Annotation of the results obtained in 2023
The dynamics of self-oscillating systems with external adaptive influence, the phase of which depends on the system variable, has been studied. In this context, the following models were studied: van der Pol oscillator, radiophysical generator of quasi-periodic oscillations, chaotic system (Rössler system). Using the method of charts of dynamical regimes and charts of Lyapunov exponents, the planes of amplitude versus frequency parameter with an increase in the adaptability parameter were studied. A picture of complication and evolution of periodic, two- and three-frequency quasiperiodic, chaotic regimes with increasing adaptability parameter has been revealed. An artificial neural network with a “machine association” type architecture has been developed and studied, designed to identify synchronous regimes of nonlinear dynamical systems with discrete time based on the analysis of pairs of sequences generated by these systems. Various types of synchronization have been tested, including complete synchronization and remote synchronization. Examples of synchronization with shift and inversion are considered. In all cases, the network showed high accuracy in determining the presence of synchronous regimes. A laboratory model of a generator that demonstrates the birth of hyperbolic chaos in the form of bursting oscillations has been developed. The characteristics of bursting oscillations correspond to neural-type models. It is shown that important when implementing the laboratory model is the selection of parameters taking into account the non-zero mismatch of the operational amplifiers. A circuit model of a genetic oscillator (repressilator) has been developed, and the correspondence between the behavior of the mathematical model and the circuit model has been shown. A circuit model of “quorum sensing” communication, characteristic of the interaction of repressive agents, has been developed. For an ensemble of repressilators, a new type of inhomogeneous solutions has been discovered that can stabilize and dominate at a certain level of interaction. A family of such cycles with different rotation numbers is shown. The dynamics of a system of second-order differential equations has been studied. These equations describe the parametric interaction of the oscillatory modes in the presence of quadratic and cubic nonlinearity of general form. A comparison of their dynamics with the dynamics of approximating models was carried out in order to determine their possibilities when modeling systems of the above type. Using the Lagrange formalism, systems of second-order differential equations describing the parametric interaction of three or more oscillatory modes were constructed for various resonant conditions and types of nonlinearity characterized the interaction between modes. A self-oscillator based on two circuits is considered. One of the circuits includes negative conductivity. The condition of parametric resonance is satisfied between the circuits. The circuits interact on quadratic nonlinearity, and a delayed feedback is added to the nonlinearity chain. A numerical study of the Kirchhoff equations, modeling in the Multisim and a numerical study of the analytically obtained three-dimensional approximating model were carried out. For small networks of spin-transfer oscillators with field coupling and RLC or RC load, a method for suppressing multistability (coexistence of synchronous and asynchronous regimes) was investigated. Suppression is carried out by adjusting the load parameters and pre-charging the capacitors. In the case of two oscillators, fine tuning of the load circuits is not required, and as they increase, the mechanism becomes less rough. To suppress multistability, more precise tuning of the chain parameters is required for a specific set of frequency mismatches of the ensemble oscillators. It is shown for the first time that the homoclinic “butterfly” bifurcation, which plays a key role in the scenario for the birth of the classical pseudohyperbolic Lorentz attractor, is also part of one of the scenarios for the emergence of a uniformly hyperbolic chaotic attractor, which is another type of robust chaos. The birth of a hyperbolic attractor is the result of the superposition of two mechanisms, the first of which is associated with the existence of a homoclinic “butterfly” bifurcation, which ensures a consistent return of the trajectory to the saddle, and the second is associated with the effect of angle multiplication when the trajectory passes near the equilibrium position. Several confidential communication schemes based on the synchronization of hyperbolic chaotic generators have beenproposed and studied.A novel method for decoding of an information mixed with the chaotic signal of the hyperbolic type is developed. Method is based on taking into account of thetime homogeneity of the function of correlation between the dynamics of the robust transmitter and receiver. This method helps to improve, significantly, quality of the extraction of an information,when significant non-identity of the receiver and transmitter is taking place.Problems of the inability to achieve absolute identity of the subsystems of communication scheme and full synchronization between them at one upon a time became the main argument for criticizing the chaotic communication and a reason for loss of the interest in this task. A new technique based onthe the hyperbolic chaos and its inherent properties can revivethis promising area of technical applications of the chaos. As a result of an experimental and numerical study on the example of two types of multi-circuit generators (with a linear and nonlinear mean-field control circuit), it was shown that multi-frequency quasi-periodic oscillations can occur in such generators. The maximum number of incommensurate components in dynamical mode is determined by the number of circuits in the generator. The destruction of a two-frequency torus corresponds to the Afraimovich-Shilnikov scenario, the destruction of multi-frequency tori is associated with the appearance of saddle tori.

 

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Annotation of the results obtained in 2021
Study within the framework of the Project was carried out in the following areas: research of networks and ensembles of oscillators; application of nonlinear dynamics to the analysis of biological systems; development and research of models of parametric interaction of modes obtained within the framework of an approximate description of active distributed systems; modeling of dynamic phenomena in nanoelectronics systems; development of new schemes for generating robust chaos, development of approaches to its analysis and applications; study of the dynamics of systems demonstrating complex behavior and multistability. As part of the first stage of work on the Project, the following results were obtained. Using the example of autonomous and non-autonomous ensembles of quasiperiodic generators with dissipative coupling, details of synchronization scenarios of multi-frequency quasiperiodic oscillations were revealed. For an autonomous system, a classification of various types of synchronization was performed, the features of the occurrence and destruction of multi-frequency quasiperiodic oscillations were revealed. Scenarios have been found in which chaotic oscillations occur, characterized by several positive Lyapunov exponents. It was shown that the variation of control parameters makes it possible to control chaos, meaning its stabilization or development into a hyperhaotic regime. The results can be used to develop new types of generators of signal with complex structure. In order to study the scenarios of transition to chaos through the destruction of multi-frequency quasiperiodicity, a model of a multi-circuit generator was developed, which is an ensemble of van der Pol oscillators connected through the mean field. The regimes typical for this system were revealed, namely periodic oscillations, four types of multi-frequency quasiperiodic oscillations, chaos and hyperhaos. Bifurcation scenarios of chaotic behavior development were described and the features typical for the observed chaotic attractors were revealed. The results can be used for engineering new high-dimensional chaos generators. Using several examples of models (the Henon map and the Toda oscillator under quasiperiodic excitation, a quasiperiodically excited RL-diode circuit), a theoretical and experimental study of the destruction of a two-frequency torus and the occurrence of chaotic dynamics with an additional zero exponent was carried out. The general mechanisms (patterns) of this scenario were revealed, the theoretical results were verified by comparison with the experimentally observed ones. The situation was studied when adaptive feedback is introduced into the system (one of the influence frequencies becomes dependent on a dynamic variable), which removes the system from the class of quasiperiodically excited systems. It was shown that in this case the additional Lyapunov exponent, close to zero, was preserved. We have studied a model of a network of spin-transfer oscillators with field coupling, given by the Landau-Lifshitz-Hilbert-Slonchevsky equations, which are coupled through magnetic fields introduced in the form of an additive to the effective field. For the case when the network has a small star-like structure (one central oscillator is connected to three others that have no connections with each other), the existence of regions in the parameter space where the synchronization mode is implemented through an intermediary (peripheral oscillators are synchronized with each other, but not synchronized with the central oscillator) was shown. The proposed system can be used as a low-power source of microwave radiation with the possibility of implementing complex oscillatory modes. Due to the fact that the interaction is carried out only by means of a magnetic field in the absence of electrical contact, such a system can be used to implement transformer communication between nodes of nanoscale microwave devices. Using the Lagrange formalism, the equations for the generalized Rabinovich-Fabrikant model describing the three-mode interaction in the presence of a general cubic nonlinearity were obtained. The bifurcation analysis of equilibrium positions and limit cycles was carried out. It was shown that the dynamics of the generalized model depends on the signature of characteristic expressions presented in the equations. The comparison with the dynamics of the Rabinovich–Fabrikant model was carried out and the parameter ranges where there is a complete or partial coincidence of dynamics were indicated. The resulting model is new and represents a natural extension of the well–known Rabinovich-Fabrikant model, it is universal and allows to model systems of various physical nature (including radio-engineering) in which there are a three-mode interaction and a general cubic nonlinearity. Genetic oscillator models have been developed, which allow the simplest implemention of a circuit model based on analog modeling. The classical model of a repressilator with inhibition in the form of a Hill function was used as a base. A mathematical model with inhibition using an exponential function was proposed, as well as a variant of the model including the autoinduser. A radiophysical model of the repressilator has been developed in the MultiSim circuit modeling package. An electronic circuit was constructed on the basis of a system of equations, which is a mathematical model of two coupled repressillators. A study of both one oscillator and coupled genetic oscillators was carried out, the conditions for the excitation of self-oscillations were revealed. The possibility of occurrence of burst and spike modes in a system demonstrating bifurcation associated with the blue sky catastrophe was shown. As a result of this bifurcation, a hyperbolic chaotic Smale-Williams attractor arises. At the same time, at the threshold of bifurcation, the observed attractor can be classified as burst one. With the change of parameters, the burst attractor destroys and turns into a spike attractor. Several variants of the circuit model of a generator with a blue sky catastrophe in the MultiSim software package based on a mathematical model have been developed. A laboratory layout corresponding to the developed scheme has been created. In a physical experiment, the possibility of the birth of hyperbolic chaos in the form of burst attractors as a result of the blue sky catastrophe was demonstrated. A geometric model of a flow system with a Smale-Williams attractor in the Poincare section was constructed. This is non-smooth four-dimensional system, the phase space of which consists of two regions in which the dynamics is described by different sets of differential equations for two complex variables. This is the first example of an exactly solvable system of differential equations with a Smale-Williams type attractor in the Poincare section. A hyperbolic attractor of the Smale-Williams solenoid type has been discovered in the model of complex Shimizu-Morioka equations (with an additional modification by introducing a term providing the necessary stretching of the argument of the complex variable). The Shimizu-Morioka complex equations are a physically reasonable model obtained by replacing variables from the complex Lorentz equations describing the dynamics of a single-mode laser. The reduction of the Shimizu-Morioka complex system for the case of purely real variables demonstrates the bifurcation of the homoclinic "butterfly" and the Lorentz attractor. The obtained results were presented at Russian and international conferences, published in peer-reviewed scientific journals, including journals from the first quartile. The results are also presented on the website of the research group http://sgtnd.narod.ru/

 

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Annotation of the results obtained in 2022
Research within the framework of the project is carried out in the following areas: research of networks and ensembles of oscillators; application of nonlinear dynamics to the analysis of biological systems; development and research of models of parametric interaction of modes of active distributed systems; simulation of dynamic phenomena in nanoelectronics systems; development and application of robust chaos generation schemes; study of the dynamics of systems demonstrating complex behavior and multistability. As part of the second phase of the project, the following results were obtained. The dynamics of three coupled oscillators capable of separately demonstrating quasi-periodic oscillations is studied. The bifurcation analysis is carried out, the space of parameters is analyzed. Invariant tori of different dimensions, quasi-periodic bifurcations of tori, and resonant Arnold's web based on tori of different dimensions are revealed. The analogy and differences from the case of three coupled van der Pol oscillators are considered. In particular, it is shown that four-frequency tori also arise when all three individual generators are in the limit cycle mode. Systems of spin-transfer oscillators with uniaxial symmetry and field coupling are studied. A theoretical analysis of a system of two such oscillators is carried out, the bistability of the synchronous and nonsynchronous solutions is analyzed, and estimates are obtained for the region of their existence. For two oscillators loaded on RLC chains, a method for controlling bistability is proposed: regardless of the initial conditions of the oscillators, they can be forced to always choose synchronous or non-synchronous modes. In a system of three oscillators, the effect of remote synchronization was discovered and studied — two peripheral oscillators are synchronized, although they do not interact directly with each other, but are coupled only with the central one. The regime of "structurally stable quasi-periodicity" is revealed: in the region of lack of synchronization, when the trajectories are wound on invariant tori, there are no high-order resonances characteristic of such cases. This regime is interesting from a practical point of view, since it should make it possible to obtain in the experiment a good approximation to the true quasi-periodicity in the mathematical sense without parasitic locking of the tuned frequencies into resonances. An idea of the universal picture of regimes for models of different nature (radiophysical generator, neuron model) demonstrating a bifurcation associated with a blue sky catastrophe is obtained. It is shown that characteristic period addition bifurcations are observed in the leech neuron model. The features of the transformation of the burst attractor are studied, it is shown that as the distance from the bifurcation line of the blue sky catastrophe, the duration of the burst segment on the temporal realization decreases. A circuit model of a radiophysical generator with scalable dynamic variables is obtained. The proposed scaling makes it possible to avoid the manifestation in the scheme of modes of limiting the amplitudes of operational amplifiers that destroy the chaotic attractor. The values of non-zero bias of operational amplifiers are determined, which do not significantly change the attractor. A picture of the dynamic regimes of a single genetic oscillator with an exponential inhibition function is described. A comparative analysis of the model with the classical model with Hill's nonlinearity was carried out, as a result of which the correspondence of the models was revealed. The circuit model was corrected, as a result of which the circuit model demonstrates characteristic dynamic modes. The system of three interacting repressors was studied. It is shown that for a sufficiently large coupling, the characteristic hyperchaotic attractor is destroyed as a result of the crisis and a stable asymmetric regime dominates: two oscillators are completely synchronous to each other, and the third one is different. For this model, a numerical study of the dynamics was carried out, bifurcation diagrams of the model were studied. Generalized models of parametric interaction of three wave modes of the active medium in the presence of quadratic and cubic nonlinearities are obtained. These models are studied and compared with previously known special cases. Equations are obtained for chains of coupled non-identical Josephson junctions loaded on an RLC circuit. A numerical simulation of a system of three elements that are not identical in terms of critical currents has been performed. The structure of the parameter space of the system is an Arnold's web, in the nodes of which there are areas of complete synchronization, the dynamics is represented by resonant limit cycles. The mechanism of the appearance of a hyperchaotic attractor has been studied. The existence of a cascade of several secondary Neimark-Sacker bifurcations of resonant cycles preceding the appearance of hyperchaos is demonstrated. The reverse cascade of bifurcations of absorption by a chaotic attractor of a set of saddle-focus cycles and saddle sets arising as a result of doublings of saddle resonant cycles leads to the appearance of Shilnikov's hyperchaotic discrete attractor. The occurrence of a hyperbolic attractor of the Smale-Williams type in the Shimizu-Morioka complex system is studied. It is shown that a hyperbolic attractor of the indicated type arises when a power-law type of perturbation is introduced in the vicinity of the points of the "butterfly" homoclinic bifurcation with a negative saddle index. In the parameter space, areas are identified where for the trajectories of the system the conditions that a hyperbolic attractor of the Smale-Williams type must satisfy are simultaneously fulfilled. A study was made of the dynamics of a complex Shimizu-Morioka system with the injection of a perturbation for the values of the parameters corresponding to the Lorenz pseudohyperbolic attractor, which exists in such a system in the case of real variables. It is shown that the pseudohyperbolicity criterion for invariant manifolds does not hold in this range of parameters, and pseudohyperbolic attractors are not observed in such a system. Several confidential communication schemes based on the synchronization of hyperbolic chaotic generators have been proposed and studied. The transmission stability is studied in the case of non-identical transmitter and receiver and in the presence of noise in the communication channel. It is shown that, compared to general chaotic generators, robust hyperbolic chaos provides more reliable, more stable and broadband transmission, “robust” synchronization of the receiver and transmitter, and more accurate information detection. A picture of the dynamic regimes of a multiloop generator with a common control scheme is obtained for sufficiently large values of the parameter responsible for excitation of oscillations in the loops. The dynamic regimes observed in numerical mathematical modeling were also found in the experiment. The experimental and simulation results are in good agreement. A mathematical model of a multiloop generator with a general nonlinearity has been developed. It is shown that periodic, quasi-periodic and chaotic self-oscillations can be observed in such a model. A study is made of a dynamic system given by an implicit mapping. The motivation for the importance of studying such systems is substantiated - they arise in the analysis of the behavior of iterative procedures of various numerical methods. Approaches to the study of such systems have been worked out. It is shown that the structure of attracting invariant sets of implicit mappings is much more complicated than traditional systems. It is shown that superstrong multistability of attracting orbits arises in certain situations. Moreover, these orbits can be both regular and strange. The results obtained are presented at Russian and international conferences, published in peer-reviewed scientific journals. Also, the results are presented on the website of the scientific team http://sgtnd.narod.ru/

 

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