Stabilization by feedback control of a novel stochastic chaotic finance model with time-varying fractional derivatives

Abstract

Chaotic systems exhibit a random behavior that may result in undesired system performance. In this context, diverse strategies have been proposed to control the chaos that appears in various areas of applications. Most of them presented in the deterministic case without taking the environmental noises into account, although many systems in practice are often-times exposed to some external disturbances that affect the structure of the considered system. The key aim in this article is to stabilize a chaotic finance model (CFM) by designing feedback controllers in the stochastic and fractional cases. Firstly, we establish a novel set of suitable hypotheses to demonstrate the uniqueness of solutions. Secondly, we discuss the Hyers–Ulam stability (HUS) and the generalized HUS for the controlled CFM under sufficient conditions. Finally, we provide several examples attached with numerical findings which clearly support the validity of theoretical findings and highlight their benefits. The numerical simulations are done based on the Euler–Maruyama method and with the help of Lagrange polynomial interpolation, enabling the authors to extract meaningful results from the model. Compared to the existing CFMs, the present study proposes a novel type of nonlinear controller for the stabilization of the fractional stochastic CFM, providing a substantial analysis based on the HUS theory and computer simulation results to verify the rapid convergence of the state variables to the origin.

Keywords

• Stochastic chaotic finance model
• Time-varying fractional order
• Feedback control
• Hyers–Ulam stability