Rough chaos generators of high and ultrahigh frequencies

Grant of the Russian Foundation for Basic Research 16-02-00135


The project is being implemented in 2016-2018
in Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences

(Saratov Branch)

Project supervisor:

Kuznetsov S.P.


Participants of the project:

Isaeva O.B.

Kruglov V.P.

Kuznetsov A.P.

Kuptsov P.V.

Ponomarenko V.I.

Popova E.S.

Sataev I.R.

Sedova Yu.V.

Stankevich N.V.

Erofeev V.S.

Jalnine A.Yu.

Rough chaos, characterized by low sensitivity to parameter variations, manufacturing imperfections, interferences that is essential for possible information and communication applications, takes place in nonlinear systems in which the attractors responsible for the chaotic dynamics relate to the category of hyperbolic or pseudohyperbolic. In the first case, we talk about systems where perturbations of the phase trajectories are represented by vectors belonging to a subspace stretching these vectors, and a subspace where the vectors undergo compression. Such systems are rough in the sense of structural stability, that is, with variation of parameters, the dynamics remain the same in the strict mathematical sense up to a change of variables. In the case of a pseudo-hyperbolic attractor there is a subspace stretching the phase volume and a subspace compressing the phase volume, and this situation corresponds to a weaker property of robustness when chaos is not destroyed with a small variation of parameters, but, in general, there is no structural stability. An alternative to rough and robust chaos is situation of quasi-attractor, when chaos can disappear with variations of parameters, being transformed into modes of regular dynamics; that is highly undesirable from the point of view of applications. The content of the study in the frame of the Project consisted in elaboration of ways for implementation of electronic devices with hyperbolic and pseudohyperbolic attractors, allowing advancement into the range of high and ultrahigh frequencies. The program of work meant testing verification methods for these types of dynamic behavior, conducting circuit simulations of the proposed devices, implementing demonstration experimental laboratory models, evaluating prospects for using the proposed schemes in secure communication, bearing in mind their advantages due to the inherent robustness.

Main results 2016

Three new versions of systems are considered, the rough chaos in which is due to the presence of hyperbolic attractors such as the Smale-Williams solenoid in the Poincaré map. The first system is made up of two oscillatory elements, one of which is self-oscillating, and its operation is based on the use of the effect of oscillation death. The second system is autonomous, and is designed by modifying the model proposed by Yu.I. Neumark; in the phase plane it has a separatrix in the form of a figure eight. Our system is made up of two Neumark subsystems, characterized by generalized coordinates x and y, and is described by equations where terms are added, because of which the system becomes self-oscillating, and at the same time the rotation angle of the vector (x, y) undergoes tripling when returning to the neighborhood of the origin at successive rounds near the separatrix. The third approach is to consider a family of autonomous systems, where the emergence of hyperbolic chaos is associated with the blue sky catastrophe, the variants of which were discussed (without presentation of concrete examples) in [Turaev D.V., Shilnikov L.P., Dokl. Akad. Nauk, 342, 1995, 596].

Departing from the problem of a geodesic flow on a surface of negative curvature, where chaotic dynamics of Anosov is realized, an electronic circuit for the generator of rough chaos has been developed. A study was carried out with a use of the NI Multisim circuit simulation package, as well as by numerically solving equations delivering varying degrees of accuracy in description of the dynamics of the system. Portraits of attractors, time dependences of generated oscillations, Lyapunov exponents, and spectra are presented, and a good correspondence of the observed dynamics of the chaos generator with the Anosov hyperbolic dynamics of the initial geodesic flow is demonstrated. Using the criterion based on statistics of the intersection angles of stable and unstable subspaces of the perturbation vectors of the reference phase trajectory on the attractor, it is shown that the hyperbolic nature of the dynamics is preserved when the parameters vary in a certain range.

Within the framework of the project, computer verification of the hyperbolicity of attractors was carried out for the first time for time-delayed systems. We used a method based on calculating distributions for the angles between expanding, contracting, and neutral manifolds for trajectories on an attractor. Three specific examples are outlined. For two of them, the previously assumed hyperbolicity was confirmed. [S.P. Kuznetsov, V.I. Ponomarenko, Tech. Phys. Lett., 34, 2008, No 9, 771; S.P.Kuznetsov, A.Pikovsky, peprint nlin. arXiv: 1011.5972.] The third example [J.D.Farmer, Physica D, 1982, 4 (3), 366] corresponds to nonhyperbolic chaos.

Techniques that are useful for recognizing the types of complex dynamical behavior and allow one to distinguish a strange nonchaotic attractor from quasiperiodic and chaotic oscillations are developed and tested.

A laboratory stand has been prepared for working with electronic devices in the high and ultrahigh frequency range, including oscilloscopes C1-104, MSO8104A (up to 1 GHz), Agilent DSOX4034A (up to 350 MHz), VSWR panoramic P2-102 (up to 2 GHz) digital converter NI 6355, multichannel digital-to-analog converter of DAC NI PXI-6723, spectrum analyzer Agilent N9320A (9kHz-3GHz).

A scheme was developed and a laboratory prototype of a rough hyperbolic chaos generator based on two alternately excited self-oscillating elements on field-effect transistors for a frequency range of 5-10 MHz was compiled. The scheme of the "electronic rotator" - a system similar in the equation form to the mathematical pendulum on a basis of the available components for the laboratory design, was implemented, and circuit simulation in the Multisim environment was povided.

In 2016 the participant of the project V.P. Kruglov defended the thesis "Finite-dimensional and distributed systems of ring structure that generate rough chaos" for the scientific degree of candidate of physical and mathematical sciences (supervisor S.P. Kuznetsov, specialty 01.04.03 - radiophysics), which partially presents the results obtained in the course of the project.

Main results 2017

A research was performed within the framework of numerical and circuit modeling of the electronic scheme proposed at the previous stage of the project, where the Anosov dynamics on the surface of negative curvature is reproduced on the attractor of the system. As elements of the electronic circuit, the phase-locked loop chains are used, which are analogs of rotators in mechanics, in which the states are characterized by a variable defined modulo 2pi and its derivative, which is a phase of the voltage-controlled generators with respect to the reference signal at fixed frequency. A comparative analysis of the dynamic behavior of models delivering different degrees of accuracy of the description is carried out.

Schemes of alternately excited oscillating elements based on field-effect transistors (lambda-diode) and on tunnel diodes, for which, as expected, it will be possible to select parameters providing hyperbolic dynamics, are developed.

A new simple scheme of chaos generator is proposed, realizing a pseudo-hyperbolic attractor of the Lorentz type, and its functioning was demonstrated through modeling in the Multisim environment. The scheme is based on an equivalent representation of the Lorentz equations in the form of a nonlinear oscillator with the help of the variable change of Yudovich (Kuznetsov S.P., Dynamical chaos, M: 2001). In the current version, the circuit uses an operational amplifier and two analog multipliers, but they can apparently be replaced by elements on transistors reproducing their properties. On the basis of the criterion of angles, for the scheme with the Lorenz attractor, the property of pseudohyperbolicity is checked, which ensures robustness of chaos and consists in the fact that one invariant subspace is compressive and the other is characterized by an exponential extension of the volume for perturbation vectors of trajectories on the attractor.

A bibliographic list was compiled on electronic chaos oscillators. In the Multisim environment, the circuits were reproduced, and chaotic dynamics were simulated providing phase portraits and spectra for a number of chaos generators described in the literature. These results are supposed to be used as a encouragement for construction of modified circuits that provide generation of rough chaos.

We have considerd ensembles of three and four oscillatory elements with a nonlinearity of sine, which can be realized, in particular, on the basis of non-identical Josephson junctions. Numerical modeling of the dynamics is carried out; areas of various behaviors in the parameter space are revealed, including quasiperiodic dynamics on tori of different dimensions and chaos.

As schemes promising for realization of complex dynamics, systems composed of van der Pol oscillators are analyzed, namely, a system of five ring-connected oscillators and a spatially developed network of star-like type. For the first system, variants with different types of connection were studied: dissipative, active, and dissipative/active with inversion of the sign of the coupling. The bifurcations observed during the transition from the five-frequency torus to the four-frequency torus are considered. A possibility of a quasi-periodic Hopf bifurcation is discovered. For the second system, non-trivial scenarios for emergence and development of the cluster synchronization are identified and described when parameters are varied.

On the basis of a method of checking transversality of the intersection of manifolds of trajectories on a chaotic attractor (the criterion of angles), the hyperbolic nature of chaotic dynamics for several time-delay systems implementable as electronic devices has been confirmed. (Kuznetsov S.P. and Pikovsky A.S., EPL, 84, 2008, 10013; Baranov S.V., Kuznetsov S.P., Ponomarenko V.I., Izvestiya VUZ. Applied Nonlinear Dynamics, 18, 2010, No 1, 11-23; Arzhanuzhina D.S., Kuznetsov S.P., Izvestiya VUZ. Applied Nonlinear Dynamics, 22, 2014, No 2, 36-49.)

For a generalized model with a bifurcation associated with a blue sky catastrophe, the hyperbolicity of attractors arising at values of the index responsible for the angle of twist of the beam of trajectories when returning to the Poincaré section, greater than or equal to 2, is confirmed on the basis of the angle criterion.

Dynamical characteristics of signasl produced by microwave oscillators of chaos in the form of gyroklistron and gyro-TWT with delayed feedback were considered. It is shown that in the regime of large supercriticality, generation of the chaotic oscillation in the millimeter wave band is possible belonging to the category of hyperchaos characterized by the presence of two or more positive Lyapunov exponents, and the spectrum has a relative width of up to 10%.

A model of a multi-loop generator with a common control scheme has been developed. Numerical simulation and laboratory experiments demonstrated a possibility of formation of chaos and hyperchaos as a result of destruction of the multifrequency quasiperiodic dynamics.

Communication schemes based on the use of generators of rough (hyperbolic) chaos are considered, their functioning is demonstrated in the framework of numerical simulation. Also in these schemes, a new efficient and noise-immune method for nonlinear intermixing of information based on phase modulation of a carrier chaotic signal is proposed. The efficiency of the schemes is demonstrated by examples of three basic systems generating robust chaos. For all these systems, methods are described for how to achieve a complete chaotic synchronization of the transmitter and receiver, and the characteristics of synchronous chaotic regimes essential for information transfer (Lyapunov exponents, phase diffusion coefficients) are calculated. Numerical simulations of the transmission of information signals with a wide spectrum were carried out.

Main results 2018

An electronic oscillator circuit with delayed feedback is proposed, in which attractor produces pseudohyperbolic chaos being similar in properties to the Lorenz attractor embedded in the infinite-dimensional phase space. A circuit simulation in the Multisim environment was carried out with demonstration of the portrait of the attractor in projection from the infinite-dimensional state space onto the phase plane. The fact that the attractor nature is similar to the Lorenz attractor is confirmed by plotting a map for consecutive maxima of one of the variables from data recorded in the process of circuit simulation. The pseudohyperbolicity property was also verified and validated for the mathematical model of the device using the angle criterion. The spectra of signals generated by the system and obtained using a virtual spectrum analyzer are continuous, which reflects the chaotic nature of the dynamics; irregularity of the spectral power density is rather small, which is an advantage of this variant of the electronic chaos generator.

A new scheme is proposed that generates hyperbolic chaos, where alternately activated self-oscillating elements are used, the excitation transfer between which is carried out in a resonant manner, due to the difference between the frequencies of small and large (relaxation) oscillations by an integer number of times, which is accompanied by phase transformation in accordance with an expanding circle map. The scheme is implemented in the Multisim environment based on two LC-circuits, each of which contains a nonlinear element in the form of a diode and an element of negative resistance on the operational amplifier, where the magnitude of the negative resistance is modulated due to the periodic variation of the drain-source conductivity of the field-effect transistor by applying control voltage to its gate. The device demonstrates chaotic dynamics corresponding to the Smale-Williams hyperbolic attractor in the stroboscopic map corresponding to the modulation period, and quasi-periodic dynamics, depending on the parameters. The advantage of the scheme is its simplicity - no need to use nonlinear elements in the connection circuit between the oscillators.

Schematic designs of rough chaos generators of ring structure with Smale-Williams attractors are proposed. One of the options is a circuit containing two oscillatory circuits, with their natural frequencies being in ratio 1:2. The circuit includes a nonlinear element on diodes and an operational amplifier, which has a quadratic characteristic at small amplitudes and saturated due to the limiting parameters of the operational amplifier. The second nonlinear element is an analog multiplier that mixes the main signal and the auxiliary one, which is a periodic sequence of radio-pulses with a rectangular envelope. The system generates a sequence of radio-pulses, in which the filling phases varies chaotically, in accordance with the Bernoulli map.

A possibility of constructing electronic chaos generators is shown whose dynamics on attractors correspond to geodesic flows on curved manifolds of dimension K=2,3,4. The system can be thought of as a set of K+1 rotators, on which a constraint is imposed, given by the algebraic relation cos Φ(1)+cos Φ(2)+ ... +cos Φ(K+1)=0. The electronic circuit is composed of elements, each of which is represented by a phase-locked loop, acting as an analogue of the rotator. In the case of K=2, the geodesic flow takes place on the Schwarz surface, which has almost everywhere negative curvature, which ensures the dynamics of the Anosov type. For the cases K=3 and 4, the corresponding geodesic flows on the three-dimensional and the four-dimensional curved manifolds, as it turned out, cannot be categorized as Anosov’s hyperbolic dynamics, since the sectional curvatures of the Riemannian manifold at the points of the trajectories take both negative and positive values [ Kuznetsov S.P., Preprint arXiv: 1810.08002 ]. However, schemes based on four and five phase locked loops have also been implemented and studied. Their functioning as generators of hyperchaos characterized by several positive Lyapunov exponents with good spectral properties was demonstrated, but the requirement of structural stability is not fulfilled for them.

Variants of circuits for chaos generators reproducing dynamics on the Lorenz pseudo-hyperbolic attractor are considered. There are circuits containing two operational amplifiers and two analog multipliers, and a circuit implementing an oscillator with inertial nonlinearity, where the Lorenz attractor takes place. The pseudo-hyperbolic nature of the generated chaos in these schemes is confirmed by the method of constructing the Lorenz map for successive maxima in the time dependence of one of the variables. In addition, a scheme is proposed that allows one to realize a four-dimensional analog of the Lorenz model, where a pseudo-hyperbolic attractor also occurs. For the mathematical model of this system, the pseudo-hyperbolic nature of the attractor is confirmed using the angle criterion. The good spectral properties of the generated chaos are demonstrated for all the mentioned variants of the schemes - a smooth dependence of the spectral power density, which has no pronounced irregularity.

To observe the hyperbolic chaos and analyze scenarios of its onset in a radio-engineering experiment, a laboratory prototype was created of a rough chaos generator. The circuit consists of two subsystems containing LC-circuits with frequencies of 24.5 and 49 kHz, supplemented with elements of negative resistance on operational amplifiers. The quadratic nonlinear connection responsible for the transfer of excitation from the first subsystem to the second is provided by an element on the analog multiplier. The impact of the second subsystem on the first one was provided by an element containing a multiplier with a product of the signal and the reference sinusoidal signal from a generator of frequency close to the natural frequency of the first LC-circuit. The modulation of the activity in the subsystems in counter-phase was ensured by varying the resistances of two field-effect transistors controlled by a signal from the frequency divider, which produced a sinusoidal voltage of period eight times longer than that of the reference signal. The types of dynamic modes were detected by observing a stroboscopic section with the modulation period on the oscilloscope screen, followed by analysis of digitized time series (phase portraits, iterative diagrams for the angular variable, power spectra, Lyapunov exponents). In the stroboscopic section, a chaotic hyperbolic attractor of the Smale – Williams type was observed, while the phase of the oscillations behaved in accordance with the Bernoulli map. A multiparameter analysis of the dynamics of the laboratory device allowed us to describe scenarios of the transition to hyperbolicity in the electronic experiment.

A hyperbolic chaos generator is designed and implemented in the radio band as a laboratory model, the circuit of which is similar in structure to the low-frequency generator taken as a prototype, as described above, but uses high-frequency analog multipliers (operating frequency up to 250 MHz) and operational amplifiers (operating frequency up to 1 GHz) . The natural frequencies of the oscillators, on the basis of which the scheme was built, were 10 MHz and 20 MHz, and the modulation frequency of the parameters of one and the other subsystem was 1.25 MHz. Diagrams are obtained for the oscillation phases at successive modulation periods, the form of which indicates the hyperbolic nature of the chaotic attractor, which is the Smale-Williams solenoid.



           Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity of chaotic dynamics in time-delay systems. Phys. Rev. E , 2016, 94, No 1, 010201.

           Jalnine A.Yu., Kuznetsov S.P. Strange nonchaotic self-oscillator. Europhysics Letters , 2016, 115, No 3, 30004.

           Kuznetsov S.P. From Anosov’s Dynamics on a Surface of Negative Curvature to Electronic Generator of Robust Chaos. Izv. Saratov Univ. (N.S.), Ser. Physics,, 2016, 16, No 3, 131–144. (In Russian.)

           Kuznetsov S.P. From Geodesic Flow on a Surface of Negative Curvature to Electronic Generator of Robust Chaos. International Journal of Bifurcation and Chaos, 2016, 26, No 14, 1650232.

           Kuptsov P.V., Kuptsova A.V. Radial and circular synchronization clusters in extended starlike network of van der Pol oscillators. Communications in Nonlinear Science and Numerical Simulation, 2017, 50, 115-127.

           Jalnine A.Yu., Kuznetsov S.P. Autonomous Strange Nonchaotic Oscillations in a System of Mechanical Rotators. Regular and Chaotic Dynamics, 2017, 22, No 3, 210–225.

           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, 551–565.

           Stankevich N.V., Kuznetsov A.P., Seleznev E.P. Quasi-Periodic Bifurcations of Four-Frequency Tori in the Ring of Five Coupled van der Pol Oscillators with Different Types of Dissipative Coupling. Technical Physics, 2017, 62, No 6, 971–974.

           Kuznetsov S.P. Chaos in three coupled rotators: From Anosov dynamics to hyperbolic attractors. Indian Academy of Sciences Conference Series, 2017, 1, No 1, 117-132.

           Rozental R.M., Ginzburg N.S., Zotova I.V., Rozhnev A.G., Isaeva O.B., Sergeev A.S. Regimes of devepoled chaos in gyrotrons and gyro-ampilifiers with delayed feedback. Memoirs of the Faculty of Physics, Lomonosov Moscow State University, 2017, No 6, 1760102. (In Russian.)

           Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity in chaotic systems with multiple time delays. Communications in Nonlinear Science and Numerical Simulation, 2018, 56, 227-239.

           Isaeva O.B., Jalnine A.Yu., Kuznetsov S.P. Chaotic Communication with Robust Hyperbolic Transmitter and Receiver. Progress In Electromagnetics Research Symposium. Proceedings: St Petersburg, Russia, 22–25 May 2017, 3129-3136. Preprint.

           Doroshenko V.M., Kruglov V.P., Pozdnyakov M.V. Robust chaos in systems of circular geometry. Progress In Electromagnetics Research Symposium. Proceedings: St Petersburg, Russia, 22–25 May 2017, 3122-3128.

           Isaeva O.B., Savin D.V., Seleznev E.P., Stankevich N.V. Hyperbolic chaos and quasiperiodic dynamics in experimental nonautonomous systems of coupled oscillators. Progress In Electromagnetics Research Symposium. Proceedings: St Petersburg, Russia, 22–25 May 2017, 3109-3113.

           Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity in chaotic systems with multiple time delays. Communications in Nonlinear Science and Numerical Simulation, 2018, 56, 227-239.

           Kuznetsov A.P., Sataev I.R., Sedova Yu.V. Dynamics of Three and Four Non-identical Josephson Junctions. Journal of Applied Nonlinear Dynamics, 2018, 7, No 1, 105-110.

           Kuznetsov S.P. Simple electronic chaos generators and their circuit simulation. Izvestiya VUZ, Applied Nonlinear Dynamics, 2018, 26, no. 3, 35–61. (In Russian.)

           Kuptsov P.V., Kuznetsov S.P. Lyapunov analysis of strange pseudohyperbolic attractors: angles between tangent subspaces, local volume expansion and contraction //Regular and Chaotic Dynamics, 23, 2018, Nos 7-8, 908–932.

           Doroshenko V.M., Kruglov V.P., Kuznetsov S.P. Smale-Williams Solenoids in a System of Coupled Bonhoeffer-van der Pol Oscillators. Russian Journal of Nonlinear Dynamics, 2018, 14, no. 4, 435-451.

           Rozental R.M., Isaeva O.B., Ginzburg N.S., Zotova I.V., Sergeev A.S., Rozhnev A.G., Tarakanov V.P. Automodulation and chaotic regimes of generation in a two-resonator gyroklystron with delayed feedback. Известия вузов. Прикладная нелинейная динамика, 2018, 26, no. 3, 78–98. (In Russian.)

           Kuznetsov S.P., Kuptsov P.V. Lorenz Attractor in a System with Delay: an Example of Pseudogyperbolic Chaos. // Izv. Saratov Univ. (N. S.), Ser. Physics, 16, 2018, iss. 3, 162-176.

           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. 77-95. (In Russian.)

Conference Abstracts

           Arzhanukhina D.S., Kuznetsov S.P. Schematic simulating systems of ring structure with chaotic dynamics corresponding to the Smale-Williams attractor. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XI All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2016, p.9-10. (In Russian.)

           Doroshenko V.M., Kuznetsov S.P. Chaos generator with Smale-Williams attractor on the basis of the oscillation death effect. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XI All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2016, p.39-40. (In Russian.)

           Kuznetsov A.P., Kuznetsov S.P., Stankevich N.V. A family of models with the blue sky catastrophes. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XI All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2016, p.188-189. (In Russian.)

           Khadzhieva L.M., Kruglov V.P., Kuznetsov S.P. Smale-Williams attractor in an autonomous system with a "figure-eight" homoclinic. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XI All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2016, p.259-260. (In Russian.)

           Zakharov S.A., Kuznetsov A.P., Stankevich N.V. Mutual synchronization of quasiperiodic oscillations. Materials of the XI International School-Conference "Chaotic self-oscillation and formation of structures" (CHAOS-2016), October 3-8, 2016, Saratov. Saratov: OOO "Publishing Center" Nauka", p.87. (In Russian.)

           Kruglov V.P., Kuznetsov S.P., Khadzhieva L.M. Smale-Williams solenoid in a system of two coupled oscillators with a homoclinic "figure-eight". Materials of the XI International School-Conference "Chaotic self-oscillation and formation of structures" (CHAOS-2016), October 3-8, 2016, Saratov. Saratov: OOO "Publishing Center" Nauka ", 2016, p.93-94. (In Russian.)

           Kuptsov P.V., Kuznetsov S.P. Hyperbolic chaos in system with multiple delays: numerical test via angle criterion. The International Scientific Workshop "Recent Advances in Hamiltonian and Nonholonomic Dynamics" (Moscow, Dolgoprudny, Russia, 15-18 June 2017). Book of Abstracts. Moscow – Izhevsk: Publishing Center "Institute of Computer Sciences", 2017, p.49-50.

           Erofeev V.S., Kuznetsov S.P., Seleznev E.P. HF hyperbolic chaos generator. Proceedings of the XII All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2017, p.57-58. (In Russian.)

           Kuptsov P.V., Kuznetsov S.P. Lyapunov analysis of hyperbolic chaos in systems with several delays. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2017, p.146-147. (In Russian.)

           Kuznetsov A.P., Sataev I.R., Sedova Yu.V. Dynamics of small-dimensional ensembles of Josephson related contacts. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XII All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2017, p.241-242. (In Russian.)

           Kuznetsov S.P. Generation of hyperbolic, pseudo-hyperbolic and quasi-hyperbolic chaos. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2018, p.153-154.(In Russian.)

           Stankevich N.V., Kuznetsov A.P., Schegoleva N.A. Coupled quasi-periodic oscillators: Chaos generation. Nanoelectronics, nanophotonics and nonlinear physics. Proceedings of the XIII All-Russian Conference of Young Scientists. “Techno-Decor”, Saratov, 2018, p.300-301.

           Isaeva O.B., Rozental R.M., Ginzburg N.S., Zotova I.V., Sergeev A.S., Rozhnev A.G. Dynamics of a distributed gyroclystron model with delayed feedback. Proceedings of the XVII Winter School-Workshop on Radiophysics and Microwave Electronics, February 5–10, 2018, Saratov. Publishing Center "Science". P.27-28. (In Russian.)

           Isaeva O.B., Savin D.V., Seleznev E.P. Analysis of the dynamics of rough chaos generators in time series. Proceedings of the XVII Winter School-Workshop on Radiophysics and Microwave Electronics, February 5–10, 2018, Saratov. Publishing Center "Science". P.44. (In Russian.)