Mathieu function: Difference between revisions

Revision as of 07:37, 6 August 2023

Revision as of 08:00, 6 August 2023

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Parametrize Mathieu equation as <math>\ddot x + k(1-m \cos(t))x = 0</math>, where <math>k \in \R, m \geq 0</math>. The regions of stability and instability are separated by curves <ref>{{Cite journal |last=Butikov |first=Eugene I. |date=April 2018 |title=Analytical expressions for stability regions in the Ince–Strutt diagram of Mathieu equation |url=https://pubs.aip.org/aapt/ajp/article/86/4/257-267/1057663 |journal=American Journal of Physics |language=en |volume=86 |issue=4 |pages=257–267 |doi=10.1119/1.5021895 |bibcode=2018AmJPh..86..257B |issn=0002-9505}}</ref>

<math display=block>m(k) = \begin{alignedcases}

& m(k)= 2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, \quad \text { for } \quad& k<0 ; \\[4pt]

& m(k)= \frac{1}{4}( \left[\sqrt{(9-4 k)(13-20 k)}-(9-4 k)), \quadright], \text { for } \quad& k<\frac{1}{4} ; \\[10pt]

& m(k)= \frac{1}{4}( \left[9-4 k \mp \sqrt{(9-4 k)(13-20 k)}), \quadright], \text { for } \quad& \frac{1}{4}<k<\frac{13}{20} ; \\[6pt]

& m(k)= \sqrt{\frac{2(k-1)(k-4)(k-9)}{k-5}}, \quad \text { for } \quad& \frac{13}{20}<k<1 ; \\[2pt]

& m(k)= 2 \sqrt{\frac{k(k-1)(k-4)}{3 k-8}}, \quad \text { for } \quad& k>1 .

\end{alignedcases}</math>

== Floquet theory ==