Prof. Dmitrii Rachinskii

- PhD in Appl. Math., 1997, Moscow Institute of Physics & Technology (advisor: Mark A. Krasnoselískii)
- DSc, 2002, Institute for Control Problems, Moscow

Department of Mathematical Sciences
The University of Texas at Dallas

dmitry.rachinskiy at
Phone:  (972) 883-6697
Fax:       (972) 883-6622
Address: 800 W. Campbell Road, Richardson, TX 75080-3021
Office: FO 2.602D


Complete list of publications

Google scholar profile

Nonlinear Analysis and Dynamical Systems group at UTD


My research centers at nonlinear dynamics and analysis, in particular, dynamics of systems with hysteresis.
I am also interested in applied mathematical modeling. This interest involves me in collaboration with physicists, engineers, biologists and economists. Mathematical models I work on mostly have the form of differential equations, including ordinary, partial, functional (delay) and operator-differential systems.

If you are interested in doing a dissertation or a project with me, please come to see me.

Past PhD students

Currently advising 3 PhD students


MATH 2418 - Linear Algebra (Math Cohort)

MATH 2420 - Differential Equations with Applications

MATH 6324 - Applied Dynamical Systems

MATH 6345 - Mathematical Methods in Medicine and Biology

Current research interests & some publications

•  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 operator-differential 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. 125-291, 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, 2010-2028.

3. Appelbe B, Rachinskii D, Zhezherun A, Hopf bifurcation in a van der Poltype oscillator with magnetic hysteresis, Physica B: Condensed Matter, 403, 2008, 301-304.

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, 1404-1411.

We have developed a number of models for plastic hysteresis and for hysteresis effects in flows through porous media for geophysics and soil physics applications in collaboration with environmental engineers and hydrologists.

5. Brokate M., Rachinskii D, Global stability of Armstrong-Frederick models with periodic biaxial inputs, NoDEA Nonlinear Differential Equations Appl. 13, 2006, 385-411.

6. Applebe B, Flynn D, McNamara H, O'Kane P, Pimenov A, Pokrovskii A, Rachinskii D, Zhezherun A, Rate-independent hysteresis in terrestrial hydrology, IEEE Control Systems Magazine 29, 2009, 44-69.

7. Brokate M, McCarthy S, Pimenov A, Pokrovskii A, Rachinskii D, Energy dissipation in hydrological systems due to hysteresis, Environmental Modeling & Assessment 16, 2011, 313-333.

I am interested in hysteresis phenomena in economics, finance and population dynamics. As one example of a hysteresis effect in economics, temporary rises in unemployment generated by supply and/or demand shocks are known to have permanent (or, at least, very persistent) effects.

8. Cross R, McNamara H, Pokrovskii A, Rachinskii D, A new paradigm for modeling hysteresis in macroeconomic flows, Physica B: Condensed Matter, 403, 2008, 231-236.

MURPHYS Conference
HMM Conference

•  Laser dynamics

Mode-locked (ML) semiconductor lasers are compact reliable low cost devices which can produce short and ultra-short optical pulses at high repetition rates (tens and hundreds of GHz). Mode-locked pulses are suitable for multiple applications such as telecommunication, medical applications, machining and probing. By modeling diverse bifurcation scenarios in slow-fast 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 mode-locked semiconductor lasers, in "Nonlinear Laser Dynamics: From Quantum Dots to Cryptography", Kathy Luedge (Ed.), Chapter VIII, pp. 189-222, Wiley-VCH, 2011.

10. Rachinskii DI, Vladimirov AG, Bandelow U, Huettl B, Kaiser R, Q-switching instability in a mode locked semiconductor laser, J. Opt. Soc. Am. 23, 2006, 663-670.

11. Vladimirov A, Pimenov A, Rachinskii D, Numerical study of dynamical regimes in a monolithic passively mode-locked semiconductor laser, IEEE Journal of Quantum Electronics 45, 2009, 462-468.

12. Vladimirov A, Rachinskii D, Rebrova N, Huyet G, An optically injected mode locked laser, Phys. Rev. E 88, 2011, 066202.

More on synchronization, excitability, oscillations...

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, 190-199.

•  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 real-world 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 supply-demand 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 Prandtl-Ishlinskii 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, 2012,54-64.

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/s00285-012-0550-9.

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.

We also use operator-differential models with hysteretic switches to examine how the long-time memory of benign or adverse environmental conditions (or changes in behavior in response to danger or epidemics) can improve fitness of species and affect evolution of populations.

19. Friedman G, Gurevich P, McCarthy S, Rachinskii D, Switching behaviour of two-phenotype 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, 1-30.

•  Topological degree methods in analysis of bifurcations and canards

21. Krasnosel'skii AM, Rachinskii DI, Subharmonic bifurcation from infinity, J. Differential Equations 226, 2006, 30-53.

22. Bouse E, Krasnosel'skii AM, Pokrovskii A, Rachinskii DI, Non-local branches of cycles, bi-stability, 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, 989-1011.

24. Pokrovskii A, Rachinskii D, Sobolev V, Zhezherun A, Topological degree in analysis of canard-type trajectories in 3-D systems, Applicable Analysis: Int. J. 90, 2011, 1123-1139.

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, 467-482.