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

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Abstract
In this thesis we consider dissipative dynamics of FrenkelKontorova (FK) models, one of the most important physical models, for example in the solid state physics. FK model generalizes onedimensional 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 nonautonomous FK model, that means in the case when the equations depend on time. In particular we consider the case of Ratchet dynamics (nonautonomous 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 semiflow, 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 nontransversal intersections of solutions), and show that the number of zeroes of a difference of two solutions of a nonautonomous FK model is nonincreasing. In particular we consider spacetime invariant measures, weak $\omega$limit sets, and space time attractors as unions of weak $\omega$limit sets. We show that the spacetime attractor is equal to the union of supports of spacetime invariant measures. We introduce the notion of a transversal spacetime attractor, as the attractor for which any two configurations in the attractor can not intersect nontransversally. 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 spacetime 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 ergodictheoretical sense.
Item Type:  Thesis (Doctoral thesis) 

Supervisor:  Slijepčević, Siniša 
Date:  2015 
Number of Pages:  80 
Subjects:  NATURAL SCIENCES > Mathematics NATURAL SCIENCES > Mathematics > Applied Mathematics and Mathematic Modeling NATURAL SCIENCES > Mathematics > Other Mathematical Disciplines 
Divisions:  Faculty of Science > Department of Mathematics 
Depositing User:  Iva Prah 
Date Deposited:  01 Sep 2015 13:06 
Last Modified:  01 Sep 2015 13:06 
URI:  http://digre.pmf.unizg.hr/id/eprint/4071 
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