Prof. Dmitrii Rachinskii  PhD in Appl. Math., 1997, Moscow Institute of Physics & Technology (advisor: Mark A. Krasnosel’skii) Department of Mathematical Sciences dmitry.rachinskiy at utdallas.edu 

Nonlinear Analysis and Dynamical Systems group at UTD Research If you are interested in doing a dissertation or a project with me, please come to see me. Currently advising 3 PhD students Teaching MATH 2420  Differential Equations with Applications MATH 6324  Applied Dynamical Systems MATH 6345  Mathematical Methods in Medicine and Biology
• Systems with hysteresis Hysteresis underpins magnetic recording technologies, allows neuralnetworks to learn and eliminates undesirable switching in electronic circuits. It strongly affects properties of smart materials which are used in actuation, sensing, energy harvesting and nanopositioning technologies. It can also cause energy losses in transformers and power electronics systems. Hysteresis can be understood as a specific type of memory, which is present, and can be sometimes modeled in a uniform way, in materials, living systems and networks. Multidisciplinary research in hysteresis uses a variety of mathematical tools; I work on developing operatordifferential models of systems with hysteresis using the theory of hysteresis operators. 1. Brokate M, Pokrovskii AV, Rachinskii DI, Rasskazov O, Differential equations with hysteresis via a canonical example, in "The Science of Hysteresis", IsaakMayergoyz and Giorgio Bertotti (Eds.), Vol. I, Chapter II, pp. 125291, Elsevier, Academic Press, 2006. 2. Krejci P, O'Kane P, Pokrovskii A, Rachinskii D, Properties of solutions toa class of differential models incorporating Preisach hysteresis operator, Physica D: Nonlinear Phenomena, 241, 2012, 20102028. 3. Appelbe B, Rachinskii D, Zhezherun A, Hopf bifurcation in a van der Poltype oscillator with magnetic hysteresis, Physica B: Condensed Matter, 403, 2008, 301304. 4. Amann A, Brokate M, McCarthy S, Rachinskii D, Temnov G, Characterization of memory states of Preisach operator with stochastic inputs, Physica B: Condensed Matter, 407, 2012, 14041411.
5. Brokate M., Rachinskii D, Global stability of ArmstrongFrederick models with periodic biaxial inputs, NoDEA Nonlinear Differential Equations Appl. 13, 2006, 385411. 6. Applebe B, Flynn D, McNamara H, O'Kane P, Pimenov A, Pokrovskii A, Rachinskii D, Zhezherun A, Rateindependent hysteresis in terrestrial hydrology, IEEE Control Systems Magazine 29, 2009, 4469. 7. Brokate M, McCarthy S, Pimenov A, Pokrovskii A, Rachinskii D, Energy dissipation in hydrological systems due to hysteresis, Environmental Modeling & Assessment 16, 2011, 313333.
8. Cross R, McNamara H, Pokrovskii A, Rachinskii D, A new paradigm for modeling hysteresis in macroeconomic flows, Physica B: Condensed Matter, 403, 2008, 231236.
• Laser dynamics Modelocked (ML) semiconductor lasers are compact reliable low cost devices which can produce short and ultrashort optical pulses at high repetition rates (tens and hundreds of GHz). Modelocked pulses are suitable for multiple applications such as telecommunication, medical applications, machining and probing. By modeling diverse bifurcation scenarios in slowfast delay differential models of ML lasers, we attempt to understand experimentally observed properties of these lasers and optimize parameters of the mode locked pulses using synchronization, suppression of noise and other techniques. 9. Vladimirov A, Rachinskii D, Wolfrum M, Modeling of passively modelocked semiconductor lasers, in "Nonlinear Laser Dynamics: From Quantum Dots to Cryptography", Kathy Luedge (Ed.), Chapter VIII, pp. 189222, WileyVCH, 2011. 10. Rachinskii DI, Vladimirov AG, Bandelow U, Huettl B, Kaiser R, Qswitching instability in a mode locked semiconductor laser, J. Opt. Soc. Am. 23, 2006, 663670. 11. Vladimirov A, Pimenov A, Rachinskii D, Numerical study of dynamical regimes in a monolithic passively modelocked semiconductor laser, IEEE Journal of Quantum Electronics 45, 2009, 462468. 12. Vladimirov A, Rachinskii D, Rebrova N, Huyet G, An optically injected mode locked laser, Phys. Rev. E 88, 2011, 066202.
13. Goulding D, Hegarty SP, Rasskazov O, Melnik S, Hartnett M, Greene G, McInerney JG, Rachinskii D, Huyet G, Excitability in a quantum dot semiconductor laser with optical injection, Phys. Rev. Lett. 98, 2007, 153903. 14. Amann A, Mortell PM, O'Reilly EP, Quinlan M, Rachinskii D, Mechanism of synchronization in frequency dividers, IEEE Transactions on Circuits and Systems 56, 2009, 190199. • Dynamics on complex networks Dynamical processes on complex random networks are used to model a wide variety of phenomena such as spreading of opinions through a population, propagation of infectious diseases, neural signaling in the brain, and cascading defaults in financial systems. Similar dynamical processes on regular lattices are used for modeling phase transitions and critical phenomena in statistical mechanics (the Ising model), avalanches and propagation of cracks in earthquake fault systems, percolation phenomena and hysteresis effects. A relationship between the network topology and dynamics is a complex problem for many realworld and randomly generated networks. We have recently examined a relation between the network topology and the network response to varying inputs for a class of networks with binary nodes of a specific type. We are now attempting to use the results to mimic the creation of financial bubbles in price dynamics models and supplydemand models (there is also a mechanical interpretation as a system with Maxwell's frictions). This is a work in progress. 15. Krejci P, Lamba H, Melnik S, Rachinskii D, Dynamics on networks of PrandtlIshlinskii nodes: an explicit solution, arXiv. • Population dynamics I enjoy mathematical modeling in collaboration with biologists, ecologists and epidemiologists, e.g., modeling complex dynamics of seasonal diseases in birds and parasitism in marine populations.
16. O'Regan SM, Flynn D, Kelly TC, O'Callaghan MJA, Pokrovskii AV, Rachinskii D, The response of woodpigeon (Columba palumbus) to relaxation of intraspecific competition: A hybrid modelling approach, Ecological Modelling 224, 2012,5464. 17. O'ReganSM, Kelly, TC, Korobeinikov A, O'Callagha MJA, Pokrovskii AV, Rachinskii D, Chaos in a seasonally perturbed SIR model: avian inuenza in a seabird colony as a paradigm, J. of Mathematical Biology, doi:10.1007/s0028501205509. 18. O'Grady EA, Culloty SC, Kelly TC, O'Callaghan MJA, Rachinskii D, A preliminary threshold model of parasitism in the Cockle Cerastoderma edule using delayed exchange of stability, arXiv.
19. Friedman G, Gurevich P, McCarthy S, Rachinskii D, Switching behaviour of twophenotype bacteria in varying environment, arXiv. 20. Pimenov A, Kelly TC, Korobeinikov A, O'Callaghan MJA, Pokrovskii A, Rachinskii D, Memory effects in population dynamics: spread of infectious disease as a case study, Mathematical Modelling of Natural Phenomena 7, 2012, 130. • Topological degree methods in analysis of bifurcations and canards 21. Krasnosel'skii AM, Rachinskii DI, Subharmonic bifurcation from infinity, J. Differential Equations 226, 2006, 3053. 22. Bouse E, Krasnosel'skii AM, Pokrovskii A, Rachinskii DI, Nonlocal branches of cycles, bistability, and topologically persistent mixed mode oscillations, Chaos 18, 2007, 015109. 23. Kozyakin VS, Krasnosel'skii AM, Rachinskii DI, Asymptotics of Arnold tongues in problems at infinity, Discrete & Continuous Dynamical Systems A 20, 2008, 9891011. 24. Pokrovskii A, Rachinskii D, Sobolev V, Zhezherun A, Topological degree in analysis of canardtype trajectories in 3D systems, Applicable Analysis: Int. J. 90, 2011, 11231139. 25. O'Grady E, Krasnosel'skii A, Pokrovskii A, Rachinskii D, Periodic canard trajectories with multiple segments following the unstable part of critical manifold, Discrete & Continuous Dynamical Systems B 18, 2013, 467482.
