Browsing by Author "Kravetc, Pavel"
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Item Applications of Topological and Perturbation Methods to Analysis of Periodic Solutions in Delay Differential Equations: Classification of Symmetries, Asymptotic Approximation and Stabilization(2019-08) Kravetc, Pavel; Rachinskiy, Dmitry; Krawcewicz, Wieslaw; Makarenkov, OlegThe present dissertation contains several interconnected results regarding periodic solutions to delay differential equations (DDEs). First of all, we formulate and prove a theorem that guarantees the occurrence of the Hopf bifurcation of relative periodic solutions from a relative equilibrium in a general Γ × S 1 - equivariant system of functional differential equations (FDEs) using the method based on twisted equivariant degree with one free parameter. This theorem also allows to classify the symmetries of the relative periodic solutions. The theoretical result is illustrated through the series of examples including D8- and S5-symmetric coupling of identical mode-locked semiconductor lasers and D8-symmetric configuration of coupled electro-mechanical oscillators with hysteresis. The latter example shows the possibility to adapt the proposed method for the settings with weakened conditions on the smoothness. Secondly, we perform the analysis of a rather broad class of slow-fast delayed models of population dynamics, that exhibit the behavior similar to the aforementioned mode-locked semiconductor lasers. In particular, we study the mechanism of formation of pulsating periodic solutions as well as develop a nonlocal method for their asymptotic approximations. Finally, we develop a noninvasive delay feedback (Pyragas) control to make a neutrally stable periodic orbit of a Hamiltonian system exponentially stable. More specifically, we establish different sufficient conditions for the stabilization of orbits with small and large amplitudes. We also present a discussion of how these conditions agree with each other.Item Chaos In Saw Map(World Scientific Publ Co Pte Ltd, 2019-02) Begun, Nikita; Kravetc, Pavel; Rachinskiy Dmitry I.; Rachinskii, Dmitry I.We consider the dynamics of a scalar piecewise linear "saw map" with infinitely many linear segments. In particular, such maps are generated as a Poincare map of simple two-dimensional discrete time piecewise linear systems involving a saturation function. Alternatively, these systems can be viewed as a feedback loop with the so-called stop hysteresis operator. We analyze chaotic sets and attractors of the "saw map" depending on its parameters.Item Periodic Pulsating Dynamics of Slow-Fast Delayed Systems with a Period Close to the Delay(Cambridge Univ Press, 2017-12-22) Kravetc, Pavel; Rachinskii, Dmitry I.; Vladimirov, A.; Kravetc, Pavel; Rachinskii, Dmitry I.We consider slow-fast delayed systems and discuss pulsating periodic solutions, which are characterised by specific properties that (a) the period of the periodic solution is close to the delay, and (b) these solutions are formed close to a bifurcation threshold. Such solutions were previously found in models of mode-locked lasers. Through a case study of population models, this work demonstrates the existence of similar solutions for a rather wide class of delayed systems. The periodic dynamics originates from the Hopf bifurcation on the positive equilibrium. We show that the continuous transformation of the periodic orbit to the pulsating regime is simultaneous with multiple secondary almost resonant Hopf bifurcations, which the equilibrium undergoes over a short interval of parameter values. We derive asymptotic approximations for the pulsating periodic solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realisation of the bifurcation scenario is highlighted.