This paper is concerned with dynamic and entropy analyses of a hyperchaotic financial system, as well as with its hyperchaos suppression and synchronization. The dynamic behaviour of the system is analyzed for several parameters and initial conditions making use of bifurcation diagrams, Lyapunov exponents and phase portraits. Moreover, entropy from resulting time series is also characterized by estimating ordinal pattern distributions. These analyses have been able to determine and locate accurately chaotic and periodic attractors in the system, thus enabling successful design of its control. In general, financial systems are not always completely synchronized; therefore, some robust synchronization technique should be considered. This study proposes a novel fuzzy disturbance-observer based integral terminal sliding mode control method for the hyperchaotic financial system. The presented control technique guarantees robustness against uncertainties, external disturbances and control input saturation. Fuzzy rules are employed to adaptively tune the gains of the proposed control scheme. Also, the fuzzy inference engine avoids the chattering problem in the system response. Simulation results illustrate the efficient performance of the proposed control technique in the presence of dynamic uncertainties, external disturbances and control input saturation.

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A financial hyperchaotic system with coexisting attractors: Dynamic investigation, entropy analysis, control and synchronization

In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lu system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.

Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization

A novel integer-order chaotic financial system is proposed by considering market confidence into a three-dimensional financial system. A four-dimensional fractional-order financial system is presented by introducing fractional calculus into the new integer-order system. The 0–1 test algorithm is employed to identify chaos. A robust controller is designed to stabilize the fractional-order chaotic system in a finite time. This finite-time control scheme can keep the original structure of system as much as possible and can be applied to stabilizing other chaotic systems including dynamic economic systems. Numerical simulations validate the main results of this work.

Finite-time stabilizing a fractional-order chaotic financial system with market confidence

This paper mainly focuses on the analysis of a hyperchaotic financial system as well as its chaos control and synchronization. The phase diagrams of the above system are plotted and its dynamical behaviours like equilibrium points, stability, hyperchaotic attractors and Lyapunov exponents are investigated. In order to control the hyperchaos, an efficient optimal controller based on the Pontryagin’s maximum principle is designed and an adaptive controller established by the Lyapunov stability theory is also implemented. Furthermore, two identical financial models are globally synchronized by using an interesting adaptive control scheme. Finally, a fractional economic model is introduced which can also generate hyperchaotic attractors. In this case, a linear state feedback controller together with an active control technique are used in order to control the hyperchaos and realize the synchronization, respectively. Numerical simulations verifying the theoretical analysis are included.

New aspects of the adaptive synchronization and hyperchaos suppression of a financial model

This manuscript mainly focuses on the construction, dynamic analysis and control of a new fractional-order financial system. The basic dynamical behaviors of the proposed system are studied such as the equilibrium points and their stability, Lyapunov exponents, bifurcation diagrams, phase portraits of state variables and the intervals of system parameters. It is shown that the system exhibits hyperchaotic behavior for a number of system parameters and fractional-order values. To stabilize the proposed hyperchaotic fractional system with uncertain dynamics and disturbances, an efficient adaptive sliding mode controller technique is developed. Using the proposed technique, two hyperchaotic fractional-order financial systems are also synchronized. Numerical simulations are presented to verify the successful performance of the designed controllers.

On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system

In this paper, a new hyperchaotic finance system which is constructed based on a chaotic finance system by adding an additional state variable is presented. The basic dynamical behaviors of this hyperchaotic finance system are investigated, such as the equilibrium, stability, hyperchaotic attractor, Lyapunov exponents, and bifurcation analysis. Furthermore, effective speed feedback controllers and linear feedback controllers are designed for stabilizing hyperchaos to unstable equilibrium points. Numerical simulations are given to illustrate and verify the results.

Dynamic analysis and control of a new hyperchaotic finance system

This paper presents a new type synchronization called modified function lag projective synchronization (MFLPS), where the drive and response systems could be synchronized up to a desired scale function matrix with time-delay. With MFLPS it achieves self-synchronization of a financial hyperchaotic system when the parameters are known and unknown, respectively. The corresponding numerical simulations are performed to verify and illustrate the analytical results.

Modified function lag projective synchronization of a financial hyperchaotic system

The application of the nonlinear chaotic dynamic system in economics and finance has expanded rapidly in the last decades. This paper considers the localization of all compact invariant sets of a new three-dimensional autonomous nonlinear finance chaotic system. The boundedness of the new finance chaotic system is the first time being investigated. Based on the iteration theorem and the first-order extremum theorem, a new method is proposed, too. The comparison of our method with the traditional method is presented as well. More specifically, the compact invariant sets are analyzed in three aspects: First of all, a localization of the new finance chaotic system by two frusta and an ellipsoidal used by traditional methods is discussed. Second, a localization of the new finance chaotic system by two frusta and a parabolic cylinder is provided. Third, localization of the new finance chaotic system according to superposition of the ellipsoidal, parabolic cylinder, and two frusta are presented, and the boundedness of the chaotic attracter is smaller than in the classical methods. Numerical simulations are given to indicate the effectiveness of the proposed method.

Localization of compact invariant sets of a new nonlinear finance chaotic system

in this paper, speed feedback controllers, linear feedback controllers and adaptive controllers are designed for stabilizing chaos of a modified chaotic finance system to unstable equilibrium points. The Lyapunov theorem and Routh-Hurwitz criterion are used to analyze the gain's scope of the controlled system and study the asymptotic stability at the equilibrium point. The effectiveness of the three different methods is illustrated by theoretical analysis and numerical simulations.

Yuxiu Li

Papers

Dynamic analysis and control of a new hyperchaotic finance system

Modified function lag projective synchronization of a financial hyperchaotic system

Localization of compact invariant sets of a new nonlinear finance chaotic system

Stabilization of a Modified Chaotic Finance System