{"title":"从数学和物理学角度看各向同性星体差分系统的动力学特性","authors":"Joan C. Artés, J. Llibre, N. Vulpe","doi":"10.3390/appliedmath4010004","DOIUrl":null,"url":null,"abstract":"The following differential quadratic polynomial differential system dxdt=y−x, dydt=2y−yγ−12−γy−5γ−4γ−1x, when the parameter γ∈(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)≥0 is the mass inside the sphere of radius r of the star, y=4πr2ρ where ρ is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ∈R∖{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4πr2ρ of an isotropic star when r decreases.","PeriodicalId":503400,"journal":{"name":"AppliedMath","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Dynamics of the Isotropic Star Differential System from the Mathematical and Physical Point of Views\",\"authors\":\"Joan C. Artés, J. Llibre, N. Vulpe\",\"doi\":\"10.3390/appliedmath4010004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The following differential quadratic polynomial differential system dxdt=y−x, dydt=2y−yγ−12−γy−5γ−4γ−1x, when the parameter γ∈(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)≥0 is the mass inside the sphere of radius r of the star, y=4πr2ρ where ρ is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ∈R∖{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4πr2ρ of an isotropic star when r decreases.\",\"PeriodicalId\":503400,\"journal\":{\"name\":\"AppliedMath\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-01-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"AppliedMath\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3390/appliedmath4010004\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"AppliedMath","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/appliedmath4010004","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
当参数 γ∈(1,2]时,下列微分二次多项式微分方程系 dxdt=y-x, dydt=2y-yγ-12-γy-5γ-4γ-1x 模拟了具有线性气压状态方程的各向同性恒星的结构方程、其中,x=m(r)/r,m(r)≥0 是恒星半径 r 球内的质量;y=4πr2ρ,ρ 是恒星的密度;t=ln(r/R),R 是恒星的半径。首先,我们对参数γ∈R∖{1}的所有取值下,这些二次多项式微分系统的波恩卡莱圆盘中所有拓扑非等价相位肖像进行了分类。其次,利用所获得的不同相位肖像信息,我们对各向同性恒星在 r 减小时 m(r)/r 和 4πr2ρ 的可能极限值进行了分类。
Dynamics of the Isotropic Star Differential System from the Mathematical and Physical Point of Views
The following differential quadratic polynomial differential system dxdt=y−x, dydt=2y−yγ−12−γy−5γ−4γ−1x, when the parameter γ∈(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)≥0 is the mass inside the sphere of radius r of the star, y=4πr2ρ where ρ is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ∈R∖{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4πr2ρ of an isotropic star when r decreases.