Resonance terms
Those terms in the Taylor–Fourier series
![]() | (1) |
![]() |
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whose indicators and
satisfy a linear relation as follows:
![]() | (2) |
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
![]() | (3) |
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
.
References
[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 |
Comments
References
[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) |
Resonance terms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resonance_terms&oldid=24553