Perturbation Analysis of Parametric Resonance

Coexistence The special case when all the independent solutions of a linear, T -periodic ODE are T -periodic. Hill’s equation A second order ODE of the form ẍ + p(t)x = 0, with p(t) T -periodic. Instability pockets Finite domains, usually intersections of instability tongues, where the trivial solution of linear, T -periodic ODEs is unstable. Instability tongues Domains in parameter space where the trivial solution of linear, T periodic ODEs is unstable. Mathieu equation An ODE of the form ẍ + (a + b cos(t))x = 0. Parametric resonance Resonance excitation arising for special values of coefficients, frequencies and other parameters in T -periodic ODEs. Quasi-periodic A function of the form ∑n i=1 fi(t) with fi(t) Ti-periodic, n finite, and the periods Ti independent over R. Sum resonance A parametric resonance arising in the case of at least three frequencies in a T -periodic ODE.