{"title":"The Darboux Polynomials and Integrability of Polynomial Levinson-Smith Differential Equations","authors":"M. Demina","doi":"10.1142/s0218127423500359","DOIUrl":null,"url":null,"abstract":"We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.","PeriodicalId":13688,"journal":{"name":"Int. J. Bifurc. Chaos","volume":"36 1","pages":"2350035:1-2350035:16"},"PeriodicalIF":0.0000,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Int. J. Bifurc. Chaos","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0218127423500359","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.
给出了非退化近无穷多项式Levinson-Smith微分方程Liouvillian可积性的充分必要条件。这些方程推广了lisamadard方程,并用于描述自持续振荡。我们的结果对方程中出现的多项式的任意次都是有效的。我们发现了一些新的Liouvillian可积亚族。在非简并或代数简并的近无穷多项式Levinson-Smith方程中,我们推导了不可约达布多项式阶上的一个变量的上界。对非简并或代数简并的近无穷Rayleigh-Duffing-van der Pol方程,即三次Levinson-Smith方程,进行了Liouvillian第一积分的完全分类。