Heteroclinic Points Research Articles

Criterion for the existence of a connected characteristic space of orbits in a gradient-like diffeomorphism of a surface

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the "source-sink" form, that is, by singling out a dual attractor-repeller pair, consisting of the attracting and repelling sets for all other trajectories of the system. A choice of an attractor-repeller dual pair so that the space of orbits in their complement (the characteristic space of orbits) is connected paves the way for finding complete topological invariants of the dynamical system. In this way, in particular, several classification results for Morse-Smale systems were obtained. Thus, a complete topological classification of Morse-Smale 3-diffeomorphisms is essentially based on the existence of a connected characteristic space of orbits associated with the choice of a one-dimensional dual attractor-repeller pair. For Morse-Smale diffeomorphisms with heteroclinic points on surfaces, there are examples in which the characteristic spaces of orbits are disconnected in all cases. In this paper, we prove a criterion for the existence of a connected characteristic space of orbits for gradient-like (without heteroclinic points) diffeomorphisms on surfaces. This result implies, in particular, that any orientation-preserving diffeomorphism admits a connected characteristic space. For an orientable surface of any kind, we also construct an orientation-changing gradient-like diffeomorphism that does not have a connected characteristic space. On any non-orientable surface of any kind, we also construct a gradient-like diffeomorphism which does not admit a connected characteristic space.

The structure of continuum-wise expansive measures

In [13] it was introduced and studied the notion of continuum-wise expansive measure. Here we extend the results for expansive measures obtained in [8] to the cw-expansive measure context. More precisely, we show that this set is a Gδσ subset of the space of Borel probability measures of the underlying space and that the cw-expansive measures with invariant support are weak* dense on it. We also study the homeomorphisms of compact metric spaces f:X→X for which the cw-expansive measures are weak* dense in the set of Borel probability measures. We show that the set of heteroclinic points of f has empty interior and X has no isolated points.

Fast growth of the number of periodic points arising from heterodimensional connections

We consider $C^{r}$-diffeomorphisms ($1 \leq r \leq +\infty$) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily $C^{r}$-small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order $r$, provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that $C^{r}$-generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.

Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection

According to the Nielsen – Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_{1}$) periodic homeomorphism; $T_{2}$) reducible non-periodic homeomorphism of algebraically finite order; $T_{3}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_{4}$) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_{1}$, $T_{2}$, $T_{4}$ only. Moreover, all representatives of the class $T_{4}$ have chaotic dynamics, while in each homotopy class of types $T_{1}$ and $T_{2}$ there are regular diffeomorphisms, in particular, Morse – Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse – Smale diffeomorphism with a finite number of heteroclinic orbits on a two-dimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and non-trivial. It is proved that if the heteroclinic points of a Morse – Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type $T_{1}$. The orbit space of non-trivial heteroclinic domains consists of a finite number of two-dimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transversally intersecting knots. That whether the Morse – Smale diffeomorphisms belong to types $T_{1}$ or $T_{2}$ is uniquely determined by the total intersection index of such knots.

Stable and Completely Unstable Periodic Points of Diffeomorphism of a Plane with a Heteroclinic Contour

The diffeomorphism of a plane into itself with three hyperbolic points is studied this paper. It is assumed that the heteroclinic points lie at the intersections of the unstable manifold of the first point and the stable manifold of the second point, of the unstable manifold of the second point and the stable manifold of the third point, of the unstable manifold of the third point and the stable manifold of the first point. The orbits of fixed and heteroclinic points form a heteroclinic contour. The case when stable and unstable manifolds intersect non-transversally at heteroclinic points is investigated. The points of tangency of finite order are firstly distinguished among the points of non-transversal intersection of a stable manifold with an unstable manifold; in this paper, such points are not considered. Diffeomorphism with a heteroclinic contour was studied in the works of L.P. Shilnikov, S.V. Gonchenko, and other authors, and it was assumed that the points of non-transversal intersection of stable and unstable manifolds are points of tangency of finite order. It follows from the works of these authors that a diffeomorphism exists for which there are stable and completely unstable periodic points in the neighborhood of the heteroclinic contour. It is assumed in the current paper that the points of non-transversal intersection of stable and unstable manifolds are not the points of tangency of finite order. It is demonstrated that two countable sets of periodic points may lie in the neighborhood of such a heteroclinic contour. One of these sets consists of stable periodic points whose characteristic exponents are separated from zero, and another set consists of completely unstable periodic points whose characteristic exponents are also separated from zero.

Stable and completely unstable periodic points of diffeomorphism of a plane with a heteroclinic contour

«Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия» — научно-теоретический рецензируемый журнал, публикующий исследования в области математики, механики и астрономии, а также математического моделирования.

Topological classification of Morse–Smale diffeomorphisms on 3-manifolds

The topological classification of even the simplest Morse–Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possibility of “wild” behavior of separatrices of saddle points. Another difference between Morse–Smale diffeomorphisms in dimension 3 and their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may not be only a point, as in the 2-dimensional case, but also a curve, compact or noncompact. The problem of topological classification of Morse–Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2016 by C. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, and O. Pochinka. The present article is devoted to completing the topological classification of the set MS(M3) of orientation-preserving Morse–Smale diffeomorphisms f on a smooth closed orientable 3-manifold M3. The complete topological invariant for a diffeomorphism f∈MS(M3) is the equivalence class of its scheme Sf which contains information on the periodic data and the topology of embedding of 2-dimensional invariant manifolds of the saddle periodic points of f into the ambient manifold.

Axisymmetric vortex breakdown: a barrier to mixing

In this paper, we investigate mixing in flows dominated by bubble-type vortex breakdown. Three-dimensional Navier–Stokes equations in cylindrical polar coordinates are solved to simulate flow in a cylindrical container. We have found that in steady regime of the flow, the vortex breakdown bubble is axisymmetric and apparent non-axisymmetric features observed in experiments are artifacts of imperfections in experimental set-ups. We also find that the heteroclinic manifold joining hyperbolic points of the vortex breakdown bubble is stable in the absence of any perturbation and no chaotic advection was found within vortex breakdown bubble. This makes the vortex breakdown bubble impermeable to outer fluid and hence, the vortex breakdown bubble inhibits mixing. We conclude that symmetry is a barrier to mixing.

On the set of expansive measures

We prove that the set of expansive measures of a homeomorphism of a compact metric space is a [Formula: see text] subset of the space of Borel probability measures equipped with the weak* topology. Next that every expansive measure of a homeomorphism of a compact metric space can be weak* approximated by expansive measures with invariant support. In addition, if the expansive measures of a homeomorphism of a compact metric space are dense in the space of Borel probability measures, then there is an expansive measure whose support is both invariant and close to the whole space with respect to the Hausdorff metric. Henceforth, if the expansive measures are dense in the space of Borel probability measures, the set of heteroclinic points has no interior and the space has no isolated points.

Dynamical characterization of transport barriers in nontwist Hamiltonian systems.

The turnstile provides us a useful tool to describe the flux in twist Hamiltonian systems. Thus, its determination allows us to find the areas where the trajectories flux through barriers. We show that the mechanism of the turnstile can increase the flux in nontwist Hamiltonian systems. A model which captures the essence of these systems is the standard nontwist map, introduced by del Castillo-Negrete and Morrison. For selected parameters of this map, we show that chaotic trajectories entering in resonances zones can be explained by turnstiles formed by a set of homoclinic points. We argue that for nontwist systems, if the heteroclinic points are sufficiently close, they can connect twin-islands chains. This provides us a scenario where the trajectories can cross the resonance zones and increase the flux. For these cases the escape basin boundaries are nontrivial, which demands the use of an appropriate characterization. We applied the uncertainty exponent and the entropies of the escape basin boundary in order to quantify the degree of unpredictability of the asymptotic trajectories.

Some properties of positive entropy maps

AbstractWe prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

Bowen Measure From Heteroclinic Points

Abstract We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another, i.e., heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

Plume spreading in groundwater by stretching and folding

This paper proposes a new approach to the hydraulics of in situ groundwater remediation. In situ remediation promotes reactions between an injected treatment solution and the contaminated groundwater, but without a hydraulic mechanism to promote spreading, the laminar flows characteristic of porous media will keep the two fluids in approximately the same relative configuration as they travel through the aquifer, limiting the opportunity for reactions to occur. To address this fundamental limitation, this paper borrows a key result from the fluid mechanics literature: Spreading in laminar flows is optimized by chaotic advection. Previous studies have applied this result to groundwater remediation using the pulsed dipole model, but that model depends on reinjection of fluid, which presents a number of theoretical and practical limitations. Accordingly, this paper proposes a new conceptual model for plume spreading by chaotic advection, using an engineered sequence of extractions and injections of clean water at an array of wells, which generates plume spreading by stretching and folding the fluid interface between the injected treatment solution and the contaminated groundwater but does not require reinjection. The paper includes an overview of the analytical techniques—Poincaré sections, periodic points, stable and unstable manifolds, heteroclinic points, and Lyapunov exponents—used to demonstrate chaotic advection in the limiting case in which diffusion is negligible. Numerical simulations show that spreading by stretching and folding is complimentary to spreading resulting from aquifer heterogeneity.

A generalized flux function for three-dimensional magnetic reconnection

The definition and measurement of magnetic reconnection in three-dimensional magnetic fields with multiple reconnection sites is a challenging problem, particularly in fields lacking null points. We propose a generalization of the familiar two-dimensional concept of a magnetic flux function to the case of a three-dimensional field connecting two planar boundaries. In this initial analysis, we require the normal magnetic field to have the same distribution on both boundaries. Using hyperbolic fixed points of the field line mapping, and their global stable and unstable manifolds, we define a unique flux partition of the magnetic field. This partition is more complicated than the corresponding (well-known) construction in a two-dimensional field, owing to the possibility of heteroclinic points and chaotic magnetic regions. Nevertheless, we show how the partition reconnection rate is readily measured with the generalized flux function. We relate our partition reconnection rate to the common definition of three-dimensional reconnection in terms of integrated parallel electric field. An analytical example demonstrates the theory and shows how the flux partition responds to an isolated reconnection event.

Lagrangian topology of a periodically reoriented potential flow: Symmetry, optimization, and mixing

Scalar transport in closed potential flows is investigated for the specific case of a periodically reoriented dipole flow. Despite the irrotational nature of the flow, the periodic reorientations effectively create heteroclinic and/or homoclinic points arising from the joining of stable and unstable manifolds. For scalar advection, Lagrangian chaos can be achieved with breakdown of the regular Hamiltonian structure, which is governed by symmetry conditions imposed by the dipole flow. Instability envelopes associated with period-doubling bifurcations of fixed points govern which regions of the flow control parameter space admit global chaos. These regions are further refined via calculation of Lyapunov exponents. These results suggest significant scalar transport enhancement is possible within potential flows, given appropriate programming of stirring protocols.

On the existence of chaotic circumferential waves in spinning disks

We use a third-order perturbation theory and Melnikov's method to prove the existence of chaos in spinning circular disks subject to a lateral point load. We show that the emergence of transverse homoclinic and heteroclinic points lead, respectively, to a random reversal in the traveling direction of circumferential waves and a random phase shift of magnitude pi for both forward and backward wave components. These long-term phenomena occur in imperfect low-speed disks sufficiently far from fundamental resonances.

Intersections of Lagrangian submanifolds and the Mel’nikov 1-form

We make explicit the geometric content of Mel’nikov’s method for detecting heteroclinic points between transversally hyperbolic periodic orbits. After developing the general theory of intersections for pairs of families of Lagrangian submanifolds N ε ± , with N 0 + = N 0 − and constrained to live in an auxiliary family of submanifolds, we explain how the heteroclinic orbits of a given Hamiltonian system are detected by the zeros of the Mel’nikov 1-form. This 1-form admits an integral expression which is non-convergent in general. We discuss different solutions to this convergence problem.

Stability and instability in the anisotropic Kepler problem

We study the degree of order and chaos in the anisotropic Kepler problem. The Lyapunov characteristic number (LCN) is constant for all orbits in the chaotic domain with a fixed mass ratio µ. It increases from zero as the mass ratio µ increases from µ = 1, reaches a maximum for µ = 4 and then decreases slowly to zero for a very large µ. We found the characteristics of many families of the periodic orbits of multiplicities up to 17. Several families have a stable part for µ close to µ = 1 (namely, µ < 1.748). We conjecture that there are stable orbits of high multiplicities for larger values of µ. The stable orbits are surrounded by islands of stability consisting of closed invariant curves. The rotation number along these invariant curves decreases from the periodic orbits outwards. Higher-order islands are enclosed in every main island around a low-order periodic orbit. In the case µ = 1.1, we find a large number of islands of multiplicities from 3 up to 17. Then we calculate the asymptotic curves from unstable periodic orbits. These curves go through the collision orbits but continue beyond these points. The intersections of the stable and unstable asymptotic curves are homoclinic and heteroclinic points. The corresponding orbits are the main characteristics of chaos.

Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit

We analyze homoclinic orbits near codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions. The main motivation for this study is a self-organized periodic replication process of travelling pulses which has been observed in reaction–diffusion equations. We establish conditions for existence and uniqueness of countably infinite families of curve segments of 1-homoclinic orbits which accumulate at codimension-1 or -2 heteroclinic cycles. The main result shows the bifurcation of a number of curves of 1-homoclinic orbits from such codimension-2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points. In addition, a leading order expansion of the associated curves in parameter space is derived. Its coefficients are periodic with one frequency from the imaginary part of the leading stable Floquet exponents of the periodic orbit and one from the winding number.

Chaos in the case of two fixed black holes

We study the structure of chaos in the relativistic problem of particles moving in the field of two fixed black holes M1 and M2 by considering the asymptotic curves of some simple unstable periodic orbits on a surface of section. These curves consist of infinite arcs reaching the two black holes. Most orbits starting along these curves (asymptotic orbits) escape, i.e., they fall into the black holes M1 (type I orbits) and M2 (type II orbits). The number of the remaining orbits after n iterations (intersections with the surface of section) decreases exponentially with n. The asymptotic curves intersect at infinite homoclinic and heteroclinic points. We study in detail the forms of the asymptotic orbits with emphasis on the homoclinic and heteroclinic orbits. The homoclinic and heteroclinic intersections are confined in certain intervals along the asymptotic curves. Every homoclinic and heteroclinic orbit is the limit of infinite more homoclinic and heteroclinic orbits. Between an asymptotic orbit falling on the black hole M1 and another orbit falling on the black hole M2 there are infinite homoclinic and heteroclinic orbits and infinite transitions between type I and type II orbits. Therefore the orbits of type I and II form fractal sets. The nonasymptotic curves falling on the black holes M1 and M2 also form fractal sets. The black holes act as attractors and the areas on the surface of section are not conserved despite the fact that the system is conservative.