Controlling chaos and codimension-two bifurcation in a discrete fractional-order Brusselator model

Abstract

This paper explores the qualitative behavior of a discrete fractional–order Brusselator model. We analyze the local dynamics of the model around its fixed point and determine its topological classification. We perform the bifurcation analysis for both codimension-one and codimension-two cases to examine the system behavior near critical parameter values. Using normal form theory and center manifold theorem (CMT), we prove that the model exhibits period-doubling bifurcation around its interior fixed point. We also study the existence and direction of Neimark–Sacker bifurcation using normal form theory. For codimension-two bifurcation, we show that the model undergoes 1:2, 1:3, and 1:4 resonances by applying normal form theory and suitable affine transformations. The system displays a rich variety of bifurcations, including quasi–periodicity, periodic orbits, chaotic behavior, and resonance bifurcation. Furthermore, the existence of chaos is discussed in the sense of Marotto, and a novel chaos control method is proposed for discrete Brusselator model using an extended pole–placement approach. This modified approach is more suitable for codimension-two bifurcation situations. Numerical simulations are used to illustrate the theoretical discussion.