# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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\{\begin{array}{cc}& {x}^{\text{'}\text{'}}+{a}^{2}\left(t\right)x=0,\hfill \\ & a\left(t\right)=\left\{\begin{array}{cc}\pi +\epsilon ,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}2nT\le t<2nT+{T}_{1},\hfill \\ \pi -\epsilon ,\hfill & \text{if}\phantom{\rule{4.pt}{0ex}}2nT+{T}_{1}\le t<2nT+{T}_{1}+{T}_{2},\hfill \end{array}\right\\phantom{\rule{1.em}{0ex}}\left(n=0,1,2,...\right),\hfill \end{array}\right\$

is investigated, where ${T}_{1}>0$, ${T}_{2}>0$ $\left(T:=\left({T}_{1}+{T}_{2}\right)/2\right)$ and $\epsilon \in \left[0,\pi \right)$. We say that a parametric resonance occurs in this equation, if for every $\epsilon >0$ sufficiently small there are ${T}_{1}\left(\epsilon \right),{T}_{2}\left(\epsilon \right)$ such that the equation has solutions with amplitudes tending to $\infty$, as $t\to \infty$. The period $2{T}_{*}$ of the parametric excitation is called a critical value of the parametric resonance if ${T}_{*}={T}_{1}\left(\epsilon \right)+{T}_{2}\left(\epsilon \right)$ with some ${T}_{1},{T}_{2}$ for all sufficiently small $\epsilon >0$. 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 $x=0,{x}^{\text{'}}=0$, and to give a sufficient condition for the parametric resonance in the asymmetric case ${T}_{1}\ne {T}_{2}$.

##### MSC:
 70J40 Parametric resonances (linear vibrations) 39A60 Applications of difference equations