AbstractWe survey the mathematics of non-linear Hamiltonian oscillations with emphasis being laid on the more recently discovered Kolmogorov instability. In the context of radial adiabatic oscillations of stars this formalism predicts a Kolmogorov instability even at low oscillation energies, provided that sufficiently high linear asymptotic modes have been excited.Numerical analysis confirms the occurrence of this instability. It is found to show up already among the lowest order modes, although high surface amplitudes are then required (ǀδrǀ/R ~ 0.5 for an unstable fundamental mode – first harmonic coupling). On the basis of numerical evidence we conjecture that in the Kolmogorov unstable regime the enhanced coupling due to internal resonance effects leads to an equipartition of energy over all interacting degrees of freedom. We also indicate that the power spectrum of such oscillations is expected to display two components: A very broad band of overlapping pseudo-linear frequency peaks spread out over the asymptotic range, and a strictly non-linear 1/f-noise type component close to the frequency origin.It is finally argued that the Kolmogorov instability is likely to occur among non-linearly coupled non-radial stellar modes at a surface amplitude much lower than in the radial case. This lends support to the view that this instability might be operative among the solar oscillations.
Numerical analysis of nonlinear forced vibrations of a cylindrical shell with combinational internal resonance in a fractional viscoelastic medium