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. 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:
The use of the special form of dispersion functions appearing in various physical applications allows to find the solutions of frequency resonance condition.
The set of resonances for given dispersion function and the form of resonance conditions is partitioned into non-intersecting resonance clusters; dynamics of each cluster can be studied independently.
Each resonance cluster can be represented by its NR-diagram which is a plane graph of the special structure. This representation allows to reconstruct uniquely 3a) dynamical system describing time-dependent behavior of the cluster, and 3b) the set of its polynomial conservation laws which are generalization of Manley–Rowe constants of motion for the simplest clusters
Dynamical systems describing some types of the clusters can be solved analytically.
These theoretical results can be used directly for describing real-life physical phenomena or various wave turbulent regimes in the theory of wave turbulence.
Nonlinear resonance shift
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. When the amplitude of the 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.
