{"title":"Saturation recovery leads to cusp type Bogdanov–Takens bifurcations of codimensions 3","authors":"","doi":"10.1016/j.aml.2024.109282","DOIUrl":null,"url":null,"abstract":"<div><p>We reconsider an SIS epidemic model studied by Cui et al. <span><span>[1]</span></span>. The model is shown that saturation recovery leads to backward bifurcation, Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. However, for the case when the Bogdanov–Takens bifurcation of codimension 2 is degenerate, the types and codimensions of Bogdanov–Takens bifurcation have not been investigated. In this paper we prove that this same model can undergo cusp type Bogdanov–Takens bifurcations of codimensions 3. Hence, more complex new phenomena, including degenerate Hopf bifurcation, degenerate homoclinic bifurcation and saddle–node bifurcation of limit cycles, exhibit. Furthermore, we get the bifurcation diagram of codimension 3 Bogdanov–Takens bifurcation with cusp type of the SIS epidemic model.</p></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":2.9000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0893965924003021","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We reconsider an SIS epidemic model studied by Cui et al. [1]. The model is shown that saturation recovery leads to backward bifurcation, Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. However, for the case when the Bogdanov–Takens bifurcation of codimension 2 is degenerate, the types and codimensions of Bogdanov–Takens bifurcation have not been investigated. In this paper we prove that this same model can undergo cusp type Bogdanov–Takens bifurcations of codimensions 3. Hence, more complex new phenomena, including degenerate Hopf bifurcation, degenerate homoclinic bifurcation and saddle–node bifurcation of limit cycles, exhibit. Furthermore, we get the bifurcation diagram of codimension 3 Bogdanov–Takens bifurcation with cusp type of the SIS epidemic model.
期刊介绍:
The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.