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The construction and several control problems of a new hyperchaotic finance system are investigated in this paper. Firstly, a new four-dimensional hyperchaotic finance system is introduced, based on which a new hyperchaos is then generated by setting parameters. And the qualitative analysis is numerically studied to confirm the dynamical processes, for example, the bifurcation diagram, Poincaré sections, Lyapunov exponents, and phase portraits. Interestingly, the obtained results show that this new system can display complex characteristics: chaotic, hyperchaotic, and quasiperiodic phenomena occur alternately versus parameters. Secondly, three single input adaptive controllers are designed to realize the control problems of such system: stabilization, synchronization, and coexistence of antisynchronization and complete synchronization, respectively. It is noted that the designed controllers are simpler than the existing ones. Finally, numerical simulations are provided to demonstrate the validity and the effectiveness of the proposed theoretical results.

It is well known that the classical chaotic attractor was firstly found by Lorenz in 1963 [

It is well known that the OGY method [

For the economic systems, the chaotic behavior in those systems was first found in 1985 [

Motivated by the above discussions, a four-dimensional hyperchaotic finance system is presented, which can generate double-wing chaotic and hyperchaotic attractors with three equilibrium points. In comparison with the most existing results, this model has simple structure and can display complex dynamics: chaotic, hyperchaotic, and quasiperiodic phenomena occur alternately. Furthermore, some basic dynamic properties of the new hyperchaotic finance system regarding equilibria, dissipation, Lyapunov exponent, Lyapunov dimension, bifurcation diagram, and Poincaré sections are investigated. Then, three control problems: stabilization, synchronization, and coexistence of antisynchronization and complete synchronization are derived with simple yet physically implementable controllers. Finally, numerical simulations are provided to demonstrate effectiveness and the validity of the proposed theoretical results. In conclusion, the main contributions of this paper are given as follows:

A new hyperchaotic finance system is firstly introduced, based on which a new hyperchaos is then generated by setting the parameters.

Dynamic properties of the new generated hyperchaotic finance system are investigated extensively in Section

Three control problems of the hyperchaotic finance system are investigated extensively in Section

The rest of this paper is organized as follows. In Section

A dynamic model of finance has been reported in [

In order to generate a new hyperchaos, we set

Since the hyperchaotic system (

The divergence of this four-dimensional hyperchaotic system (

The equilibria of the system (

Bifurcation diagram is a useful method to show the dynamical processes of a system with respect to a parameter. In this subsection, the parameter

Bifurcation diagram of the system (

Time course of the system (

For the chaotic systems, Lyapunov exponent is not only an important index to distinguish chaotic attractor, but also a quantitative description of the sensitivity about the initial values. In addition, Poincaré section is also a critical approach to analyze chaotic system; for example, if there exists a closed curve, the movement is quasicycle; if there only exists one fixed point or a few discrete points, the movement is cycle; and if there exists a collection of distribution points along a segment of a line or a curve, the movement is chaos. Here, the Lyapunov exponent and Poincaré sections of the system (

Figure

Lyapunov exponent of system (

Poincaré sections of system (

3D projection of the strange attractor of system (

Time course of

By Wolf algorithm, the Lyapunov exponents of this four-dimensional hyperchaotic finance system (

In this section, three control problems, stabilization, complete synchronization, and coexistence of antisynchronization and complete synchronization, of the new hyperchaotic finance system (

For convenience, let

Since system (

According to Section

Consider system (

Thus, the controller is designed as

In [

For convenience, systems (

Then, the following conclusion is presented.

For any initial values, the orbits

The time derivative of

Numerical simulation is carried out with the following initial conditions:

The figure shows that the orbit

In this subsection, the complete synchronization of two hyperchaotic finance systems with different initial conditions are investigated, and some new results are presented in the following.

Make system (

Let

Since system (

Consider system (

Thus, the designed controller is given as

Similarly, systems (

Next, a conclusion is obtained.

For any initial values, the orbits

The time derivative of

Similarly, numerical simulation is carried out with the following initial conditions:

The figure shows that the states,

In this section, the coexistence of antisynchronization and complete synchronization in the two hyperchaotic finance systems is investigated, and some new results are obtained in the following.

According to the results in [

Since system (

Consider the combination system (

Therefore, the designed controller is expressed as

Similarly, systems (

Next, a conclusion is presented as follows.

For any initial values, the orbits

The time derivative of

Similarly, numerical simulation is carried out with the following initial conditions:

The figure shows that the orbits

The figure shows that states

In this paper, a new four-dimensional hyperchaotic finance system has been introduced, based on which a new hyperchaos has been generated. This system has simple algebraic structure and can display complex characteristics: chaotic, hyperchaotic, and quasiperiodic phenomena occur alternately versus parameters. Then, the dynamic properties including symmetry, dissipation, equilibrium point, Lyapunov exponent, bifurcation diagram, and Poincaré sections are studied. Furthermore, three control problems of such system have been achieved by three single input controllers, respectively. Compared to the existing ones, the designed controllers in this paper are simpler. Finally, the validity and the effectiveness of the obtained theoretical results have been verified by the provided numerical simulations.

The author declares that there are no financial and personal relationships with other people or organizations that can inappropriately influence this work; there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this article. The author declares that there are no conflicts of interest regarding the publication of this paper.

The author would like acknowledge the School of Business Administration at Andrews University for providing wonderful research environment. In particular, the help of Dr. Ralph Trecartin, Dr. Jerry Chi, and the colleagues during the current research is highly appreciated. This paper is supported by the Social Science Foundation of Shandong Province (17CLYJ43).