Predicting nonlinear phenomena through fractional derivatives and advanced Riccati mapping: bifurcation and solitary wave behavior in nonlinear systems

This paper introduces a new mathematical analysis for approximating and predicting nonlinear dynamical systems through fractional calculus coupled with diffusion-reaction equations. The idea is to discuss specific aspects of system behaviors, memory effects, and historical dependencies for fractional-order systems. We apply the Generalized Riccati Equation Mapping Method (GREMM) for imaging the nonlinear system and developing more insights into the complex systems’ behaviors as well as memory effects, especially in the case of fractional-order systems. This methodology, thus, allows us to get a better understanding. More than that, the current approaches introduced to understand the behavior of sophisticated systems do so far better compared to the traditional methodologies to study and make a prognosis on the behavior of solitary waves in these systems. The extended GREMM is also important for the analysis as it allows for the investigation of complex wave processes, such as solitary waves and solitons. We begin by describing kink waves and pulse waves, as well as linear and nonlinear periodic waves, and how they behave. We also discuss bifurcation analysis in which a relatively small change in the value of the parameters leads to large changes in the behavior, up to chaos by introducing fractional derivatives, and the method assists in the advancement of the analysis, the prediction, and the control of many processes in science and engineering disciplines, such as biology, chemistry, and physics. It provides better approximations. As compared to other methods, this method can provide better predictions of nonlinear phenomena for controlling power interactions and any other system demonstrating nonlinear behavior.