View source for Resonance terms

Those terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816301.png" /> in the Taylor–Fourier series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816303.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816304.png" /></td> </tr></table>

whose indicators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816306.png" /> satisfy a linear relation as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816307.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816308.png" /> are constant coefficients, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816309.png" /> is the scalar product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163011.png" />; the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163013.png" /> are usually the eigenvalues and the basis of frequencies of a specific system of ordinary differential equations; the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163014.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163016.png" /> and it is defined by the role of the series (1) in the problem under analysis.

If in a linear system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163018.png" /> are purely imaginary and in (2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163019.png" />, 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|normal form]] in which the series contains only resonance terms (see [[#References|[1]]]). Thus, for a [[Hamiltonian system|Hamiltonian system]] in a neighbourhood of a fixed point, the Hamiltonian function is reducible to the form (1) where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163020.png" /> and (2) is fulfilled with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163021.png" />, whence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163022.png" /> is the vector of eigenvalues of the linearized system (see ). In this case, the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163024.png" />, 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 [[#References|[1]]]). For a point transformation with multipliers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163025.png" /> the indices of the resonance terms of the series (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163026.png" /> satisfy the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163027.png" />; if one assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163029.png" />, then (2) is obtained with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r08163030.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.D. [A.D. Bryuno] Bruno, "Local methods in nonlinear differential equations" , Springer (1978) (Translated from Russian) {{MR|0993771}} {{ZBL|0674.34002}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> 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</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) {{MR|}} {{ZBL|0692.70003}} {{ZBL|0572.70001}} {{ZBL|0647.70001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian) {{MR|}} {{ZBL|1049.34001}} {{ZBL|0744.34001}} {{ZBL|0659.58012}} {{ZBL|0602.58020}} {{ZBL|0577.34001}} {{ZBL|0956.34502}} {{ZBL|0956.34501}} {{ZBL|0956.34503}} {{ZBL|0237.34008}} {{ZBL|0135.42601}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Avez, "Ergodic problems of classical mechanics" , Benjamin (1968) (Translated from Russian) {{MR|0232910}} {{ZBL|0167.22901}} </TD></TR></table>