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## Chaos and hyperchaos arising from the destruction of multifrequency tori

Chaos, Solitons and Fractals. 2021. Vol. 147. Article 110998.

Appearance of chaotic dynamics as a result of multi-frequency tori destruction is carried out on the example of a model of a multimode generator. Quasiperiodic bifurcations occurring with multi-frequency tori are discussed in the context of the Landau-Hopf scenario. Structure of the parameter space is studied, areas with various chaotic dynamics, including chaos and hyperchaos, are revealed. Scenarios of the development of chaotic dynamics are described, the features of chaotic signals of various types are revealed.

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

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

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

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

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

Stankevich N., Dvorak A., Astakhov V. et al., Regular and Chaotic Dynamics 2018 Vol. 23 No. 1 P. 120-126

The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three diﬀerent routes of transition to chaotic dynamics associated with diﬀerent bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained ...

Added: December 2, 2019

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

Letellier C., Stankevich N., Rössler O., International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2022 Vol. 32 No. 2 Article 2230004

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety ...

Added: February 24, 2022

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

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

Sataev I. R., Stankevich N., Chaos 2021 Vol. 31 No. 2 Article 023140

We study the hyperchaos formation scenario in the modified Anishchenko–Astakhov generator. The scenario is connected with the existence of sequence of secondary torus bifurcations of resonant cycles preceding the hyperchaos emergence. This bifurcation cascade leads to the birth of the hierarchy of saddle-focus cycles with a two-dimensional unstable manifold as well as of saddle hyperchaotic sets resulting ...

Added: February 26, 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

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

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

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

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

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

Stankevich N., Volkov E., Chaos 2020 Vol. 30 No. 4 P. 043122-1-043122-9

The dynamics of three three-dimensional repressilators globally coupled by a quorum sensing mechanism was numerically studied. This number (three) of coupled repressilators is sufficient to obtain such a set of self-consistent oscillation frequencies of signal molecules in the mean field that results in the appearance of self-organized quasiperiodicity and its complex evolution over wide areas of ...

Added: April 17, 2020

Крылосова Д. А., Селезнев Е. П., Stankevich N., Chaos, Solitons and Fractals 2020 Vol. 134 P. 109716

The dynamics of a non-autonomous oscillator in which the phase and frequency of the external force depend on the dynamical variable is studied. Such a control of the phase and frequency of the external force leads to the appearance of complex chaotic dynamics in the behavior of oscillator. A hierarchy of various periodic and chaotic ...

Added: March 6, 2020

Кузнецов А. П., Седова Ю. В., Stankevich N., Журнал технической физики 2021 Т. 91 № 11 С. 1619-1624

Исследована возбуждаемая гармоническим сигналом система двух диссипативно связанных генераторов,
способных демонстрировать автономные квазипериодические колебания. Представлены ляпуновские карты,
выявляющие режимы инвариантных торов разной размерности и хаоса. Представлены фазовые портреты в
стробоскопическом сечении и двойном сечении Пуанкаре. Обсуждено сосуществование различных режимов,
в частности, бифуркации инвариантных торов. ...

Added: October 6, 2021

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

Kuznetsov A., Kuznetsov S., Shchegoleva N. et al., Physica D: Nonlinear Phenomena 2019 Vol. 398 P. 1-12

A problem of synchronization of quasiperiodic oscillations is discussed in application to an example of coupled systems with autonomous quasiperiodic dynamics. Charts of Lyapunov exponents are presented that reveal characteristic domains on the parameter plane such as oscillator death, complete synchronization, phase synchronization of quasiperiodic oscillations, broadband synchronization, broadband quasiperiodicity. Features of each kind of ...

Added: December 2, 2019