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Difference between revisions of "Monodromy matrix"

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A constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064670/m0646701.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064670/m0646702.png" /> which is the value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064670/m0646703.png" /> of the [[Fundamental matrix|fundamental matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064670/m0646704.png" />, normalized at zero, of a linear system of differential equations
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A constant $(n\times n)$-matrix $X(\omega)$ which is the value at $t=\omega$ of the [[Fundamental matrix|fundamental matrix]] $X(t)$, normalized at zero, of a linear system of differential equations
  
<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/m/m064/m064670/m0646705.png" /></td> </tr></table>
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$$\dot x=A(t)x,\quad t\in\mathbf R,\quad x\in\mathbf R^n,$$
  
with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064670/m0646706.png" />-periodic matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064670/m0646707.png" /> that is summable on each compact interval in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064670/m0646708.png" />.
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with an $\omega$-periodic matrix $A(t)$ that is summable on each compact interval in $\mathbf R$.
  
  

Latest revision as of 17:01, 12 August 2014

A constant $(n\times n)$-matrix $X(\omega)$ which is the value at $t=\omega$ of the fundamental matrix $X(t)$, normalized at zero, of a linear system of differential equations

$$\dot x=A(t)x,\quad t\in\mathbf R,\quad x\in\mathbf R^n,$$

with an $\omega$-periodic matrix $A(t)$ that is summable on each compact interval in $\mathbf R$.


Comments

References

[a1] J.K. Hale, "Ordinary differential equations" , Wiley (1969)
How to Cite This Entry:
Monodromy matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monodromy_matrix&oldid=32876
This article was adapted from an original article by Yu.V. Komlenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article