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On the critical values of parametric resonance in Meissner’s equation by the method of difference equations. (English) Zbl 1197.70013
Summary: The second-order linear differential equation $$\left\{\aligned &x''+a^2 (t)x=0,\\ &a(t)= \cases \pi+\varepsilon, &\text{if }2nT\le t<2nT+ T_1,\\ \pi-\varepsilon, &\text{if }2nT+T_1\le t<2nT+T_1+T_2, \endcases\quad (n=0,1,2,\ldots), \endaligned\right.$$ 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 $T_1\ne T_2$.

70J40Parametric resonances (linear vibrations)
39A60Applications of difference equations
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