Non-local criteria for the existence and stability of periodic oscillations in autonomous hamiltonian systems

Abstract

The conditions under which single-parameter families of periodic solutions (the existence in a sufficiently small neighbourhood of the origin of coordinates follows from the Lyapunov theorem (see /1/)) can be continued in a parameter to the boundary of the given domain, in particular to a certain isoenergetic surface, are found. These conditions, which can be verified by the use of the Hessian of a Hamilton function, also ensure the orbital stability of solutions to a first approximation. Bilateral estimates of the oscillation periods are obtained, and it is established that any solution with a period which satisfies such an estimate belongs to the corresponding family. As an example, the non-linear oscillations of a string with lumped masses are examined. The well-known non-local results relevant to the periodic oscillations of autonomous Hamiltonian systems are, as a rule, theorems on the existence of periodic solutions (see reviews /2–4/). One group of papers establishes the existence of periodic solutions with a specified value of the Hamiltonian, and other papers, establish solutions with a specified period; in the first case assumptions and made regarding the form of the corresponding constant energy surface; and in the second assumptions are made regarding the behaviour of the Hamiltonian in the vicinity of the equilibrium configuration and at infinity. The majority of the results were obtained by variational methods, the desired periodic solutions being identified with the stationary points of certain functionals. The discussion in the present paper is based on other concepts.