Stability Of Periodic Solutions Of Lagrange Equations And Planar Hamiltonian Systems

We are mainly interested in the Lyapunov stability of periodic solutions of La-grange equations and planar nonlinear Hamiltonian systems. This paper is divided intofour parts.In Chapter 1, we introduce the historical background and some recent results ob-tained in the literature. We also state some important basic results.In Chapter 2, we study the Lyapunov stability of elliptic periodic solutions of La-grange equations. First we give some reasonable estimates of the periodic solutions ofErmakov-Pinney equations when the linearized equation is in the first stability zone.These estimates can also give the estimates of the rotation numbers of the Hill equa-tions. The results concerning the lower bounds of the rotation numbers are completelynew in the literature. By using these estimates, we prove that two classes of nonlinear,scalar, time-periodic, Lagrange equations will have twist periodic solutions, one classbing regular, including , another class beingsingular, .In Chapter 3, we try to extend the analytical method in studying the stability ofperiodic solutions of Lagrange equations to the nonlinear planar Hamiltonian systems.First, we establish two important facts on linear planar Hamiltonian systems. Oneis the reduction from ellipticity to R-ellipticity. Another is the relation between thestability of linear systems and the existence of periodic solutions of the generalizedErmakov-Pinney equations. Based on these two basic facts and the Birkhoff normalforms of area-preserving mappings, we compute the twist coefficients of planar non-linear Hamiltonian systems. Such twist coefficients play an important role in studyingthe Lyapunov stability of periodic solutions. For some special nonlinear systems, westate and prove the stability results. As an example, the stability of the equilibrium ofthe one-dimensionalΦ-Laplacian is given.As can be seen in Chapter 2 and 3, the existence and the estimates of periodic so- lutions of singular equations play important roles in the stability theory. Therefore, wedevelop some existence results for second order non-autonomous dynamical systems inChapter 4. The first one is based on a nonlinear alternative principle of Leray–Schauderand the result is applicable to the case of a strong singularity as well as the case of aweak singularity. The second one is based on Schauder's fixed point theorem and theresult sheds some new light on problems with weak singularities and proves that insome situations weak singularities may help create periodic solutions. The third exis-tence result is concerned with the nontrivial periodic solutions and the proof is basedon a well-known fixed point theorem in cones.