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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 being eigen-frequencies of the linear part of some nonlinear partial differential equation. Here is a vector with the integer subscripts 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.
Main notions and results of the theory of nonlinear resonances are:
Nonlinear effects may significantly modify the shape of the resonance curves of harmonic oscillators. First of all, the resonance frequency is shifted from its "natural" value according to the formula
where is the oscillation amplitude and is a constant defined by the anharmonic coefficients. Second, the shape of the resonance curve is distorted (foldover effect). When the amplitude of the (sinusoidal) external force reaches a critical value instabilities appear. The critical value is given by the formula
where is the oscillator mass and is the damping coefficient. Furthermore, new resonances appear in which oscillations of frequency close to are excited by an external force with frequency quite different from
Generalized frequency response functions, and nonlinear output frequency response functions  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.