Periodic Solutions And Related Dynamics Of Infinite Lattices And Second Order Equations With Indefinite Weight

Lattices and second order equations with indefinite weight are two important models of dynamical systems and differential equations. In this paper, we study the periodic solutions and related dynamics of the solutions of lattices and second order equations with indefinite weight. We will study them in the following parts: the existence of periodic solutions of a class of sub-linear lattices with nearest neighbour interaction; the existence of periodic solutions and periodic impact solutions for some class of super-linear equations with indefinite weight; the existence and multiplicity of global-defined bouncing solutions and chaotic dynamics of a class of super-linear oscillator with indefinite weight.In Chapter 1, we introduce the background and the recent reserches of these topics. Then state the main results of this paper.In Chapter 2, we are concerned with the existence of T—periodic solutions of some non-conservative systems of classic particals periodically pertured with nearest neighbor coupling and the restoring forces are sub-linear about the distance between particles. By using a priori bounds and topological degree, over the mean values of the external forces, we find a necessary and sufficient conditions for the existence of periodic solutions in the case of finite systems and at the same time, we prove the existence of infinite periodic solutions in the case of infinite systems.In Chapter 3, we consider the existence of periodic solutions to a class of second order non-conservative super-linear equations with indefinite weight. First of all, we present a new so called bend-twist fixed point theorem. Secondly, we investigate the dynamics of the solutions on the phase plane and we find that, the solutions with big norms have a strong oscillations in the intervals where the weight is positive as well as the blow-up phenomene are appear in the intervals where the weight is negative. Finally, based on the qualitative properties of solutions, we apply the bend-twist theorem on a series of topological quadrangles suitable constructed and we prove the existence of infinite periodic solutions for the equations.In Chapter 4, by using the qualitative analysis in phase-plane, we consider the existence and the multiplicity of periodic bouncing solutions to a forced super-linear oscillators with indefinite weight. Firstly, by using a truncation function, we define a new impact equations such that the dynamics of the solutions is simple in a neighbourhood of the origin so that we can avoid the arguments for complicated dynamical behaviour of the solutions produced by the forced term nearby the origin. Secondly, we will introduce a new coordinate transformation, transform the impact phase-plane from right half plane to the whole plane. Thus we can use the similar arguments in In Chapter 3 to construct a seies of topological quadrangles and apply the bend-twist theorm to obtain the existence of infinite periodic solutions. These periodic solutions has large norm, therefore we prove the existence of infinite periodic bouncing solutions for the original impact oscillators.In Chapter 5, we discuss the existence of the global defined solutions and chaotic dynamics of solutions to a class of super-linear impact oscillators with indefinite weight. At first, doing as in Chapter 4, we introduce a new coordinate transformation to translate the impact systems into a new equal systems which defined in the whole phase-plane. Second, by using the qualitative method, we get the existence of solutions that defined in the intervals in which the weight change sign for finite times. And in this case, the solutions have different impact times given beforehand in the intervals where the weight take positive number or negative number. This fact help to obtain the existence of global defined solutions. Latter, by some delicate analysis for the dynamics of solutions in the phase plane, we prove that, for each infinite dimentional non-negative integer vector there is a global defined impact solution such that the impact times of the solution in each positive weight interval or negative weight interval, are the same as the number in the corresponding component. When the weight is periodic, above results imply that the impact oscillator exhibits chaotic-like dynamics.