# Dinamika neautonomnoga Frenkel-Kontorovina modela

Rabar, Braslav (2015) Dinamika neautonomnoga Frenkel-Kontorovina modela. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this thesis we consider dissipative dynamics of Frenkel-Kontorova (FK) models, one of the most important physical models, for example in the solid state physics. FK model generalizes one-dimensional elastically connected chains of particles in a periodic potential, with a constant or periodic uniform force. The thesis focuses on development of the theory for non-autonomous FK model, that means in the case when the equations depend on time. In particular we consider the case of Ratchet dynamics (non-autonomous dynamics without an external force); with a number of open problems, for example existence of transport. We first show in the thesis existence of solutions on appropriate function spaces. We demonstrate existence of a semi-flow, smoothness and analyticity of the solution depending on the initial condition and the vector field. We then define a synchronized solution, and show that for every mean spacing there exist at least one synchronized solution. The key idea needed to describe the dynamics is zeroes of a difference of two solutions. We distinguish regular and singular zeroes (transversal and non-transversal intersections of solutions), and show that the number of zeroes of a difference of two solutions of a non-autonomous FK model is non-increasing. In particular we consider space-time invariant measures, weak $\omega$-limit sets, and space time attractors as unions of weak $\omega$-limit sets. We show that the space-time attractor is equal to the union of supports of space-time invariant measures. We introduce the notion of a transversal space-time attractor, as the attractor for which any two configurations in the attractor can not intersect non-transversally. The key result are sufficient, verifiable conditions for an attractor to be transversal, for example analyticity for the Ratchet system. We distinguish two dynamical phase: depinned and pinned phase, and rigorously introduce the notion of transport (for the Ratchet system). For transversal space-time attractors, we give a weak, general sufficient condition for the existence of transport. The conjecture that transport exists for specific systems remains open; however the sufficient condition gives a possibility of fast numerical verification of it for a specific system. Finally, we show that for the Ratchet system, the synchronized solutions are stable in an ergodic-theoretical sense.