Nanasaheb Phatangare, Krishnat D. Masalkar, S. Kendre
{"title":"具有边界摄动的分段光滑哈密顿系统极限环的分岔","authors":"Nanasaheb Phatangare, Krishnat D. Masalkar, S. Kendre","doi":"10.7153/dea-2022-14-34","DOIUrl":null,"url":null,"abstract":"In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\\dot{X}=\\begin{cases} (H_y^+,-H_x^+) & \\mbox{if}~y>\\varepsilon f(x)\\\\ (H_y^-,-H_x^-) & \\mbox{if}~y<\\varepsilon f(x) \\end{cases}$ equals to the number of positive zeros of $f$ when at $\\varepsilon=0$ the system has a period annulus around the origin.","PeriodicalId":179999,"journal":{"name":"Differential Equations & Applications","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2019-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation\",\"authors\":\"Nanasaheb Phatangare, Krishnat D. Masalkar, S. Kendre\",\"doi\":\"10.7153/dea-2022-14-34\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\\\\dot{X}=\\\\begin{cases} (H_y^+,-H_x^+) & \\\\mbox{if}~y>\\\\varepsilon f(x)\\\\\\\\ (H_y^-,-H_x^-) & \\\\mbox{if}~y<\\\\varepsilon f(x) \\\\end{cases}$ equals to the number of positive zeros of $f$ when at $\\\\varepsilon=0$ the system has a period annulus around the origin.\",\"PeriodicalId\":179999,\"journal\":{\"name\":\"Differential Equations & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-12-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Equations & Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7153/dea-2022-14-34\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations & Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7153/dea-2022-14-34","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation
In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\dot{X}=\begin{cases} (H_y^+,-H_x^+) & \mbox{if}~y>\varepsilon f(x)\\ (H_y^-,-H_x^-) & \mbox{if}~y<\varepsilon f(x) \end{cases}$ equals to the number of positive zeros of $f$ when at $\varepsilon=0$ the system has a period annulus around the origin.