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 is 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
• 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. Rus. J. Nonlin. Dyn., 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. Rus. J. Nonlin. Dyn., 2017, 13, No 3, 331–348. (In Russian.)
• R.M. Rozental, O.B. Isaeva, Ginzburg N.S., Zotova I.V., Sergeev A.S, Rozhnev A.G. Characteristics of chaotic regimes in a spacedistributed gyroklystron model with delayed feedback. Rus. J. Nonlin. Dyn., 2018, 14, No 2. (Accepted.)
• 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.)
Publications 2018
• 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: Preprint arXiv:1710.07828 [nlin.CD], pp. 110.
• Stankevich N.V., Astakhov O.V., Kuznetsov A.P., Seleznev E.P. Excitation of chaotic and quasiperiodic oscillations in a multicontour generator with a common control circuit. ZhTP Letters, 44, 2018, вып. 10, 4654.
• 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.