A Bifurcation Phenomenon for the Periodic Solutions of the Duffing Equation without Damping Terms

Abstract

We investigate a bifurcation phenomenon for the periodic solutions of the Duffing equation without damping terms:

d²u/dt²(t) + κ u(t) + αu³(t) = f_λ(t),  t ∈ R          (1.1)
where κ and α are positive constants and f_λ(t) (λ > 0) is a given family of T-periodic external force parameterized by λ. We show an existence of not only T and exact 2T-periodic solutions but also exact mT-periodic solutions (m ≥ 3) bifurcated from a specific T-periodic solution.