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  Mathieu equation for case of negative instability

+ 3 like - 0 dislike

Let's assume Mathieu equation:
\tag 1 \frac{d^{2}y(t)}{dt^{2}} + \omega^{2}(t)y(t) = 0, \quad \omega^{2}(t) = A(t) - 2q(t)cos(2t)

Here $A(t), q(t)$ are slowly decreased with time functions (i.e., $|\dot{A}(t)| <A(t), |\dot{q}(t)| < q(t)$), and in initial moment of time $A(t_{0}) < 2q(t_{0})$. Eq. $(1)$ describes, for example, magnetic field evolution in presence of axion field and preheating in an expanding Universe.

I look for exponentially growing solutions of Eq. $(1)$; more precisely, I need to determine the Floquet exponent $\mu$, $y(t) \sim e^{\int \limits_{}^{t} \mu (t)dt}$. If I temporary assume that $q(t), A(t)$ are constans, then $(1)$ is reduced to Mathieu equation for case $A - 2q < 0$, called Mathieu equation with negative instability. Unfortunately, I haven't found an expression for Floquet exponent for this case.

Can someone give the reference in which this case is treated? I note that it is different from case which is described in famous article by Kofman, Linde and Starobinsky, where always $A - 2q > 0$.

asked Sep 19, 2015 in Mathematics by NAME_XXX (1,060 points) [ revision history ]

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