# Nonlinear resonance > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Nonlinear_resonance > Markdown URL: https://mediated.wiki/source/Nonlinear_resonance.md > Source: https://en.wikipedia.org/wiki/Nonlinear_resonance > Source revision: 1102929944 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Physical phenomenon}}In [physics](/source/physics), '''nonlinear resonance''' is the occurrence of [resonance](/source/resonance) in a [nonlinear system](/source/nonlinear_system). In nonlinear resonance the system behaviour – [resonance frequencies](/source/resonance_frequency) and [modes](/source/normal_mode) – depends on the [amplitude](/source/amplitude) of the [oscillation](/source/oscillation)s, while for [linear system](/source/linear_system)s this is independent of amplitude. The mixing of modes in non-linear systems is termed [resonant interaction](/source/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](/source/force) coincides with the [eigen-frequency](/source/natural_frequency) of the system (linear and nonlinear resonance correspondingly). Vibrational modes can interact in a [resonant interaction](/source/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: : \omega_n=\omega_{1}+ \omega_{2}+ \cdots + \omega_{n-1}, with possibly different \omega_i=\omega(\mathbf{k}_i), being eigen-frequencies of the linear part of some nonlinear [partial differential equation](/source/partial_differential_equation). The \mathbf{k}_i is the [wave vector](/source/wave_vector) associated with a mode; the integer subscripts i being indexes into Fourier harmonics – or [eigenmode](/source/eigenmode)s – see [Fourier series](/source/Fourier_series). Accordingly, the frequency resonance condition is equivalent to a [Diophantine equation](/source/Diophantine_equation) with many unknowns. The problem of finding their solutions is equivalent to the [Hilbert's tenth problem](/source/Hilbert's_tenth_problem) that is proven to be algorithmically unsolvable. Main notions and results of the theory of nonlinear resonances are:{{Citation | last=Kartashova | first=E. | year=2010 | title=Nonlinear Resonance Analysis | publisher=Cambridge University Press | isbn=978-0-521-76360-8 }} # The use of [dispersion relation](/source/dispersion_relation)s \omega=\omega(\mathbf{k}) 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](/source/soliton) of the [KdV equation](/source/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](/source/resonant_interaction)), they are "bound" to the fundamental.{{cite journal |first=P. A. E. M. |last=Janssen |year=2009 |title=On some consequences of the canonical transformation in the hamiltonian theory of water waves |journal=J. Fluid Mech. |volume=637 |pages=1–44 |doi=10.1017/S0022112009008131 |bibcode=2009JFM...637....1J |s2cid=122752276 }} # Each collection of bound modes (resonance cluster) can be represented by its [NR-diagram](/source/NR-diagram) which is a plane graph of the special structure. This representation allows to reconstruct uniquely 3a) [dynamical system](/source/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](/source/Manley%E2%80%93Rowe_relations) for the simplest clusters ([triads](/source/three-wave_equation) and quartets). # Dynamical systems describing some types of the clusters can be solved analytically; these are the [exactly solvable model](/source/exactly_solvable_model)s. # 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](/source/wave_turbulence). Many more examples are provided in the article on [resonant interaction](/source/resonant_interaction)s. ==Nonlinear resonance shift== thumb|Foldover effect [Nonlinear effects](/source/Anharmonicity) may significantly modify the shape of the [resonance](/source/resonance) curves of [harmonic oscillator](/source/harmonic_oscillator)s. First of all, the resonance frequency \omega is shifted from its "natural" value \omega_0 according to the formula :\omega=\omega_0+\kappa A^2, 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 F_\mathrm{crit} instabilities appear. The critical value is given by the formula :F_\mathrm{crit}=\frac{4 m^2\omega_0^2\gamma^3}{3\sqrt{3}\kappa}, 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 Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013 allow the user to study complex nonlinear behaviors in the frequency domain in a principled way. These functions reveal resonance ridges, [harmonic](/source/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. ==See also== * [Duffing equation](/source/Duffing_equation) * [Wave turbulence](/source/Wave_turbulence) ==Notes and references== ===Notes=== {{Reflist}} ===References=== * {{Citation | last1=Landau | first1=L. D. | author-link1=Lev Landau | last2=Lifshitz | first2=E. M. | author-link2=Evgeny Lifshitz | year=1976 | title=Mechanics | edition=3rd | publisher=Pergamon Press | isbn=0-08-021022-8 | id=(hardcover).and (softcover) | url-access=registration | url=https://archive.org/details/mechanics00land }} * {{Citation | last1=Rajasekar | first1=S. | last2=Sanjuan | first2=M. A. F. | author-link2=Miguel A. F. Sanjuan | year=2016 | title=Nonlinear Resonances | edition=1st | publisher= Springer |isbn=978-3-319-24886-8 |id=(ebook)}} ==External links== * {{Citation | last=Elmer | first=Franz-Josef | url=https://elmer.unibas.ch/pendulum/nonres.htm | title=Nonlinear Resonance | publisher=University of Basel | date=July 20, 1998 | access-date=27 October 2010 }} Category:Mechanical vibrations --- Adapted from the Wikipedia article [Nonlinear resonance](https://en.wikipedia.org/wiki/Nonlinear_resonance) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Nonlinear_resonance?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.