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## Twin Heteroclinic Connections of Reversible Systems

Regular and Chaotic Dynamics. 2024. Vol. 29. No. 1. P. 40-64.

We examine smooth four-dimensional vector fields reversible under some smooth involution L that has a smooth two-dimensional submanifold of fixed points. Our main interest here is about the orbit structure of such system near two types of heteroclinic connections involving saddle-foci and heteroclinic orbits connecting them. In both cases we found families of symmetric periodic orbits, multi-round heteroclinic connections and countable families of homoclinic orbits of saddle-foci. All this says about very complicated orbit structure near such connections. An example of non-variational stationary Swift-Hohenberg equation is considered where such structure has been found numerically.

Publication based on the results of:

Yu. V. Bakhanova, S. V. Gonchenko, Gonchenko A. S. et al., Journal of difference equations and applications 2023 Vol. 29 No. 9-12 P. 1184-1201

We describe scenarios for the emergence of Shilnikov attractors, i.e. strange attractors containing a saddle-focus with two-dimensional unstable manifold, in the case of threedimensional flows and maps. The presented results are illustrated with various specific examples ...

Added: May 30, 2022

Kazakov A., Bakhanova Y., Козлов А. Д. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2019 Т. 27 № 5 С. 7-52

The main goal of the present paper is an explanation of topical issues of the theory of spiral chaos of three-dimensional flows, i.e. the theory of strange attractors associated with the existence of homoclinic loops to the equilibrium of saddle-focus type, based on the combination of its two fundamental principles, Shilnikov’s theory and universal scenarios ...

Added: October 18, 2019

Gonchenko S. V., Gonchenko A. S., Kazakov A. et al., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2018 Vol. 28 No. 11 P. 1830036-1-1830036-29

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the ...

Added: October 26, 2018

Gonchenko A. S., Gonchenko S., Lobachevskii Journal of Mathematics 2021 Vol. 42 No. 14 P. 3352-3364

We give a short review on discrete homoclinic attractors. Such strange attractors contain only one saddle fixed point and, hence, entirely its unstable invariant manifold. We discuss the most important peculiarities of these attractors such as their geometric and homoclinic structures, phenomenological scenarios of their appearance, pseudohyperbolic properties etc. ...

Added: February 10, 2023

Bialy M., Fierobe C., Glutsyuk A. et al., Arnold Mathematical Journal 2022 Vol. 8 P. 411-422

This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021. ...

Added: February 1, 2024

Gonchenko S., Gordeeva O. V., Russian Journal of Nonlinear Dynamics 2024 Vol. 20 No. 1

We consider two-dimensional diffeomorphisms with homoclinic orbits to nonhy- perbolic fixed points. We assume that the point has arbitrary finite order degeneracy and is either of saddle-node or weak saddle type. We consider two cases when the homoclinic orbit is transversal and when a quadratic homoclinic tangency takes place. In the first case we give ...

Added: December 12, 2023

Trifonov K., / Cornell University. Series arXiv "math". 2020. No. 3454820.

An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian and two non-symmetric heteroclinic orbits permuted by the involution. This is a co- dimension one structure and therefore ...

Added: December 26, 2020

Gonchenko M. S., Kazakov A., Samylina E. et al., Regular and Chaotic Dynamics 2022 Vol. 27 No. 2 P. 198-216

—We consider reversible nonconservative perturbations of the conservative cubic H´enon maps H± 3 : ¯x = y, y¯ = −x + M1 + M2y ± y3 and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues e±i2π/3. It follows from [1] that this resonance is degenerate for M1 = ...

Added: May 30, 2022

D. A. Baranov, Kosolapov E. S., O. V. Pochinka, Siberian Mathematical Journal 2023 Vol. 64 No. 4 P. 687-699

It is known that Morse–Smale diffeomorphisms with two hyperbolic periodic orbits exist
only on the sphere and they are all topologically conjugate to each other. However, if we allow three
orbits to exist then the range of manifolds admitting them widens considerably. In particular, the
surfaces of arbitrary genus admit such orientation-preserving diffeomorphisms. In this article we find
a ...

Added: July 19, 2023

Vera Ignatenko, Discrete and Continuous Dynamical Systems 2018 Vol. 38 No. 7 P. 3637-3661

A one-parameter family of Mackey-Glass type differential delay equations is considered. The existence of a homoclinic solution for suitable parameter value is proved. As a consequence, one obtains stable periodic solutions for nearby parameter values. An example of a nonlinear functions is given, for which all sufficient conditions of our theoretical results can be verified ...

Added: May 25, 2018

Shykhmamedov A., Karatetskaia E., Kazakov A. et al., Nonlinearity 2023 Vol. 36 No. 7 P. 3501-3541

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades ...

Added: October 5, 2023

Osenkov E., Pochinka O., Russian Journal of Nonlinear Dynamics 2024 Vol. 20 No. 1 P. 167-178

In this paper, we consider a class of Morse – Smale diffeomorphisms defined on a closed 3-manifold (not necessarily orientable) under the assumption that all their saddle points have the same dimension of the unstable manifolds. The simplest example of such diffeomorphisms is the well-known “source-sink” or “north pole – south pole” diffeomorphism, whose non-wandering ...

Added: March 29, 2024

Grines V., Zhuzhoma E. V., Medvedev V. et al., Труды Средневолжского математического общества 2016 Т. 18 № 1 С. 12-16

We consider the class of continuous Morse-Smale flows defined on a topological closed manifold $M^n$ of dimension n which is not less than three, and such that the stable and unstable manifolds of saddle equilibrium states do not have intersection. We establish a relationship between the existence of such flows and topology of closed trajectories ...

Added: June 8, 2016

Kazakov A., / Cornell University. Series math "arxiv.org". 2017. No. 1801.00150.

n 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, ...

Added: January 15, 2018

Kazakov A., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 8-9 С. 729-738

In this paper, a new scenario of the appearance of mixed dynamics in two-dimensional reversible diffeomorphisms is proposed. The key point of the scenario is a sharp increase of the sizes of both strange attractor and strange repeller which appears due to heteroclinic bifurcations of the invariant manifolds of saddle fixed points belonging to these ...

Added: October 26, 2018

Golikova L., Зинина С. Х., Известия высших учебных заведений. Прикладная нелинейная динамика 2021 Т. 29 № 6 С. 851-862

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the
diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be ...

Added: December 3, 2021

Kazakov A., Козлов А. Д., Журнал Средневолжского математического общества 2018 Т. 20 № 2 С. 187-198

In the paper a new method of constructing of three-dimensional flow systems with different chaotic attractors is presented. Using this method, an example of three-dimensional system possessing an asymmetric Lorenz attractor is obtained. Unlike the classical Lorenz attractor, the observed attractor does not have symmetry. However, the discovered asymmetric attractor, as well as classical one, ...

Added: October 26, 2018

Kazakov A., Gonchenko S. V., Turaev D. V. et al., Physica D: Nonlinear Phenomena 2017 Vol. 350 P. 45-57

A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits ...

Added: October 13, 2017

Kazakov A., Gonchenko A. S., Gonchenko S. V. et al., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2014 Vol. 24 No. 8 P. 1440005-1440030

We give a qualitative description of two main routes to chaos in three-dimensional maps. We discuss Shilnikov scenario of transition to spiral chaos and a scenario of transition to discrete Lorenz-like and figure-eight strange attractors. The theory is illustrated by numerical analysis of three-dimensional Henon-like maps and Poincar´ e maps in models of nonholonomic mechanics ...

Added: March 29, 2015

Bakhanova Y., Kazakov A., Karatetskaia E. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2020 Т. 28 № 3 С. 231-258

The main goal is to construct a classification of such attractors and to distinguish among them the classes of pseudohyperbolic attractors which chaotic dynamics is preserved under perturbations of the system. The main research method is a qualitative method of saddle charts, which consists of constructing an extended bifurcation diagram on the plane of the ...

Added: September 16, 2020

Kulagin N., Lerman L., Malkin A., Communications in Nonlinear Science and Numerical Simulation 2021 Vol. 93 Article 105525

Solitons and cavitons (the latter are localized solutions with singularities) for the nonlocal Whitham equations are studied. The fourth order differential equation for traveling waves with a parameter in front of the fourth derivative is reduced to a reversible Hamiltonian system defined on a two-sheeted four-dimensional space. Solutions of the system which stay on one ...

Added: September 16, 2020

Sirotin V., Arkhipova M., Dubrova T. A. et al., Bielsko-Biala : University of Bielsko-Biala Press, 2016

The main attributes of modern enterprises should be the flexibility and the ability of forecasting the future. Constant adaptation to the changing environment and the rapidity of undertaking certain actions which are conditioned by specific situations determine the rules for the future position of market competition. Effective and efficient adjustment of the company in line ...

Added: November 2, 2016

Pahomov F., Известия РАН. Серия математическая 2016 Т. 80 № 6 С. 173-216

Полимодальная логика доказуемости
GLP была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории свободной GLP-алгебры, порожденной константами 0, 1 [1]. В этой статье для любого натурального n решается аналогичный вопрос для логик GLPn, являющихся ...

Added: December 4, 2017

Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016