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I am using the Mathieu equation $$x''(t) + a x(t) + 2 q \cos(2 t) x(t) = 0,$$ as a model of a physical system. The conditions on the parameters $a$ and $q$ for stable solutions to exist are well known and are even implemented as built-in functions in systems such as Mathematica.
Now, I would like to extend my model to the form $$x''(t) + c x^2(t)+ a x(t) + 2 q \cos(2 t) x(t) = 0,$$ and establish the stability criteria for this modified model.
Is this a known model, or can it be transformed to one?
Does it even make sense to talk about stability conditions for this case -- as far as I can see Floquet's theorem does not apply?

Although the most general form would be most interesting, it would also be interesting to know the stability criteria for the less general case of $a=0$.

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I am not sure of an answer, but note that your new model is a non-linear ODE. In general (unless there exists a nice algebraic transform that maps it to a linear ODE) the stability analysis for non-linear ODEs depends not only on the coefficients of the equation, but also the initial values prescribed.

As an example, for initial data with $|x|$ very small, your equation is a small perturbation of the Mathieu equation, and one might expect behaviour similar to the Mathieu equation. But for initial data with $|x|$ very large, your equation behaves more like $x''(t) = -cx^2$, then in the case where $x < 0$ and $x' < 0$ initially with large $|x|$, the solution behaves like $(t_0 - t)^{-2}$ and blows up in finite time.

To put it in other words, I am not sure if the notion of stability analysis is even useful. For a linear ODE, the characteristic behaviours are, roughly speaking, limited to exponential decay, exponential blowup, and oscillation (corresponding roughly to the $t\to\infty$ behaviour of $x'' = \pm x$ with increasing or decreasing initial data). For a non-linear ODE, vastly different behaviour can manifest.

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Thanks. I thought I remembered something along these lines -- thus my question about whether it even makes sense to talk about stability in this case. Good point with the asymptotic behavior. For my immediate needs, I expect to get by by numerically finding the basin of attraction for the origin for a specific set of parameters. – Janus Jan 26 '11 at 1:58

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