Some Applications Of Variational Method In Mathematical And Physical Problems

In chaotic dynamics, detailed prediction of dynamical value is impossible due tothe sensitive dependence on the initial condition. The meaningful thing is thedescription of the dynamical averages. In these nonlinear systems, the traditional wayfails. In this thesis, we will firstly introduce periodic orbit theory, which provides a newway to calculate dynamical averages. In this theory, periodic orbits will play theessential role as far as their properties such as topological invariant are concerned. Wemay find that the averages can be expressed in terms of short unstable periodic orbits.After that, we introduce the application of that theory in a spatiotemporally chaoticsystem, the Kuramoto-Sivashinsky equation (KSe). Using Mutipoint shootingmethod, we can locate all the short periodic orbits in the KSe. In order to overcomethe difficulties of finding periodic orbits in a high dimensional system, we introduce anew effective method to locate periodic orbits: the variational method.Applying the variational method, this thesis studied the important applications ofthe method in finding the important organizing orbits in the phase space of a chaoticsystem, such as periodic orbits, homoclinic and heteroclinic orbits. Firstly, wesystematically studied periodic steady solutions of the KSe from a dynamical point ofview. At fixed system size L=43.5, important equilibria are identified and shown toorganize the dynamics. At fixed integration constant c=0.40194, four simplest cyclesare identified and used as basic building blocks to initialize the search for longer cycles,and the corresponding symbolic dynamics are constructed to classify all the shortperiodic orbits. The probation of the return map on a chosen Poincaré section shows thecomplexity of the dynamics and the bifurcation of building blocks provides a chart forlooking for the possible cycles of given periods.Then we propose a variational method for determining homoclinic and heteroclinicorbits including spiral-shaped ones in nonlinear dynamical systems. With a guessedloop, a loop evolution equation serves to tinker it into a true connecting orbit. Theadvantage of the method is that we do not need linearization for searching simpleconnecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamicalsystems are also presented. In particular, several heteroclinic orbits of the steady-state Kuramoto-Sivashinsky equation are found, which display interesting topologicalstructure, that is closely related to the corresponding periodic orbits.