Research On Periodic Solutions Or Almost Periodic Solutions For Several Classes Of Differential Systems

It is well known that Henri Poincare has established the qualitative theory of the ordinary differential equations between1881and1886. The qualitative theory in which shapes of integral curves and properties of critical points are investigated, whose core idea is focused on avoiding solving common solutions of the differential equations but directly studying properties of integral curves defined by equations, and which aims to indirectly obtain properties of solutions by just starting from equations themselves. The theory of periodic solution or almost periodic solution as an important component of the qualitative theory, which is widely applied to prac-ticalities such as astronomy, communication, service system, automatic control and electronics. For instance, three-body problem in astronomy and physics, the problem of Kolmogorov system in ecology, triple-molecular model in chemistry, population competition and cooperation model in biology, etc. Consequently, research on peri-odic solutions or almost periodic solutions of differential equations has dual value of theory and practice.It is obviously that nontrivial periodic solutions can be used to portray various kinds of classic nonlinear differential equations. On the basis of the fact that a closed trajectory of a self-sustained oscillation occurring in a vacuum tube circuit appears as one limit cycle, and beginning with efforts from Van der Pol, Lienard and Andronov, the study about the existence, nonexistence, uniqueness, stability and other properties of isolated periodic solutions has been infiltrated into the qualitative theory of differential equations. Almost periodic phenomenon which can be more easily seen than periodic one in the natural and social sciences. The solutions of differential equations which are implicated in some problems such as the movement of spheres, ecological environment, and the law of supply and demand for the market in daily life, which just have strong law of almost period. In sum, discussion about periodic solutions or almost periodic solutions of ordinary differential equations are not only helpful to enrich the concept system of the qualitative theory, but also will play a crucial role in dealing with practical problems.This thesis mainly studies periodic solutions or almost periodic solutions for several classes of ordinary differential equations, and the following is detailed struc-ture of the paper.In chapter one, the historical background of periodic solutions or almost peri-odic solutions problems for ordinary differential equations and the existing research result is introduced. In the sequel, the study work in the thesis is mainly summa-rized.In chapter two, the existence of nontrivial periodic solutions for a class of the generalized Lienard systems is investigated. First of all, the existence and uniqueness of solutions corresponding to the initial value problem is discussed. Secondly, the simple conditions of the existence for nontrivial periodic solutions in some existing papers are optimized, and the results of which are extended and improved by using the geometrical theory of differential equations.In chapter three, limit cycle and bifurcation for a class of quadratic systems (II) is discussed. By introducing the theory of the point at infinity, comparison theorem and Poincare-Bendixson theorem, the existence of limit cycle and bifurcation within a wider range is given by applying the geometric theory of differential equations. In the sequel, the existing results are extended and improved.In chapter four, limit cycle for a class of nonpolynomial planar differential equations is analyzed. Using Liapunov method theory, some sufficient conditions for determining the origin to be the focus or center are obtained. Two practical examples are given to illustrate the effectiveness of new theoretical results. In chapter five, the existence of limit cycles and the type of critical points at infinity for a class of (n+1)th polynomial systems is explored. The center and focus problem is analyzed by computing the focal values. The existence of limit cycles and the fact which there is at most one limit cycle is investigated based on the rotated vector field theory and the generalized Lienard system theory. According to the Poincare transformation, the type of critical points at infinity is discussed.In chapter six, periodic solutions and almost periodic solutions to a class of quadratic differential systems are interpreted. The existence of periodic solutions for the system with periodic coefficients is analyzed. The existence, uniqueness, and global attractivity of almost periodic solutions for the other system with almost periodic coefficients is proved.