Equilibrium Properties Of FrenkelKontorova Model And Chaos Control In Conservative Systems

In this thesis, we surnmarize our recent contribution to Frenkel--Kontorovamodel and chaos control in conservative systems, which correspond to two different kinds of subjects in the mutual intersect between nonlinear science andcondensed matter physics, and that between nonlinear science and statisticalphysics.1. We introduce Frenkel--Kontorova--Devonshire (FKD) model to studycommensuraterincommensurate (CI) transitions, where the usual quadraticpotential is replaced by phenomenological potential identical in form to a sixth-degree Landau free energy function. As a function of a singIe parameter, theinteratomic potential changes smoothly from a quadratic single well to a sym-metric triple--welI and finaIly to a symmetric double--welI. By use of effectivepotential method with the numerical implementation, we studied the phasestructures of FKD model where the interactions are of different trip1e--wellpotentials, the external potential are of sinusoidal and piecewise parabolic po-tentials. According to the redefined winding number and rotation number,we analyze the periodicity of the phase diagrams and find some complex butregular phase structures, which are closely related to the period of the exter-nal potential. We also compared FKD model of triple--well interactions withFK model of double--well interactions, and discussed the evolutional behaviorsfrom the triple--well to double--well interactions.2. We proposed two onedimensional diatomic chain (i. e. complex-lattice) Frenkel--Kontorova models to simulate the 2 x N reconstruction inthe dimerized overlayer systems with vacancies. Model l describes the systemwhich contains an arbitrary number of vacancies per unit cell, the interatomicinteractions and the bonding of each atom to the substrate are taken to be har-monic potential. This is an exactly solvable model, so we presented its analyticexpression of the ground states. Model 2 describes the system which containsonly a single vacancy per unit cell, the interatodric interactions is harmonicpotential and the bonding of each atom to the substrate is taken to sinusoidalpotential. We determined its ground states by use of the gradient method.We develop Frenkel--Kontorova models to analyze the microscopic origins ofvacancy-line interactions in a pseudomorphic adsorbate system. These resultscan explain the 2 x N reconstruction in the dimerized overlayer systems, suchas Ga/Si(l12) and Ge/Si(l00).3. We modified the constant periodic pulse method in which the periodicpulse is independent of the system variables, and extended to control Hamil-tonian chaos. The procedure is as follows. First we calculate the finite timeLyapunov exponents for the initial points distributed uniformly in the phasespace, and obtain the finite time convergence zones in different time lengths,Then we select the stable segment that satisfies the desired dynamical features,and determine the intensity and interval of the perturbation. By successiveaction of the constant periodic perturbations on the system variables, the stable segment of the chaotic orbit can be made to form a closed periodic orbit.This method can stabilize the nonlinear dynadrical systems into any desiredperiodic orbit and does not require any previous knowledge of the system. Themethod is robust against the presence of external noise. It is more importantthat the Jacobian deterdrinant is invariant under the action of the perturbationon the system variables, i.e., this does not change the conservative property ofthe system. TherefOre, the method is suitable for both dissipative systems andconservative systems. For conservative systems, because the volume of phasespace is constant and thus there are no attractors, the stable periodic orbitcannot be obtained in these two cases. What we obtain is the quasiperiodicorbit surrounding the fixed points of the stable periodic orbit. If we want tostabilize the system into the periodic...