Lyapunov coefficients for degenerate Hopf bifurcations and an application in a model of competing populations

This paper focuses on the study of codimension one and two Hopf bifurcations and the pertinent Lyapunov stability coefficients for a general reaction–diffusion system. We display algebraic expressions for the first and second Lyapunov coefficients for the infinite dimensional system subject to Neuma...

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Bibliographic Details
Published in:	Journal of mathematical analysis and applications Vol. 455; no. 1; pp. 1 - 51
Main Authors:	Ferreira, Jocirei D., Nieva, Aida P. González, Yepez, Wilmer Molina
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01-11-2017
Subjects:	Degenerate Hopf bifurcation Lyapunov coefficients Predator–prey system Reaction diffusion system Spatially homogeneous bifurcation Spatially non-homogeneous bifurcation Spatially non-homogeneous bifurcation Predator–prey system Reaction diffusion system Spatially homogeneous bifurcation Lyapunov coefficients Degenerate Hopf bifurcation
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Summary:	This paper focuses on the study of codimension one and two Hopf bifurcations and the pertinent Lyapunov stability coefficients for a general reaction–diffusion system. We display algebraic expressions for the first and second Lyapunov coefficients for the infinite dimensional system subject to Neumann boundary conditions. As an application, a special subset of a three-dimensional Lotka–Volterra dynamical system with diffusions subject to Neumann boundary conditions is analyzed. The main goal is to perform a detailed local stability analysis for the proposed predator–prey model to show the existence of multiple spatially homogeneous and non-homogeneous periodic orbits, due to the occurrence of codimension one Hopf bifurcation.
ISSN:	0022-247X 1096-0813
DOI:	10.1016/j.jmaa.2017.05.040