Singularity analysis of response bifurcation for a coupled pitch–roll ship model with quadratic and cubic nonlinearity

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Based on the coupling of roll and pitch motion of ships, a mathematical model with quadratic and cubic nonlinear terms is presented. Primary resonance is discussed by the averaging method when the frequency of excitation acting only on the pitch mode is near the natural frequencies of pitch mode and 1:2 internal resonance exists. By means of the slow-flow equations, the singularity analysis of the approximate solutions is investigated by means of bifurcation theory with constraint, from which one can find the effects of system parameters and excitation on the response. A total of 15 kinds of different persistent bifurcation modes were found. In terms of these results with stability analysis, the response can be classified into uncoupled periodic, coupled periodic and complex motions such as quasi-periodic or chaotic motions. By singularity analysis, saturation and jumping phenomena are also included in the present result which can be regarded as an extension of the available published works. The numerical simulations are in good agreement with the approximate analytical prediction.