Resonant, nonplanar motion of a stretched string

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Abstract
The weakly nonlinear, nonplanar, resonant response of a stretched string to simple harmonic, planar excitation is examined in a four‐dimensional phase space in which the coordinates are slowly varying amplitudes of a sinusoidal motion of the dominant mode at the driving frequency. There are two similarity parameters, α and β, which measure damping and resonant offset, respectively. The bifurcation points β=β(α), of which there may be as many as eight, of the equilibrium points in the phase space are determined. Two of these points are Hopf bifurcations, which suggests the possibility of bifurcation to a strange attractor, as in the mathematically similar problem for a spherical pendulum. In the present problem, however, there is at least one stable equilibrium point for each pair of α and β, and it appears that the solution of the phase‐plane equations always terminates at such a point (which corresponds to harmonic motion of the string). This conclusion is supported by the results of numerical integration. It is shown in an appendix that the equations considered here also describe a large class of nonlinearly coupled, identical oscillators.

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