THE BEHAVIOUR OF HARMONIC RESONANT STEADY SOLUTIONS TO A MODEL DIFFERENTIAL EQUATION
A. J. Roberts · The Quarterly Journal of Mechanics and Applied Mathematics · 1981
Perturbation solutions of wave phenomena sometimes involve divisions by zero at certain values of the parameters. These zero divisors correspond to the occurrence of harmonic resonances. A wave-number perturbation technique is applied to a model equation to analyse the structure of the steady solutions near harmonic resonances. Resonances higher than third-harmonic resonance are also considered. It is shown that in general there exist regions in the parameter plane near each resonance where up to three different solutions exist. The relationship between the multiple steady solutions that occur near resonance and the ordinary fixed wave-number perturbation solution is explored. The ordinary perturbation is found to jump from one branch of the solutions to another as the parameters are varied.