Nonlinear oscillators at resonance with periodic forcing

TL;DR

The paper studies nonlinear second-order oscillators with a resonant linear part and periodic forcing, focusing on the existence of 2π-periodic solutions. It unifies and extends prior results (Lazer, Leach, Frederickson, Alonso, Ortega, Korman, Li) by deriving a necessary condition that links the forcing coefficients with the nonlinearities F and g, and shows that no universal sufficiency condition exists for the full equation. When the resonance threshold is not met to guarantee periodic solutions, the dynamics exhibit blow-up: all solutions become unbounded in the long time, a fact proven via energy methods and Ortega–Alonso style arguments. The work also provides a general unboundedness result using a Lyapunov-like function and a Poincaré-type map, clarifying the dichotomy between periodic behavior and unbounded growth at resonance.

Abstract

In this note we unify the results of A.C. Lazer and P.O. Frederickson [3], A.C. Lazer [6], A.C. Lazer and D.E. Leach [7], J.M. Alonso and R. Ortega [1], and P. Korman and Y. Li [4] on periodic oscillations and unbounded solutions of nonlinear equations with linear part at resonance and periodic forcing. We give conditions for the existence and non-existence of periodic solutions, and obtain a rather detailed description of the dynamics for nonlinear oscillations at resonance, in case periodic solutions do not exist.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Proposition 1
• Proposition 2
• Proposition 3
• Lemma 2.1
• Lemma 2.2
• Proposition 4
• Proposition 5
• Lemma 2.3