Fractional-Order Financial System and Fixed-Time Synchronization

Abstract

This study is concerned with the dynamic investigation and fixed-time synchronization of a fractional-order financial system with the Caputo derivative. The rich dynamic behaviors of the fractional-order financial system with variations of fractional orders and parameters are discussed analytically and numerically. Through using phase portraits, bifurcation diagrams, maximum Lyapunov exponent diagrams, 0–1 testing and time series, it is found that chaos exists in the proposed fractional-order financial system. Additionally, a complexity analysis is carried out utilizing approximation entropy SE and C0 complexity to detect whether chaos exists. Furthermore, a synchronization controller and an adaptive parameter update law are designed to synchronize two fractional-order chaotic financial systems and identify the unknown parameters in fixed time simultaneously. The estimate of the setting time of synchronization depends on the parameters of the designed controller and adaptive parameter update law, rather than on the initial conditions. Numerical simulations show the effectiveness of the theoretical results obtained.

Global Existence and Fixed-Time Synchronization of a Hyperchaotic Financial System Governed by Semi-Linear Parabolic Partial Differential Equations Equipped with the Homogeneous Neumann Boundary Condition

“…As pointed out above, several papers have already been published concerning the synchronization problem for (hyper)chaotic financial systems. Among them, as will be seen later, are the results in References [30][31][32], which are closer to the synchronization result in this paper. Yousefpour et al [30] designed an adaptive terminal sliding mode control, equipped with a radial basis function neural network estimator, for the response system to the time-fractional-order (in the Grünwald-Letnikov sense) counterpart of the financial system (2), and came up with a criterion to guarantee that the the time-fractional-order (in the Grünwald-Letnikov sense) counterpart of (2) and its response system with the designed control implemented achieves finite-time synchronization.…”

“…We come up with a synchronization control for the response system corresponding to the drive financial system (3), and provide two criteria ensuring that the drive system (3) and its response system with the proposed control implemented achieve fixed-time synchronization. To the authors' knowledge, among the results in the vast references concerning synchronization problems for (hyper)chaotic financial systems, only the results in References [30][31][32], whose main contributions were introduced briefly above, are highly close to our fixed-time synchronization results in this paper.…”

“…As mentioned in Reference [31], the control designed in Reference [31] has many advantages over the linear feedback control used in several other references; among them, it ensures that the synchronization errors remain always within the predefined output constraints. Almost in the same time period, He, Peng, and Zheng [32] considered the fixed-time synchronization problem for time-fractional-order counterparts of the financial system (1), more precisely, they designed an appropriate control for the response system to the time-fractional-order (in Caputo's sense) counterpart of the system (1), and provided a criterion to guarantee that the the time-fractional-order (in Caputo's sense) counterpart of (1) and its response system with the designed control implemented achieved fixedtime synchronization.…”

“…The stabilization problem has already been extensively investigated for (hyper)chaotic financial systems; the interested readers can consult, for instance, References [13,14,[25][26][27], for some intuition and new observations concerning this topic. Analogously to the stabilization problem, the synchronization problem for (hyper)chaotic financial systems has also been investigated in several references; see References [28][29][30][31][32][33]. Synchronization is one of the most interesting collective behaviors of dynamical systems, and therefore has aroused tremendous interest in many application fields, such as secure communication, biological systems, and information processing; see References [26,[34][35][36][37] and the references cited therein.…”

Ultimate Boundedness and Finite Time Stability for a High Dimensional Fractional-Order Lorenz Model

“…In recent years, interest in studying fractional-order dynamical systems has increased. Modeling many systems with fractional order equations is a necessity to study the behavior of dynamical systems in more realistic applications [1][2][3][4][5][6][7][8]. Integer order calculus is a special case in fractional calculus that is approximate to the real system in the mathematical model.…”