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:
Main notions and results of the theory of nonlinear resonances are:
The use of dispersion relations appearing in various physical applications allows finding the solutions of the frequency resonance condition.
The set of resonances for a given dispersion function and the form of resonance conditions is partitioned into non-intersecting resonance clusters; dynamics of each cluster can be studied independently (at the appropriate time-scale). These are often called "bound waves", which cannot interact, as opposed to the "free waves", which can. A famous example is the soliton of the KdV equation: solitons can move through each other, without interacting. When decomposed into eigenmodes, the higher frequency modes of the soliton do not interact (do not satisfy the equations of the resonant interaction), they are "bound" to the fundamental.
Each collection of bound modes (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; these are generalization of Manley–Rowe constants of motion for the simplest clusters (triads and quartets).
Dynamical systems describing some types of the clusters can be solved analytically; these are the exactly solvable models.
These theoretical results can be used directly for describing real-life physical phenomena (e.g. intraseasonal oscillations in the Earth's atmosphere) or various wave turbulent regimes in the theory of wave turbulence. Many more examples are provided in the article on resonant interactions.
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
Nonlinear frequency response functionsEdit
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.
^Kartashova, E. (2010), Nonlinear Resonance Analysis, Cambridge University Press, ISBN 978-0-521-76360-8
^Janssen, P. A. E. M. (2009). "On some consequences of the canonical transformation in the hamiltonian theory of water waves". J. Fluid Mech. 637: 1–44. Bibcode:2009JFM...637....1J. doi:10.1017/S0022112009008131. S2CID 122752276.
^Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013