Abstract
In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.57 for the incommensurate order case. Also, the period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.
Highlights
Investigating chaos in dynamical systems is one of the most interesting topics which have been carried out extensively in different scientific fields such as medicine [1], biology [2], mathematics [3], and many others
The dynamics of a financial system with fractional order as well as the robust chaos control in this system are studied analytically and numerical simulations are performed to confirm the analytical results. e existence of chaos in this study is validated by a positive Lyapunov exponent and by an analytical condition existing in the literature. e fractional-order system exhibits rich dynamics behaviors such as periodic and chaotic behaviors
Numerical simulations revealed that chaos exists in this fractional-order system for derivation orders less than 3. e lowest derivation order found to have chaos in the commensurate fractional-order case is 2.49 and 2.57 for the incommensurate fractional-order case
Summary
Investigating chaos in dynamical systems is one of the most interesting topics which have been carried out extensively in different scientific fields such as medicine [1], biology [2], mathematics [3], and many others. E sliding mode control is a powerful technique to robustly control uncertain dynamical systems subject to uncertainties and external disturbances [40,41,42]. In [38], a fractionalorder sliding mode controller was designed to eliminate the chaotic behavior in an economical system in the presence of model uncertainties and external disturbances. Motivated by the above discussions, in this paper, chaos in the financial system presented by Liao et al [37] with fractional order and robust control of this chaotic behavior are investigated. A sliding mode control law is designed to control the chaos in this fractionalorder financial system with or without uncertainties and external disturbances.
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