Nonlinear resonance

In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. 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.

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). The frequency condition of nonlinear resonance reads

with possibly different ${\displaystyle \omega _{i}=\omega (\mathbf {k} _{i}),}$ being eigen-frequencies of the linear part of some nonlinear partial differential equation. Here ${\displaystyle \mathbf {k} _{i}}$ is a vector with the integer subscripts ${\displaystyle i}$ being indexes into Fourier harmonics – or eigenmodes – see Fourier series. 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.

...
Wikipedia