Monotone twist maps and periodic solutions of systems of Duffing type

Abstract

The theory of twist maps is applied to prove the existence of many harmonic and subharmonic solutions for certain Newtonian systems of differential equations. The method of proof leads to very precise information on the oscillatory properties of these solutions.

Multiple periodic solutions of Hamiltonian systems confined in a box

“…It has been shown, indeed, that the singularities of this type provide a behavior of the solutions which resembles the situation encountered while dealing with superlinear systems (see, e.g. [1,2,3,6,7,8,9,10,11,12,13]). In the same spirit, the existence of subharmonic solutions can also be easily obtained, but we will avoid such a discussion, for briefness.…”

“…Now, we will provide the existence of a guiding curve γ which controls the solutions of system (1). As a consequence, we will see that this curve guides also the solutions of system (7), when p is chosen large enough.…”

“…Hence, by [7, Theorem 1.2], system (7) has at least two T -periodic solutions z(t),z(t), with rot z, [0, T ] = rot z, [0, T ] = K. By Proposition 2 again, it has to be that z(t),z(t) ∈ R(p K ), for every t ∈ [0, T ], so that these are indeed solutions of the original system (1). The proof of Theorem 1.1 is thus completed.…”

“…The above theorem thus provides the existence of infinitely many T -periodic solutions of system (1). Its proof will be carried out in Section 2, by the use of a generalized version of the Poincaré -Birkhoff theorem [7,Theorem 1.2], recently obtained by the first author and A. J.…”

A Poincaré–Birkhoff theorem for Hamiltonian flows on nonconvex domains

“…The efforts to generalize this theorem to higher dimensions go back to Birkhoff himself [7], and have been later continued in many works, including [2,8,13,16,18,23,24,29,31]. Out of these extensions, we shall be particularly interested in the version due to Moser and Zehnder [24,Theorem 2.21,p.…”