Complete solution of the modified Cherry oscillator problem
In 1925, T.M. Cherry demonstrated that linear stability analysis is an insufficient method for determining the stability of a system with respect to small amplitude perturbations. The example he used consisted of two nonlinearly coupled oscillators, one possessing positive energy, the other negative energy, with frequencies omega(1) = 2 x omega(2) allowing third-order resonance. Cherry presented a two-parameter solution set for his example which would, however, allow a four-parameter solution set. A three-parameter solution set for the resonant three-oscillator case was obtained which would allow a six-parameter solution set. Nonlinear instability could therefore be proven only for a very small part of the phase space of the oscillators. This paper gives the complete solution for the three-oscillator case and shows that, except for a singular case, all initial conditions, especially those with arbitrarily small amplitudes, lead to explosive behaviour. This is true of the resonant case. The non-resonant oscillators can sometimes also become explosively unstable, but only if the initial amplitudes are not infinitesimally small.
Bibliographic Reference: Report: IPP 6/293 EN (1990) 19 pp.
Availability: Available from Max-Planck-Institut für Plasmaphysik, 8046 Garching bei München (DE)
Record Number: 199011270 / Last updated on: 1994-12-01
Original language: en
Available languages: en