Adaptive Integral Sliding Mode Controller Design for the Regulation and Synchronization of a Novel Hyperchaotic Finance System with a Stable Equilibrium

Abstract

Chaos and hyperchaos have important applications in finance, physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. Control and synchronization of chaotic and hyperchaotic systems are important research problems in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. This work proposes a novel 4-D hyperchaotic finance system. We show that the hyperchaotic finance system has a unique equilibrium point, which is locally exponentially stable. Also, we show that the Lyapunov exponents of the hyperchaotic finance system are \(L_1 = 0.0365\), \(L_2 = 0.0172\), \(L_3 = 0\) and \(L_4 = -0.8727\). The Kaplan-Yorke dimension of the hyperchaotic finance system is derived as \(D_{KY} = 3.0615\), which shows the high complexity of the system. Next, an adaptive integral sliding mode control scheme is proposed for the global regulation of all the trajectories of the hyperchaotic finance system. Furthermore, an adaptive integral sliding mode control scheme is proposed for the global hyperchaos synchronization of identical hyperchaotic finance systems. The adaptive control mechanism helps the control design by estimating the unknown parameters. Numerical simulations using MATLAB are shown to illustrate all the main results derived in this work.