We consider continuous and discrete reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. In particular, such problems describe chemical reactions and biological processes in which diffusive and nondiffusive substances interact according to hysteresis law. As a result various spatial and spatio-temporal patterns may appear.
We found a broad class of initial data (transverse functions) for which a solution exists, is unique, and continuously depends on initial data. As far as we know, this is the first uniqueness result for such systems with hysteresis.
For the nontransverse case we consider the discretization of the problem and consider it as a systems of coupled differential equation with hysteresis. We analyse patterns which appears in this system and leads to nonexistence of a solution for the original reaction-diffusion equation. It is interesting to mention that for 2-dimensional space those patterns depends on the type of discretization.
This is joint work with P. Gurevich.
Chimera states represent remarkable spatio-temporal patterns, where phase-locked oscillators coexist with drifting ones. Surprisingly, they can arise in arrays of coupled identical oscillators without any sign of asymmetry as a manifestation of internal nonlinear nature of dynamical networks. We discuss the appearance of chimera states for repulsively coupled phase oscillators of the Kuramoto-Sakaguchi type, i.e., when the phase lag parameter $\alpha > \pi/2$ and hence the network coupling works against synchronization. We find that chimeras exist in wide domain of the parameter space as a cascade of the states with increasing number of regions of irregularity---the so-called chimera's heads. We also study the origin of the chimera states and show that they grow from so-called multi-twisted states. Three typical scenarios of the chimera birth are reported: 1) via saddle-node bifurcation on invariant circle, also known as SNIC or SNIPER, 2) via blue-sky catastrophe, when two periodic orbits – stable and saddle - approach each other creating a saddle-node periodic orbit, and 3) via homoclinic transition, when the unstable manifold comes back crossing the stable manifold of a saddle.
This talk is about generalizations of the classical problem of maximizing the entropy when the expectation values of a set of measurable functions are constrained, see:
I. Csiszár and F. Matúš (2012) Minimization of entropy functionals revisited. Proceedings ISIT 2012, Cambridge, MA, USA, 150-154. http://staff.utia.cas.cz/matus/Boston2012.ps
