Nonlinear resonance

In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is independent of amplitude.

The mixing of modes in non-linear systems is termed resonant interaction.

Generically two types of resonances have to be distinguished – linear and nonlinear.

From the physical point of view, they are defined by whether or not external force coincides with the eigen-frequency of the system (linear and nonlinear resonance correspondingly).

being eigen-frequencies of the linear part of some nonlinear partial differential equation.

is the wave vector associated with a mode; the integer subscripts

Accordingly, the frequency resonance condition is equivalent to a Diophantine equation with many unknowns.

The problem of finding their solutions is equivalent to the Hilbert's tenth problem that is proven to be algorithmically unsolvable.

Main notions and results of the theory of nonlinear resonances are:[1] Nonlinear effects may significantly modify the shape of the resonance curves of harmonic oscillators.

Second, the shape of the resonance curve is distorted (foldover effect).

Generalized frequency response functions, and nonlinear output frequency response functions [3] allow the user to study complex nonlinear behaviors in the frequency domain in a principled way.

These functions reveal resonance ridges, harmonic, inter modulation, and energy transfer effects in a way that allows the user to relate these terms from complex nonlinear discrete and continuous time models to the frequency domain and vice versa.