### ?

## Multistability in a three-dimensional oscillator: tori, resonant cycles and chaos

Nonlinear Dynamics. 2018. Vol. 94. No. 4. P. 2455-2467.

Stankevich N., Volkov E.

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 of different winding numbers, torus and chaos forms the multistability.

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

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

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. P., Seleznev E. P., 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, ...

Added: May 12, 2021

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., 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

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

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

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

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

Stankevich N., Mosekilde E., Koseska A., European Physical Journal: Special Topics 2018 Vol. 227 P. 747-756

Complex biochemical networks are commonly characterised by the coexistence of multiple stable attractors. This endows living systems with plasticity in responses under changing external conditions, thereby enhancing their probability for survival. However, the type of such attractors as well as their positioning can hinder the likelihood to randomly visit these areas in phase space, thereby ...

Added: December 2, 2019

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

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

A. Kilina, Panteleeva P., Stankevich N., Communications in Nonlinear Science and Numerical Simulation 2024 Vol. 135 Article 108041

A non-autonomous model of the Anishchenko–Astakhov generator in the regime of periodic and chaotic self-oscillations is considered. A periodic sequence of short pulses is considered as an external force. It is shown that the synchronization picture is close in structure to the classical synchronization picture observed in a two-dimensional system, but the pulse action leads ...

Added: May 3, 2024

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

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

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

Garashchuk I., Sinelshchikov D., Kudryashov N. A., Regular and Chaotic Dynamics 2018 Vol. 23 P. 257-272

Contrast agent microbubbles, which are encapsulated gas bubbles, are widely used to enhance ultrasound imaging. There are also several new promising applications of the contrast agents such as targeted drug delivery and noninvasive therapy. Here we study three models of the microbubble dynamics: a nonencapsulated bubble oscillating close to an elastic wall, a simple coated ...

Added: December 16, 2019

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

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

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

Stankevich N., Nonlinear Dynamics 2024 Vol. 112 No. 4 P. 2949-2967

Stabilization by a periodic pulsed force of trajectories running away to infinity in the three-dimensional R ̈ossler system at a threshold of a saddle-node bifurcation, birth of equilibrium states is studied. It is shown that the external pulsed action stabilizes dynamical regimes in a fairly wide range of external signal parameters. Stabilized regimes can be ...

Added: December 12, 2023

Garashchuk I., Kazakov A., Sinelshchikov D. et al., Chaos 2019 Vol. 29 No. 6 P. 063131-1-063131-16

We study nonlinear dynamics of two coupled contrast agents that are micrometer size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here, we consider a model of two such bubbles interacting via the ...

Added: December 16, 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