# Rachinskiy, Dmitry I.

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Professor Rachinskiy's research "centers at nonlinear dynamics, in particular, dynamics of systems with memory." He also works in applied mathematical modeling." Learn more about Dr. Rachinskiy on his Home page.

Please note: At UT Dallas Dr. Rachinskiy's name is spelled with a "y," but in his publications his name is spelled Rachinskii. Treasures retains this latter spelling in all the records of his publications.

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### Recent Submissions

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.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 TASEP Modelling Provides a Parsimonious Explanation for the Ability of a Single uORF to Derepress Translation during the Integrated Stress Response(eLIFE Sciences Publications Ltd) Andreev, D. E.; Arnold, Maxim; Kiniry, S. J.; Loughran, G.; Michel, A. M.; Rachinskiy, Dmitry I.; Baranov, P. V.; 0000 0003 5341 0057 (Rachinskiy DI); 0000-0002-4500-8394 (Arnold, M); Arnold, Maxim; Rachinskiy, Dmitry I.Translation initiation is the rate-limiting step of protein synthesis that is downregulated during the Integrated Stress Response (ISR). Previously, we demonstrated that most human mRNAs that are resistant to this inhibition possess translated upstream open reading frames (uORFs), and that in some cases a single uORF is sufficient for the resistance. Here we developed a computational model of Initiation Complexes Interference with Elongating Ribosomes (ICIER) to gain insight into the mechanism. We explored the relationship between the flux of scanning ribosomes upstream and downstream of a single uORF depending on uORF features. Paradoxically, our analysis predicts that reducing ribosome flux upstream of certain uORFs increases initiation downstream. The model supports the derepression of downstream translation as a general mechanism of uORF-mediated stress resistance. It predicts that stress resistance can be achieved with long slowly decoded uORFs that do not favor translation reinitiation and that start with initiators of low leakiness. © Andreev et al.Item Timing Jitter of Passively-Mode-Locked Semiconductor Lasers Subject to Optical Feedback: A Semi-Analytic Approach(American Physical Society, 2015-11) Jaurigue, Lina; Pimenov, Alexander; Rachinskiy, Dmitrii L.; Schoell, Eckehard; Luedge, Kathy; Vladimirov, Andrei G.; Rachinskii, Dmitrii L.We study the effect of delayed coherent optical feedback on the pulse timing jitter in passively-mode-locked semiconductor lasers with the help of a semi-analytical method which we develop to calculate the timing fluctuations in these lasers. Through the proposed method physical insights into the feedback dependence of the timing jitter are gained and the greatly reduced computation times allow for the investigation of the dependence of timing fluctuations over greater parameter domains. We show that resonant feedback leads to a reduction in the timing jitter and that a frequency-pulling region forms about the main resonances, within which a timing jitter reduction is observed. The width of these frequency-pulling regions increases linearly with short feedback delay times. We derive an analytic expression for the timing jitter, which predicts a monotonic decrease in the timing jitter for resonant feedback of increasing delay lengths, when timing jitter effects are fully separated from amplitude jitter effects. For long feedback cavities the decrease in timing jitter scales approximately as 1/τ with increasing feedback delay time tau. This behavior is not related to the stability of the system but is instead due to the influence of the noise, on the timing jitter, being reduced since the solution space is larger for increasing τ.Item Effect of Dynamical Instability on Timing Jitter in Passively Mode-Locked Quantum-Dot Lasers(Optical Soc Amer, 2014-12-08) Pimenov, A.; Habruseva, T.; Rachinskii, Dmitry I.; Hegarty, S. P.; Huyet, G.; Vladimirov, A. G.; 0000 0003 5341 0057 (Rachinskii, DI); Rachinskii, Dmitry I.We study the effect of noise on the dynamics of passively mode-locked semiconductor lasers both experimentally and theoretically. A method combining analytical and numerical approaches for estimation of pulse timing jitter is proposed. We investigate how the presence of dynamical features such as wavelength bistability in a quantum-dot laser affects timing jitter.Item Analytical Solution for a Class of Network Dynamics with Mechanical and Financial Applications(Published By The American Physical Society Through The American Institute Of Physics, 2014-09-29) Krej?©, P.; Lamba, H.; Melnik, S.; Rachinskii, Dmitry I.; 0000 0003 5341 0057 (Rachinskii, DI); Rachinskii, Dmitry I.We show that for a certain class of dynamics at the nodes the response of a network of any topology to arbitrary inputs is defined in a simple way by its response to a monotone input. The nodes may have either a discrete or continuous set of states and there is no limit on the complexity of the network. The results provide both an efficient numerical method and the potential for accurate analytic approximation of the dynamics on such networks. As illustrative applications, we introduce a quasistatic mechanical model with objects interacting via frictional forces and a financial market model with avalanches and critical behavior that are generated by momentum trading strategies.Item Bistability and Hysteresis in an Optically Injected Two-Section Semiconductor Laser(Amer Physical Soc, 2014-05-08) Pimenov, A.; Viktorov, E. A.; Hegarty, S. P.; Habruseva, T.; Huyet, G.; Rachinskii, Dmitry I.; Vladimirov, A. G.; 0000 0003 5341 0057 (Rachinskii, DI); Rachinskii, Dmitry I.The effect of coherent single frequency injection on two-section semiconductor lasers is studied numerically using a model based on a set of delay differential equations. The existence of bistability between different continuous-wave and nonstationary regimes of operation is demonstrated in the case of sufficiently large linewidth enhancement factors.Item Hysteresis Can Grant Fitness in Stochastically Varying Environment(Public Library of Science, 2014-07-28) Friedman, Gary; McCarthy, Stephen; Rachinskii, Dmitry I.; 0000 0003 5341 0057 (Rachinskii, DI); Rachinskii, Dmitry I.Although the existence of multiple stable phenotypes of living organisms enables random switching between phenotypes as well as non-random history dependent switching called hysteresis, only random switching has been considered in prior experimental and theoretical models of adaptation to variable environments. This work considers the possibility that hysteresis may also evolve together with random phenotype switching to maximize population growth. In addition to allowing the possibility that switching rates between different phenotypes may depend not only on a continuous environmental input variable, but also on the phenotype itself, the present work considers an opportunity cost of the switching events. This opportunity cost arises as a result of a lag phase experimentally observed after phenotype switching and stochastic behavior of the environmental input. It is shown that stochastic environmental variation results in maximal asymptotic growth rate when organisms display hysteresis for sufficiently slowly varying environmental input. At the same time, sinusoidal input does not cause evolution of memory suggesting that the connection between the lag phase, stochastic environmental variation and evolution of hysteresis is a result of a stochastic resonance type phenomenon.Item Periodic Canard Trajectories with Multiple Segments Following the Unstable Part of Critical Manifold(2012-11-01) Krasnosel'skii, Alexander M.; O'Grady, Edward; Pokrovskii, Alexei V.; Rachinskii, Dmitry I.; 0000 0003 5341 0057 (Rachinskii, DI); Rachinskii, Dmitry I.We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on n scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes n fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has n portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.