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. (c) 2021 Elsevier Ltd. All rights reserved.

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 them has an additional zero Lyapunov exponent, which strictly follows from the structure of mathematical models. In the process of research, the influence of feedback is studied, in which the frequency of one of the harmonics of external forcing becomes dependent on a dynamic variable. Charts of dynamic regimes were constructed, examples of typical oscillation modes were given, and the spectrum of Lyapunov exponents was analyzed. Numerical simulations confirm that chaos resulting from the cascade of torus doubling has a close to the zero Lyapunov exponent, beside the trivial zero exponent.

The purpose of this work is to review of works in which experimental studies of the regularities of chaotic dynamics revealed theoretically in works of S.P. Kuznetsov were carried out. Methods. The research methods used are primarily based on the construction of experimental schemes; they correspond most closely to the mathematical models proposed and theoretically and numerically investigated by S.P. Kuznetsov. These are systems of radio engineering oscillators with various types of communication and impact, autogenerators with various types of feedback. Results. The transition to chaos in the electron beam - backward electromagnetic wave system is investigated using the example of a backward wave tube. On the example of coupled nonlinear radio engineering oscillators with in-phase excitation, the discovered S.P. Kuznetsov, universal regularities and similarity laws for coupled systems with period doubling. The paper presents results of experimental study of radiophysical devices, on the example of which it was possible to verify the universal laws of the critical behavior of two unidirectionally coupled systems with period doublings. The results of joint with S.P. Kuznetsov of experimental studies, which for the first time in the world presented convincing arguments for the existence of a transition to chaos through the birth of a strange non-chaotic attractor. An experimental system with delayed feedback is presented for theoretical regularities testing that appear on the threshold of the transition to chaos. The experimentally developed by S.P. Kuznetsov's scheme of an auto-generator of hyperbolic chaos, which, apparently, is the world's first known example of a physical system with rough chaos.

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 oscillations is observed. The paper studies the structure of the space of control parameters. It is shown that in the dynamics of the system there are oscillatory modes similar to those of a non-autonomous oscillator with a potential in the form of a periodic function, but there are also significant differences.

A numerical study of van der Pol oscillators coupled in a ring is presented. The study of the characteristics of chaotic dynamics by analyzing the Fourier spectra and the spectrum of Lyapunov exponents is carried out. It is shown that the chaotic dynamics arising from the destruction of multi-frequency tori have a broadband spectrum.

An ensemble of van der Pol oscillators coupled in a ring has been numerically simulated. Peculiarities of chaotic dynamics have been studied by analysis of Fourier spectra and Lyapunov exponents. It is established that chaotic dynamics appearing upon the breakup of multifrequency tori is characterized by a broadband spectrum.

The dynamics of a non-autonomous oscillator in which the phase of external force linearly depends on a dynamic variable has been investigated. This control of the external-drive phase leads to the formation of a hierarchy of various periodic and chaotic oscillations. The structure of the space of control parameters has been studied. It is established that dynamics of the system exhibits oscillatory regimes analogous to those of a non-autonomous oscillator with potential in the form of a periodic function.

AbstractThe dynamics of a non-autonomous oscillator in which the phase of external force linearly depends on a dynamic variable has been investigated. This control of the external-drive phase leads to the formation of a hierarchy of various periodic and chaotic oscillations. The structure of the space of control parameters has been studied. It is established that dynamics of the system exhibits oscillatory regimes analogous to those of a non-autonomous oscillator with potential in the form of a periodic function.

We propose a multicircuit oscillator with a common control scheme driving self-sustained oscillations in each circuit, which can exhibit quasi-periodic and chaotic oscillations. The proposed oscillator was investigated by experimental and numerical methods that confirmed the possibility of exciting multifrequency quasi-periodic, chaotic, and hyperchaotic oscillations with several positive Lyapunov exponents.

AbstractWe propose a multicircuit oscillator with a common control scheme driving self-sustained oscillations in each circuit, which can exhibit quasi-periodic and chaotic oscillations. The proposed oscillator was investigated by experimental and numerical methods that confirmed the possibility of exciting multifrequency quasi-periodic, chaotic, and hyperchaotic oscillations with several positive Lyapunov exponents.

In the present paper a model of coupled neuron with excitory and inhibitory coupling is considered. It is shown that the in-phase solution appears as a result of a sequence of period-doubling bifurcations of the degenerate period-l cycle. Multistability between in-phase and out-of-phase solutions is shown in the system.

In the present paper in radiophysical experiments formation of multi-modes chaotic attractor, based on the multi-dimensional quasiperiodic torus is observed. Character phase portraits are shown.

Квазипериодические бифуркации четырехчастотных торов в кольце пяти связанных осцилляторов ван дер Поля с различными видами диссипативной связи

We suggest a new technique of fold Poincare section, allowing one to visualize an invariant curve of a multi-frequency invariant torus in a physical experiment. Details of the technique are presented, along with examples of its application to various experimental studies. Examples of how an invariant curve is visualized in double-, triple- and four-fold Poincare sections are shown. (C) 2016 Elsevier B.V. All rights reserved.

Предложена оригинальная математическая модель сердечно-сосудистой системы человека, воспроизводящая основной сердечный ритм, процессы его регуляции, артериальное давление и учитывающая влияние на кровообращение процесса дыхания.Включение в модель контуров вегетативной регуляции, демонстрирующих автоколебательную автономную динамику, позволило воспроизвести наблюдаемые в экспериментах эффекты фазовой синхронизации этих контуров.Адекватность модели подтверждена качественным и количественным соответствием ее спектральных и статистических характеристик свойствам ре

Five van der Pol oscillators connected into a ring have been investigated numerically. We have considered different types of coupling: dissipative and active, as well as dissipative and active with a single coupling sign reversal. Bifurcations observed upon a transition from a five- to four-frequency torus have been investigated. The possibility of quasi-periodic Hopf bifurcation in this system has been established.

In the present paper in radiophysical experiments two different ways of destruction of invariant torus are shown. In the first case formation of chaotic attractor from four-frequency quasiperiodic torus involving torus doubling bifurcations is presented. In the second transformation of two-frequency torus into hyperbolic chaotic attractor was observed.