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## On Shilnikov attractors of three-dimensional flows and maps

Journal of difference equations and applications. 2022. Vol. 1. P. 1-18.

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

Publication based on the results of:

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

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

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 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

Kazakov A., Баханова Ю. В., Коротков А. Г., Журнал Средневолжского математического общества 2017 Т. 19 № 2 С. 13-24

Investigations of spiral chaos in generalized Lotka-Volterra systems and Rosenzweig-MacArthur systems that describe the interaction of three species are made in this work. It is shown that in systems under study the spiral chaos appears in agreement with Shilnikov's scenario, that is when changing a parameter in system a stable limit cycle and a saddle-focus ...

Added: October 13, 2017

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

Kazakov A., Гонченко А. С., Гонченко С. В. et al., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 10 С. 867-882

We study dynamical properties of a Celtic stone moving along the plane. Both one- and two-parameter families of the corresponding nonholonomic models are considered, in which bifurcations are studied that lead to changing types of stable motions of the stone as well as to the onset of chaotic dynamics. It is shown that multistability phenomena ...

Added: October 26, 2018

Barinova M., Grines V., Pochinka O. et al., Chaos 2021 Vol. 31 No. 6 Article 063112

This paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any AA-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering ...

Added: June 10, 2021

Bakhanova Y., Bobrovskii A., Burdygina T. et al., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 2 P. 157-164

We study spiral chaos in the classical Rössler and Arneodo-Coullet-Tresser systems. Special attention is paid to the analysis of bifurcation curves that correspond to the appearance of Shilnikov homoclinic loop of saddle-focus equilibrium states and, as a result, spiral chaos. To visualize the results, we use numerical methods for constructing charts of the maximal Lyapunov ...

Added: October 14, 2021

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

Burov A. A., Nikonov V., International Journal of Non-Linear Mechanics 2021 Vol. 137 Article 103791

The motion of a heavy bead on a smooth ring is considered. The ring is rotating uniformly around an inclined diameter. The integrability of the equations is studied using the method of splitting separatrices. The Poincaré maps for the period are constructed. Some classes of periodic motion are identified, and their stability is investigated. The reaction is calculated, and ...

Added: September 29, 2021

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

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

Гонченко А. С., Гонченко С. В., Kazakov A., Regular and Chaotic Dynamics 2013 Vol. 18 No. 5 P. 521-538

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of ...

Added: March 29, 2015

Kazakov A., Борисов А. В., Пивоварова Е. Н., Regular and Chaotic Dynamics 2016 Vol. 21 No. 7-8 P. 885-901

This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario ...

Added: January 30, 2017

Spiral attractors as the root of a new type of "bursting activity" in the Rosenzweig-MacArthur model

Bakhanova Y., Kazakov A., Korotkov A. et al., European Physical Journal: Special Topics 2018 No. 227 P. 959-970

We study the peculiarities of spiral attractors in the Rosenzweig-MacArthur model, that relates to the life-science systems and describes dynamics in a food chain prey-predator-superpredator. It is well-known that spiral attractors having a \teacup" geometry are typical for this model at certain values of parameters for which the system can be considered as slow-fast system. ...

Added: October 24, 2018

Malkin M., Gonchenko S., Li M. -., Dynamical Systems 2018 Vol. 33 No. 3 P. 441-463

Consider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold Ws(O), one-dimensional unstable manifold Wu(O), and with the saddle value σ different from 1. We assume that Ws(O) and Wu(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we ...

Added: March 12, 2020

Kazakov A., Борисов А. В., Пивоварова Е. Н., Нелинейная динамика 2017 Т. 13 № 2 С. 277-297

This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario ...

Added: October 13, 2017

Korotkov A., Kazakov A., Леванова Т. А. et al., Communications in Nonlinear Science and Numerical Simulation 2019 Vol. 71 P. 38-49

We investigated the phenomenological model of ensemble of two FitzHugh–Nagumo neuron-like elements with symmetric excitatory couplings. The main advantage of proposed model is the new approach to model the coupling which is implemented by smooth function that approximates rectangular function and reflects main important properties of biological synaptic coupling. The proposed coupling depends on three ...

Added: October 18, 2019

Kazakov A., Borisov A. V., Sataev I. R., Regular and Chaotic Dynamics 2014 Vol. 19 No. 6 P. 718-733

In this paper we consider the motion of a dynamically asymmetric unbalanced ball
on a plane in a gravitational field. The point of contact of the ball with the plane is subject
to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the
nonholonomic reversible system of 6 differential equations. In the case ...

Added: March 29, 2015

Kazakov A., Борисов А. В., Кузнецов С. П., Успехи физических наук 2014 Т. 184 № 5 С. 493-500

Based on the results of numerical simulations we discuss and illustrate dynamical phenomena characteristic for the rattleback, a solid body of convex surface moving on a rough horizontal plane, which are associated with the lack of conservation for the phase volume in the nonholonomic mechanical system. Due to local compression of the phase volume, behaviors ...

Added: October 22, 2015

Kuryzhov E., Karatetskaia E., Mints D., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 2 P. 165-174

We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble ...

Added: September 8, 2021

Kazakov A., Гонченко С. В., Гонченко А. С. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2017 Т. 25 № 2 С. 4-36

We consider important problems of modern theory of dynamical chaos and its applications. At present, it is customary to assume that in the finite-dimensional smooth dynamical systems three fundamentally different forms of chaos can be observed. This is the dissipative chaos, whose mathematical image is a strange attractor; the conservative chaos, for which the whole ...

Added: October 13, 2017

Karatetskaia E., Шыхмамедов А. И., Kazakov A., Chaos 2021 Vol. 31 Article 011102

A Shilnikov homoclinic attractor of a three-dimensional diffeomorphism contains a saddle-focus fixed point with a two-dimensional unstable invariant manifold and homoclinic orbits to this saddle-focus. The orientation-reversing property of the diffeomorphism implies a symmetry between two branches of the one-dimensional stable manifold. This symmetry leads to a significant difference between Shilnikov attractors in the orientation-reversing ...

Added: September 8, 2021