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Reaction-diffusion systems: hysteresis and nonlocal interactions

DFG Collaborative Research Center 910 on Control of self-organizing nonlinear systems

The project deals with reaction-diffusion systems involving hysteresis, or, more generally, bistability. The models under consideration have applications to a large number of biological, chemical, physical, and economic processes. Besides the nontrivial issue of well-posedness, we are interested in the qualitative description of solutions. The analysis will be in terms of emerging spatio-temporal patterns that are influenced primarily by the interplay between diffusion and spatially distributed hysteresis. We will also develop theoretical concepts for the stability analysis and control of these patterns by temporally and spatially nonlocal feedback.

Principal Investigator

PD Dr. Pavel Gurevich


Dr. Nikita Begun: My main research interests include Dynamical systems, Differential equations, Reaction-diffusion equations, Hyperbolic attractors and systems with dry friction.

Mark Curran: I am interested in the well-posedness of reaction-diffusion equations with a nonlinear term involving spatially distributed hysteresis. In particular, I am dealing with this problem where the spatial domain is a subset of Rn with n greater than one.

Adem Güngör (Masters student-Student Assistant)

Dr. Eyal Ron (Former PhD Student): I am interested in delay differential equations and hysteresis operators. Research questions invole long-term behaviour, such as periodic solutions and patterns. Specifically, the question of controlling these phenomenon using delay terms (e.g. Pyragas control) or hysteresis operators is of great interest.

Konstantinos Zemas (Former masters student-Student Assistant)

Preprints and Publications

Gurevich P., Tikhomirov S.
Error estimates for Riemann sums of some singular functions
arXiv:1703.03203 (2017)

Gurevich P., Reichelt S. 
Pulses in FitzHugh--Nagumo systems with rapidly oscillating coefficients
Multiscale Model. Simul. 16(2), (2018), pp. 833–856 

Gurevich P., Tikhomirov S. 
Spatially discrete reaction-diffusion equations with discontinuous hysteresis. 
Ann. Inst. H. Poincaré Anal. Non Linéaire Vol. 35, No. 4, (2018), pp. 1041-1077,  https://doi.org/10.1016/j.anihpc.2017.09.006 

Gurevich P., Tikhomirov S. 
Rattling in spatially discrete diffusion equations with hysteresis. 
Multiscale Model. Simul. 15(3) (2017), pp. 1176-1197 

Begun N., Pliss V. A., Sell G.
On the Stability of Weakly Hyperbolic Invariant Sets

Journal of Differential Equations, Vol. 262, No. 4, pp. 3194-3213 (2017).

Arnold M., Begun N., Gurevich P., Kwame E., Lamba H., Rachinskii D.
Dynamics of discrete time systems with a hysteresis stop operator.
SIAM Journal on Applied Dynamical Systems. Vol. 16, No. 1, pp. 91-119 (2017) 

Ron E. 
Hysteresis-Delay Differential Equations

PhD Thesis, Free University of Berlin (2016).

Gurevich P.
Asymptotics of parabolic Green's functions on lattices. 
Algebra i Analiz. 28, No. 5, pp. 21-60 (2016), English transl.: St. Petersburg Math. J. 28, pp. 569-596 (2017). 

Curran M., Gurevich P., Tikhomirov S.
Recent advances in reaction-diffusion equations with non-ideal relays. In: "Control of Self-Organizing Nonlinear Systems", edited by E. Schöll, S. H. L. Klapp, P. Hövel (Springer, Berlin, Heidelberg, 2016). p. 211-234. 

Begun N., Pliss V. A., Sell G.
On the stability of hyperbolic attractors of systems of differential equations
Differential Equations, Vol. 52, No. 2, pp. 139-148 (2016) 

Gurevich P., Vaeth M. 
Stability for semilinear parabolic problems in L_2, W^{1,2}, and interpolation spaces. 
Z. Anal. Anwendungen. 35, No. 3 (2016), 333-357. 
(Extended version: pdf)

Gurevich P., Rachinskii D. 
Asymptotics of sign-changing patterns in hysteretic systems with diffusive thresholds. 
Asymptotic Analysis. Vol. 96, pp. 1-22 (2016)

Friedman G., Gurevich P., McCarthy S., Rachinskii D. 
Switching behaviour of two-phenotype bacteria in varying environment. 
J. Physics: Conference Series. Vol. 585, 012012 (2015). 

Gurevich P., Stark H., Zeitz M.
Feedback control of flow vorticity at low Reynolds numbers. 
The European Physical Journal E (EPJE). Vol. 38:22 (2015).

Begun N.
Perturbations of Weakly Hyperbolic Invariant Sets of Two-Dimensional Periodic Systems.
Vestnik St. Petersburg University: Mathematics, Vol. 48, No 1, pp. 1-8 (2015) 

Gurevich P., Rachinskii D.
Pattern formation in parabolic equations containing hysteresis with diffusive thresholds. 
J. Math. Anal. Applications. Vol. 424 No. 2, pp. 1103-1124 (2015) 

Begun N., Kryzhevich S.
One-dimensional chaos in a system with dry friction: analytical approach 
Meccanica, Vol. 50, No. 8, pp. 1935-1948 (2015) 

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