A global solution curve for a class of periodic problems, including the pendulum equation

Abstract

Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic solutions with any number of Fourier coefficients arbitrarily prescribed.

Global solution curves for several classes of singular periodic problems

“…It follows that w(t) ≡ 0, and then µ = 0. ♦ Remark A similar result holds if (2.3) is replaced by |h(t)| ≤ ω √ ω 2 + c 2 , see J. Cepicka et al [2], or Lemma 2.3 in [8]. However, for singular problems we need a one-sided condition (2.3).…”

“…ByL 2 T andH 2 T we denote the respective subspaces of L 2 T and H 2 T , consisting of functions of zero average on (0, T ), i.e., T 0 u(t) dt = 0. The following lemma we proved in [8], by using Fourier series.…”

A Global Solution Curve for a Class of Free Boundary Value Problems Arising in Plasma Physics

“…We show that ξ 1 is a global parameter, and then we study the curve µ = µ(ξ 1 ), yielding a multiplicity result. The continuation approach of this paper is similar to our paper on pendulumlike equations, see [9]. In addition to its conceptual clarity, this approach is suitable for numerical computation of all solutions of (1.1).…”

“…We use continuation techniques to study the solution curves for the problem (1.1), similarly to our approach to the pendulum-like equations [9]. One can think of the problem (1.1) as being "at resonance", i.e., some conditions on p(x) are necessary for its solvability.…”

A global solution curve for a class of periodic problems, including the relativistic pendulum

“…We give conditions under which ξ is a global parameter, which means that for each ξ ∈ (−∞, ∞) there is a unique pair (µ, u(t)) solving (1.1). To establish that, we continue solutions of (1.1) back in k on curves of fixed average ξ, similarly to author's recent papers [10], [9] and [11], and at k = 0 we have a complete description of T -periodic solutions, in particular we have the existence and uniqueness of T periodic solutions of any average, which implies the uniqueness of (µ, u(t)). We then study properties of the curve µ = µ(ξ), depending on g(u).…”