The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system | Zendy

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The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

Author(s) -

Xin Wei,

Junjie Wei

Publication year - 2018

Publication title -

nonlinear analysis modelling and control

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.734

H-Index - 32

eISSN - 2335-8963

pISSN - 1392-5113

DOI - 10.15388/na.2018.5.7

Subject(s) - center manifold , hopf bifurcation , mathematics , autocatalysis , bifurcation , stability (learning theory) , neumann boundary condition , eigenvalues and eigenvectors , boundary (topology) , instability , control theory (sociology) , reaction–diffusion system , mathematical analysis , nonlinear system , physics , mechanics , classical mechanics , computer science , kinetics , control (management) , quantum mechanics , machine learning , artificial intelligence

This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions.

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