Applied differential equations and critical self-organization

This seminar presents ongoing research in applied differential equations and critical self-organization theory. One research area is dynamical systems, for which several results are shown, e.g. dynamical-system integration through solution of polynomial equations, detection of invariant tori and Hamiltonian separatrix-chaos beyond KAM theory, and compact solitons stabilization by billiards-integrability conditions identification.

Another research topic is integration of second-order elliptic and hyperbolic PDEs by spectral methods, application examples being electromagnetism and acoustics in heterogeneous continua.

A detailed discussion is given on critical self-organization of strongly correlated systems, one example being fractal-pattern formation during fracture in heterogeneous materials. A more detailed example is describing hybrid smooth–nonsmooth (mesoscopic) modeling of critical self-organization during irreversible (plastic) deformation in metal crystals, where plasticity is addressed as a quantized process, which enables the numerical derivation of experimentally observed critical exponents. Slow-process assumptions reduce the dynamical system to a set of nonlinear algebraic equations, which are solved in polynomial time using a homotopy approach/dissipative quasi-convexification.

Several developed 1st and 2nd order local-minimization/nonlinear-solver algorithms are shown.

Bio: Nathan Perchikov got his MSc degree in the direct track at Tel Aviv Universitywith specialization in continuous optimization algorithms. He later obtained his PhD degree in dynamical systems at the Technion. Subsequently, he was a postdoctoral researcher at the Sorbonne Université in Paris, France, working on critical self-organization in strongly-correlated high-symmetry systems and at the Max-Planck-Institut in Düsseldorf, Germany, working on spectral methods for PDEs. He is currently a visiting lecturer at the Technion.