THE BEHAVIOR OF A NONLINEAR HAMILTONIAN SYSTEM WITH ONE DEGREE-OF-FREEDOM AT THE BOUNDARY OF A PARAMETRIC RESONANCE DOMAIN

The non-linear oscillations of a nearly-integrable time-periodic Hamil tonian system with one degree of freedom are studied in the neighbourh ood of its equilibrium position. The case when the multipliers of the linearized system are multiple is considered. Using a canonical change of variables the Hamiltonian function is reduced to a simpler form re flecting the resonant nature of the problem under consideration. An ap proximate result is considered in detail; some of the results are exte nded to the complete system. A rule is established which enables one t o use the nature of the dependence of the non-linear oscillation frequ encies on the amplitude in the unperturbed system to distinguish betwe en the boundaries of the parametric resonance domain at which the equi librium position is stable from those boundaries at which it is unstab le. In the unstable case an estimate is given of the size of the equil ibrium neighbourhood to which the trajectory of a perturbed system is confined. The existence of stable periodic motions is demonstrated in the neighbourhood of an unstable equilibrium position. The stochastic behaviour of the system is discussed. A number of examples of the appl ication of the general results to specific problems in mechanics are c onsidered.