Physical phenomenonIn 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. The mixing of modes in non-linear systems is termed resonant interaction.
Description.
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). Vibrational modes can interact in a resonant interaction when both the energy and momentum of the interacting modes is conserved. The conservation of energy implies that the sum of the frequencies of the modes must sum to zero: 
formula_1
with possibly different formula_2 being eigen-frequencies of the linear part of some nonlinear partial differential equation. The formula_3 is the wave vector associated with a mode; the integer subscripts formula_4 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 resonance shift.
Nonlinear effects may significantly modify the shape of the resonance curves of harmonic oscillators.
First of all, the resonance frequency formula_6 is shifted from its "natural" value formula_7 according to the formula
formula_8
where formula_9 is the oscillation amplitude and formula_10 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 formula_11 reaches a critical value formula_12 instabilities appear. The critical value is given by the formula
formula_13
where formula_14 is the oscillator mass and formula_15 is the damping coefficient.
Furthermore, new resonances appear in which oscillations of frequency close to formula_7 are excited by an external force with frequency quite different from formula_17
Nonlinear frequency response functions.
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.
Notes and references.
Notes.
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