On the Stability of Exact Subharmonic Solutions of the Duffing Equation

This paper is concerned with the classical Duffing equation which describes the motion of a nonlinear oscillator with an elastic force that is odd with respect to the value of deviation from its equilibrium position, and in the presence of an external periodic force. The equation depends on three dimensionless parameters. When they satisfy some relation, the equation admits exact periodic solutions with a period that is a multiple of the period of external forcing. These solutions can be written in explicit form without using series. The paper studies the nonlinear problem of the stability of these periodic solutions. The study is based on the classical Lyapunov methods, methods of KAM theory for Hamiltonian systems and the computer algorithms for analysis of area-preserving maps. None of the parameters of the Duffing equation is assumed to be small.