On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in 
        
          
            
              R
              3

Han, Maoan; Llibre, Jaume; Tian, Yun

doi:10.3390/math8071137

Open AccessArticle

On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3

by

Maoan Han

¹,

Jaume Llibre

^2,*

and

Yun Tian

¹

Department of Mathematics, Shanghai Normal University, 100 Guilin Road, Xuhui District, Shanghai 200234, China

²

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

^*

Author to whom correspondence should be addressed.

Mathematics 2020, 8(7), 1137; https://doi.org/10.3390/math8071137

Submission received: 5 May 2020 / Revised: 30 June 2020 / Accepted: 5 July 2020 / Published: 12 July 2020

(This article belongs to the Special Issue Modelling and Analysis in Biomathematics)

Download Versions Notes

Abstract

Here we study 3-dimensional Lotka–Volterra systems. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. We remark that these systems with such six periodic orbits are non-competitive Lotka–Volterra systems. The proof is done using the algorithm that we provide for computing the periodic solutions that bifurcate from a zero-Hopf equilibrium based in the averaging theory of third order. This algorithm can be applied to any differential system having a zero-Hopf equilibrium.

Keywords: Lotka–Volterra differential systems; periodic orbit; Hopf bifurcation; averaging theory

Share and Cite

MDPI and ACS Style

Han, M.; Llibre, J.; Tian, Y. On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3 . Mathematics 2020, 8, 1137. https://doi.org/10.3390/math8071137

AMA Style

Han M, Llibre J, Tian Y. On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3 . Mathematics. 2020; 8(7):1137. https://doi.org/10.3390/math8071137

Chicago/Turabian Style

Han, Maoan, Jaume Llibre, and Yun Tian. 2020. "On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3 " Mathematics 8, no. 7: 1137. https://doi.org/10.3390/math8071137

APA Style

Han, M., Llibre, J., & Tian, Y. (2020). On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3 . Mathematics, 8(7), 1137. https://doi.org/10.3390/math8071137

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI