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. 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: with possibly different being eigen-frequencies of the linear part of some nonlinear partial differential equation. The is the wave vector associated with a mode; the integer subscripts 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. 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 \omega is shifted from its "natural" value \omega_0 according to the formula where A is the oscillation amplitude and \kappa 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 F reaches a critical value instabilities appear. The critical value is given by the formula where m is the oscillator mass and \gamma is the damping coefficient. Furthermore, new resonances appear in which oscillations of frequency close to \omega_0 are excited by an external force with frequency quite different from \omega_0.

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