Simple conditions for parametrically excited oscillations of generalized Mathieu equations

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• 作者：Kazuki Ishibashi up>ishibashi_kazuaoi@yahoo.co.jp" class="auth_mail" title="E-mail the corresponding authorup> ; Jitsuro Sugieup> ; up>up>jsugie@riko.shimane-u.ac.jp" class="auth_mail" title="E-mail the corresponding authorup>
• 关键词：Mathieu's equation ; Oscillation problem ; Damped linear differential equations ; Variable transformation ; Parametric excitation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：233-247
• 全文大小：421 K

文摘

The following equation is considered in this paper: where α, β and γ   are real parameters and ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303249&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303249&_rdoc=1&_issn=0022247X&md5=cbdd4368b842b49e933864e14415ac9e" title="Click to view the MathML source">γ>0$\gamma >0$. This equation is referred to as Mathieu's equation when ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303249&_mathId=si3.gif&_user=111111111&_pii=S0022247X16303249&_rdoc=1&_issn=0022247X&md5=b89fe6caa5ef12ab0463c55b890d7cc3" title="Click to view the MathML source">γ=2$\gamma =2$. The parameters determine whether all solutions of this equation are oscillatory or nonoscillatory. Our results provide parametric conditions for oscillation and nonoscillation; there is a feature in which it is very easy to check whether these conditions are satisfied or not. Parametric oscillation and nonoscillation regions are drawn to help understand the obtained results.