Modulating Chaotic Oscillations in Autocatalytic Reaction Networks Using Atangana–Baleanu Operator

Abstract

Many mathematical models describing dynamics that occur in autocatalytic reaction networks have been proved to be chaotic, exhibiting orbits with unpredictable outcomes. Is it always possible to modulate that chaos? We use Haar wavelet numerical method to investigate a fractional system modeling autocatalytic reaction networks, where particular attention is made on biochemical systems of four-component networks. The convergence of the method is detailed through error analysis. Graphical representations reveal that the dynamic of the whole system is characterized by limit-cycles followed by period-doubling bifurcations that culminate with chaos, depending on the change of the total concentration of cofactor. The behavior of the system becomes more unpredictable as the concentration of cofactor increases, but the phenomenon is shown to be regulated by an additional parameter, the order of the fractional derivative \(\gamma ,\) which plays an important role in triggering and controlling the appearance of chaos. Moreover, the chaotic behavior observed in the cascade diagram of the pure fractional case is proven to appear earlier, showing that the parameter \(\gamma \) is a valuable tool to regulate the chaos observed in some biochemical systems.