Discrete Dynamics in Nature and Society (2020-01-01)

Bifurcation Behaviors of Steady-State Solution to a Discrete General Brusselator Model

  • Ruyun Ma,
  • Zhongzi Zhao

DOI
https://doi.org/10.1155/2020/5417218
Journal volume & issue
Vol. 2020

Abstract

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We study the local and global bifurcation of nonnegative nonconstant solutions of a discrete general Brusselator model. We generalize the linear u in the standard Brusselator model to the nonlinear fu. Assume that f∈C0,∞,0,∞ is a strictly increasing function, and f′f−1a∈0,∞. Taking b as the bifurcation parameter, we obtain that the solution set of the problem constitutes a constant solution curve and a nonconstant solution curve in a small neighborhood of the bifurcation point b0j,f−1a,ab/f−1a2. Moreover, via the Rabinowitz bifurcation theorem, we obtain the global structure of the set of nonconstant solutions under the condition that fs/s2 is nonincreasing in 0,∞. In this process, we also make a priori estimation for the nonnegative nonconstant solutions of the problem.