Russian
Implementation of electronic devices manifesting different phenomena of nonlinear dynamics and their applications including analog modeling systems of different nature
Grant of Russian Science Foundation No 171201008
The project was carried out in 20172019 (Saratov Branch)

Project supervisor:

Participants of the project: 

The project aims at elaborating principles and implementing electronic devices, specifically directed to provide diverse phenomena of nonlinear dynamics, including chaos, quasiperiodic and strange nonchaotic dynamics, various types of synchronization and scenarios of emergence of the complex dynamics and bifurcations.
This also concerns to identifying areas for electronic analog simulation in application to systems of different nature (mechanical, biophysical, technical systems) and to information and communication applications (communication systems, radar systems, masking and suppressing signals, random number generation and generation of cryptographic keys).
Topicality and relevance of the work due to the need of mastering and development of material accumulated in the modern theory of dynamical systems in the physical and technical context as this material remains yet largely of abstract and mathematical character.
The main specificity of the project is that the object of study are electronic circuits built using modern element base and mathematical models of these systems targeted specifically at the implementation of definite phenomena of complex dynamics. It is supposed to consider different approaches to obtaining the rough chaos, including the class of uniformly hyperbolic, partially hyperbolic, singular hyperbolic (Lorenz type attractors), pseudohyperbolic attractors.
Exploiting numerical simulation and involving the principles and results of the modern theory, new systems of electronic nature will be proposed and investigated implementing chaos, quasiperiodic dynamics, and other types of behavior, and the possibility of analog modeling will be considered for systems of other physical nature.
For description of the dynamics mathematical models will be developed, the topography of parameter space device s will be studied for the models, which implement different types of behavior, bifurcation phenomena, chaos and synchronization. In the course of the research methods of computer studies for the phenomena of complex nonlinear dynamics will be developed and improved, including those relating to processing data of electronic experiments.
In a view of analog simulation and possible applications it is planned to put attention to such areas as parametric excitation of oscillations to control dynamics of micro and nanoscale systems, use of the chaos generators in complex with quasiperiodic oscillators for communications with encoding of the transmitted signals referring to radio and optical systems, analysis of possible significance of quasiperiodic oscillations, and chaos synchronization in model systems of biophysical and biomedical nature.
Main results 2017
For oscillator with control of external forced frequency the numerical analysis and experimental study of forced oscillations are revealed, that at low values of the driving amplitude parameter and the frequency variation parameter the periodic oscillations are observed and with increasing of these parameters transition to chaos and complication of forced oscillations are appeared. Chaotic oscillations in such a system can be interpreted as nonperiodic transitions from one local minimum of the potential function to the other. Similar oscillators can be used as elements for constructing of robust chaos generators with fine spectral properties. This scheme also has potential for use in analog modeling of systems based on the Josephson effect.
We examine dynamics of parametric electronic chaos oscillator based on two LC circuits, one of which includes negative conductivity (the active LCcircuit). The study is based on numerical computations with equations that directly describe the oscillations of voltages and currents in the oscillatory circuit, amplitude equations, and the equations represented in the form suggested by Vyshkind and Rabinovich. Results of numerical research and circuit simulation with the use of the software product Multisim are in good agreement. The proposed electronic scheme can be used for analog simulation of oscillatory and wave phenomena in systems to which the Vyshkind and Rabinovich model is applicable.
We have introduced a system governed by ordinary fourthorder differential equations in which electronic scheme constructing is possible. In the model hyperbolic chaos corresponding to different topological types of Smale –Williams solenoid occurs arising as a result of blue sky catastrophes. Electronic generators which are planned to be implemented basing on this new model will be characterized by insensitivity to variation of parameters, manufacturing errors, interferences etc., since a fundamental attribute of hyperbolic chaos is its property of roughness (structural stability).
We propose a principle of constructing a new class of systems exhibiting hyperbolic attractors and quasiperiodic dynamics, where the transfer of vibrational excitation between subsystems is resonant due to the difference frequency of small and large fluctuations at integer times. An example is the new model of this class, where the attractor of Smale – Williams solenoid type is realized on the base of two coupled oscillators of Bonhoffer – van der Pol.
For the first time, a possibility of application of the oscillation death effect for construction of system with hyperbolic attractor is demonstrated. It is identified that the oscillation phases are transformed in accordance with the Bernoulli map. A test of hyperbolic nature of attractor, based on an analysis of statistics distribution of angles between stable and unstable subspaces, is applied. It seems possible to build other models of systems with hyperbolic chaos based on the oscillation death effect, including distributed models.
We study properties of transformation for chaotic repellers, different types of attractors and quasiperiodic neutral sets at the transition from the class of complex analytical to the class of generalized unitary systems. Property of the generalized unitarity implies realization for the evolution operator the unitarity condition in its traditional meaning at preservation of the nonlinearity of this operator. This situation is possible if the evolution operator is ambiguously defined both in direct and in reverse time, namely, is determined by implicit function with special symmetry. It is demonstrated that generalized unitary systems are displayed phenomena inherent to conservative systems. We discover and analyze relationships between behavior of systems of the general form implicit and ambiguous in both directions of time, degenerate generalized unitary systems and dissipative models unambiguous in direct time.
We test a method of calculation of Lyapunov exponents from time series. The method has high accuracy and statistical significance even for local values of the Lyapunov exponents. With the help of this method we identify chaotic and hyperchaotic regimes of radiation of gyroklistron with feedback delay.
With the help of charts of dynamical regimes and Lyapunov exponents charts a region of hyperbolic chaos for the previously proposed model system of alternately activated neurons described by FitzhughNagumo equations is identified. We propose also a new model system based on one alternately activated and inactivated FitzhughNagumo neuron, supplemented by the feedback circuit delay. It is shown that in such system, which is formally infinitedimensional, all main phenomena observed for the model of two alternately excited neurons are revealed.
Quasihyperbolic Belykh attractor is considered applicable to the map, describing the dissipative rotator driven by periodic kicks, the intensity of which depends on the instantaneous angular coordinate of the rotator as a sawtoothlike function. It is shown that the smoothing of the sawtooth function leads to the destruction of quasihyperbolic nature of attractor and implies appearance of phenomena typical for quasiattractor  periodicity windows, which correspond to the dips in the graph of Lyapunov exponent versus parameter. However, at the small scale of smoothing there are regions in the parameter, where these windows are indistinguishable, and they will be effectively masked by the noise in concrete implementations of the system. In these regions radiophysical devices with similar type of attractor can be used as generators of almost robust chaos, neglecting the difference from quasihyperbolic situation.
We elaborated and experimentally studied an electronic circuit implementing the quasiperiodic dynamics and strange nonchaotic attractor in a chain of nonlinear oscillators, each of which is under quasiperiodic forcing at the inphase and not inphase excitation. In the parameter plane the picture of regions is determined and described where lines of torus doubling ending in special terminal points of codimension two are existed.
The model of multicircuit generator with a common control scheme is created. With help of numerical simulations and laboratory experiments we demonstrate the possibility of generation of chaos and hyperchaos as result of destruction of multifrequency quasiperiodic dynamics.
For study of chaotic and hyperchaotic oscillations based on a system of coupled oscillators we develop multimode system – selfoscillator containing five independent oscillating systems with a common active element. The research of multimode generator dynamics are executed, Lyapunov exponents charts are calculated, regions of parameter space with quasiperiodic oscillations including two, three and four independent frequencies are localized. We implement laboratory model of a multimode generator and carry out its experimental study including the computation of charts of dynamic regimes by using the multiple Poincare section method and the identification of multifrequency quasiperiodic oscillations.
Publications 2017
Articles
• Kuptsov P.V., Kuznetsov S.P., Stankevich N.V. A Family of Models with Blue Sky Catastrophes of Different Classes. Regular and Chaotic Dynamics, 2017, 22, No 5, 551565.
• Doroshenko V. M., Kruglov V. P., Kuznetsov S. P. Chaos generator with the Smale–Williams attractor based on oscillation death. Russian Journal of Nonlinear Dynamics, 2017, 13, No 3, 303–315. (In Russian.)
• Isaeva O. B., Obychev M. A., Savin D. V. Dynamics of a discrete system with the operator of evolution given by an implicit function: from the Mandelbrot map to a unitary map. Russian Journal of Nonlinear Dynamics, 2017, 13, No 3, 331–348. (In Russian.)
• Kuznetsov S.P., Turukina L.V. Complex dynamics and chaos in electronic selfoscillator with saturation mechanism provided by parametric decay. Izvestiya VUZ. Applied Nonlinear Dynamics, 26, 2018, No 1, 33–47. (In Russian.)
• Kuznetsov S.P. Belykh attractor in Zaslavsky map and its transformation under smoothing. Izvestiya VUZ. Applied Nonlinear Dynamics, 26, 2018, No 1, 6479. (In Russian.) English Translation.
• Rosental R.M., Isaeva O.B., Ginzburg N.S., Zotova I.V., Sergeev A.S., Rozhnev A.G. Characteristics of Chaotic Regimes in a Spacedistributed Gyroklystron Model with Delayed Feedback. Russian Journal of Nonlinear Dynamics, 2018, 14, no.2, 155–168.
Conference Abstracts
• В.М. Doroshenko, V.P. Kruglov, S.P. Kuznetsov. Generation of hyperbolic chaos on the basis of the effect of oscillation death: Numerical and circuit simulation. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.4748. (In Russian.)
• O.B. Isaeva, R.M. Rosenthal, A.G. Rozhnev. Chaotic and hyperchaotic modes of operation of gyroklistron with delayed feedback. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.8485. (In Russian.)
• V.V. Kuzmina, E.P. Seleznev. Forced oscillations of the oscillatory circuit when controlling the frequency of the driving. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.131132. (In Russian.)
• M.A. Obychev, O.B. Isaeva. Collective phenomena in a network of coupled oscillatory systems, associated with complex analytical dynamics and its destruction. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.191192. (In Russian.)
• E.S. Popova, E.P. Seleznev. The variety of vibrational regimes in a system of coupled nonlinear oscillators with a threefrequency action. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.270271. (In Russian.)
• N.V. Stankevich, O.V. Astakhov, E.P. Seleznev. Investigation of the excitation of chaotic oscillations in a multimode generator with a common control circuit. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.270271. (In Russian.)
• A.V. Syudeneva, E.P. Seleznev, N.V. Stankevich. Numerical study of forced oscillations of an oscillator with frequency response control. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.276277. (In Russian.)
• A.P. Kuznetsov, S.P. Kuznetsov, L.V. Turukina. Complex dynamics and chaos in an electronic selfoscillator with saturation provided by parametric decay. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.289290. (In Russian.)
• A.P. Kuznetsov, N.V. Stankevich, N.A. Schegolev. Dynamics of coupled quasiperiodic generators. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2017, p.328329. (In Russian.)
Main results 2018
A method for verifying hyperbolic and pseudohyperbolic nature of attractors is developed. It consists in checking the absence of tangencies between the subspaces of the perturbation vectors, expanding and compressing the phase volume in combination with the analysis of Lyapunov exponents and charts of sections of the parameter space both for finitedimensional systems and infinitedimensional systems with delay (a kind of distributed systems). A method for determining the quantitative characteristics of chaos based on time series was also developed, using which several examples of electronics systems with chaotic dynamics were analyzed as a part of the project, including evaluating the spectrum of Lyapunov exponents for time series obtained from the Multisim simulating for circuits with chaotic dynamics.
For the twoparameter analysis of bifurcation sequences that constitute the content of the saddlenode scenario of the birth of the SmaleWilliams hyperbolic attractor in multidimensional systems, an approximate onedimensional model is proposed that is the Bernoulli map with a forbidden zone. Using examples of twodimensional and fourdimensional mappings and a system of two coupled nonautonomous van der Pol oscillators, we compared bifurcation scenarios for the onset of hyperbolic chaos in an approximate onedimensional model and in multidimensional dynamic systems, for which the correspondence of the bifurcations of the birth of cycles was demonstrated.
As a result of numerical simulations and experimental study of a nonautonomous oscillator when controlled phase of the external force it was shown that introducing a linear dependence of the phase of the external force on the dynamic variable of the oscillator leads to a significant enrichment of the system dynamics, which results in the appearance of periodic and chaotic oscillation hierarchies. An increase in the amplitude of the external force leads to the expansion of areas of existence of complex oscillatory modes, as well as to the emergence of new zones of periodic oscillations in the area occupied by chaos. In this case, the oscillatory modes appear that correspond to phenomena intrinsic to a nonlinear oscillator with periodic potential (sine nonlinearity), and multistability occurs. The structure of the regions of different dynamical modes in the cross sections of the parameter space was analyzed, phase portraits, Fourier spectra were obtained, and the spectrum of Lyapunov exponents was analyzed. The results confirm the possibility of analog modeling of the class of systems with sine nonlinearity.
It is shown that hyperbolic chaos can be realized in a selfoscillating system with sine nonlinearity and with delayed feedback when controlled by a periodic modulation of the dissipation parameter.
A new model basing on a FitzHughNagumo neuron with sinusoidal modulation of the parameter and supplemented with delayed feedback circuit providing quadratic transformation of the transmitted signal was introduced into consideration. Numerical results are presented confirming the hyperbolic nature of chaos in a wide range of parameters: the spectrum of Lyapunov exponents was calculated, Lyapunov charts on parameter plane were depicted, an algorithm for calculating the intersection angles for a stable and unstable subspace was implemented, and the absence of tangencies of these subspaces was confirmed in the domain of hyperbolic dynamics.
A system of two coupled neurons FitzHughNagumo is proposed, where the modulation of parameters is carried out using specially selected periodic piecewise linear functions, and a cascade of topologically different SmaleWilliams solenoids occurs, differing in an integer parameter that is the phase expansion ratio.
For a multicircuit electronic oscillator with different values of the transmission coefficients of the amplifiers responsible for excitation of each oscillatory mode, in the context of the LandauHopf scenario, the presence of tori with different numbers of frequency components was revealed. At small values of the gain factors, tori of the highest dimensionality (in our case, the fivefrequency tori) are preserved, while areas of chaos in this case on the parameter plane are small. With increase in the transfer coefficients, a gradual destruction of the tori takes place, with emergence of the chaotic dynamics. It is shown that when multifrequency tori are destroyed, chaotic modes are typical, for which, along with a positive Lyapunov exponent, there are additional nearly zero exponents, and also formation of a hyperchaos with two or more positive Lyapunov exponents is possible. At large values of the gains, which correspond to strongly nonlinear oscillations in each circuit, only twofrequency tori remain, which, when destroyed, are transformed into a “classical” chaos with one positive and one zero Lyapunov exponent.
Using an example of coupled elements with autonomous quasiperiodic dynamics, the problem of synchronization of quasiperiodic oscillations is considered, in particular, for coupled oscillators with an equilibrium state, the parameter plane of frequency detuning versus the coupling strength is investigated, both in the case of the elements identical in the excitation parameter and in the case of the identity violation. A rich phenomenology of the system was demonstrated, which includes the resonant Arnold web, special points of codimension 2, formation of chaos and hyperchaos. A numerical bifurcation analysis of equilibrium states and cycles, as well as an approximate analysis of quasiperiodic bifurcations, were carried out, and several types of synchronization (the phase synchronization, complete synchronization, and broadband synchronization) have been revealed. It is shown that when the oscillators are identical in terms of the excitation parameter, broadband synchronization degenerates into broadband quasiperiodicity, or into partial broadband synchronization.
A circuit simulation of an electronic generator demonstrating hyperbolic chaos resulting from a blue sky catastrophe was carried out for two variants of the circuits implemented in Multisim. In accordance with this, an experimental setup has been developed, where the SmaleWilliams attractor is observed arising through the blue sky catastrophe. The nature of the attractor is confirmed by demonstrating characteristic dynamical modes before and after the blue sky catastrophe, with presentation of the Lissajous curves, diagrams for phase transformation, Fourier spectra.
A study was undertaken on the complex dynamics of parametric resonant triplets with different pumping mechanisms in systems with quadratic and cubic nonlinearity, and electronic circuits that implement these types of parametric interaction are presented, which can be used for analog modeling of parametric oscillations and waves e.g. in hydrodynamics, acoustics, plasma physics, and nonlinear optics. Equations are obtained that determine the dynamics of systems for different levels of approximation: equations that directly describe oscillations of voltages and currents in the oscillatory circuits, amplitude equations, and those, when passing to real amplitudes, reduce to VyshkindRabinovich, RabinovichFabrikant and PikovskyRabinovichTrakhtengerts thirdorder equations.
For a situation of quadratic nonlinearity, where the decay mechanism of stabilization of parametric instability takes place (the VyshkindRabinovich model), complex dynamics and chaos are studied in the analog oscillator device built on the basis of two LCcircuits, one of which includes a negative conductivity. For a system where complex dynamics arise as a result of the modulation instability in the presence of cubic nonlinearity, a comparative study of models of different levels (coupled oscillators, amplitude equations, RabinovichFabrikant model) has been carried out. The study showed that in both situations there is chaos resulting from the sequence of period doubling bifurcations. Characteristic of the RabinovichFabrikant model is a pronounced multistability, when different types of attractors coexist in the phase space, which were identified and studied by means of numerical bifurcation analysis using the MаtCont program.
An electronic circuit on switchable capacitors is proposed that implements pseudohyperbolic attractors corresponding to the threedimensional Hénon map, as well as a new circuit that implements a Lorentztype pseudohyperbolic attractor. The functioning of the systems has been demonstrated within the framework of circuit simulation in Multisim with obtaining the oscilloscope traces, portraits of attractors, oscillatory spectra for the pseudohyperbolic attractors, and comparison was provided of these results with the numerical simulation based on the corresponding differential equations.
Using the method of pseudohyperbolicity testing based on the criterion of angles between manifolds, together with the method of Lyapunov charts, the 3D Hénon map has been considered, and areas on the parameter plane were revealed, where this system demonstrates pseudohyperbolic chaos and bifurcation structures in their vicinity were described.
Cascades of bifurcations of the period adding for the mixedmode oscillation in a nonautonomous Bonhoffer – van der Pol oscillator, which allows electronic implementation, with a piecewiselinear currentvoltage characteristic, are investigated. It is shown that bifurcation structures are described adequately by a onedimensional map obtained by approximating the Poincaré map for the equations of the oscillator. It was found that these bifurcations accumulate at the points of tangent bifurcations, which terminate the cascades of bifurcations of the period adding. It is shown that the “firing numbers” characterizing the mixed mode of oscillations form the Fairy tree, and the dependence of these values on the parameter is demonstrated, which has the form of the “devil’s staircase”.
A system of two coupled Bonheoffer – van der Pol oscillators controlled by periodic variation of parameters, was proposed and investigated. The parameters that are responsible for the AndronovHopf bifurcation are modulated using specially selected periodic piecewise linear functions with a shift for the partner oscillators by half a period, so that the oscillators alternate from small oscillations to relaxation selfoscillations with frequencies relating as an integer number. It is shown that the system shows topologically different types of SmaleWilliams hyperbolic attractors and quasiperiodic dynamics depending on the parameters; portraits of attractors, diagrams illustrating the phase transformation according to the expanding circle map, diagrams of dynamic modes on different planes of parameters are depicted; Lyapunov exponents are calculated. The hyperbolic nature of attractors has been verified by numerical calculations, which confirmed the absence of tangencies of stable and unstable manifolds for trajectories on the attractor (the angle criterion).
For the Bonheoffer – van der Pol oscillator, which alternately becomes active and suppressed due to periodic modulation of the parameter by external control signal and supplemented by a delayed feedback circuit, a mathematical model is formulated as a nonautonomous secondorder equation with a retarded argument, and the possibility of implementing a hyperbolic attractor is shown. The presence of the SmaleWilliams solenoid is due to the fact that the phase conversion for the generated radiopulses corresponds to a double or triple expanding circle map due to the resonant transfer of excitation from a previous to the next stage of activity, with doubling or tripling of the phase through the harmonics of the relaxation oscillations having twice or three times smaller period, than that for small oscillations.
Functionality of the schemes of secure communication based on complete and generalized synchronization of chaotic receiver and transmitter, which used rough hyperbolic chaos generators (those with SmaleWilliams attractor, with dynamics corresponding to the Arnold cat map and with a hyperhaotic map on the torus). The simulation of the functioning of the communication schemes was demonstrated in Multisim. The functionality was tested with nonidentical transmitter and receiver units. Two types of parametric detuning were analyzed: one interpreted as amplitude decay of signal in the communication channel, and the second type is the Doppler effect in the communication channel, and it is shown that in a certain parameter range they do not interfere with the feasibility of rough generalized synchronization of the receiver and transmitter. The effect of the intermixed information signal on the transmitter dynamics is analyzed, and requirements for its characteristics are formulated, under which the structural stability for the carrier chaotic signal is not violated.
Publications 2018
Articles
• Stankevich N.V., Astakhov O.V., Kuznetsov A.P., Seleznev E.P. Exciting Chaotic and QuasiPeriodic Oscillations in a Multicircuit Oscillator with a Common Control Scheme. Technical Physics Letters, 44, 2018, no. 5, 4654. Russian.
• Kuznetsov A.P., Stankevich N.V. Dynamics of coupled generators of quasiperiodic oscillations with equilibrium state. Izvestiya VUZ. Applied Nonlinear Dynamics, 26, 2018, No 2, 4158. (In Russian.)
• Kuznetsov S.P., Sedova Yu.V. Hyperbolic Chaos in Systems Based on FitzHugh–Nagumo Model Neurons. Regular and Chaotic Dynamics, 2018, 23, №4, 329–341.
• Doroshenko V.M., Kruglov V.P., Kuznetsov S.P. SmaleWilliams Solenoids in a System of Coupled Bonhoeffervan der Pol Oscillators. Russian Journal of Nonlinear Dynamics, 2018, 14, no. 4, 435451.
• Takahashi H., Kousaka T., Asahara H., Stankevich N., Inaba N. Mixedmode oscillationincrementing bifurcations and a devil’s staircase from a nonautonomous, constrained Bonhoeffer–van der Pol oscillator. Progress of Theoretical and Experimental Physics, vol. 2018, 1 Oct. 2018, Issue 10, 103A02.
• Stankevich N.V., Astakhov O.V., Seleznev E.P. Generation of chaotic and quasiperiodic oscillations in multicontour selfgenerator. IEEE Xplore. Progress In Electromagnetics Research Symposium. Proceedings: St Petersburg, Russia, 22–25 May 2017. Date Added to IEEE Xplore: 18 Jan. 2018. Pp. 31193121.
• Seleznev E.P., Stankevich N.V. Complex dynamics of a nonautonomous oscillator with controlled phase of external force. Pis'ma ZhTF, 2019, 45, iss.2, 5962. (In Russian.)
• Kuznetsov S.P., Sedova Yu.V. Hyperbolic chaos in the Bonhoeffer  van der Pol oscillator with additional delayed feedback and periodically modulated excitation parameter //Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, 27, iss.1. 7795. (In Russian.)
• Kyznetsov A.P., Kyznetsov S.P., Turukina L.V. Complex dynamics and chaos in the RabinovichFabrikant model // Izv. Saratov Univ. (N. S.), Ser. Physics. 2019, 19, iss.1., 418. (In Russian.)
Conference Abstracts
• Kruglov V.P., Kuznetsov S.P. Hyperbolic chaos in coupled FitzHughNagumo model neurons with alternating excitation of relaxation selfoscillations. Proceedings of Volga Neuroscience Meeting. Opera Medica et Physiologica, 2018, 4 (51), 4849.
• Kruglov V.P., Doroshenko V.M., Kuznetsov S.P. Hyperbolic chaos in coupled Bonhoeffer  van der Pol oscillators functioning with excitation of relaxation selfoscillations. Nonlinear waves  2018. XVIII scientific school. February 26  March 4, 2018. Abstracts of Young Scientists. . IAP RAS, Nizhny Novgorod, 2018, p.8789. (In Russian.)
• Isaeva O.B. Analysis of the parameter space of the threedimensional Henon map using the angle criterion in the search for a pseudohyperbolic attractor. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2018, p.8485. (In Russian.)
• Kruglov V.P., Kuznetsov S.P. Bonhoeffer  van der Pol oscillators. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2018, p.139140. (In Russian.)
• Kruglov V.P., Kuznetsov S.P. Bonhoeffer  van der Pol oscillators. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2018, p.139140. (In Russian.)
• Kuznetsov S.P. Generation of hyperbolic, pseudohyperbolic and quasihyperbolic chaos. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2018, p.153154. (In Russian.)
• Sedova Yu.V., Kuznetsov S.P. On the possibility of realizing the Smale  Williams attractor in the dynamics of a neuron with delayed feedback. Nanoelectronics, nanophotonics and nonlinear physics. “TechnoDecor”, Saratov, 2018, p.262263. (In Russian.)
• Turukina L.V., Kuznetsov A.P., Kuznetsov S.P. Complex dynamics and chaos in the Rabinovich  Fabrikant model system. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2018, p.324325. (In Russian.)
• Schegolev N.A., Kuznetsov A.P., Stankevich N.V. Coupled quasiperiodic generators: a variety of modes and bifurcations. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2018, p.370371. (In Russian.)
• Schegolev N.A., Kuznetsov A.P., Stankevich N.V. Coupled quasiperiodic generators: a variety of modes and bifurcations. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII AllRussian Conference of Young Scientists. “TechnoDecor”, Saratov, 2018, p.370371. (In Russian.)
• Isaeva O.B., Kuznetsov S.P., Savin D.V., Ponomarenko V.I. Radiophysical schemes of broadband communication based on hyperbolic chaos. Proceedings of the XVII winter schoolseminar on radiophysics and microwave electronics, 5–10 February 2018, Saratov. Saratov: Publishing Center "Nauka". P.45. (In Russian.)
• Kuznetsov A.P., Kuznetsov S.P., Turyukin L.V. Complex dynamics and chaos in the electronic selfoscillator with saturation provided by parametric decay. Proceedings of the XVII winter schoolseminar on radiophysics and microwave electronics, 5–10 February 2018, Saratov. Saratov: Publishing Center "Nauka". P.9495. (In Russian.)
• Kuznetsov S.P., Seleznev E.P., Stankevich N.V. Circuit simulation of a hyperbolic chaos generator with a blue sky catastrophe in Multisim. Proceedings of the XVII winter schoolseminar on radiophysics and microwave electronics, 5–10 February 2018, Saratov. Saratov: Publishing Center "Nauka". P.4849. (In Russian.)
• Stankevich N.V., Borovkova E.I., Seleznev E.P. Features of inphase synchronization in the model of coupled neurons with different types of coupling. Dynamics of complex networks and their application in intelligent robotics. Collection of materials of the II International SchoolConference of Young Scientists. Saratov: Publishing House "Nauchnaya Kniga", 2018. P.255258. (In Russian.)
• Burashnikov V.V., Stankevich N.V., Seleznev E.P. Complex dynamics in a multiloop generator: simulation and experiment. Actual problems of physical and functional electronics. Materials of the 21st AllRussian Youth Scientific SchoolSeminar, Ulyanovsk, December 46, 2018. Ulyanovsk Technical University Publ., 2018. P.113114. (In Russian.)
Main results 2019
For the first time, an autonomous oscillatory system is proposed defined by equations with algebraic nonlinearity together with an electronic circuit of the robust chaos generator, where the trajectories on the hyperbolic attractor approximately correspond to the dynamics on a twodimensional surface of negative curvature, and approximate uniformity of expansion and compression of phase volume elements in continuous takes place, which determines good spectral properties of the signal.
An experimental laboratory device was designed and studied, which for the first time realized the generation of hyperbolic chaos in the high frequency band (tens of megahertz), which is important for development of applications of robust chaos radiophysical generators in the field of communication systems.
A new approach is proposed for constructing systems with hyperbolic chaos in the form of lattices of spaceordered elements, where the Smale–Williams attractor occurs due to alternating excitation of patterns of different spatial scales, the spatial phase of which undergoes a stretching map at each step, and a concrete example of such a system obtained by spatial discretization of the Swift–Hohenberg model has been demonstrated.
In relation to the scenario of the birth of the Smale–Williams attractor, accompanied by saddlenode bifurcations of its cycles, a model is presented in the form of a onedimensional Bernoulli map with a forbidden zone (“hole”), which describes quantitative regularities characteristic of bifurcations that compose the content of the scenario.
For a family of systems in the form of coupled elements demonstrating a blue sky catastrophe, scenarios of transition to hyperbolic chaos have been studied, involving quasiperiodic dynamics and creation of structurally stable chaos, which corresponds to the Smale–Williams solenoids with different exponents of the angular variable extension.
A new numerical method for determining the parameters corresponding to the attractor in the form of Smale–Williams solenoid is proposed and tested. This method is based on an automatic analysis of the topological nature of the map for the angular variable measured along the solenoid filaments.
The occurrence of chaotic dynamics as a result of destruction of multifrequency tori in the model of a multicontour electronic generator has been studied, for which areas in the parameter space are revealed where chaos, hyperhaos, and chaotic regimes with additional zero Lyapunov exponents arise.
For the model defined as a combination of continuous transformations of the sphere, the structure of the parameter space around the domain of the Plykin attractor, which is continuous due to its structural stability, is studied with adding parameters that determine durations of the transformation stages. Outside this domain, various types of dynamical behavior have been identified, including nonhyperbolic chaos, periodic and quasiperiodic regimes.
For the Froude pendulum model complemented by delayed feedback and periodic braking, as a result of numerical studies, the possibility of realization a robust hyperbolic chaos determined by an attractor in the form of a Smale–Williams solenoid is demonstrated, and with the use of automatic analysis of the topological nature of the map for the angular variable, the hyperbolicity region in the parameter space is determined.
Laboratory electronic devices with control of phase and frequency by means of the generated signal are designed, implemented and experimentally studied. The variety of dynamical behaviors depending on the parameters, including chaotic and periodic regimes, is demonstrated, in particular, a possibility of generating broadband chaos is confirmed, which can be significant for applications in the field of communication systems.
A study of complex dynamics of the Rabinovich and Fabrikant model describing evolution of selfmodulation and emergence of chaos in parametric interaction of the main mode and two symmetric satellites is performed. Charts of dynamical regimes on the plane of control parameters, the dependence of Lyapunov exponents on parameters, the results of numerical bifurcation analysis are presented. A generalization of the model for the case of arbitrary symmetric cubic nonlinearity is proposed, and the generalized equations are derived, starting with the model of three nonlinearly coupled oscillators in the framework of Lagrange formalism, with the transition to a reduced system of three differential equations similar to the Rabinovich–Fabrikant equations, but containing two new parameters.
In the context of neuron models, the problem of synchronization of oscillations in coupled systems with autonomous quasiperiodic dynamics is examined, for which the parameter regions corresponding to the oscillation death, full synchronization, phase synchronization of quasiperiodic oscillations, broadband synchronization, broadband quasiperiodicity are identified.A numerical study of a family of model systems of two coupled neurons is carried out, where the bifurcation transition to the burst generation regime is determined by the blue sky catastrophe in the variants when the bifurcation results in the quasi–periodic dynamics or structurally stable chaotic Smale–Williams attractor in the Poincaré section. A circuit simulation of the analog model of coupled neurons demonstrating different variants of the blue sky catastrophe is carried out and its dynamics in the Multisim environment is demonstrated.
Publications 2019
Articles
• Stankevich N.V., Kuznetsov A.P., Popova E.C., Seleznev E.P. Chaos and hyperchaos via secondary Neimark–Sacker bifurcation in a model of radiophysical generator. Nonlinear Dynamics, 97, 2019, 2355–2370. (Preprint.)
• Kuznetsov A.P., Kuznetsov S.P., Shchegoleva N.A., Stankevich N.V. Dynamics of coupled generators of quasiperiodic oscillations: Different types of synchronization and other phenomena. Physica D, 398, 2019, 112. (Preprint.)
• Kuznetsov S.P., Sedova Yu.V. Robust hyperbolic chaos in Froude pendulum with delayed feedback and periodic braking. International Journal of Bifurcation and Chaos, 29, 2019, no. 12, 19300350. (Preprint.)
• Kuznetsov S.P. Selfoscillating system generating rough hyperbolic chaos. Izvestiya VUZ, Applied Nonlinear Dynamics, 27, 2019, no. 6, 39–62. (In Russian.)
• N.V. Stankevich, A.P. Kuznetsov, E.P. Seleznev Complex dynamic in multicontour generator. Bulletin of the Russian Academy of Natural Sciences, 2019, 19, iss.2, 157160. (In Russian.)
• Kuznetsov A.P., Seleznev E.P., Stankevich N.V. Nano and biomedical technologies. Quality control. Problems and prospects. Collection of scientific articles.  Saratov, SSU, 2019, issue 3, 149163 (In Russian.)
• Isaeva O.B., Sataev I.R. Bernoulli mapping with hole and a saddlenode scenario of the birth of hyperbolic Smale – Williams attractor. Discontinuity, Nonlinearity, and Complexity, 2020, 9. (Accepted.) (Preprint.).
• Kuznetsov S.P., Sataev I.R. Parameter space arrangement of a model system nearby domain of existence of Plykin type attractor. Preprint: arXiv:1910.14394 [nlin.CD], 2019, 116.
Conference Abstracts
• Kuznetsov S.P., Seleznev E.P., Stankevich N.V. Hyperbolic chaos in coupled HFgenerators. Radar, Navigation, Communications: Proceedings of XXV International scientific and technical conference (Voronezh, April 1618, 2019). VSU Publishing House, 2019, pp.158164. (In Russian.)
• Krylosova D.A., Seleznev E.P., Stankevich N.V. Broadband chaos in nonautonomous oscillator with controlling frequency of external action. Radar, Navigation, Communications: Proceedings of XXV International scientific and technical conference (Voronezh, April 1618, 2019). VSU Publishing House, 2019, pp.243246. (In Russian.)
• Bagautdinova E.R., Kuznetsov S.P., Seleznev E.P., Stankevich N.V.Circuit simulation of a system with bifurcation of a blue sky catastrophe. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV Conference of young scientists. Saratov, publishing house "TechnoDecor", 2019, pp.1819. (In Russian.)
• Zhitar A.D., Seleznev E.P., Kuznetsov S.P. RF generators with transmission of the excitation phase. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV Conference of young scientists. Saratov, publishing house "TechnoDecor", 2019, pp.7577. (In Russian.)
• Krylosova D.A., Seleznev E.P., Stankevich N.V. Forced oscillations of an oscillator when controlling the phase of action. Nanoelectronics, nanophotonics and nonlinear physics. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV Conference of young scientists. Saratov, publishing house "TechnoDecor", 2019, pp.129130.(In Russian.)
• Kuznetsov S.P. Hyperbolic chaos in lattice models. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV Conference of young scientists. Saratov, publishing house "TechnoDecor", 2019, pp.131132. (In Russian.)
• >Kuznetsov S.P., Sedova Yu.V. Hyperbolic chaos in a selfoscillating system with nonlinearity of the sine type and delayed feedback. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV Conference of young scientists. Saratov, publishing house "TechnoDecor", 2019, pp.229230. (In Russian.)
• Turukina L.V., Kuznetsov S.P. The generalized RabinovichFabrikant system and its complex dynamics. Nanoelectronics, nanophotonics and nonlinear physics. Abstracts of XIV Conference of young scientists. Saratov, publishing house "TechnoDecor", 2019, pp.272273. (In Russian.)
• Kuznetsov S.P., Turukina L.V. The RabinovichFabrikant model and its generalization. Abstracts of 12th International School “Chaotic Oscillations and Pattern Formation”. Saratov, Publishing Center “Nauka”, 2019, pp.4647. (In Russian.)
• Isaeva O.B., Kuznetsov S.P. Investigation of roads to hyperbolic and almost hyperbolic chaos. Abstracts of 12th International School “Chaotic Oscillations and Pattern Formation”. Saratov, Publishing Center “Nauka”, 2019, pp.7576. (In Russian.)
• Kruglov V.P., Sataev I.R. SmaleWilliams solenoids in an autonomous model of coupled oscillators with a figureeight homoclinic bifurcation. Abstracts of 12th International School “Chaotic Oscillations and Pattern Formation”. Saratov, Publishing Center “Nauka”, 2019, pp.7879. (In Russian.)
• Krylosova D.A., Seleznev E.P., Stankevich N.V. Forced oscillations of a dissipative oscillator when controlling the phase and frequency of external force. Abstracts of 12th International School “Chaotic Oscillations and Pattern Formation”. Saratov, Publishing Center “Nauka”, 2019, pp.8081. (In Russian.)
• Sataev I.R., Isaeva O.B. Bernoulli mapping with a “hole” and a saddlenode scenario for the birth of the SmaleWilliams hyperbolic attractor. Abstracts of 12th International School “Chaotic Oscillations and Pattern Formation”. Saratov, Publishing Center “Nauka”, 2019, p.94. (In Russian.)
• Sedova Yu.V., Kuznetsov S.P., Kruglov V.P. Rough hyperbolic chaos in systems based on Froude pendulums. Abstracts of 12th International School “Chaotic Oscillations and Pattern Formation”. Saratov, Publishing Center “Nauka”, 2019, p.95. (In Russian.)
• Seleznev E.P. Complex dynamics of nonautonomous oscillator with controlled external force. XXXIX Dynamics Days Europe. International Conference. Rostock, Germany (Sept. 26, 2019). Book of Abstracts, p.139.
• Bagautdinova E.P., Kuznetsov S.P., Seleznev E.P., Stankevich N.V. Circuit simulation of a blue sky catastrophe in the context of bursting dynamics occurrence. The Third School for Young Scientists “Dynamics of Complex Networks and their Application in Intellectual Robotics”. IEEE Xplore.