Domains of Attraction of System of Nonlinearly Coupled Ship Motions by Simple Cell Mapping

Abstract

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 two-degree-of freedom system. The specific system examined is a system of nonlinearly coupled ship motions in regular seas. The system is described by two second-order nonlinear nonautonomous ordinary differential equations. When the frequency of the pitch mode is twice the frequency of the roll mode, and is near the encountered wave frequency, the system can have two asymptotically stable steady-state periodic solutions. The one solution has the same period as the encountered wave period and has the pitch motion only. The other solution has twice the period of the encountered wave period and has pitch and roll motions. The harmonic and second-order subharmonic solutions show up as period-1 and period-2 solutions, respectively, in a Poincaré map. We show how the method of simple cell mapping 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.