Domains of Attraction of a Forced Beam by Interpolated Mapping

A nonlinear dissipative dynamical system can often have multiple attractors. In this case it is important to study the global behavior of the system by determining the global domain of attraction of each attractor. In this paper we study the global behavior of a forced beam with two-mode interaction. The governing equation of motion is reduced to two second-order nonlinear nonautonomous ordinary differential equations. When ω₂ ≈ 3ω₁ and Ω ≈ ω₁, the system can have two asymptotically stable steady-state periodic solutions, where ω₁, ω₂ and Ω denote natural frequencies of the first and second modes and the excitation frequency, respectively. Both solutions have the same period as the excitation period. Therefore, each of them shows up as a period-1 solution in Poincare map. We show how interpolated mapping method can be used to determine the two four-dimensional domains of attraction of the two solutions in a very effective way. The results are compared with the ones obtained by direct numerical integration.