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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Sufficient Condition of the Bifurcation for a Dynamic System of Three Equations and Application

Journal of Applied Nonlinear Dynamics 12(1) (2023) 99--112 | DOI:10.5890/JAND.2023.03.007

Amine Bernoussi

Laboratory: '{E}quations aux d'{e}riv'{e}es partielles, Alg`{e}bre et G'{e}om'{e}trie spectrales, Faculty of Science, Ibn Tofail University, BP 133, 14000 Kenitra, Morocco

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Abstract

In this paper we propose a new sufficient condition of the bifurcation for nonlinear autonomous differential equations with delay in three dimension, which just depends in coefficient on the characteristic equation, who can verify them analytically at the level of the application to a dynamics system. For the application of these conditions we propose a SEIR epidemic model with delay and nonlinear incidence rate. The resulting model has two possible equilibria. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of a disease-free equilibrium is established. Finally we affirm the existence of non constant periodic solutions which bifurcate from the endemic equilibrium when the delay crosses some critical values. In the end, the numerical simulations are proposed to illustrate our results.

References

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