A Type-3 Fuzzy Approach for Stabilization and Synchronization of Chaotic Systems: Applicable for Financial and Physical Chaotic Systems

In this paper


Introduction
Chaos theory studies the mathematical formulation and behavior of dynamic systems sensitive to initial conditions. Chaos often has been seen as an interdisciplinary theory for exploring the randomness of complex chaotic systems to identify fundamental fractals, self-organization, basic patterns, fixed feedback loops, repetitions, and self-similarities. ese special features have provided good potential applications for chaotic systems. Recently, various applications have been reported for chaotic systems such as image encryption [1], chaotic maps [2], time series [3], optimization algorithms [4], medical systems [5], and secure communications [6]. e control of chaotic systems (CSs) is a complex control problem because of their complex nonlinear dynamics, high senility to the initial condition, hard dynamic perturbation in most of their applications, and stochastic dynamical behavior such as symmetry and dissipation. e controllers in this field can be classified into three classes: classical methods, neurofuzzy controllers, and hybrid control methods.
For the first class, some model-based controllers have been presented. For example, in [7], the passivity-based approach is developed using a sliding-mode controller (SMC) and it is applied for unified CSs. In [8], bifurcation analyses are presented for a CS, and by the Lyapunov method, a robust synchronization scheme is proposed. e feedback controller is designed in [9], and the stability is investigated using the Barbashin-Krasovskii approach. e adaptive SMC is studied in [10], and its accuracy is evaluated on chameleon CSs. In [11], the behavior of CSs is analyzed and by the use of the finite-time stability theorem, the stability is studied in various conditions. In [12], the SMC scheme is developed and the bifurcation diagrams are analyzed. In [13], a fixed-time convergence scheme is presented for memductance-based CSs and the robustness is examined under bounded perturbations. In [14], the bifurcation of butterflyfish CSs is investigated and a linear control method is suggested for synchronization. e model-based adaptive controller is developed in [15] to investigate the stability of a generator of nuclear spin CSs. e projective control systems and synchronization of CSs are studied in [16].
e FLSs and neural networks (NNs) are widely used to cope with uncertainties. For chaotic systems, some neurofuzzy methods have also been studied. For example, in [17], NNs are used for prediction problems in Hyperjerk CSs and their physical circuit implementation is investigated. e finite-time synchronization of CSs is studied in [18] and, by analyzing the convergence time, an FLS-based controller is presented. In [19], the synchronization problem is studied by the use of NNs and the designed synchronized scheme is applied for a secure communication system. In [20], a Takagi-Sugeno FLS is used for modeling and a sampled-data control system is developed for synchronization. e FLSbased SMC is formulated in [21], and the better performance of FLS-based schemes is shown by applying a gyroscope CS. e backstepping SMC is suggested in [22] for stabilizing CSs in the presence of time delay, and FLSs are used to estimate some nonlinearities. In [23], based on reinforcement learning and FLSs, a synchronization scheme is developed. e performance of FLS-based synchronization methods is evaluated and analyzed in [24]. In [25], improvement of the synchronization accuracy and the convergence speed is studied by the use of FLSs. In [26,27], the superiority of FLS-based controllers in robotic applications is studied. e financial CSs are widely used in economic problems [28][29][30]. e dynamics of these classes of chaotic systems are much more complex because of the existence of various unpredictable factors. e stabilization and synchronization of financial CSs have been rarely studied. For example, the integral SMC is designed in [31] and the stabilizing conditions are studied. Similarly, the terminal SMC is developed in [32] and the control of a financial CS is analyzed. In [33], an H ∞ -based control system is designed and FLSs are used to investigate the robustness. e risk assessment of financial CSs is investigated in [34], and an FLS-based method is presented. In [35], an FLS-based system is proposed for forecasting applications. e finite-time control of hypercritic financial systems is investigated in [36], and some adaptation rules are suggested for stabilizing. A neuro-fuzzy-based controller is designed in [37] to guarantee the stability of the financial CS. e literature reviewing show that (i) In most studies, the controller is designed for a special case of CS and the designed controller cannot be applied for a CS with different parameters and dynamics [7][8][9] (ii) e stability of most reviewed controllers is not guaranteed under perturbations and uncertainties [30,38] (iii) Some type-1 and type-2 neuro-fuzzy controllers have been developed for CSs, but most of them cannot handle the high uncertainties in CS dynamics [33][34][35][36][37] (iv) Most of the previous studies are optimized in an offline scheme [33][34][35][36][37] (v) e robustness against unknown perturbations in CS dynamics needs more studies [9,36,37] Regarding the above discussion, a new type-3 fuzzybased controller is developed for CSs. Most recently, the type-3 FLSs with strong uncertainty modeling capability have been developed. In various studies, it has been shown that the T3-FLSs result in much better performance in a high-noisy environment [39]. Some T3-FLS-based controllers have been developed MEMS gyroscopes and a class of fractional-order CSs [39][40][41]. In these studies, the T3-FLSbased controllers are developed for first-order fractionalorder CSs and the special cases of MEMS gyroscopes. To the best of our knowledge, stable and robust controllers based on T3-FLSs have not been studied for financial CSs. As mentioned earlier, this class of CSs exhibits a complex stochastic behavior and their dynamics is perturbed in most applications. In this study, a new control scenario is suggested on basis of T3-FLSs. e designed T3-FLSs are optimized by the Lyapunov learning rules. e robustness is analyzed in both stabilization and synchronization problems, and an adaptive compensator is developed to deal with AEs. e main contributions are (i) e designed controller does not depend on the parameters and dynamics of the case study CS. It can be easily applied for the various cases of CSs. (ii) e stability is ensured under perturbations and uncertainties. (iii) A T3-FLS-based approach is developed for better handling the uncertainties. (iv) e suggested controller is tuned in an online scheme. In other words, the free parameters of the T3-FLS are tuned at each sample time. is approach can handle unpredicted perturbations.
(v) e robustness is studied, and a compensator is developed.

Problem Formulation
e following financial CSs are considered [42]: where b denotes the investment cost, a represents the saving amount, c is elasticity of commercial demands, _ χ 1 is the interest rate, _ χ 2 represents investment demand, and _ χ 3 is the price exponent. e bifurcation and the Lyapunov spectrum analysis have been studied in [43,44]. e phase portraits are depicted in Figure 1 that shows the chaotic attractors.
2 Complexity e variation of the investment cost, the saving amount, and elasticity of commercial demands are assumed to be unknown, and also the dynamics is unknown. e suggested T3-FLS is used to cope with variations of parameters and dynamic uncertainties and perturbations. e suggested control diagram is shown in Figure 2. We see that T3-FLSs are optimized such that all states are stabilized and the outputs track the reader system. e compensators deal with the perturbations.

Remark 1.
e case study CS 1 is considered in this paper because it is so popular in the field of economic plants, and it exhibits more complex chaotic behavior. However, the suggested controller does not depend on the dynamics, and it can be easily applied for the various cases of CSs.

Type-3 FLS
e neuro-fuzzy systems and learning methods are extensively used to tackle uncertainties. In this paper, a new stronger approach is formulated to cope with uncertainties in the complex dynamics of CSs. e suggested FLS scheme is depicted in Figure 3. All uncertainties are tackled using optimized T3-FLSs. e computations are given as follows: (1) e inputs of the T3-FLS ψ i are χ 1 , χ 2 , and χ 3 .

Complexity
Similarly, for Ω (3) In the next step, considering the memberships in step 2, the rule firings should be computed. We have R rules. e r − th rule is written as where x � r , � x r denote the rule parameters. (4) By the use of the product T-norm and the simple type-reduction of [45], the output of ψ i is written as and ς i are where R is number of rules and � x ir and x � ir represent the r − th rule parameters. ς �ir and � ς il are where n ] denotes slice numbers and

Stabilization
In this section, the stabilizing controller is designed and its stability conditions are analyzed. e main results are presented in eorem 1.
where ι and k are constants, ψ i denotes the T3-FLS, χ and f are the input vector and the rule parameter vector of ψ i , respectively, and E i is the upper bound of AE.
Proof. By applying the controllers u i , i � 1, 2, 3 (17) into (1), we have From (31), _ χ is written as e optimal T3-FLSs (ψ * i (χ|f * i )) are defined as T3-FLSs with optimal parameters (f * i ). e optimal T3-FLSs are added and subtracted into (32); then, we have By simplification, the dynamics of χ i , i � 1, 2, 3 in (33) is rewritten as _ By the use of the vector from (5) we have where f i � f * i − f i . For stability investigation, to prove the results of eorem 1, a Lyapunov candidate is considered as e time derivative of (38) yields Substituting from _ χ i , i � 1, 2, 3 (26)- (28) into (39), we obtain 6 Complexity Equation (40) can be rewritten as Equation (11) is simplified as Considering the tuning rules (18), from (42), we obtain From (43), we have By applying the u ci from (9), we have Considering the fact that We can write _ V < 0, and then, the proof is completed. □

Synchronization
In this section, the synchronization controller is designed and its stability conditions are analyzed.
where ι and k are constants, ψ i denotes the T3-FLS, s i is defined as s i � χ i − r i , the input vector s is defined as s � [s 1 , . . . , s n ] T and n represents the state number, f is the rule parameter vector of ψ i , respectively, and E i is the upper bound of AE.
Proof. By applying the controllers u i , i � 1, 2, 3 (28), the dynamics of (47) is written as From (31), _ s is written as Similar to the proof of eorem 2, the optimal T3-FLSs are added and subtracted into (32); then, we have e dynamics of (33) is rewritten as By the use of the vector form (5), we have _ where To prove the results of eorem 2, a Lyapunov candidate is considered as e time derivative of (38) yields Substituting _ s i , i � 1, 2, 3, from (35)-(37), we obtain Equation (40) can be rewritten as Equation (29) is simplified as Considering the tuning rules (29), we obtain 8 Complexity From (43), we have By applying u ci from (9), we have Considering the fact that we can write, _ V < 0, and then, the proof is completed. □

Simulation
By various simulations, the feasibility and accuracy of the stabilizing and synchronization scheme are verified.

Stabilization.
In this section, the dynamics of (1) is stabilized by controller (17), tuning rules (18), and compensator (19). e simulation conditions are described in Table 1. e initial conditions are as follows: χ 1 (0) � −0.20, χ 2 (0) � 1.50, and χ 3 (0) � 0.30. e dynamics is unknown. A general view on the simulation scheme is depicted in Figure 5. e trajectories of outputs χ 1 , χ 2 , and χ 3 are depicted in Figure 6. We see that signals χ 1 , χ 2 , and χ 3 are well converged to zero. e corresponding control signals u 1 , u 2 , and u 3 are given in Figure 7, and the outputs of T3-FLSs (ψ 1 , ψ 2 , and ψ 3 ) are depicted in Figure 8. To better see the convergence to the zero point, the phase portrait is shown in Figure 9. We see that the phase trajectories have well reached the origin at a finite time.

Synchronization.
In this section, the synchronization accuracy is examined. e chaotic system (1) is considered as a leader system, and the slave financial chaotic system is assumed as follows: System (47) is synchronized with (1) by using controller (40), tuning rules (41), and compensator (42). e simulation conditions are the same as those of the previous section. e initial conditions are as follows: e trajectories of outputs χ 1 , r 1 , χ 2 , r 2 , and χ 3 , r 3 are depicted in Figure 10. We see that signals χ 1 , χ 2 , and χ 3 are well converged to r 1 , r 2 , and r 3 . e synchronization errors in Figure 11 show a good synchronization performance. e corresponding control signals u 1 , u 2 , and u 3 are given in Figure 12, and the outputs of T3-FLSs (ψ 1 , ψ 2 , and ψ 3 ) are depicted in Figure 13. To better see the synchronization, the phase portrait is shown in Figure 14. We see that the phase trajectories of the slave system have well reached the leader system at a finite time.

High Noisy Condition and Comparison.
In this section, a high perturbation is applied to the system as shown in Figure 15. e simulation conditions are the same as those of the previous examination. e trajectories of outputs χ 1 , r 1 , χ 2 , r 2 , and χ 3 , r 3 are depicted in Figure 16. We see that the perturbations are well tackled, and the signals χ 1 , χ 2 , and χ 3 are well converged to r 1 , r 2 , and r 3 . e synchronization errors are shown in Figure 17 that shows a desired synchronization under high-level disturbances. e corresponding control signals u 1 , u 2 , and u 3 are given in Figure 18, and the phase portrait is shown in Figure 19. We see that similar to the previous examination, the phase trajectories of the slave system have well reached the leader system at a finite time in the presence of noisy conditions.

Complexity
To examine the superiority of the suggested control scenario versus other similar approaches, a comparison is presented. e root-mean-square of synchronization errors (RMSEs) for the designed controller are compared with those the type-1 FLS-based controller (T1-FLC) [46], type-2 FLS-based controller (T2-FLC) [47], and generalized FLSbased controller (GT2-FLC) [48]. e comparison results are shown in Table 2. We see that the designed controller gives better synchronization accuracy in high noisy conditions.

Complexity
Remark 2. In this paper, a new T3-FLS-based controller is developed for both stabilization and synchronization of financial CSs. e simulations under various conditions show that the designed controller gives the desired efficiency. Even under the high noisy conditions and completely unknown dynamics, it is seen that the presented approach well regulates the target trajectories into reference. Also, by using the designed compensator, the robustness is well preserved under high-level perturbations. e designed controller does not depend on the model of CSs, and then, it can easily be applied to other cases of CSs.

Conclusion
In this paper, the synchronization and stabilization of financial CSs are studied and a new control system is presented. e dynamics is known and is approximated by using the designed T3-FLSs. e robustness is proved through the Lyapunov method. e accuracy of the designed controller is examined in three cases. In the first examination, the stabilization performance is investigated and it shows that all states are perfectly converged to zero at a finite time. For the second examination, the suggested controller is applied for a synchronization problem and it is shown that the case study financial chaotic system well tracks a nonidentical chaotic system; the synchronization errors have   Data Availability e data that support the findings of this study are available within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.