Robust synchronization of four-dimensional chaotic finance systems with unknown parametric uncertainties

ABSTRACT The inherent randomness in economic factors causes complex and irregular behaviour that affects financial system stability and economic growth. Such chaotic behaviour can make it difficult to synchronize financial systems. The chaotic finance system synchronization precision maintains financial stability and economic growth. In this paper, the controller design procedure assumes that the financial system is exposed to unknown bounded exogenous disturbances and model uncertainties. This research proposes a novel direct adaptive control strategy that achieves robust synchronization of two identical four-dimensional finance chaotic (FDFC) systems. The proposed controller establishes a faster, smoother synchronization error vector convergence to zero. The controller design procedure does not eliminate the closed-loop's nonlinear terms and is independent of the financial system parameters. These controller's attributes accomplish the closed-loop robust performance. Further, this controller uses real-time estimates of unknown model uncertainties and bounds to compensate for unknown exogenous disturbances. Computer simulation results and proofs of theoretical analysis based on the Lyapunov stability theory confirm that the proposed control technique compels the error vector trajectories to the origin in a short transient time with less active oscillations for all signals. The paper includes comparative computer simulations for verifying the theoretical findings.


Introduction
Chaotic dynamical systems show sensitivity to variations in the parameters and initial conditions; such systems display irregular, unpredictable, and complex behaviour [1].Several nonlinear dynamical systems in social and natural sciences exhibit chaos [1,2].
The intrinsic random nature of economic factors originates complex and irregular behaviour in a timevarying uncertain nonlinear finance system [3].Stock market price's abrupt variations, environmental interferences, exchange rate fluctuations, and interest rate alteration due to external forces, among other economic factors, change the savings amount in a financial system; the periodic dynamics of the financial system transform into chaos [4][5][6].Chaos in financial and economic systems concedes that the uncertainties in the macroeconomic operations produce disturbances that lead to financial system crises [5].The chaotic nature of the financial systems affects short-medium-long-term economic planning and predictions [6].
The chaotic synchronization of finance dynamical systems has been an active research area for the last two decades.It has produced numerous feedback controller schemes for synchronizing the financial system's chaotic behaviour.A brief literature review of this research is as follows.The paper [7] realizes the synchronization of two coupled three-dimensional chaotic finance (TDCF) systems.The closed-loop stability is investigated using the Routh-Hurwitz criterion when the finance system's parameters are known and active adaptive feedback controller for uncertain parameters.The article [8] derives a nonlinear adaptive-impulsive control strategy and employs the impulsive dynamical systems' invariant principle for synchronizing two identical four-dimensional finance chaotic (FDFC) systems.A set of sufficient conditions for impulsive synchronization using a comparative approach for impulsive functional differential equations is derived, and upper bounds for the impulsive interval necessary for stable synchronization are estimated.The article [9] discusses the robust chaotic synchronization of two uncertain FDFC systems with bounded exogenous disturbances.The authors [10] propose a time-varying delay nonlinear adaptive control technique for synchronizing two identical TDCF systems with exogenous disturbances.The paper [11] discusses the two mixed synchronization strategies for two identical TDFC systems and compares the closed loops using simulation results.Utilizing the Lyapunov stability theory, the paper [12] uses the adaptive control algorithm for synchronizing two identical uncertain FDFC systems.The authors [13] employ the linear feedback control technique to synchronize two identical TDFC systems.Ref. [14] investigates the network synchronization of many uncertain FDFC systems by estimating uncertain system parameters.The network synchronization is achieved by calculating the Lyapunov exponent of the network.The article [15] investigates the FDFC system synchronization using a conventional nonlinear control method.The authors [16] propose an adaptive control strategy for synchronizing two identical FDFC systems by estimating uncertain system parameters.Based on the Lyapunov stability theory, sufficient conditions are derived, and a linear feedback control technique is designed [17] for the global asymptotic synchronization of two identical FDFC systems.Ref. [18] achieves fixed-time synchronization of two identical FDFC systems using the drive-response system arrangement.
The following items discuss the motivations and challenges.
The FDCF systems synchronization precision is essential for maintaining financial stability and economic growth.
The abovementioned challenges motivate a statefeedback controller design that synchronizes two identical FDFC systems; developing a closed-loop structure should possess the following attributes.This article aims to achieve two key objectives regarding controller design procedure.
(1) Development of the robust closed-loop The proposed closed-loop should be insensitive to exogenous disturbances, and parameter variations should be smooth and slow.The closed-loop system's insensitivity to parameter variations and exogenous disturbances enhances feedback controllers' robustness and improves system performance, fault tolerance, and adaptability.These features allow the system to operate reliably and effectively within changing conditions and uncertainties.
(2) Synthesizing a smooth control effort The controller should synthesize a smooth control effort to make the closed-loop energy efficient.Smooth control effort does not produce fluctuations in the closed-loop dynamical system; it keeps the synchronization stability of financial systems and operates a smooth, healthy economic cycle.
Therefore, this article proposes a novel robust direct adaptive synchronization control strategy (RDASCS) that resolves the aforementioned issues.The proposed controller in this paper for synchronizing two identical FDFC systems is a novel design.The article also devises an analytical procedure for studying closed-loop stability.
The closed-loop formed by incorporating the proposed adaptive synchronization control algorithm characteristics are as follows.
(1) It develops a faster oscillation-free stable closedloop: it reduces risks, improves data accuracy, and enhances stability and processing capabilities for time-sensitive transactions.(2) The control input to the nonlinear control system does not depend on the system's nonlinear terms; this makes the closed-loop stability performance independent of the time-varying model uncertainties and exogenous disturbances.This controller attribute flourishes robust closed-loop stability, makes the system more predictable, and simplifies the controller design and analysis processes.
(3) The control signals are less oscillatory and chatterfree; such control efforts do not develop fluctuations in the financial system dynamics and economic growth operations.This feature of the controller design makes the control effort variations smoother against uncertainties or environmental changes.
The proposed algorithm improves financial system stability by reducing the unanticipated fluctuations and crashes risk.Benefits of the less active synchronization oscillations include increased efficiency and productivity and decreased investment demands.
The article provides a detailed stability analysis utilizing the Lyapunov second theorem of stability [21], ensuring a robust, asymptotic stable closed-loop.The paper discusses the computer-based simulation results verifying the theoretical findings.The article compares the closed-loop's performance with other state-ofthe-art synchronization control methodologies [13,22] using computer-based simulation results.
The breakdown of the remaining papers' structure is as follows.
Section 2 provides the relevant terminology.In Section 3, the article illustrates the four-dimensional finance chaotic system dynamics.Section 4 discusses the synchronization problem between two identical four-dimensional chaotic finance systems with unknown parametric uncertainties.Section 5 designs a novel feedback controller for synchronizing two identical four-dimensional chaotic systems in the master-slave system arrangement and provides proof of robust closed-loop stability.Section 6 gives numerical simulations with a comparative study.The article concludes in Section 7.

Dynamics of the four-dimensional chaotic finance model
The authors [3] proposed a three-dimensional finance chaotic (TDFC) system.The TDFC system consists of four sub-components: labour force, production, stock, and money.The TDFC dynamics system in matrix form is given in Equation (1).
where x 1 (t), x 2 (t), and x 3 (t) are the state variables representing the rate of interest, investment demand, and price index, alternatively [3].The state variable x 1 (t) is influenced by the goods price structural adjustment and investment market contradiction.The rate of change of state variable x 2 (t) is related to the investment and interest rates and cost investment inversion.The state variable x 3 (t) depends on inflation rates, the contradiction between the commercial market supply and demand also controls it [3].The parameters a 1 > 0, a 2 > 0, and a 3 > 0 denote the saving amount, investment cost, and commercial markets elasticity demand, alternatively.The research article [6] introduces a new state variable x 4 (t) to the TDFC system (1) that represents the average profit margin and proposes the dynamics of an FDFC system.Hence, Equation (2) describes the new FDFC system's dynamics in vector form.
The following section comprises five fundamental system characteristics that describe chaos in the finance system (2).
Vectors of the unknown time-varying model uncertainties present in the MSS (10)(11), respectively γ m and γ s are the vectors that represent the least upper bounds of γ m (x m (t)) and γ s (x s (t)), respectively Vectors of the unknown time-varying exogenous disturbances acting on the MSS (10)(11), respectively and δ m and δ s are the vectors that represent the least upper bounds of δ m (t) and δ s (t), respectively The feedback controller gains matrices Parameters of the controller (16), which are unknown The time derivative of Vector absolute, where x ∈ R n×1 and
The Jacobean matrix of system (2) is given below.
The computer simulation results in this article consider a 1 = 0.9, a 2 = 0.2, a 3 = 1.2, a 4 = 0.17, and a 5 = 0.2 [6] 2 shows that all the balancing points are saddle-focus, determining that the finance system (2) exhibits chaos.

Dissipative behaviour
The presence of a strange attractor in a nonlinear dynamical system guarantees chaos in the system [23].
The negative vector field's divergence assures a dissipative system.Following (2), let us consider the vector field v as follows.Equation ( 7) gives the divergence.

Poincare sections
This subsection discusses the continuous dynamical system's long-term stability based on the Poincare sections (maps) analysis.This procedure transforms the continuous dynamical system into a discrete state-space system having one dimension less than the original system [25].Fractals in the state-variable trajectories behaviour on a Poincare section indicate chaos in the dynamical system.Figure 4(a-d) illustrates the Poincare maps of system (2).

Bifurcation graphs
The qualitative shifts in the dynamical system vector fields caused by the plant parameter variations are known as bifurcation.The values of the parameters where such qualitative changes occur are called bifurcation values [25].Figure 5(a-d) shows the bifurcation plots for the variations in the system parameter a 3 ∈ [0, 2.5] verses state variables x i (t), i = 1, 2, 3, 4.These figures confirm that the four-dimensional finance system (2) is chaotic.
The following section formulates the problem of synchronizing two identical FDFC systems described in (2).

Synchronization between two identical FDFC systems with unknown parametric uncertainties
Let x m (t) and x s (t) are the state variables vectors of master and slave FDFC systems, respectively.Equations (10 and 11) give the MSS configuration for the FDFC system (2) and u(t) ∈ R 4×1 is a control input vector for synchronizing the slave FDFC system (11) with the master system (10).Further, δ m (t) and δ s (t) represent vectors of the time-varying unknown bounded exogenous disturbances, and γ m (x m (t)) and γ s (x s (t)) denote vectors of the time-varying unknown bounded model uncertainties.Slave FDFC system:
The synchronization control methodologies [26,27] use signum functions to deal with uncertainties and improve synchronization convergence rates.Signum functions are discontinuous, and using them in feedback controller design can produce abrupt variations with finite frequency and amplitude in the control signals, known as chattering; it creates oscillations in system state variables.The reported literature [28,29] (among others) replaces the signum functions with the continuous hyperbolic tangent functions in feedback control laws to generate smooth control signals and eradicate chattering.However, despite these controller designs, control signal oscillation still exists due to high state-feedback controller gains and the external forces acting on the system.

Objective 4.1:
The aim is to construct a state-feedback control algorithm that synthesizes a chatter-free control signal vector u(t) ∈ R 4×1 , which establishes faster and smooth synchronization errors (12) convergence to zero without oscillations.This work proposes designing a robust adaptive control law that forces the synchronization error signals (12) to zero with reduced oscillations in the error trajectories and feedback control signals.The controller design methodology does not eliminate the closedloop's nonlinear terms.
The following section discusses the structure of the proposed controller to address the problem elaborated above.It develops a design procedure based on the Lyapunov stability analysis theory.

Solution to problem
This section analyzes the synchronization error (12) converges to zero, and designs the FDFC system parameters update laws.

Controller design and parameters update laws
Equation (17) designs the structure of an RDASCS, which computes a control effort u(t) ∈ R 4×1 that enforces the slave FDFC system (11) to follow the master FDFC system (10) and satisfies the above Objective 4.1. where Theorem 5.1: If the feedback controller (17) is added to the slave FDFC system in (11) and the adaptive laws in (18) estimate the unknown parameters γ (t) and δ(t), then the dynamical error system (12) converges to the vicinity of zero.
Remark 5.4: Figure 8(a-d) depicts that the synchronization speed increases with large values of k ii and ψ ii .

Numerical simulations and comparative study
Table 3 assumes the initial conditions, FDFC system parameters, controller parameters, exogenous disturbances, and model uncertainties in numerical simulations.
The state-variable trajectories x i (t), i = 1, 2, 3, 4 in Figure 6(a-d) and error vector trajectories in Figure 6(e) demonstrate that the MSS (10)(11) do not synchronize without any control effort.Example 6.1:In this example, Figure 7(a-h) simulation results are performed under the control effort computed using the proposed RDASCS (17). Figure 7(a-d) demonstrates that the corresponding state variables x m i (t) and x s i (t), i = 1, 2, 3, 4 trajectories of the MSS (10)(11) show similar behaviour after a short transient time.Figure 7(e) illustrates that the error trajectories converge to the origin smoothly in less than 0.05 s with significantly reduced fluctuations.Figure 7(f) shows the synchronization error vector convergence in the range [−0.002, 0.002].In this figure, blue trajectories are y 1 (t) = 0.00053e −0.0071 t + 0.00147e −0.18 t and y 2 (t) = −y 1 (t) represent the convergence behaviour of the e(t).These trajectories are obtained using four points from the maximum oscillations of e(t) simulation data at t = 0, 20, 40, and 80 seconds.It is obvious that lim t→∞ y 1 (t) = 0 validating lim t→∞ e(t) = 0, which is consistent with the theoretical findings in Subsection 5.3.Figure 7(g-h) depicts the convergence of the estimated parameters γ (t) and δ(t) using the parameter update laws (18).Example 6.2:This example discusses the effects of feedback controller gains on the error vector convergence behaviour.
The convergence behaviour of the errors vector for different controller gains k ii and ψ ii are shown in Figure 8(a-d).These figures demonstrate that the convergence speed of the errors vector increases, and the error oscillation's amplitude decreases when the values of k ii and ψ ii are considered large.Table 4 summarizes the data for the simulation results shown in Figure 8(a-d).

Robust analysis of the closed-loop
Example 6.3:This example studies the effects of exogenous disturbances and 10% smooth FDFC system parameters variations on the performance of the closed-loop.

Scenario 1:
When no exogenous disturbances act on the MSS (10)(11).Scenario 2: Effects of exogenous disturbances Scenario 3: Effects of 10% smooth FDFC system parameters variations a 1 = 0.9 + 0.09e −0.01t , a 2 = 0.2 + 0.02e −0.01t , Figure 9(a-c) illustrates the error trajectories convergence behaviour to zero by the proposed method ( 17) for Scenarios 1-3, alternatively.These figures demonstrate that the proposed controller is robust to exogenous disturbances and smooth FDFC system parameter variations, and the convergence time is less than 0.05 s.The robust performance of two identical hyperchaotic finance systems synchronization is helpful for predictive analytics and improved security, which can benefit financial forecasting, risk management, and protection against cyber-attacks.By analyzing the synchronized data, it is possible to identify patterns and trends that can be used to make more informed financial decisions and enhance the overall system's security.

Comparative analysis
Example 6.4 presents the computer simulation results of the proposed method (17) and compares it with the feedback controllers proposed in recent published research articles [13,22], addressing the synchronization of similar chaotic dynamic systems.The comparative study is based on synchronization errors and control effort behaviour analysis in this context.The significance of these variables is as follows.
Errors: Fast synchronization of two identical hyperchaotic finance systems with reduced fluctuations of the error signals can improve the accuracy of the financial forecasts, which often rely on precise timing.Further, it reduces the risk of errors or discrepancies between the two systems and can help to prevent losses and other financial problems.

Control efforts:
The smooth control effort is inherently robust to model uncertainties and disturbances in the system; the synchronization process is less likely to be affected by external factors, such as disturbances or variations in the system parameters.Further, it reduces the system's overall energy consumption and makes synchronization more efficient.Therefore, to make the comparison systematic, the analysis describes these two variables in detail and summarizes essential performance indicators in Tables 5-7.The following describes these indicators.

Example 6.4:
The state-feedback control input vector u(t) ∈ R 4×1 synthesized by the sliding mode control (SMC) strategy (33) and adaptive controller (34) reported in [22] and [13], respectively.For the benchmark, initial conditions, model uncertainties, exogenous disturbances, FDFC system (2) parameters, and controller parameters are selected as given in Table 3.
Figure 7(e) depicts the error vector convergence behaviour accomplished by the proposed RDASCS (16), and Figure 10(a and b) illustrate the error vector behaviour by the controllers (33) and (34), respectively.These figures demonstrate that the proposed RDASCS (17) establishes the synchronization in 0.05 seconds with less active oscillations.Figure 10   the adaptive controller in (34) remain oscillatory, as illustrated in Figure 11(c), and it diverges after 15 s.
Table 5 provides essential performance indicators regarding errors convergence time, steady-state oscillation range, and control signals amplitude range shown in Figures 7(e), 10(a-b) and 11(a-c).
Table 6 summarizes the standard error performance indices-based comparative simulation studies of synchronization errors [30] for the simulation duration T s = 100 seconds.This simulation analytic data verifies that the proposed RDASCS (17) performs better than the other controllers (33, 34) in all the performance indices.
The synchronization energy error behaviour for the proposed algorithm (17), SMC strategy (33), and the adaptive controller (34) are defined as η(t) = η 1 (t) = η 2 (t) = 1 2 e T (t)e(t).Simulation results in Figure 12(a) show that the proposed controller (17) brings the system synchronization error energy to zero steady-state in less than 0.046 s, while η 1 (t) and η 2 (t) do not converge.Figure 12(b) illustrates that η 1 (t) converges in 1.5 s, and η 2 (t) does not converge; it remains oscillating until the end of the simulation time, two seconds.
Table 7 compares the energy dissipation rate regarding the Lyapunov function gradient.
The inequality (35) is obtained using Table 6.The time gradient of a constant synchronization error energy function is zero.V(t) stays at zero after 0.044 s, indicating the closed-loop remains at zero energy level after convergence.V1 (t) = 0 shows that controller (33) takes 2 s to keep the closed-loop at zero after convergence but oscillates in the steady state.V2 (t) = 0 demonstrates that the closed-loop will not reach zero as the energy gradient function becomes zero before the convergence; it shows that the controller (34) is not successfully bringing the system's synchronization error energy to zero.Therefore, the control signals shown in Figure 11(c) are ineffective in attaining zero synchronization error energy function.In this case, the control signal energy develops sustained oscillations in the synchronization error states.
The inequality (35) and the simulation results in Figure 13(a and b) illustrate that the proposed RDASCS (17) furnishes faster convergence than the other two controllers as the energy dissipation initial rate is high.In the vicinity of zero, the energy dissipation rate is low.This characteristic improved closed-loop stability, reduced computation time, and increased synchronization accuracy.

Conclusions
This paper discusses the chaotic behaviour of a financial system using theoretical analysis and numerical simulations.Then, it proposes a novel adaptive control algorithm for synchronizing two identical fourdimensional chaotic financial systems supported by a detailed stability analysis of the designed closed-loop.The computer simulation results and theoretical analysis show that the proposed adaptive controller achieves faster and smoother synchronization error convergence to the origin for the unknown time-varying model with uncertainties and exogenous disturbances.The Lyapunov second stability theorem guarantees the closed-loop's robust performance against bounded time-varying unknown exogenous disturbances and un-modelled dynamics.The article also includes theoretical and computer simulation analysis demonstrating that the designed closed-loop behaviour is lesser oscillatory and that control effort is chatter-free and smoother than the other state-of-the-art controllers selected for the comparative study.The proposed robust synchronization control of two identical hyperchaotic finance systems can be beneficial in improving the security, stability, prediction, performance, and flexibility of financial systems.
Timely buffeting-free synchronization can improve financial systems' behaviour.Therefore, pre-assigned settling time and buffeting suppression synchronization algorithms should be developed in future.The proposed article can provide a paradigm for the analysis.

( 1 )
The controller should avoid the closed-loop's nonlinear terms cancellation.(2) It should reduce the state error vector trajectories and control signal oscillations.(3) It should show robustness to time-varying unknown model uncertainties and exogenous disturbances.(4) It should achieve faster error vector convergence rates in the origin's direction.

Figure 4 Figure 7 4 Figure 10
Figure No/ Errors convergence time
behaviour, and the error vector does not achieve a steady-state and oscillates in the range [−0.2, 0.15].The error signals from the adaptive controller (34) lose synchronization behaviour after 15 s, as shown in Figure 10(b).

Figure 11 (Figure 11 .
Figure 11(a) illustrates the proposed controller (17) transient behaviour showing oscillations in the range [−0.8.0.8]; it takes appropriate action to compel the state errors vector to the origin.The control signals of the SMC strategy (33) exhibit chattering behaviour and oscillate in the range [−1.5, 1.5], i.e. the steady state does not establish.The control input signals of

Figure 13 (
Figure13(a)  shows the behaviour of energy dissipation rate functions V(t) and Vi (t), i = 1, 2 accomplished by controllers(17), (33), and (34), alternatively for 0.05 s and Figure13(b) for 3 s.The time gradient of a constant synchronization error energy function is zero.V(t) stays at zero after 0.044 s, indicating the closed-loop remains at zero energy level after convergence.V1 (t) = 0 shows that controller (33) takes 2 s to keep the closed-loop at zero after convergence but oscillates in the steady state.V2 (t) = 0 demonstrates that the closed-loop will not reach zero as the energy gradient function becomes zero before the convergence; it shows that the controller (34) is not successfully bringing the system's synchronization error energy to zero.Therefore, the control signals shown in Figure11(c) are ineffective in attaining zero synchronization error energy function.In this case, the control signal energy develops sustained oscillations in the synchronization error states.The inequality (35) and the simulation results in Figure13(a and b) illustrate that the proposed RDASCS(17) furnishes faster convergence than the other two controllers as the energy dissipation initial rate is high.In the vicinity of zero, the energy dissipation rate is low.This characteristic improved closed-loop stability, reduced computation time, and increased synchronization accuracy.

Table 1
describes the terminology.

Table 4 .
Comparison of the computer simulation results.

Table 5 .
Comparison of the computer simulation results.

Table 7 .
Comparison of the rate of dissipation of the energy function.