $C^\infty$ surface diffeomorphisms and diffeomorphisms away from homoclinic tangencies.
Considering random noise in finite dimensional parameterized families of diffeomorphisms of a compact finite dimensional boundaryless manifold M, we show the existence of time averages for almost every orbit of each point of M, imposing mild conditions on the families. Moreover these averages are given by a finite number of physical absolutely continuous stationary probability measures. We use this result to deduce that situations with infinitely many sinks and Henon-like attractors are not stable under random perturbations, e.g., Newhouse's and Colli's phenomena in the generic unfolding of a …
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 understanding of the existence and structure of attractors as well as dynamical and topological implications of the existence of a global partially hyperbolic splitting. The main contributions are new examples of dynamics without attractors where we get a quite complete description of the dynamics around some wild homoclinic classes and two results on dynamical cohere…
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…
Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincaré map of $X$ at some transverse section. We remark that this universality is approximate …
We study nonlinear dynamics of two coupled contrast agents that are micro-meter size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here we consider a model of two such bubbles interacting via the Bjerknes force and exposed to an external ultrasound field. We demonstrate that in this five-dimensional nonlinear dynamical system various types of complex dynamics can occur, namely, we observe periodic, quasi-periodic, chaotic an…
Many central problems in geometry, topology, and mathematical physics lead to questions concerning the long-time dynamics of solutions to ordinary and partial differential equations. Examples range from the Einstein field equations of general relativity to quasilinear reaction-advection-diffusion equations of parabolic type. Specific questions concern the convergence to equilibria, the existence of periodic, homoclinic, and heteroclinic solutions, and the existence and geometric structure of global attractors. On the other hand, many solutions develop singularities in finite time. The singular…
