DYNAMICS AND BIFURCATIONS OF THE MAPPINGS DERIVED FROM GENERALIZED ?? SINE-GORDON EQUATION

• Main Author: ZAKARIA, LA
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/22876
• Institution: Institut Teknologi Bandung

The sine-Gordon equation is a partial differential equations which is known to have<br /> soliton solutions, and hence it is called one of the soliton equations. A discrete<br /> version of the equation could be obtained in various ways. Here, we will follow a<br /> method that use the so called Lax-pair. By restricting to the traveling wave solution,<br /> we derive an integrable ordinary difference equation.<br /> The aim of this research is to study the dynamics and bifurcation of the ordinary<br /> difference equations (or discrete dynamical systems) derived from traveling wave<br /> solution. To achieve this goal, we introduce a number of the parameters in<br /> the original Lax pair to obtain a generalized system. By using the compability<br /> condition, the discrete dynamical systems that depend on four parameters are<br /> obtained. Two of the parameters are bifurcation parameters while the others are<br /> not. The non bifurcational parameters determine the dimension of the system.<br /> In this dissertation, we will study the dynamical systems of low-dimensional (two<br /> and three-dimensional). The integral of the system is obtained by computing the<br /> monodromy matrix along a staircase on the lattice. By studying the level sets of<br /> the integral, we describe the dynamics of the system through the fourteen normal<br /> form of the integral. We observe an interesting local bifurcation of critical point<br /> in the system, namely: the period doubling bifurcation, where two 2-period points<br /> are created from a critical point. We have observed also a nonlocal bifurcation<br /> involving collision of homoclinic and heteroclinic connection between saddle type<br /> critical points.