Periodic motion in perturbed elliptic oscillators revisited
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We analytically study the Hamiltonian system in \(\mathbb{R}^{4}\) with Hamiltonian $$\begin{aligned} H= \frac{1}{2} \bigl(p_{x}^{2}+p_{y}^{2} \bigr)+\frac{1}{2} \bigl(\omega_{1}^{2} x ^{2}+\omega_{2}^{2} y^{2} \bigr)- \varepsilon V(x,y) \end{aligned}$$ being \(V(x,y)=-(x^{2}y+ax^{3})\) with \(a\in\mathbb{R}\), where \(\varepsilon\) is a small parameter and \(\omega_{1}\) and \(\omega_{2}\) are the unperturbed frequencies of the oscillations along the \(x\) and \(y\) axis, respectively. Using averaging theory of first and second order we analytically find seven families of periodic solutions in every positive energy level of \(H\) when the frequencies are not equal. Four of these seven families are defined for all \(a\in\mathbb{R}\) whereas the other three are defined for all \(a\ne0\). Moreover, we provide the shape of all these families of periodic solutions. 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Nonlinear Dyn. 70, 951 (2012b)MathSciNetCrossRefGoogle ScholarCopyright information© Springer Science+Business Media Dordrecht 2016Authors and AffiliationsM. Corbera1J. Llibre2Email authorC. Valls31.Escola Politècnica SuperiorUniversitat de Vic—Universitat Central de Catalunya (UVic-UCC)VicSpain2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain3.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal About this article CrossMark Print ISSN 0004-640X Online ISSN 1572-946X Publisher Name Springer Netherlands About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s10509-016-2927-5_Periodic motion in perturbed ellip", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s10509-016-2927-5_Periodic motion in perturbed ellip", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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