Stable Periodic Points Research Articles

ОСОБЕННОСТИ СИНХРОНИЗАЦИИ ДИНАМИКИ ЧИСЛЕННОСТИ В СИСТЕМЕ ТРЕХ МИГРАЦИОННО СВЯЗАННЫХ ПОПУЛЯЦИЙ

The article is devoted to the study of oscillations (cycles) synchronization in a system of three populations coupled by migration in a circle. We consider a discrete time model of population dynamics, which is a system of three identical dissipatively connected logistic maps. One-dimensional bifurcation diagrams (trees) were constructed, supplemented with the capture index of the population numbers cycles phases at adjacent sites. We conducted numerical experiments that show phase multistability – the cycles coexistence with different phases. Using qualitative methods for studying dynamic systems, a complete phase portrait of the model is constructed, showing that in the phase space there are several periodic points corresponding to elements of synchronous (in-phase) and asynchronous (out-of-phase) cycles. The 2- and 3-cycle stability conditions are investigated. It is shown that these two cycles are represented by three options: 1) a fully synchronous regime, when the abundances in the three populations coincide at any point in time; 2) a partially synchronous regime, when abundances coincide only for two populations, 3) a non-synchronous (out-of-phase) regime, when all three numbers have different values. For a 2-cycle, the third regime is unstable, possible as part of a long transition process. We found that for the 3-cycle, besides to the synchronous and partially synchronous regime, it is possible a stable out-of-phase behavior of three populations. It is shown that stable and unstable periodic points lie on certain surfaces (invariant manifolds), which separate the areas of attraction for regimes with different degrees of phase synchronization.

Review of the research on the qualitative theory of differential equations at St. Petersburg University. II. Locally qualitative analysis of essentially nonlinear systems

This article is the second one in a series of articles devoted to a review of the results of scientific research that have been carried out at the Department of Differential Equations of St.Petersburg University over the past three decades and continue to be carried out at the present time. The first article talked about studies of stable periodic points of diffeomorphisms with homoclinic points and systems of differential equations with weakly hyperbolic invariant sets. This paper presents the results of locally qualitative analysis of essentially nonlinear systems in the neighbourhood of the zero solution, obtained by employees and graduates of the department. A system is said to be essentially nonlinear if the Taylor series expansion of its right-hand sides does not contain linear terms. The study of such systems, firstly, is complicated by a more complex picture of the behavior of solutions compared to quasilinear systems. Secondly, there are not even theoretical formulas for the general solution of a nonlinear first-approximation system, the presence of which is so helpful in the quasi-linear case. All this complicates analysis and significantly limits technical capabilities. Therefore, almost any new results and any new methods of working with such systems are of great interest. One of the most effective tools for working with essentially nonlinear systems turned out to be the logarithmic Lozinsky norms. In a sense, they are an analogue of the characteristic exponents (eigenvalues) used in the theory of quasilinear systems. Research conducted at the department has demonstrated the wide possibilities of using logarithmic norms in a wide variety of problems.

Different Types of Stable Periodic Points of Diffeomorphism of a Plane with a Homoclinic Orbit

A diffeomorphism of the plane into itself with a fixed hyperbolic point is considered; it is assumed that there is a non-transversal homoclinic point. Stable and unstable manifolds are tangent to each other at the homoclinic point; there are various ways of tangency of stable and unstable manifolds. Diffeomorphisms of the plane with a non-transversal homoclinic point were analyzed in the studies of Sh. Newhouse, L.P. Shil’nikov, and other authors under the assumption that this point is a tangency point of finite order. It follows from the studies of these authors that an infinite set of stable periodic points can lie in a neighborhood of a homoclinic point; the presence of such a set depends on the properties of the hyperbolic point. In this paper, we assume that a homoclinic point is not a point at which the tangency of a stable and an unstable manifold is a tangency of finite order. There is a countable number of types of periodic points lying in the neighborhood of the homoclinic point; points belonging to the same type are called n-pass, where n is a natural number. In this paper, it is shown that, if the tangency is not finite-order, the neighborhood of a non-transversal homoclinic point can contain an infinite set of stable single-pass, two-pass, or three-pass periodic points with characteristic exponents bounded away from zero.

Multi-pass stable periodic points of diffeomorphism of a plane with a homoclinic orbit

A diffeomorphism of a plane into itself with a fixed hyperbolic point and a nontransversal point homoclinic to it is studied. There are various ways of touching a stable and unstable manifold at a homoclinic point. Periodic points whose trajectories do not leave the vicinity of the trajectory of a homoclinic point are divided into a countable set of types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns that lie outside a sufficiently small neighborhood of the hyperbolic point. Earlier in the articles of Sh. Newhouse, L. P. Shil’nikov, B. F. Ivanov and other authors, diffeomorphisms of the plane with a nontransversal homoclinic point were studied, it was assumed that this point is a tangency point of finite order. In these papers, it was shown that in a neighborhood of a homoclinic point there can be infinite sets of stable two-pass and three-pass periodic points. The presence of such sets depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point with a finite order of tangency of a stable and unstable manifold. It is shown in the paper that for any fixed natural number n, a neighborhood of a nontransversal homolinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents separated from zero.

Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit

A diffeomorphism of the plane into itself with a fixed hyperbolic point is considered; the presence of a nontransverse homoclinic point is assumed. Stable and unstable manifolds touch each other at a homoclinic point; there are various ways of touching a stable and unstable manifold. In the works of Sh. Newhouse, L. P. Shilnikov and other authors, studied diffeomorphisms of the plane with a nontranverse homoclinic point, under the assumption that this point is a tangency point of finite order. It follows from the works of these authors that an infinite set of stable periodic points can lie in a neighborhood of a homoclinic point; the presence of such a set depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point at which the tangency of a stable and unstable manifold is a tangency of finite order. Allocate a countable number of types of periodic points lying in the vicinity of a homoclinic point; points belonging to the same type are called n-pass (multi-pass), where n is a natural number. In the present paper, it is shown that if the tangency is not a tangency of finite order, the neighborhood of a nontransverse homolinic point can contain an infinite set of stable single-pass, double-pass, or three-pass periodic points with characteristic exponents separated from zero.

Transient Chaos, Hyperchaotic Dynamics, and Transport Properties in a Bailout Embedding Web Map

In this work, we show that the bailout embedding method is responsible for creating different dynamical behaviors and for destroying intrinsic features present in mixed phase spaces of the area-preserving Hamiltonian maps, where the sticking to regular (or resonant) islands degrades chaotic properties. In particular, the base map chosen for the study is the two-dimensional (2D) Web Map (WM). The four-dimensional (4D) embedded Web Map dynamics is governed by four-parameters: ([Formula: see text]) in the WM control the nonlinearity and the type of symmetry structures (crystalline or quasi-crystalline) in phase space, respectively; ([Formula: see text]) in the embedding equations determine the mass density ratio and dissipation, respectively. For specific parameter combinations we explore the existence of transient chaos phenomenon, hyperchaotic dynamics and control the degradation of the underlying diffusive behaviors observed in phase space of the WM. If the WM is subjected to large enough dissipation through the embedding equations, stable periodic points (inside resonance islands) become sinks attracting almost all the surrounding orbits, destroying all invariant curves which divide the phase space into chaotic and regular domains. As area-preserving maps obtained from Hamiltonian flows usually share the crucial property that resonance islands can be found immersed in chaotic sea (characterizing the mixed phase space) for appropriated parameter combinations, the results obtained here for the 4D embedded WM should be considered generic for such whole class of nonlinear systems.

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.

Stability of Periodic Points of a Diffeomorphism of a Plane in a Homoclinic Orbit

We considered a self-diffeomorphism of the plane with a fixed hyperbolic point at the origin and a non-transverse point homoclinic to it. Periodic points located in a sufficiently small neighborhood of the homoclinic point are divided into single-pass and multi-pass points depending on the location of the orbit of the periodic point with respect to the orbit of the homoclinic point. It follows from the works of W. Newhouse, L.P. Shil’nikov, B.F. Ivanov and other authors that for a certain method of tangency of the stable and unstable manifolds there can be an infinite set of stable periodic points in a neighborhood of a non-transverse homoclinic point, but at least one of the characteristic exponents of these points tends to zero with increasing period. Previous works of the author imply that for a different method of tangency of the stable and unstable manifolds there can be an infinite set of stable single-pass periodic points, the characteristic exponents of which are bounded away from zero in the neighborhood of a non-transverse homoclinic point. It is shown in this paper that under certain conditions imposed primarily on the method of tangency of the stable and unstable manifolds there can be a countable set of two-pass stable periodic points, the characteristic exponents of which are bounded away from zero in any neighborhood of a non-transverse homoclinic point.

Multidimensional diffemorphisms with stable periodic points

Diffeomorphisms of a multidimensional space into itself with a hyperbolic fixed point are discussed in this paper. It is assumed that at the intersection of stable and unstable manifolds, there are points that are different from the hyperbolic point. Such points are called homoclinic and are divided into transversal and non-transversal, depending on the behavior of stable and unstable manifolds. It follows from articles by S. Newhouse, L. P. Shil’nikov, B. F. Ivanov, and others, that with a certain method of tangency of a stable manifold with an unstable one, the neighborhood of a non-transversal homoclinic point contains an infinite number of stable periodic points, but at least one of the characteristic exponents at these points tends to zero with increasing period. The present study is a continuation of previous studies by the author. In previously published papers, restrictions were imposed on the eigenvalues of the Jacobi matrix of the original diffeomorphism at a hyperbolic point. More precisely, it was assumed that either all eigenvalues are real and the Jacobi matrix is diagonal, or the matrix has only one real eigenvalue less than one in modulus, while all other eigenvalues are various complex integers greater than one in modulus. Within this framework, conditions are obtained for the presence of an infinite set of stable periodic points with characteristic exponents separated from zero in an arbitrary neighborhood of a non-transversal homoclinic point. It is assumed in this paper that the Jacobi matrix of a diffeomorphism has an arbitrary set of eigenvalues at a hyperbolic point. In this case, the conditions are obtained for the existence of an infinite set of stable periodic points whose characteristic exponents are separated from zero in the neighborhood of the non-transversal homoclinic point. The conditions are imposed, first of all, on the method of tangency of a stable manifold with an unstable one; however, the proof of the theorem essentially uses the properties of the eigenvalues of the Jacobi matrix at a hyperbolic point.

Stability of Periodic Points of Diffeomorphisms of Multidimensional Space

We study the diffeomorphism of a multidimensional space into itself with a hyperbolic fixed point at the origin and a nontransversal homoclinic point. From the works of Sh. Newhouse, B.F. Ivanov, L.P. Shilnikov, and other authors, it follows that there is a method of tangency for the stable and unstable manifold such that the neighborhood of a nontransversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of those points tends to zero as the period increases. In this paper, we study diffeomorphisms such that the method of tangency for the stable and unstable manifold differs from the case studied in the works of the abovementioned authors. This paper continues previous works of the author, where diffeomorphisms are studied such that their Jacobi matrices at the origin have only real eigenvalues. In those previous works, we find conditions such that the neighborhood of a nontransversal homoclinic point of the studied diffeomorphism contains an infinite set of stable periodic points with characteristic exponents separated from zero. In the present paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has real eigenvalues and several pairs of complex conjugate eigenvalues. Under this assumption, we find conditions guaranteeing that a neighborhood of a nontransversal homoclinic point contains an infinite set of stable periodic points with characteristic exponents separated from zero.

Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

Hierarchical collapse of regular islands via dissipation

In this work we investigate how regular islands localized in a mixed phase-space of generic area-preserving Hamiltonian systems are affected by a small amount of dissipation. Mainly we search for a universality (hierarchy) in the convergence of higher-order resonances and their periods when dissipation increases. One very simple scenario is already known: when subjected to small dissipation, stable periodic points become sinks attracting almost all the surrounding orbits, destroying all invariant curves which divide the phase-space in chaotic and regular domains. However, performing numerical experiments with the paradigmatic Chirikov–Taylor standard mapping we show that this presumably simple scenario can be rather complicated. The first, not trivial, scenario is what happens to chaotic trajectories, since they can be attracted by the sinks or by chaotic attractors, in cases when they exist. We show that this depends very much on how basins of attraction are formed as dissipation increases. In addition, we demonstrate that higher-order resonances are usually first affected by small dissipation when compared to lower-order resonances from the conservative case. Nevertheless, this is not a generic behaviour. We show that a local hierarchical collapse of resonances, as dissipation increases, is related to the area of the islands from the conservative case surrounding the periodic orbits. All observed resonance destructions occur via the bifurcation phenomena and are quantified here by determining the largest finite-time Lyapunov exponent.

To the question of stability of periodic points of three-dimensional diffeomorphisms

Self-diffeomorphisms of three-dimensional space with a hyperbolic fixed point at the origin and a nontransversal point homoclinic to it are considered. It is assumed that the Jacobian matrix of the initial diffeomorphism has complex eigenvalues at the origin. It is shown that, under certain conditions imposed mainly on the character of tangency of the stable and unstable manifolds, a neighborhood of the nontransversal homoclinic point contains an infinite set of stable periodic points whose characteristic exponents are bounded away from zero.

To the question of stability of periodic points of three-dimensional diffeomorphisms

Self-diffeomorphisms of three-dimensional space with a hyperbolic fixed point at the origin and a nontransversal point homoclinic to it are considered. It is assumed that the Jacobian matrix of the initial diffeomorphism has complex eigenvalues at the origin. It is shown that, under certain conditions imposed mainly on the character of tangency of the stable and unstable manifolds, a neighborhood of the nontransversal homoclinic point contains an infinite set of stable periodic points whose characteristic exponents are bounded away from zero.

Impulsive Multisynchronization of Coupled Multistable Neural Networks With Time-Varying Delay.

This paper studies the synchronization problem of coupled delayed multistable neural networks (NNs) with directed topology. To begin with, several sufficient conditions are developed in terms of algebraic inequalities such that every subnetwork has multiple locally exponentially stable periodic orbits or equilibrium points. Then two new concepts named dynamical multisynchronization (DMS) and static multisynchronization (SMS) are introduced to describe the two novel kinds of synchronization manifolds. Using the impulsive control strategy and the Razumikhin-type technique, some sufficient conditions for both the DMS and the SMS of the controlled coupled delayed multistable NNs with fixed and switching topologies are derived, respectively. Simulation examples are presented to illustrate the effectiveness of the proposed results.

Weak dissipative effects on trajectories from the edge of basins of attraction

The purpose of this work is to present convergence properties of regular and chaotic conservative trajectories under small dissipation. It is known that when subjected to dissipation, stable periodic points become sinks attracting the surrounding trajectories which belong to rational/irrational tori, while chaotic trajectories converge to a chaotic attractor, if it exists. Using the standard map and a mixed plot we show that this simple scenario can be rather complicated and strongly depends on the dissipation intensity. For small dissipations the huge amount of attractors of the phase-space generates a complicated and intricate dynamics where trajectories are steered to their attractors based on the local (non)hyperbolicity, measured by the Lyapunov vectors. Dissipation creates holes (or attracting channels) on the torus from the conservative limit and allows trajectories to penetrate it. These holes are regions of large local hyperbolicity and are related to sticky channels reported recently. For stronger dissipation sinks tend to attract all trajectories, prevailing over the chaotic attractor.

Determining the nature of orbits in a three‐dimensional galaxy model hosting a BL Lacertae object

AbstractA three‐dimensional dynamical model for a galaxy hosting a BL Lacertae object is constructed. The model consists of a logarithmic potential representing an elliptical host galaxy with a bulge of radius cb and a dense massive nucleus. Using numerical experiments, we try to distinguish between regular and chaotic motion in both 2D and 3D system. In particular, we investigate how the basic parameters of our model, such as the mass of the nucleus, the internal perturbation and the flattening parameters influence the amount and the degree of chaos. Interesting correlations are presented for both 2D and 3D dynamical models. Our numerical results are explained and supported using elementary theoretical arguments and analytical calculations. Of particular interest is the local integral of motion which have been found to exist in the vicinity of stable periodic points. The obtained numerical outcomes of the present research are linked and also compared with several data derived from observations. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Устойчивые периодические точки гладких диффеоморфизмов многомерного пространства

Устойчивые периодические точки гладких диффеоморфизмов многомерного пространства

Stable periodic points of infinitely smooth diffeomorphisms

Stable periodic points of infinitely smooth diffeomorphisms