Nonlinear interaction among oscillation modes of accretion tori

Idea (Kluzniak & Abramowicz)◮ Torus oscillating radially δr(t) ∝ cos(ω r t)◮ Equation for vertical displacement:◮ Mathieu equation◮ Parametric resonanceδ¨z + ω 2 z [r 0 + δr(t)]δz = 0δ¨z + ω 2 z [1 + ɛ cos(ω z t)]δz = 0ω z= n , first possible: n = 3ω r 2