# Mathieu Equation

Discussion in 'Boat Design' started by netjaws, Dec 1, 2005.

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### netjawsJunior Member

Trying to investigate parametric roll, and don't actually need to SOLVE the Mathieu equation - just need to know whether the solution will be bounded or unbounded.

Any helps?

Joshua N Straume

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### GuillermoIngeniero Naval

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### netjawsJunior Member

p and q?

Interestingly, that first article is what posed my question!

That is - how do we find whether the solution is bounded or unbounded, without actuallly solving it?

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### GuillermoIngeniero Naval

Good question. I don't know.

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### tspeerSenior Member

I think Guillermo has put you on the right road - the Christian paper says the imaginary part of the solution's exponent has to be zero for a bounded solution, and the boundary curves shown on the Ince-Strutt diagram _are_ the conditions that separate the bounded from the unbounded solutions.

You might try looking for a Lyapunov function for the problem and finding the bounds on that. Zounes and Rand use the function L(phi) = p*phi and present a number of ways of finding the stability contours. Zhang and Li use the same Lyapunov function, too, so I suspect it's the way to go.

It looks like a really hairy problem!

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