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## On Examples of Pseudohyperbolic Attractors in Flows and Maps

Lobachevskii Journal of Mathematics. 2021. Vol. 42. No. 14. P. 3451-3467.

M. Kainov, A. Kazakov

In this paper we give some known examples of pseudohyperbolic attractors of systems of differential equations and diffeomorphisms and also describe our numerical method for the verification of strange attractors on pseudohyperbolicity. By means of this method we give numerical evidence of the pseudohyperbolicity of the Lorenz attractor in the Lyubimov–Zaks model, the wild spiral attractor of Turaev and Shilnikov type in a four-dimensional Lorenz system, various discrete attractors of Lorenz type in three-dimensional H\'enon maps, and the figure-eight attractor in the nonholonomic model of Chaplygin top.

Publication based on the results of:

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

Tatyana A. Alexeeva, Barnett W., Kuznetsov N. et al., Chaos, Solitons and Fractals 2020 Vol. 140 Article 110239

Forecasting and analyses of the dynamics of financial and economic processes such as deviations of macroeconomic aggregates (GDP, unemployment, and inflation) from their long-term trends, asset markets volatility, etc., are challenging because of the complexity of these processes. Important related research questions include, first, how to determine the qualitative properties of the dynamics of these ...

Added: October 21, 2020

Malkin M., Safonov K., Applied Mathematics and Nonlinear Sciences 2020 Vol. 5 No. 2 P. 293-306

We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps T_{c,epsilon}(x)=(-1+c|x|^{1-epsilon}) sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to ...

Added: October 31, 2020

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

Kuptsov P., Kuptsova A. V., Stankevich N., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 1 P. 5-21

We suggest a universal map capable of recovering the behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter ...

Added: April 3, 2021

Малкин М. И., Safonov K., Chaos 2021 Vol. 31 Article 043107

This paper deals with one-dimensional factor maps for the geometric model of Lorenz-type attractors in the form of two-parameter family of Lorenz maps on the interval 𝐼=[−1,1]I=[−1,1] given by 𝑇𝑐,𝜈(𝑥)=(−1+𝑐⋅|𝑥|𝜈)⋅𝑠𝑖𝑔𝑛(𝑥)Tc,ν(x)=(−1+c⋅|x|ν)⋅sign(x). This is the normal form for splitting the homoclinic loop with additional degeneracy in flows with symmetry that have a saddle equilibrium with a one-dimensional unstable manifold. Due ...

Added: September 21, 2021

Skripchenko A., Hubert P., Avila A., / Cornell University. Series math "arxiv.org". 2014. No. 1412.7913.

We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on diffusion rate of these sections using the connection between Novikov's problem and systems of isometries - some natural generalization of interval exchange transformations. ...

Added: January 27, 2015

Stankevich N., Kazakov A., Gonchenko S., Chaos 2020 Vol. 30 Article 123129

The generalized four-dimensional Rössler system is studied. Main bifurcation scenarios leading to a hyperchaos are described phenomenologically and their implementation in the model is demonstrated. In particular, we show that the formation of hyperchaotic invariant sets is related mainly to cascades (finite or infinite) of nondegenerate bifurcations of two types: period-doubling bifurcations of saddle cycles with a ...

Added: January 18, 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

Stankevich N., Volkov E., Nonlinear Dynamics 2018 Vol. 94 No. 4 P. 2455-2467

The emergence of multistability in a simplethree-dimensionalautonomousoscillatorisinvestigatedusingnumericalsimulations,calculationsofLyapunov exponents and bifurcation analysis over a broad area of two-dimensional plane of control parameters. Using Neimark–Sacker bifurcation of 1:1 limit cycle asthestartingregime,manyparameterislandswiththe coexisting attractors were detected in the phase diagram,includingthecoexistenceoftorus,resonantlimit cycles and chaos; and transitions between the regimes were considered in detail. The overlapping between resonant limit cycles ...

Added: December 2, 2019

Fougeron C., Skripchenko A., Monatshefte fur Mathematik 2021 Vol. 194 No. 4 P. 767-787

We introduce a new strategy to prove simplicity of the spectrum of Lyapunov exponents that can be applied to a wide class of Markovian multidimensional continued fraction algorithms. As an application we use it for Selmer algorithm in dimension 2 and for the Triangle sequence algorithm and show that these algorithms are not optimal.
There is ...

Added: February 10, 2021

Гонченко А. С., Korotkov A., Samylina E., Дифференциальные уравнения и процессы управления 2022 № 2 С. 187-204

We study the problem on the existence of Lorenz-like attractors and repellers in three-dimensional time-reversible systems, as well as the structure of bifurcation scenarios for their emergence. In this connection, we consider a system that is the flow normal form for reversible bifurcations of a fixed point with the triplet (-1,-1,+1) of multipliers. The bifurcation ...

Added: August 28, 2023

Stankevich N., Kuznetsov A., Popova E. et al., Nonlinear Dynamics 2019 Vol. 97 P. 2355-2370

Using an example of a radiophysical generator model, scenarios for the formation of various chaotic attractors are described, including chaos and hyperchaos. It is shown that as a result of a secondary Neimark–Sacker bifurcation, a hyperchaos with two positive Lyapunov exponents can occur in the system. A comparative analysis of chaotic attractors born as a ...

Added: December 2, 2019

Gonchenko S., Karatetskaia E., Kazakov A. et al., Chaos 2022 Vol. 32 No. 12 Article 121107

We describe new types of Lorenz-like attractors for three-dimensional flows and maps with symmetries. We give an example of a three-dimensional system of differential equations, which is centrally symmetric and mirror symmetric. We show that the system has a Lorenz-like attractor, which contains three saddle equilibrium states and consists of two mirror-symmetric components that are ...

Added: January 31, 2023

Stankevich N., Volkov E., Chaos 2021 Vol. 31 No. 10 Article 103112

We investigate the dynamics of three identical three-dimensional ring synthetic genetic oscillators (repressilators) located in different cells and indirectly globally coupled by quorum sensing whereby it is meant that a mechanism in which special signal molecules are produced that, after the fast diffusion mixing and partial dilution in the environment, activate the expression of a ...

Added: October 12, 2021

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

Kuznetsov A. P., Sedova Y. V., Stankevich N., Chaos, Solitons and Fractals 2023 Vol. 169 Article 113278

The interaction of a system with quasi-periodic autonomous dynamics and a chaotic Rossler system is studied. We have shown that with the growth of the coupling, regimes of two-frequency and three-frequency quasiperiodicity, a periodic regime and a regime of oscillation death sequentially arise. With a small coupling strength, doubling bifurcations of three-frequency tori are observed ...

Added: March 3, 2023

Kruglov V., Krylosova D., Sataev I. R. et al., Chaos 2021 Vol. 31 No. 7 Article 073118

Transition to chaos via the destruction of a two-dimensional torus is studied numerically using an example of the Hénon map and the Toda oscillator under quasiperiodic forcing and also experimentally using an example of a quasi-periodically excited RL–diode circuit. A feature of chaotic dynamics in these systems is the fact that the chaotic attractor in ...

Added: July 15, 2021

Kuznetsov N. V., Mokaev T. N., Alexeeva T. A., Ekaterinburg : Институт математики и механики УрО РАН им. Н.Н. Красовского, 2019

Added: October 30, 2019

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

Stankevich N., Shchegoleva N. A., Sataev I. R. et al., Journal of Computational and Nonlinear Dynamics 2020 Vol. 15 No. 11 P. 111001

Using an example a system of two coupled generators of quasiperiodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involve saddle tori occurring at their ...

Added: September 4, 2020

Кузнецов А. П., Stankevich N., Щеголева Н. А., Известия высших учебных заведений. Прикладная нелинейная динамика 2021 Т. 29 № 1 С. 136-159

The purpose of this study is to describe the complete picture of synchronization of two coupled generators of quasi-periodic oscillations, to classify various types of synchronization, to study features of occurrence and destruction of multi-frequency quasi-periodic oscillations. Methods. The object of the research is systems of ordinary differential equations of various dimensions. The work uses the fourth-order Runge–Kutta ...

Added: February 2, 2021

Alexeeva T., Kuznetsov N., Mokaev T., Chaos, Solitons and Fractals 2021 No. 152 Article 111365

Cyclicality and instability inherent in the economy can manifest themselves in irregular fluctuations, including chaotic ones, which significantly reduces the accuracy of forecasting the dynamics of the economic system in the long run.
We focus on an approach, associated with the identification of a deterministic endogenous mechanism of irregular fluctuations in the economy.
Using of a mid-size ...

Added: September 21, 2021

Avila A., Hubert P., Skripchenko A., Inventiones Mathematicae 2016 Vol. 206 No. 1 P. 109-146

We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on the diffusion rate of these sections using the connection between Novikov’s problem and systems of isometries—some natural generalization of interval exchange transformations. Using ...

Added: November 9, 2016