Exact Multiplicity and Stability of Periodic Solutions for Duffing Equation with Bifurcation Method

Abstract

Under some $$L^p$$ -norms( $$p\in [1,\infty ]$$ ) assumptions for the derivative of the restoring force, the exact multiplicity and the stability of $$2\pi $$ -periodic solutions for Duffing equation are considered. The nontrivial $$2\pi $$ -periodic solutions of it are positive or negative, and the bifurcation curve of it is a unique reversed S-shaped curve. The class of the restoring force is extended, comparing with the class of $$L^{\infty }$$ -norm condition. The proof is based on the global bifurcation theorem, topological degree and the estimates for periodic eigenvalues of Hill’s equation by $$L^p$$ -norms( $$p\in [1,\infty ]$$ ).