Hopf Bifurcation , Positively Invariant Set , and Physical Realization of a New Four-Dimensional Hyperchaotic Financial System

1Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China 2School of Mathematics and Statistics, Inner Mongolia University of Financial and Economics, Hohhot 100124, China 3School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China 4Department of Electrical and Electronics Engineering, Faculty of Technology, Sakarya University, 54187 Sakarya, Turkey


Introduction
In recent years, economic dynamics become very prominent in the mainstream economics, and stochastic analysis is a popular way to interpret financial time series based on the existing information.Random exogenous shocks is usually considered to be the reason of a periodic behavior of the economic system by employing stochastic analyses, caused by factors outside the system.The shortcoming of the stochastic approach is incapable to illustrate the dynamics of financial systems [1].
An effective and rapid control method is very important for the government when some chaotic phenomenon appears [2][3][4][5].A classical three-dimensional financial dynamic model describing the time variations of state adjustment has been reported [3], which includes the interest rate, the price index, and the investment demand.Later, another threedimensional financial chaotic risk dynamic system was constructed for management process of financial markets and proved to be controlled effectively [6].Then Holyst and Urbanowicz [7] used the method of delayed feedback control (DFC) and proved that a chaotic financial system can be stabilized on various periodic orbits [8].Reference [9] discussed the complex behaviors of a financial system with time-delayed feedback by numerical simulation.Normal form of a financial system with delaying has been derived, which is associated with Hopf and double Hopf bifurcations and makes the financial system more complicated [10,11].In addition, the positively invariant sets of the dynamic system have a basic significance in the state constraints and control constraints, which are widely applied in different kinds of field: general 3-body problem [12], delay differential-equations [13], stability of dynamic system [14], robust attitude control schemes [15], relative motion control of the spacecraft [16], interconnected and time-delay systems [17], and permanent magnet synchronous motor system [18].
On the other hand, high dimensional control system is now a research hot topic, which has been applied in many fields.Zhang et al. introduced a three-dimensional ultimate bound and positively invariant set in which parameters are positive [19].Wang et al. showed a smooth fourdimensional quadratic autonomous hyperchaotic system, which generated two novel double-wing periodic, quasiperiodic, and hyperchaotic attractors [20].Wei and Zhang obtained the ultimate bound and positively invariant set is in a four-dimensional autonomous system [21].Du et al. proposed hyperchaotic Rikitake system which was a novel four-dimensional autonomous nonlinear system and assured the existence of Hopf bifurcation [22].A new fourdimensional quadratic autonomous hyperchaotic attractor was presented and the Hopf bifurcation at the equilibrium point was analyzed by Prakash and Balasubramaniam [23].Therefore, the dynamical behaviors of hyperchaotic systems are more complex than chaotic systems and will have practical meanings for carrying out this research about financial systems [24].Therefore, it is very necessary to explain complicated phenomena of financial dynamics by a thorough studying on the internal structural characteristics of hyperchaotic systems.On the viewpoint of mathematics and finance, these research points are attractive and rational.This paper constructs a new type of hyperchaotic financial system by nonlinear feedback method.
The remainder of the paper is organized as follows.Firstly, a new four-dimensional financial system is introduced by adding feedback controllers to the classic financial system in Section 2. And a sufficient condition for nonexistence of chaotic or hyperchaotic behaviors is obtained theoretically.Then the novel hyperchaotic financial system is confirmed numerically from Lyapunov exponents.Characterizations for the four-dimensional Hopf bifurcations are surveyed in Section 3. Section 4 introduces the ultimate bound and positively invariant set.The dynamic properties of the system are showed via bifurcation diagram in Section 5.In Section 6, a real contribution to engineering will be realized by an electronic circuit and oscilloscope in real time.Section 7 gives some conclusions.

Financial System from Classic Financial Model
2.1.Formulation of System.A financial dynamic system with different factors is reported in [3].With the proper dimensions and appropriate coordinates, a simplified financial model is proposed: where "" shows the interest rate that is defined as the price or cost of money for borrowing and return to lending."" implies the investment demand, which will be directly affected by the interest rate.In addition, the supply and demand of the commercial goods and inflation rate cause changes of ""; let  > 0,  > 0, and  > 0, respectively, be the saving amount, the cost per investment, and the elasticity of demand in commercial market.This paper is dedicated to greatly survey the dynamical behaviors of a four-dimensional financial system which expanded from a known three-dimensional financial system and also proposes a simplified mathematical system.In this paper, we design the controlled system as follows: where  denotes control input and economically state intervention to balance the economic environment.For example, United States interest rates are determined by the Federal Reserve with considering short term economic targets.Regulatory agencies of open markets meet at certain intervals to monitor the economic and financial situation and decide on monetary policies.To decrease inflation and increase the purchasing power of the consumers, government will increase the interest rate.As a result of this, control input is the factor that interacts with all variables.This cross relation between interest rate, inflation (this also represents the price and goods and services which are denoted as "z" in the study), and government regulations can be expressed from (2d), in which d, k, and m mean corresponding amplitudes.Therefore, it will be expected to study some complex dynamical behaviors about the proposed four-dimensional autonomous system (2a), (2b), (2c), and (2d).

Nonexistence of Chaotic or Hyperchaotic
Behaviors.In addition, chaotic solutions do not exist for certain parameter values in system (2a), (2b), (2c), and (2d).We get three positive Lyapunov exponents, and it will be beneficial for us to find hyperchaos.More precisely, the main results are obtained as follows.
The proof is complete.

An Analysis of System Bifurcation (2a), (2b), (2c), and (2d).
Now we calculate the first Lyapunov coefficient in relation to Hopf bifurcation by employing the projection method [25].Let  1 be the first Lyapunov coefficient related to Hopf bifurcation.Firstly, consider the differential equation in which  ∈  4 denotes vectors representing phase variables and  ∈  6 denotes control parameters.Suppose  is a class of  ∞ in  4 ×  6 and there is an equilibrium point in (18)  =  0 at  =  0 .Let variable  −  0 be , and we can write as where  =   (0,  0 ) and, for  = 1, 2, 3, The following considerations are supposed:  has a pair of complex eigenvalues on the imaginary axis  2,3 = ± ( 0 > 0), which are the only eigenvalues with Re  = 0,   is the generalized eigenspace of  with regard to  2,3 , and ,  ∈  3 are vectors such that in which  Τ is the transposition of the matrix .
Vector  ∈   can be showed as  =  +   with  = ⟨, ⟩ ∈ .The two-dimensional center manifold related to the eigenvalues  2,3 is parameterized by  and , with an immersion of the form  = (, ), where  :  2 →  3 is a Taylor expansion of the form with ℎ  ∈  3 and ℎ  = ℎ  .Substituting ( 23) into (19), the following differential equation could be reached: By solving the system of linear equations defined by the coefficients of ( 19), we can get the complex vectors ℎ  .Given the coefficient , system equation ( 19) can be written on the chart  for a central manifold as where  21 ∈ .
A Hopf bifurcation point ( 0 ,  0 ) is an equilibrium point of ( 18), a pair of purely imaginary eigenvalues ± 0 ( > 0) and another eigenvalue with nonzero real part only exists in the Jacobian matrix .A two-dimensional center manifold is well defined at a Hopf point, which is invariant under the flow produced by (18) and can continue with arbitrarily high class of differentiability to nearby parameter values.
If the parameter-dependent complex eigenvalues cross the imaginary axis with nonzero derivative, then a Hopf point is called transversal.In the area of a transversal Hopf point with  1 ̸ = 0, the dynamic behavior of system ( 18) can decrease to a family of parameter-dependent continuations of the center manifold and is topologically orbital equivalent to the complex normal form where  ∈  and , , and  1 are real functions with arbitrarily higher order derivatives, which are continuations of 0,  0 , and the first Lyapunov coefficient at the Hopf point [23].When  1 < 0 ( 1 > 0), we can find one family of stable (unstable) periodic orbits on this manifolded family, contracting to an equilibrium point at the Hopf point.
The remaining part of this section employs the fourdimensional Hopf bifurcation theory and uses symbolic computations to carry out the analysis of parametric variations concerning dynamical bifurcations.Because the system has only one equilibrium, the bifurcation of system (2a), (2b), (2c), and (2d) will be our only concern, and then we get Theorem 3.
Proof.As to the parameters (, , , , ) = (0.9, 1.5, 0.2, 0.05, 0.005) and  =  0 = 0.7068, we have Under this condition, it is easy to know the transverse condition   ( =  0 ) < 0. Accordingly, there exists a Hopf bifurcation at  0 .The stability of  0 can be decided by the value of the first Lyapunov coefficient  1 and  1 can show the stability of  0 and the occurred periodic orbits.Making use of the notation of the section above, the multilinear symmetrical functions could be written as (, , ) = (0, 0, 0, 0) .

(31d)
Then the following value is computed: Therefore, the theorem is proved.Near the unstable equilibrium point  0 , we can find a stable periodic solution for  <  0 .
Proof.Because of symmetry,  1 will be considered.Since the parameters satisfy (, , , , ) = (0.9, 1.5, 0.2, 0.05, 0.2) and  =  0 = 0.5139, we have Under this condition, it is easy to know the transverse condition   ( =  0 ) < 0. Accordingly, Hopf bifurcation appears.There exists a Hopf bifurcation at  0 .The value of the first Lyapunov coefficient  1 can decide the stability of  0 .It demonstrates the stability of the periodic orbits and the equilibrium point.Making use of the notation obtained from the section above, the multilinear symmetric functions could be written as (, , ) = (0, 0, 0, 0) .
Then the following values are computed: Therefore, the theorem is justified.We can find a stable periodic solution close to the unstable equilibrium point  1 for  <  0 .
If  is positive definite and  is negative definite, then ṗ = 0 must be a bounded sphere in  4 ; and now that chaotic systems are bounded, () reaches the maximum or minimum values under the solution set of ( 18), whereas it is necessary that ṗ = 0. So, in order to assess the ultimate bounds of system equation ( 18), the following optimization problem should only be solved: If the above discussed conditions are satisfied, then the above optimization problem has a solution supposing  min ≤ () ≤  max , and then set Ω = { ∈  4 |  min ≤ () ≤  max } would be our desired ultimate bounds.
Making use of the above method, we talk about the ultimate bound and positively invariant set for system (2a), (2b), (2c), and (2d) with  = 0.

Numerical Simulation
In the previous twenty years, different tools were developed to tap the periodic responses of nonlinear dynamical systems.The periodic responses can be spotted by several traditional criteria such as bifurcation diagrams and phase portraits.In this section, we use these techniques to illustrate the existence of the periodic motions for the financial system.Based on the four-dimensional nonlinear financial system (2a), (2b), (2c), and (2d), numerical simulations have been conducted in this paper.Using MATLAB software, a numerical approach is utilized to show the nonlinear dynamic behavior in the financial system.The red in Figure 1 shows the hyperchaotic attractors for the four-dimensional financial system (2a), (2b), (2c), and (2d) with parameter values  = 0.9,  = 0.2,  = 1.5,  = 0.2,  = 0.05, and  = 0.005 and the initial condition (0, 1, −0.5, 0). Figure 1(a) is the phase portrait in (, , ) space and  = 0, and Figure 1(b) is the phase portrait in (, , ) space and  = 0.It shows that the corresponding Lyapunov exponents are  1 = 0.03003,  2 = 0.01448,  3 = −0.0003,and  4 = −1.2318,and the system has complex dynamics.Figure 2 implies the bifurcation set of system (2a), (2b), (2c), and (2d) when the values of parameters are set as  = 0.9,  = 1.5,  = 0.2,  = 0.05,  ∈ [−1, 1], and  ∈ [−1, 1], and the bifurcation set divides the space into three regions with three solutions above the bifurcation set, one solution in the bifurcation set, and none under the set, which is consistent with the theoretical results.Figure 3 illustrates the bifurcation curve when  = 0.9,  = 1.5,  = 0.2,  = 0.05, and  = 0.005, and the point of Hopf bifurcation is  = 0.68.

Physical Realization for the 4D Financial
System (2a), (2b), (2c), and (2d) In this section, the 4D hyperchaotic financial system will be applied for its application using the electronic circuit which has been designed on the oscilloscope.The hyperchaotic system is studied by using an electronic circuit design.The numerical simulation and the oscilloscope outputs were obtained for the similar shaped phase portraits [28][29][30].

Conclusion
In our paper, the proposed financial system, which is generated from three-dimensional classical financial system, greatly expands the list of hyperchaotic financial attractors.The sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically, and the solutions of equilibria are obtained.For each equilibrium, its stability and existence of Hopf bifurcation are validated.Besides, based on its corresponding first Lyapunov coefficient, the analytic proof of the existence of periodic solutions is exhibited.Then the research has got the ultimate bound and positively invariant set for the proposed hyperchaotic financial system.The direction and stability of the bifurcating periodic solutions can be determined, and some numerical solutions are obtained to verify the theoretical results.In addition, the hyperchaotic financial system is extended to an electronic circuit implementation for real-time application.
We can understand the functions of financial policies and reveal the true geometrical structure of the attractors by the results of this paper.Since the global dynamics and geometrical structure of this system are not presented completely, more detailed theoretical simulation and investigations are expected in the forthcoming study.

Figure 8 :
Figure 8: The circuit of the hyperchaotic circuit.