Abstract
In this paper, a fractional-order model of a financial risk dynamical system is proposed and the complex behavior of such a system is presented. The basic dynamical behavior of this financial risk dynamic system, such as chaotic attractor, Lyapunov exponents, and bifurcation analysis, is investigated. We find that numerical results display periodic behavior and chaotic behavior of the system. The results of theoretical models and numerical simulation are helpful for better understanding of other similar nonlinear financial risk dynamic systems. Furthermore, the adaptive fuzzy control for the fractional-order financial risk chaotic system is investigated on the fractional Lyapunov stability criterion. Finally, numerical simulation is given to confirm the effectiveness of the proposed method.
Highlights
Dynamical analysis of the new fractional-order financial risk chaotic system was described by the phase portraits, Lyapunov exponents spectrum, and bifurcation diagram
We found that the chaotic behavior exists for the new fractional-order financial risk system in the range q1 ∈ [0.63, 1], q2 ∈ [0.9, 1], q3 ∈ [0.9, 1], and q ∈ [0.944, 1]
Periodic behavior exists for the fractional-order financial risk system in the range of q1 ≤ 0.62, q2 < 0.9, q3 < 0.9, and q ≤ 0.943
Summary
Chaotic systems have received more attention due to their potential applications in economics and management, such as equity market indices: cases from the United Kingdom [1], monetary aggregates [2], business cycle [3], firm growth and R&D investment [4], chaotic behavior in foreign direct investment, and foreign capital investments [5, 6]. 2, we present the properties and dynamics of a new fractional-order financial risk chaotic system and investigate the properties numerically via Lyapunov exponents and bifurcation diagram. Let the derivative order q3 vary from 0.7 to 1 Complexity of the fractional-order financial risk chaotic system with derivative q and control parameter r varying is analyzed, where the step size of q is 0.001 and in the range of q ∈ [0.9, 1]. 3.2 Controller design and stability analysis Adaptive fuzzy control of the commensurate fractional-order model of financial risk chaotic system (4) is as follows:. All signals in a closed loop system (8) are bounded
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