Dynamics of a coupled system of Duffing's equations

In this paper the dynamic response of a two-degree-of-freedom coupled system of Duffing's equations is investigated. The approach used in the mathematical analysis and numerical computations are presented. For an autonomous system, in which the excitation forcing term is not included, we show that the existence of an equilibrium solution depends solely on the system parameters. Moreover, only a single point attractor exists when the damping coefficients are present. Otherwise, the dynamical motion for the coupled system can be described by a quasiperiodic motion. When a periodically forcing term is introduced, Hopf bifurcation may occur for a certain range of parameters. A numerical algorithm for computing the tired points of the corresponding Poincare' map is discussed, and it is used to demonstrate the existence of Hopf bifurcations. An example showing the coexistence of harmonic and quasiperiodic motions, and chaos in a forced coupled system is given.