Resonance terms

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Those terms in the Taylor–Fourier series


whose indicators and satisfy a linear relation as follows:


Here are constant coefficients, is the scalar product of and ; the constants and are usually the eigenvalues and the basis of frequencies of a specific system of ordinary differential equations; the constant is independent of and and it is defined by the role of the series (1) in the problem under analysis.

If in a linear system


all are purely imaginary and in (2) , then the total resonance term of the series (1) coincides with the average of this series along the solutions of the system (3). A system of ordinary differential equations in a neighbourhood of invariant manifolds can be reduced to a normal form in which the series contains only resonance terms (see [1]). Thus, for a Hamiltonian system in a neighbourhood of a fixed point, the Hamiltonian function is reducible to the form (1) where and (2) is fulfilled with , whence is the vector of eigenvalues of the linearized system (see ). In this case, the terms , , are sometimes called secular (for them (2) is fulfilled trivially), and the remaining terms of the series (1) for which (2) is fulfilled are called the resonance terms.

The separation of resonance terms, derived in problems with a small parameter, can often be based on a normal form (see [1]). For a point transformation with multipliers the indices of the resonance terms of the series (1) with satisfy the relation ; if one assumes that and , then (2) is obtained with .


[1] A.D. [A.D. Bryuno] Bruno, "Local methods in nonlinear differential equations" , Springer (1978) (Translated from Russian)
[2a] A.D. [A.D. Bryuno] Bruno, "The analytic form of differential equations" Trans. Moscow Math. Soc. , 25 (1971) pp. 131–288 Trudy Moskov. Mat. Obshch. , 25 (1971) pp. 119–262
[2b] A.D. [A.D. Bryuno] Bruno, "The analytic form of differential equations" Trans. Moscow Math. Soc. , 26 (1972) pp. 199–238 Trudy Moskov. Mat. Obshch. , 26 (1972) pp. 199–239



[a1] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a2] V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian)
[a3] A. Avez, "Ergodic problems of classical mechanics" , Benjamin (1968) (Translated from Russian)
How to Cite This Entry:
Resonance terms. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.D. Bryuno (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article