On the critical values of parametric resonance in Meissner's equation by the method of difference equations

L. Hatvani, Bolyai Institute, University of Szeged, Hungary

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 13., pp. 1-10.

Abstract: In this paper the second order liner differential equation
\begin{equation*}
\left\{\begin{array}{l}
x'' + a^2 (t) x=0,\\
a(t) = \left\{\begin{array}{ll} \pi+\varepsilon, &\textrm{if\ $2nT\le
t<2nT+T_1,$}\\
\pi-\varepsilon, &\textrm{if\ $2nT+T_1\le t<2nT+T_1+T_2,\quad
(n=0,1,2,\ldots),$}
\end{array}\right.
\end{array}\right.
\end{equation*}
is investigated, where $T_1>0$, $T_2>0$ ($T:=(T_1+T_2)/2$) and $\varepsilon \in [0,\pi)$. We say that a parametric resonance occurs in this equation if for every $\varepsilon >0$ sufficiently small there are $T_1(\varepsilon)$, $T_2(\varepsilon)$ such that the equation has solutions with amplitudes tending to $\infty$, as $t\to\infty$. The period $2T_*$ of the parametric excitation is called a critical value of the parametric resonance if $T_*=T_1(\varepsilon)+T_2(\varepsilon)$ with some $T_1$, $T_2$ for all sufficiently small $\varepsilon>0$. We give a new simple geometric proof for the fact that the critical values are the natural numbers. We apply our method also to find the most effective control destabilizing the equilibrium $x=0$, $x'=0$, and to give a sufficient condition for the parametric resonance in the asymmetric case $T_1\ne T_2$.

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