Nikita V. Barabash

Bio: Nikita V. Barabash is an academic researcher from Volga State University of Water Transport. The author has contributed to research in topics: Attractor & Mathematics. The author has an hindex of 3, co-authored 9 publications receiving 22 citations.

A Lorenz-type attractor in a piecewise-smooth system: Rigorous results.

Abstract: Chaotic attractors appear in various physical and biological models; however, rigorous proofs of their existence and bifurcations are rare. In this paper, we construct a simple piecewise-smooth model which switches between three three-dimensional linear systems that yield a singular hyperbolic attractor whose structure and bifurcations are similar to those of the celebrated Lorenz attractor. Due to integrability of the linear systems composing the model, we derive a Poincare return map to rigorously prove the existence of the Lorenz-type attractor and explicitly characterize bifurcations that lead to its birth, structural changes, and disappearance. In particular, we analytically calculate a bifurcation curve explicit in the model's parameters that corresponds to the formation of homoclinic orbits of a saddle, often referred to as a "homoclinic butterfly." We explicitly indicate the system's parameters that yield a bifurcation of two heteroclinic orbits connecting the saddle fixed point and two symmetrical saddle periodic orbits that gives birth to the chaotic attractor as in the Lorenz system. These analytical tasks are out of reach for the original nonintegrable Lorenz system. Our approach to designing piecewise-smooth dynamical systems with a predefined chaotic attractor and exact solutions may open the door to the synthesis and rigorous analysis of hyperbolic attractors.

Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs.

Abstract: Non-smooth systems can generate dynamics and bifurcations that are drastically different from their smooth counterparts. In this paper, we study such homoclinic bifurcations in a piecewise-smooth analytically tractable Lorenz-type system that was recently introduced by Belykh et al. [Chaos 29, 103108 (2019)]. Through a rigorous analysis, we demonstrate that the emergence of sliding motions leads to novel bifurcation scenarios in which bifurcations of unstable homoclinic orbits of a saddle can yield stable limit cycles. These bifurcations are in sharp contrast with their smooth analogs that can generate only unstable (saddle) dynamics. We construct a Poincare return map that accounts for the presence of sliding motions, thereby rigorously characterizing sliding homoclinic bifurcations that destroy a chaotic Lorenz-type attractor. In particular, we derive an explicit scaling factor for period-doubling bifurcations associated with sliding multi-loop homoclinic orbits and the formation of a quasi-attractor. Our analytical results lay the foundation for the development of non-classical global bifurcation theory in non-smooth flow systems.

Partial synchronization in the second-order Kuramoto model: An auxiliary system method

Abstract: Partial synchronization emerges in an oscillator network when the network splits into clusters of coherent and incoherent oscillators. Here, we analyze the stability of partial synchronization in the second-order finite-dimensional Kuramoto model of heterogeneous oscillators with inertia. Toward this goal, we develop an auxiliary system method that is based on the analysis of a two-dimensional piecewise-smooth system whose trajectories govern oscillating dynamics of phase differences between oscillators in the coherent cluster. Through a qualitative bifurcation analysis of the auxiliary system, we derive explicit bounds that relate the maximum natural frequency mismatch, inertia, and the network size that can support stable partial synchronization. In particular, we predict threshold-like stability loss of partial synchronization caused by increasing inertia. Our auxiliary system method is potentially applicable to cluster synchronization with multiple coherent clusters and more complex network topology.

Chaotic driven maps: Non-stationary hyperbolic attractor and hyperchaos

Abstract: In this paper we study simple examples of non-autonomous maps having different changing in time chaotic attractors. We present the definition of non-stationary hyperbolic attractor of the driven maps. We rigorously prove the existence of non-stationary hyperbolic attractor in 2D driven map and introduce a hyperchaotic attractor for autonomous 3D map of master-slave structure. Our analysis is based on the auxiliary systems approach and the construction of invariant cones.

Synchronization Thresholds in an Ensemble of Kuramoto Phase Oscillators with Randomly Blinking Couplings

Abstract: We consider a network of Kuramoto phase oscillators with randomly blinking couplings. Applicability of the averaging method for small switching intervals is rigorously substantiated. Using this method, we analytically estimate the threshold coupling force for synchronizing the ensemble oscillators. The threshold synchronization is studied as a function of the switching interval for various network sizes. The effect of preserving synchronization for a significant increase in the switching interval is found, which is the key feature of the system since a slight increase in this interval usually leads to the synchronization failure. The intermittent-synchronization possibility for small network sizes and large switching intervals is shown. An increase in the network size is shown to result in a stability increase due to the decreasing probability of appearance of uncoupled configurations. The regions corresponding to the global synchronization of oscillators are singled out in the system-parameter space.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Abstract: Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Blinking coupling enhances network synchronization.

Abstract: This paper studies the synchronization of a network with linear diffusive coupling, which blinks between the variables periodically. The synchronization of the blinking network in the case of sufficiently fast blinking is analyzed by showing that the stability of the synchronous solution depends only on the averaged coupling and not on the instantaneous coupling. To illustrate the effect of the blinking period on the network synchronization, the Hindmarsh-Rose model is used as the dynamics of nodes. The synchronization is investigated by considering constant single-variable coupling, averaged coupling, and blinking coupling through a linear stability analysis. It is observed that by decreasing the blinking period, the required coupling strength for synchrony is reduced. It equals that of the averaged coupling model times the number of variables. However, in the averaged coupling, all variables participate in the coupling, while in the blinking model only one variable is coupled at any time. Therefore, the blinking coupling leads to an enhanced synchronization in comparison with the single-variable coupling. Numerical simulations of the average synchronization error of the network confirm the results obtained from the linear stability analysis.

A Lorenz-type attractor in a piecewise-smooth system: Rigorous results.

Abstract: Chaotic attractors appear in various physical and biological models; however, rigorous proofs of their existence and bifurcations are rare. In this paper, we construct a simple piecewise-smooth model which switches between three three-dimensional linear systems that yield a singular hyperbolic attractor whose structure and bifurcations are similar to those of the celebrated Lorenz attractor. Due to integrability of the linear systems composing the model, we derive a Poincare return map to rigorously prove the existence of the Lorenz-type attractor and explicitly characterize bifurcations that lead to its birth, structural changes, and disappearance. In particular, we analytically calculate a bifurcation curve explicit in the model's parameters that corresponds to the formation of homoclinic orbits of a saddle, often referred to as a "homoclinic butterfly." We explicitly indicate the system's parameters that yield a bifurcation of two heteroclinic orbits connecting the saddle fixed point and two symmetrical saddle periodic orbits that gives birth to the chaotic attractor as in the Lorenz system. These analytical tasks are out of reach for the original nonintegrable Lorenz system. Our approach to designing piecewise-smooth dynamical systems with a predefined chaotic attractor and exact solutions may open the door to the synthesis and rigorous analysis of hyperbolic attractors.

"Cognitive" modes in small networks of almost identical chemical oscillators with pulsatile inhibitory coupling.

Abstract: The Lavrova-Vanag (LV) model of the periodical Belousov-Zhabotinsky (BZ) reaction has been investigated at pulsed self-perturbations, when a sharp spike of the BZ reaction induces a short inhibitory pulse that perturbs the BZ reaction after some time τ since each spike. The dynamics of this BZ system is strongly dependent on the amplitude Cinh of the perturbing pulses. At Cinh > Ccr, a new pseudo-steady state (SS) emerges far away from the limit cycle of the unperturbed BZ oscillator. The perturbed BZ system spends rather long time in the vicinity of this pseudo-SS, which serves as a trap for phase trajectories. As a result, the dynamics of the BZ system changes qualitatively. We observe new modes with packed spikes separated by either long “silent” dynamics or small-amplitude oscillations around pseudo-SS, depending on Cinh. Networks of two or three LV-BZ oscillators with strong pulsatile coupling and self-inhibition are able to generate so-called “cognitive” modes, which are very sensitive to small changes in Cinh. We demonstrate how the coupling between the BZ oscillators in these networks should be organized to find “cognitive” modes.The Lavrova-Vanag (LV) model of the periodical Belousov-Zhabotinsky (BZ) reaction has been investigated at pulsed self-perturbations, when a sharp spike of the BZ reaction induces a short inhibitory pulse that perturbs the BZ reaction after some time τ since each spike. The dynamics of this BZ system is strongly dependent on the amplitude Cinh of the perturbing pulses. At Cinh > Ccr, a new pseudo-steady state (SS) emerges far away from the limit cycle of the unperturbed BZ oscillator. The perturbed BZ system spends rather long time in the vicinity of this pseudo-SS, which serves as a trap for phase trajectories. As a result, the dynamics of the BZ system changes qualitatively. We observe new modes with packed spikes separated by either long “silent” dynamics or small-amplitude oscillations around pseudo-SS, depending on Cinh. Networks of two or three LV-BZ oscillators with strong pulsatile coupling and self-inhibition are able to generate so-called “cognitive” modes, which are very sensitive to small chan...

Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs.

Abstract: Non-smooth systems can generate dynamics and bifurcations that are drastically different from their smooth counterparts. In this paper, we study such homoclinic bifurcations in a piecewise-smooth analytically tractable Lorenz-type system that was recently introduced by Belykh et al. [Chaos 29, 103108 (2019)]. Through a rigorous analysis, we demonstrate that the emergence of sliding motions leads to novel bifurcation scenarios in which bifurcations of unstable homoclinic orbits of a saddle can yield stable limit cycles. These bifurcations are in sharp contrast with their smooth analogs that can generate only unstable (saddle) dynamics. We construct a Poincare return map that accounts for the presence of sliding motions, thereby rigorously characterizing sliding homoclinic bifurcations that destroy a chaotic Lorenz-type attractor. In particular, we derive an explicit scaling factor for period-doubling bifurcations associated with sliding multi-loop homoclinic orbits and the formation of a quasi-attractor. Our analytical results lay the foundation for the development of non-classical global bifurcation theory in non-smooth flow systems.