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Those terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081630/r0816301.png" /> in the Taylor–Fourier series
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<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>
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{{TEX|auto}}
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{{TEX|done}}
  
<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>
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Those terms  $  f _ {PQ} X  ^ {P}  \mathop{\rm exp} \{ i \langle  Q, Y \rangle \} $
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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/r0816304.png" /></td> </tr></table>
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$$ \tag{1 }
 +
f( X, Y)  = \sum f _ {PQ} X  ^ {P}  \mathop{\rm exp} \{ i \langle  Q, Y \rangle \} ,
 +
$$
  
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:
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$$
 +
P  \in  \mathbf Z  ^ {m} ,\  P  \geq  0 ,\  Q  \in  \mathbf Z  ^ {n} ,
 +
$$
  
<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>
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$$
 +
X  ^ {P}  = x _ {1} ^ {p _ {1} } \dots x _ {m} ^ {p _ {m} } ,
 +
$$
  
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.
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whose indicators  $  P $
 +
and  $  Q $
 +
satisfy a linear relation as follows:
 +
 
 +
$$ \tag{2 }
 +
\langle  P, \Lambda \rangle + i \langle  Q, \Omega \rangle  = c.
 +
$$
 +
 
 +
Here  $  f _ {PQ} $
 +
are constant coefficients, $  \langle  Q, Y\rangle $
 +
is the scalar product of $  Q $
 +
and $  Y $;  
 +
the constants $  ( \lambda _ {1} \dots \lambda _ {m} ) = \Lambda $
 +
and $  ( \omega _ {1} \dots \omega _ {n} ) = \Omega $
 +
are usually the eigenvalues and the basis of frequencies of a specific system of ordinary differential equations; the constant $  c $
 +
is independent of $  P $
 +
and $  Q $
 +
and it is defined by the role of the series (1) in the problem under analysis.
  
 
If in a linear system
 
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>
+
$$ \tag{3 }
 +
\dot{x} _ {j}  = \lambda _ {j} x _ {j} ,\ \
 +
j = 1 \dots m,\ \
 +
\dot{y} _ {k}  = \omega _ {k} ,\ \
 +
k = 1 \dots n,
 +
$$
  
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.
+
all $  \lambda _ {j} $
 +
are purely imaginary and in (2) $  c= 0 $,  
 +
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 $  n= 0 $
 +
and (2) is fulfilled with $  c= 0 $,  
 +
whence $  \Lambda = ( \lambda _ {1} \dots \lambda _ {l} , - \lambda _ {1} \dots - \lambda _ {l} ) $
 +
is the vector of eigenvalues of the linearized system (see ). In this case, the terms $  p _ {j} = p _ {j+} l $,  
 +
$  j = 1 \dots l $,  
 +
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" />.
+
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 $  ( \mu _ {1} \dots \mu _ {m} ) = M $
 +
the indices of the resonance terms of the series (1) with $  n= 0 $
 +
satisfy the relation $  M  ^ {P} = 1 $;  
 +
if one assumes that $  \Lambda = \mathop{\rm ln}  M $
 +
and $  \omega _ {1} = 1 $,  
 +
then (2) is obtained with $  c= 0 $.
  
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:11, 6 June 2020


Those terms $ f _ {PQ} X ^ {P} \mathop{\rm exp} \{ i \langle Q, Y \rangle \} $ in the Taylor–Fourier series

$$ \tag{1 } f( X, Y) = \sum f _ {PQ} X ^ {P} \mathop{\rm exp} \{ i \langle Q, Y \rangle \} , $$

$$ P \in \mathbf Z ^ {m} ,\ P \geq 0 ,\ Q \in \mathbf Z ^ {n} , $$

$$ X ^ {P} = x _ {1} ^ {p _ {1} } \dots x _ {m} ^ {p _ {m} } , $$

whose indicators $ P $ and $ Q $ satisfy a linear relation as follows:

$$ \tag{2 } \langle P, \Lambda \rangle + i \langle Q, \Omega \rangle = c. $$

Here $ f _ {PQ} $ are constant coefficients, $ \langle Q, Y\rangle $ is the scalar product of $ Q $ and $ Y $; the constants $ ( \lambda _ {1} \dots \lambda _ {m} ) = \Lambda $ and $ ( \omega _ {1} \dots \omega _ {n} ) = \Omega $ are usually the eigenvalues and the basis of frequencies of a specific system of ordinary differential equations; the constant $ c $ is independent of $ P $ and $ Q $ and it is defined by the role of the series (1) in the problem under analysis.

If in a linear system

$$ \tag{3 } \dot{x} _ {j} = \lambda _ {j} x _ {j} ,\ \ j = 1 \dots m,\ \ \dot{y} _ {k} = \omega _ {k} ,\ \ k = 1 \dots n, $$

all $ \lambda _ {j} $ are purely imaginary and in (2) $ c= 0 $, 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 $ n= 0 $ and (2) is fulfilled with $ c= 0 $, whence $ \Lambda = ( \lambda _ {1} \dots \lambda _ {l} , - \lambda _ {1} \dots - \lambda _ {l} ) $ is the vector of eigenvalues of the linearized system (see ). In this case, the terms $ p _ {j} = p _ {j+} l $, $ j = 1 \dots l $, 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 $ ( \mu _ {1} \dots \mu _ {m} ) = M $ the indices of the resonance terms of the series (1) with $ n= 0 $ satisfy the relation $ M ^ {P} = 1 $; if one assumes that $ \Lambda = \mathop{\rm ln} M $ and $ \omega _ {1} = 1 $, then (2) is obtained with $ c= 0 $.

References

[1] A.D. [A.D. Bryuno] Bruno, "Local methods in nonlinear differential equations" , Springer (1978) (Translated from Russian) MR0993771 Zbl 0674.34002
[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) Zbl 0692.70003 Zbl 0572.70001 Zbl 0647.70001
[a2] V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian) 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
[a3] A. Avez, "Ergodic problems of classical mechanics" , Benjamin (1968) (Translated from Russian) MR0232910 Zbl 0167.22901
How to Cite This Entry:
Resonance terms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resonance_terms&oldid=24553
This article was adapted from an original article by A.D. Bryuno (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article