Ap. Markeyev, THE BEHAVIOR OF A NONLINEAR HAMILTONIAN SYSTEM WITH ONE DEGREE-OF-FREEDOM AT THE BOUNDARY OF A PARAMETRIC RESONANCE DOMAIN, Journal of applied mathematics and mechanics, 59(4), 1995, pp. 541-551
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.