General function approximation of a class of cascade chaotic fuzzy systems

Abstract

This paper presents an innovative architecture called cascade chaotic fuzzy system (CCFS) for the function approximation and chaotic modeling. The proposed model can dominate complications in the type-2 fuzzy systems and increase the chaotic performance of a whole framework. The proposed cascade structure is based on combining two or more one-dimensional chaotic maps. The combination provides a new chaotic map with more high nonlinearity than its grain maps. The fusion of cascade chaotic structure into the neurons of the membership layer of a conventional fuzzy system makes the CCFS more capable of confronting nonlinear problems. Based on the General Function Approximation and Stone-Weierstrass theorem, we show that the proposed model has the function approximation property. By analyzing the bifurcation diagram and applying the CCFS to the problem of chaotic modeling, the new model is investigated. Simulation results and analysis are demonstrated to illustrate the concept of general function approximation.

Keywords

Chaotic fuzzy system function approximation chaotic neural network oscillatory neuron

1 Introduction

Two main approaches to fuzzy system design have been followed in research. Type-1 fuzzy sets in which the membership functions are definite and Type-2 Fuzzy Sets in which the membership functions are uncertain and fuzzy [1]. In other words, Type II fuzzy sets are extended type-I fuzzy sets. Because of their higher degree of freedom, these sets are used to model and minimize uncertainties and increase the accuracy of the approximation. In parallel with research on the structures of fuzzy systems, numerous studies have been done in the field of function approximation of fuzzy systems [2]. The approximation capabilities of fuzzy systems were first investigated in [3] from the application point of view. In this first study, it is shown that a fuzzy system can approximate any arbitrary continuous function on a compact universe of discourse. In [4], approximation capabilities of Mamdani-type controllers were studied. In [5], an investigation of MISO 1 fuzzy systems, including Gaussian membership function, Larsen inference engine, and centroid defuzzification was done. The mentioned fuzzy system could approximate continuous functions with the desired accuracy. Besides, in [6], the Stone-Weierstrass theorem was applied to the previously described fuzzy systems to exhibit the density of the system in C (Iⁿ). Different types of fuzzy systems have been investigated for general approximation capability so far [7]. For instance, in [7] studying the fuzzy systems with various MF 2 and defuzzification methods has been done. It has been proven that fuzzy systems with predefined parameters, including the center of area defuzzification, are universal approximators without considering the number of rules and construction methods. An investigation of the approximation capability of Sugeno-type controllers was done in [8]. The results were interesting. Any continuous function could be approximated through fuzzy systems with only two rules and normalized fuzzy sets in the antecedent parts. Furthermore, the MF of the antecedents should be achieved through a straight conversion of the function to be modeled. So, if there is no adequate knowledge about the system under study, as in most real-world problems, this conclusion does not lead to a meaningful help for achieving the parameters of the fuzzy system. Although the first research on universal approximation mainly focused on Mamdani controllers, the same achievements were acquired for Sugeno controllers from the late 1990 s [9]. Sugeno fuzzy systems, including linear functions in the consequent part [9] and fully overlapped MFs, are proved to be universal approximators. In [10], a grid-based FS with high order Takagi-Sugeno rules is presented for function approximation while reducing the number of rules. In [11], a fuzzy recurrent wavelet neural network based on the TSK fuzzy system is presented, and the abilities of system identification and function approximation are studied. Moreover, the approximation capability of the type-2 fuzzy logic was studied in various papers [12]. Another technique for function approximation based on the fuzzy approach is proposed in [13]. The method is combined with the reinforcement learning approach in multi-agent systems. Also, the creation of helpful tools incorporating fuzzy logic and neural systems, which leads to the approximation of any continuous function with arbitrary accuracy, was presented in [14]. Various combining methods can be considered in this area. T1FNN 3 [15], and IT2FNN 4 [16] are some outstanding structures, and their efficiency and universal approximation properties were reported in research. However, research around the function approximation capability of chaotic fuzzy systems are rare and should be followed.

Considering the outstanding FSs studied in the function approximation research, the most critical limitations of these FSs, are as follows: 1- crisp and predefined primary membership function for type-1 fuzzy systems, 2-blurring all points of input space leading to the unessential increased number of generated T2FSs and increased complexities of type-2 computation, 3-limitation of upper and lower membership functions to convex membership function and ignoring non-convex fuzzy systems, decreasing the inference speed. According to the fact that the human brain originates the fuzzy theory, it is probable that its limitation can also be resolved by integrating the functional properties of the human brain into the fuzzy theory. As reported by the neuroscience researchers [17], fuzziness and chaotic dynamics are the two significant properties of the human brain, whereas fuzziness incorporates in the proper decision making and chaotic dynamics incorporate in the processing of large amounts of information immediately [18]. Hence, combining these two features in the unique structure can generate a robust and flexible intelligent network that includes approximate reasoning of fuzzy logic and chaotic behavior of chaotic systems. Compared to T1FS and T2FS, chaotic fuzzy systems are more compatible with brain function. Therefore, they could be more effective in decreasing the time of information processing. Nevertheless, these systems have not been analyzed carefully yet.

Regarding the critical role of chaos and chaotic dynamics in the brain, these dynamic systems have received much attention. In particular, chaotic maps, as traditional dynamical systems, are the basis of many studies [19, 20]. Ergodicity and unpredictability are the significant properties of these maps. The first feature leads to a comprehensive search without considering duplicate points, and the latter generates dissimilar sequences using different initial values or parameters of the chaotic maps. Regarding this remarkable feature, chaotic maps are practical tools in various domains such as cryptography, mathematics, computer science [21, 22]. There are two classes of chaotic maps, one-dimensional (1-D) and high-dimensional (HD) chaotic maps [22, 23]. The former are mathematical functions that track the single variable progress upon a discrete phase in time and the latter simulates the progress of two or more variables. From the structure point of view, these maps are understandable and are simple to construct. The chaotic properties of these maps are strong, and a variety of applications can benefit from these maps. However, there is a severe drawback of the 1-D chaotic maps, such as limitation of chaotic span, few parameters, and easy predictability of function output with low complexity [23]. HD chaotic map outperforms the 1-D chaotic map in security and application points of view [24]. Their chaotic behavior is much more complex, so the prediction of their routes is more complex, too. However, this complexity increases the computational cost of the system, and therefore some application potential of the HD chaotic maps, especially in the real-time domain, is restricted.

Cascade Chaotic System (CCS) is proposed to remove the restricted efficiency of 1-D chaotic map and high-computation cost of HD chaotic map. In general, CCS proposes a 1-D structure that utilizes the features of 1-D and HD Chaotic maps by connecting two one-dimensional chaotic functions in series. “The output of the first seed map is linked to the input of the second seed map. The output of the second one is fed back into the input of the first one for recursive iterations, and it is also the output of CCS” [22]. Generation of a new chaotic map using any two 1-D chaotic maps is one of the most promising features of the CSS framework. The newly generated chaotic maps have outstanding chaotic features in comparison with their primary seed maps.

The main contribution of this paper is presenting a new framework to overcome the mentioned limitations of classic fuzzy systems and providing a framework for function approximation and chaotic modeling. In the proposed structure, a coupled chaotic map (CCM) is served as a bifurcating fuzzy system (BFS) in a unique structure. The BFS is homogeneous to coupled chaotic maps in which membership function is determined by one of the CCM variables. Besides, BFS is homogenous to an oscillatory neuron biologically, where the AF 5 of the input neuron is the membership function of the BFS. So, the bifurcation diagram of a chaotic map can exemplify a fuzzy system with blurred parameters. If each point of the input space is regarded as a bifurcation parameter, the bifurcation diagram can generate bifurcating blurring type-1 fuzzy systems. Besides, this structure can provide a broad scope of fuzzy systems from type-1 or type-2 to chaotic or non-chaotic and from convex or non-convex to normal or non-normal fuzzy systems. Therefore, according to the generality of this concept, it could be efficient in increasing the speed of information processing compared to type-2 fuzzy systems and remove the limitation of these systems because of their flexible bounds.

The second category of innovations concentrates on the chaotic performance of the framework. To date, applying tools such as cascade systems and neural networks, to the chaotic fuzzy systems has not been investigated. To the best of our knowledge, there are limited studies in this area. Therefore, providing a general, flexible structure is one of the scopes of this paper. This paper proposes a new chaotic fuzzy system, namely cascade chaotic fuzzy system (CCFS). Ergodicity properties of chaos search plus chaos entropy make the system more robust. Using the remarkable features of CCS and embedding it into a conventional fuzzy system, the CCFS is generated. This combination provides a more complex and highly nonlinear behavior. Fusion of cascade chaotic structure into the membership layer of the FS increases the capability of the CCFS to confront nonlinear problems because of its high nonlinearity weight. An impressive chaotic search with maximum entropy is done for the membership values of the fuzzy system that covers the boundaries and ignores the duplicate points. This efficient search seems to improve the total performance and reduce the time complexity of the fuzzy system. The weights of nonlinearity in the proposed architecture are analogous to the weights of nonlinearity in the modeled system. This feature is suitable when complex systems such as weather forecasting systems, traffic time series, and encryption systems are modeled. In short, the notable contributions of this paper are as follows:

Introducing the CCFS model as a general framework of function-independent chaotic fuzzy system.

Improving the chaotic behavior of the proposed fuzzy model by using the cascade structure in the membership layer of the fuzzy system.

Designing the proposed architecture in two phases of chaos generation and chaos application, the final tool is not affected by the complexities of the chaos generation process.

Introducing the novel chaotic fuzzy function approximator and using the Stone-Weierstrass theorem to exhibit the function approximation capability of the proposed model.

Designing a T2FS through a chaotic type of T1FS by applying the Lee oscillator and excitatory and inhibitory neurons as chaos generators in FS makes the overall system more robust.

The rest of the paper is organized as follows. Section 2 gives a brief review of the chaotic fuzzy system, its inspiration, and the development of CFS 6 , especially in the field of chaotic membership function. Cascade chaotic system and its analysis based on the Lyapunov exponent are considered in section 3. The proposed model is introduced in section 4. The bifurcation behavior of the CCS is presented in this section. Moreover, the applied oscillator, along with its equations and the detailed description of the CCFS in the layered format, are presented in this section. Section 5 proves the approximation capability of the proposed method by utilizing the Stone-Weierstrass theorem. Section 6 applies the CCFS to the problem of chaotic system identification and forecasting to demonstrate the notion of function approximation. Finally, section 7 presents the conclusion and proposed notes about future works.

2 Review on Chaotic fuzzy systems

The interaction between Chaos theory and fuzzy systems has received much attention in the last two decades. The study on this category displays two main zones. One is to incorporate fuzzy systems and chaos theory in order to simulate and examine chaotic systems [25, 26], and to forecast the chaotic time series [27]. This zone has been widely investigated and applied in different applications. The other zone is the hybridization of chaotic maps and fuzzy systems, called chaotic fuzzy systems. The survey of this zone is far limited and can be divided into four categories. Although, investigation of function approximation has not been studied in either type. In the first category, a chaotic map is added to a membership function of a fuzzy system [26, 28]. In the second category, the input of a chaotic map is a fuzzy system. Chaotic iteration of a fuzzy system and chaotic mapping are instances of this category [29]. In these researches, the primary fuzzy system is converted to a different Fuzzy system after several iterations of the chaotic map. In the first and second categories, the structure of the asymptotic set is highly chaotic. The third category is dedicated to the chaotic generation of the MF 7 parameters. In fact, in this class, the parameters of a fuzzy membership function are determined through a chaotic map. For instance, as mentioned in [30], for the generation of triangular fuzzy MF parameters (a, b, c), using a logistic map, a logistic map f(x) = λx(1 - x) should be adjusted to MF domain changes, first. Considering J for the domain changes of MF, f(x) can be rewritten by: f(x) = λx(J - x)/J; in which the dynamics of f is determined by the bifurcation parameter λ. Second, three distinct values of λ (for example, λ₁ =3.87, λ₂ = 3.9, and λ₃ = 3.93) should be selected in which f has chaotic behaviors. Third, three chaotic numbers should be produced through the mentioned values of λ. Finally, the parameters a, b, and c are specified as a minimum, middle, and a maximum of the above chaotic numbers, respectively. In this class of chaotic fuzzy systems, the achieved set has no blurred membership range. In the fourth and the last category, a paired chaotic function is used as a membership function. There is limited research in this area. LOCFM presented by Wong et al. [31] is the only model proposed in this category. The shoulder chaotic membership function presented in the Wong paper is defined as follows [31]: $\begin{matrix} u^{'} = tanh (a_{1} u - a_{2} v + I), \\ v^{'} = tanh (b_{1} u - b_{2} v), \\ w = tanh (I), \\ z = m . ((u^{'} - v^{'}) . e^{- {kI}^{2}} + w) + n, \end{matrix}$ (1) in which u, v, w, and z denote the variables for excitatory, inhibitory, input, and output neurons, respectively. a₁, a₂, b_1, and b₂ are the constant values indicating the weight parameters. I is the external signal, and k is the decadence constant. Tanh() is a hyperbolic tangent function; m and n are the slope and intercept parameters for the shoulder function. These sets are the improved presentation of the Lee chaotic oscillator. These sets are achieved by combining fuzzy membership functions and chaotic oscillators. Hence, it seems that developing the integration of chaotic maps and fuzzy membership functions and presenting a function independent general structure for chaotic fuzzy systems is needed. This is the other motivation followed in this paper. Moreover, according to the lack of research around the function approximation property of chaotic fuzzy systems, examining the GFA 8 property of the proposed chaotic fuzzy system is followed, too.

3 Cascade chaotic system (CCS)

Cascade chaotic structure is inspired by cascade structure in electronic circuits and based on a combination of two or more 1-D chaotic maps it can produces a new 1-D chaotic map. The architecture of CCS is shown in Fig. 1. G(x) And F(x) denote two-grain maps connected through the CCS. An input vector is fed to the CCS structure, and the output of G(x) is calculated and imported to F(x) [22]. Similarly, the output of F(x) is calculated and fed back to the G(x) in the recursive mode. The F(x)’s output is the CCS output in recursive and non-recursive mode [22]:

Fig. 1

The general architecture of cascade chaotic system.

$x_{n + 1} = Γ (x_{n}) = F (G (x_{n})),$ (2) where F(x) and G(x) are two-grain maps and can be any 1-D chaotic map, these two chaotic maps can be the same or different due to problem consideration. If F(x) and G(x) are identical, CCS is substituted with the structure that a chaotic function cascaded with itself: $x_{n + 1} = F (F (x_{n})) or x_{n} + 1 = G (G (x_{n})) .$ (3)

If F(x) and G(x) are not identical, CCS is substituted with a new structure and defined by the following [22]: $x_{n + 1} = F (G (x_{n})), or x_{n + 1} = G (F (x_{n})) .$ (4)

Various new chaotic maps (NCMs) are generated using distinct grain maps. This versatility, along with more parameters of the CCS structure, yields a consistent framework with complex and high nonlinear chaotic behavior. As mentioned before, the CCS structure can be developed into three or more layers using different cascaded grain maps. Although the result of CCS with more layers has a much more complicated output and also better chaotic behavior, this developed cascaded structure confronted with severe problems such as time delay and difficulty in hardware implementation besides the saturation phenomena in the output. The CCS output series have the anatomy of the first map, second map, or both. The CCS consists of all parameters of its grain maps, providing complex properties of CCS output. For demonstrating and analyzing the chaotic properties of CCS, due to the difficulty of direct analyzing [32], Lyapunov exponent (LE) is adopted. LE of a dynamical system is the divergence rate of infinitesimally close trajectories in the phase space [33]. It can be applied to demonstrate the chaotic behavior of a dynamic system. Suppose that δZ₀ is the initial separation of two trajectories in the phase space, the divergence of these two trajectories is calculated after time t using the following equation [22]: $| δ Z (t) | \approx e^{λ t} | δ Z 0 |,$ (5) where λ denotes the Lyapunov exponent, in the previously described CCS equation (Equation (2)), suppose that x₀ and y₀ are near the initial value of CCS, x_n, and y_n denote the forenamed path after the nth iteration. Therefore, LEs of Γ(x) is described by [22]: $λ_{Γ (x)} = lim_{n \to \infty} \frac{1}{n} \sum_{i = 0}^{n - 1} \ln | \frac{dF}{dx} | x_{i} | + lim_{n \to \infty} \frac{1}{n} \sum_{i = 0}^{n - 1} \ln | \frac{dG}{dx} | x_{i} | .$ (6)

Similarly, LEs of grain maps F(x) and G(x) are given by: $λ_{F (x)} = lim_{n \to \infty} \frac{1}{n} \sum_{i = 0}^{n - 1} \ln | \frac{dF}{dx} | x_{i} |,$ (7) $λ_{G (x)} = lim_{n \to \infty} \frac{1}{n} \sum_{i = 0}^{n - 1} \ln | \frac{dG}{dx} | x_{i} | .$ (8)

Thus, Equation (6) can be rewritten as follows [22]: $λ_{Γ (x)} = λ_{F (x)} + λ_{G (x)} .$ (9)

Hence, based on the above equation, LE of cascade chaotic structure is defined based on the summation of LE values of its two-grain maps. If λ_Γ(x) is bigger than zero, the trajectories of two initially nearby orbits significantly diverge as the iteration number increases and CCS changes to chaotic behavior. The more positive the LE exponent, the higher the divergence, leading to better chaotic efficiency. Based on the analysis and experiments reported in [22], chaotic behaviors of CCS are as follows: If λ_F(x) λ_F(x) and λ_G(x) are positive, then λ_Γ(x) is positive and bigger than the LE of its two-grain maps. It means that if both grain maps are chaotic, then the CCS is also chaotic and has superior chaotic performance than its grain maps. If λ_F(x) and λ_G(x) are negative, then λ_Γ(x) is negative too, and there is not any chaotic behavior in the CCS output sequences. If λ_F(x) is negative and λ_G(x) is positive or vice versa, then λ_Γ(x) is positive whenλ_F(x) + λ_G(x) is positive, and λ_Γ(x) is negative when λ_F(x) + λ_G(x) is negative. It means that if there is only one chaotic grain map, the chaotic output of CCS depends on the value of λ_F(x) + λ_G(x); if it is positive, the CCS behavior is chaotic, and if it is negative, the CCS behavior is non-chaotic. In short, it could be said that CCS behavior is chaotic when there is at least one chaotic grain map. Also, if both of the grain maps are chaotic, the chaotic performance of each grain map is dominated by the chaotic performance of the CCS. Overall, it seems that the cascade structure could enhance the chaotic properties of primary functions.

In short, the limitations of conventional fuzzy systems are described in section 1. These restrictions include the limited performance of type-1 fuzzy systems and computation complexity of type-2 fuzzy systems. According to these limitations and the lack of research around the hybridization of chaos theory and fuzzy systems, to construct a general framework independent of the function type (as in LOCFM), this paper proposes a new fuzzy system benefited from the ability of cascade structure in system modeling. It seems that using inherent features of chaos like ergodicity, entropy, and variability that can be enhanced through cascade structures, the outstanding results of type-2 fuzzy systems can be achieved through type-1 fuzzy systems, and it is the main contribution of this paper. Hence, this paper uses cascade chaotic model for generating a chaotic fuzzy system. In fact, after generating chaotic properties in the neuronal oscillation model, the cascade structure is used for enhancing it and forwarding it to the inference engine of the fuzzy model. The following section describes the model in detail.

4 Cascade chaotic fuzzy system (CCFS): Proposed method

This section describes the framework in four steps. The primary motivation for creating the overall structure, along with its inspiration, is described in the first step. The second part demonstrates the chaotic fuzzy system based on the Lee oscillator and embedding the oscillator dynamics in the membership layer. Besides, equations of the oscillator, along with its communications in various conditions, are investigated. In the third part, the cascade structure is analyzed through the bifurcation diagram, and the efficiency of the CCS is depicted. In the fourth part, the whole structure of the CCFS is characterized in the layered format.

4.1 Philosophy

As mentioned in section 1, the human brain’s operational and structural specification is a solution to problems in fuzzy system modeling. Embedding these features into the classical fuzzy system is one of the main approaches of this paper leading to a new chaotic fuzzy system (CFS) with divarication and bifurcation. Neuroscience research has revealed that the human brain can process an immense amount of information forthwith due to the chaotic dynamics [34]. Controlled chaos in the human brain is more than a random event that results from the brain’s complexity –involving thousands of connections. Instead, it could be called the critical feature that distinguishes the brain from an AI machine. The human brain can process the uncertainties in the shortest possible time by generating new information or the chaotic search among the old information. Moreover, based on neuroscience studies, the brain provides different neuronal signals by chaotic rules. Then, combining these signals results in a fuzzy feeling [35]. The chaos theory models the feeling, and uncertainty is modeled by fuzzy logic. Therefore, it seems that there is a close relationship between chaotic dynamics and fuzzy modeling. According to the latest finding in neuroscience, neuronal models are a suitable framework to unfold this relation [31], and chaotic oscillators are one of the best neuronal models to emulate the behavior and specification of the neurons. As reported in [36], the advanced LEE model, is an instance of these oscillators. This improved model also encompasses excitatory and inhibitory neurons exhibiting oscillatory dynamics. Altogether, the dynamics of this oscillatory model make it possible to add chaotic properties into the fuzzy system. This feature and attempts to improve it through cascade arrangement are the most important contributions of this paper.

4.2 LEE oscillator

Based on neuroscience studies, neuron behavior is interactive oscillations among excitatory and inhibitory neurons [36]. Due to the properties of chaotic oscillators such as nonlinear dynamics and hierarchy of a system, these oscillators are inherently able to predict and model highly complex and nonlinear systems. The lee oscillator comprises four fundamental elements u, v, w, and z, in a one-layer recurrent neural network. The generalized neurotic dynamics of these neurons are given as follows [31]: $\begin{matrix} u (t + 1) = f [a_{1} . u (t) - a_{2} . v (t) + a_{3} z (t) + a_{4} x - θ_{u}], \\ v (t + 1) = f [b_{3} z (t) - b_{1} . u (t) - b_{2} . v (t) + b_{4} x - θ_{v}], \\ w (t + 1) = μ_{A} (x), \\ z (t) = [u (t) - v (t)] . e^{- k (x - m_{A})^{2}} + w (t), \end{matrix}$ (10) in which u(t), v(t), w(t) and z(t) indicate excitatory, inhibitory, input, and output neurons, respectively; f(x) is the activation function for network neurons; a_1 - 4 and b_1 - 4 are the weight parameters of the neuronal connections. It should be noted that in the presented project, the task of these parameters is just putting the oscillator in the chaotic mode. After finding the values of chaos mode, they are regarded as constants and not parameters. K is a decadence constant, θ_u and θ_v are the thresholds for excitatory and inhibitory neurons, respectively. If a type-1 fuzzy system is imported as an activation function of the input neuron w(t) and an output of the network is regarded as a membership value indicated by μ, then the LEE oscillator becomes a chaotic fuzzy system with the ability to bifurcate. μ_A(x) and m_A express a membership function and mean of the membership function, respectively. Finally, x denotes the input. As mentioned above, μ in the oscillator’s output expresses the chaotic membership value, so it should be in the range of 0 and 1. Figure 2 depicts the neuronal structure of the LEE oscillator that served as a fuzzifier of CCFS. As shown in Fig. 2, the network comprises four neurons, with three neurons interacting with each other having a recursive relation. This interaction and recursive relation produce a chaotic dynamic. The generated chaos in the LEE oscillator is the result of inter-communication between the neurons at a particular input interval and under the stimulation of an input stimulus. Without the input stimulus, the inter-neuronal connections will result from the input neuron plus the recursive connection of the other neuron and the recursive connection from the past behavior of the neuron itself. This recursive relationship of the neuron itself reflects the effect of past behavior on the system. If the stimulus input is applied, the network dynamics will change. The input neuron, the stimulus input, the recurrent relation of the neuron itself, and the other neuron’s recurrent relation lead to the generation of chaos behavior. According to the neuronal structure of the Lee oscillator and by importing the Gaussian function to the oscillator, the chaotic Gaussian membership function is achieved. It is noteworthy that any other function can be applied in the oscillator. Here the Gaussian function is considered as an example. The chaotic Gaussian MF can be expressed by the following equations [31]:

Fig. 2

Neuronal structure of LEE oscillator.

$\begin{matrix} u^{'} = a_{1} u - a_{2} v + a_{3} z + a_{4} . I, \\ v^{'} = b_{3} z - b_{1} u - b_{2} v + b_{4} . I, \\ z^{'} = e^{- \frac{{[(u^{'} . v^{'} . k) - c]}^{2}}{2 σ^{2}}}, \end{matrix}$ (11) in which u, v, and z indicate excitatory, inhibitory, and output neurons, respectively; a₁-a₄ and b₁-b₄ are the weights for the neuronal connections. I is the external signal, and k is the decadence constant. C and σ are the Gaussian function parameters, controlling the center of the peak and the curve’s width.

4.3 Bifurcation behavior of Cascade Chaotic System

As mentioned in section 3, the nonlinearity features and chaotic performance of the CCS are improved. In this section, the bifurcation behavior of CCS is studied using the Gaussian seed map. The behavior of the Gaussian seed map and cascade Gaussian map are evaluated and compared using their bifurcation diagrams. To illustrate and compare the chaotic behavior of the Gaussian seed map and cascade chaotic Gaussian map, the Gaussian map is inserted in the chaotic ambiance. This chaotic ambiance is achieved through the chaotic Lee oscillator. Based on the neurotic dynamics of the Lee oscillator and considering the Gaussian function, a chaotic Gaussian map is generated and expressed by Equation (11). Assume that in Equation (11), parameters are set as below: $\begin{matrix} a_{1} = 0 {. 7856; a}_{2} = 0 {. 6681; a}_{3} = 0 {. 1502; a}_{4} = 0 . 825; \\ b_{1} = - 0 {. 2043; b}_{2} = 0 {. 5078; b}_{3} = 0 {. 919; b}_{4} = 0 . 4852; K = 0 . 4188 . \end{matrix}$

In this study, the chaotic mode has been chosen based on the mentioned reasons, and the above values were selected accordingly. In other words, these four parameters are tuned so that the oscillator exhibits chaotic behavior. After finding the values related to chaos mode, they are regarded as constants and not parameters. So, the above values can be used in the various implementations of the CCFS model without further tuning. Figure 3 Shows the 1-D bifurcation diagram of a chaotic Gaussian map based on the Lee oscillator. Now, this structure is inserted in the cascade platform, and CCS is provided upon a chaotic Gaussian seed map. Figure 4 depicts the 1-D bifurcation diagram of the Gaussian CCS. As shown in figures 3 and 4 , the Gaussian CCS has a wider chaotic span with the same parameters. It means that chaotic behavior provided by CCS not only inherits but also boosts the chaotic performance of their grain maps. These findings confirm the issues related to Lyapunov exponents of CCS and its seed maps, raised in section 3. Based on these results and a similar outcome from [22], the nonlinear term of CCS is enhanced compared to the chaotic seed map. So, it seems that this structure with high nonlinearity weights can be appropriate for analyzing and modeling highly nonlinear systems because the nonlinearity weights in the proposed system are analogous to the nonlinearity weights in the modeled system.

Fig. 3

Bifurcation diagram of Chaotic Gaussian Function.

Fig. 4

Bifurcation diagram of Cascade Chaotic Gaussian Function.

4.4 Cascade chaotic fuzzy system (CCFS): Proposed structure

The proposed cascade structure is depicted in fig. 5. Two hyperchaotic systems are chained to one another and practically act as F and G in Fig. 1. In the proposed structure, each seed map of the CCS is a hyperchaotic system (according to the definition of the hyperchaotic system presented in [37]) with multiple parameters, capable of modeling high nonlinearity and further uncertainties. This property can be strengthened by cascading the structure, producing more complex mapping, and modeling more complex behaviors. This figure presents the cascade model based on the LEE oscillator. Two oscillators are chained to one another. The output of the first oscillator is the input of the second oscillator, and the output of the second oscillator is the entire output of the model that can be fed back to the input of the

Fig. 5

Neuronal structure of the cascade hyperchaotic model based on the Lee oscillator.

$\begin{matrix} u^{″} = a_{1} u^{'} - a_{2} v^{'} + a_{3} z^{'} + a_{4} . I, \\ v^{″} = b_{3} z^{'} - b_{1} u^{'} - b_{2} v^{'} + b_{4} . I, \\ z^{″} = e^{- \frac{{[(u^{″} . v^{″} . k) - c]}^{2}}{2 σ^{2}}}, \end{matrix}$ (12) in which u, v, and z indicate excitatory, inhibitory, and output neurons, respectively. z^′ is the output of the first part of the structure, regarded as an input of the second part. a_1 - 4 and b_1 - 4 are the weights parameters; k is a decadence constant; I is the input stimulus. It should be noted that although the proposed oscillatory neural network has four nodes, only its output enters the second oscillator, and so does the second oscillator. Since the oscillator variables interact with each other, chaos is generated by this interaction. Thus, there is a performance impact of all four nodes on the output node, which is transferred to the next oscillator. Figure 6 shows the entire structure of the CCFS model in five layers. The input layer is the first layer where input signals are fed to the fuzzy system. Nodes in the input layer indicate input linguistic variables. The second layer is the membership layer, where the chaotic structure is applied for computing the membership values using Equation (12). The third layer is the inference layer, where the firing strength of each rule is computed according to the type of inference engine. The product inference engine is used here, according to simplicity in the implementation and training process:

Fig. 6

Structure of the CCFS.

$F^{i} = \prod_{j = 1}^{n} μ_{f_{j}} (x_{j}) .$ (13)

The fourth layer is the normalization layer, where the weights of this layer and the predecessor are the centers of the sets of fuzzy output: $w^{l} (x) = u {(x)}^{l} F_{l}, where u {(x)}^{l} = \frac{1}{\sum_{i = 1}^{M} F_{l}},$ (14) in which l = 1, 2 . . . , M. M is the number of rules. The above computation must be done for each M rule of the fuzzy rule base. Finally, the fifth layer is the output layer, where a defuzzifier is applied to compute the final output, as the center of the set used here: $f (x) = \sum_{l = 1}^{M} {\bar{y}}^{l} w^{l} (x),$ (15) in which ${\bar{y}}^{l}$ is the center of the consequent. The output of chaotic structure enters the fuzzy inference system after examining, presenting, and modifying neuronal activation functions. In the fuzzy system operation, the proposed system is neither fuzzy type-1 nor fuzzy type-2. Despite the type-2 fuzzy system that continuously covers a Min to Max interval and considers all Memberships, the presented chaotic fuzzy system scans all space and provides samples of the entire space. Then picks the output and sends it to FIS. In fact, obtaining membership values is dynamically followed.

In short, CCFS is the embedding of chaotic structure in all neurons of the membership layer. In the proposed architecture, cascade chaotic maps (Lee oscillator) make the membership function layer. This combination could enhance the nonlinearity weights of each neuron’s function. In other words, the CCFS is achieved by adding a type-1 membership value into the interaction of excitatory and inhibitory neurons. Therefore, a chaotic membership function is generated in the whole model and provides the bifurcation with the x operator. In the proposed model, turning to a chaotic membership function makes the system capable of searching the search space for the values of more dynamic and diverse membership without redundancy, which accomplishes the task of the T2FS membership function. The following section investigates the function approximation capability of the proposed chaotic fuzzy system.

5 CCFS as a universal approximator

Based on the specification of the proposed model and the description presented in section 4, it is possible to show and prove that the proposed model (CCFS) possess unlimited power to approximate any continuous function on a compact set using the GFA theorem [38] and Stone-Weierstrass theorem [39].

5.1 Stone-weierstrass theorem

Let Z be a set of continuous real functions in the compact set U. if 1) Z is an algebra, that is, Z is closed under the addition, multiplication, and scalar multiplication; 2) Z is point separator, that is, for each x, yɛU, x ≠ y, there exists a fɛZ such that f(x) ≠ g(x); 3) Z vanishes at no points, that is for each xɛU there exists a fɛZ such that f(x) ≠ 0. Then, the uniform closure of Z contains all real continuous functions on U, such that (Z, d_∞) is dense in (C [U] , d_∞) [39]_.

5.2 General function approximation (GFA) theorem

Suppose that input is a universal set U, a closed set in Rⁿ. Then, for any given real function g(x) in U and arbitrary ɛ > 0, there exists a fuzzy system f ɛY (set of fuzzy systems with a predefined structure) like Equation (17) such that: $\sup | f (x) - g (x) | < 0, x ɛ U$ (16) in which f(x) is the output of fuzzy system with product fuzzy inference and center of sets defuzzification as shown below: $f (x) = \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} [(\prod_{i = 1}^{n} μ_{A_{i}}^{l} (x_{i}))]}{\sum_{l = 1}^{M} [(\prod_{i = 1}^{n} μ_{A_{i}}^{l} (x_{i}))]},$ (17) where i = 0, 1, 2, . . . , n, l = 1, 2, . . . , M (M is the number of fuzzy rules in the rule base, with M≥1). ${\bar{y}}^{l}$ is the center of the lth set. $μ_{A_{i}}^{l} (x_{i})$ is the proposed chaotic membership function.

5.3 Applying the Stone-Weierstrass Theorem to the Proposed Cascade Chaotic Fuzzy System

Considering the chaotic membership function introduced in section 4, the chaotic equation is as follows: $μ_{A_{i}}^{l} (x_{i}) = \exp [- {((\overset{'}{u} \overset{'}{v} k) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}],$ (18) where i = 0, 1, 2, . . . , n, l = 1, 2, . . . , M (M is the number of fuzzy rules in the rule base, with M≥1). ${\bar{x_{i}}}^{l}$ and $σ_{i}^{l}$ are the parameters of a simple Gaussian MF and determine the form of a membership function. $\overset{´}{u}$ and $\overset{´}{v}$ are as follows: ${\begin{matrix} \overset{´}{u} = a_{1} u - a_{2} v + a_{3} z + a_{4} I, \\ \overset{´}{v} = b_{3} z - b_{1} u - b_{2} v . \end{matrix}$ (19) considering I as an excitation input and parameters b_i and a_i as follows, we have: ${\begin{matrix} a_{1} = 1 \\ a_{2} = 1 \\ a_{3} = 1 \\ a_{4} = 1 \end{matrix}}, {\begin{matrix} b_{1} = - 1 \\ b_{2} = - 1 \\ b_{3} = - 1 \end{matrix}} \overset{we have}{\Rightarrow} {\begin{matrix} u^{'} = u - v + z + x_{i,} \\ v^{'} = - z + u + v . \end{matrix}}$ (20) so the output of the proposed fuzzy system with the product fuzzy inference engine, center of sets defuzzification, and chaotic Gaussian membership function is as follows: $f (x) = \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} [(\prod_{i = 1}^{n} \exp [- {((\overset{'}{u} \overset{'}{v} k) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}])]}{\sum_{l = 1}^{M} [(\prod_{i = 1}^{n} \exp [- {((\overset{'}{u} \overset{'}{v} k) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}])]},$ (21) where f : U ⊂ Rⁿ → R, x_{i=(x₁,x₂,…,x_n)} ∈ U, M is the fuzzy rules number. ${\bar{y}}^{l}$ is the center of the lth set. As stated before, $\overset{´}{v}$ and $\overset{´}{v}$ have been expressed in Equation (20). Now suppose Y is the set of all fuzzy systems in the form of Equation (21). By using Stone-Weierstrass Theorem, it is shown that (Y, d_∞) is dense in (C [U] , d_∞), where C[U] is the set of all real functions defined in the compact set U. We show that Y is an algebra, Y distinguishes points on U, and Y vanishes at no point of U.∥A. Proof of Part 1 of the Stone-Weierstrass theorem (Y is an algebra)∥Lemma 1. Y is closed under addition∥Proof: Let f₁, f₂ɛY, so the output of each one can be written as: $f_{1} (x) = \frac{\sum_{l_{1} = 1}^{M_{1}} {\bar{y_{1}}}^{l_{1}} [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}])]}{\sum_{l_{1} = 1}^{M_{1}} [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}])]},$ (22) $f_{2} (x) = \frac{\sum_{l_{2} = 1}^{M_{2}} {\bar{y_{2}}}^{l_{2}} [(\prod_{i = 1}^{n} \exp [- {((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}])]}{\sum_{l_{2} = 1}^{M_{2}} [(\prod_{i = 1}^{n} \exp [- {((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}])]} .$ (23) ∥So, the summation of f₁(x) and f₂(x) is achieved by: $\frac{\sum_{l_{1} = 1}^{M_{1}} \sum_{l_{2} = 1}^{M_{2}} ({\bar{y}}_{1}^{l_{1}} + {\bar{y}}_{2}^{l_{2}}) [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}] - [{((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}])]}{\sum_{l_{1} = 1}^{M_{1}} \sum_{l_{2} = 1}^{M_{2}} [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}] - [{((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}])]} .$ ∥By factorizing common expressions, we will have: $= \frac{\sum_{l_{1} = 1}^{M_{1}} \sum_{l_{2} = 1}^{M_{2}} ({\bar{y}}_{1}^{l_{1}} + {\bar{y}}_{2}^{l_{2}}) [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}] - [{((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}])]}{\sum_{l_{1} = 1}^{M_{1}} \sum_{l_{2} = 1}^{M_{2}} [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}] - [{((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}])]} .$ (24) ∥Since $\exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}] - [{((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}]$ is the same form as Equation (18) $(μ_{A_{i}}^{l} (x_{i}) = \exp [- {((\overset{'}{u} \overset{'}{v} k) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}])$ , and $({\bar{y}}_{1}^{l_{1}} + {\bar{y}}_{2}^{l_{2}})$ can also be considered as the center of a fuzzy set, so f₁(x) + f₂(x) can be the same form as Equation (21) (the main equation of chaotic fuzzy system). Hence, f₁ + f₂ɛY and therefore, Y is closed under addition.∥Lemma 2. Y is closed under multiplication∥Proof: Considering f₁(x) and f₂(x) as Equations (22) and (23) and factorizing of common expressions, we will have: $\begin{matrix} f_{1} (x) \times f_{2} (x) = Eq . (22) Eq . (23) = \\ \frac{\sum_{l_{1} = 1}^{M_{1}} \sum_{l_{2} = 1}^{M_{2}} ({\bar{y}}_{1}^{l_{1}} . {\bar{y}}_{2}^{l_{2}}) [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}] - [{((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}])]}{\sum_{l_{1} = 1}^{M_{1}} \sum_{l_{2} = 1}^{M_{2}} [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}] - [{((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}])]}, \end{matrix}$ (25) in which, $\exp [- {((u'_{1_{i}}^{l_{1}} v'_{1_{i}}^{l_{1}} k_{1_{i}}^{l_{1}}) - {\bar{x_{1_{i}}}}^{l_{1}})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}] - [{((u'_{2_{i}}^{l_{2}} v'_{2_{i}}^{l_{2}} k_{2_{i}}^{l_{2}}) - {\bar{x_{2_{i}}}}^{l_{2}})}^{2} / - (2 σ_{2_{i}}^{l_{2}})^{2}]$ can be rewritten in the same form as the Equation (18) $(μ_{A_{i}}^{l} (x_{i}) = \exp [- {((\overset{'}{u} \overset{'}{v} k) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}])$ through the straightforward algebraic operation. Also, ${\bar{y}}_{1}^{l_{1}} . {\bar{y}}_{2}^{l_{2}}$ can be considered as the center of a fuzzy set, so f₁(x) × f₂(x) can be the same form as Equation (21) (the main equation of chaotic fuzzy system). Hence, f₁ × f₂ɛY and therefore, Y is closed under multiplication.∥Lemma 3. Y is closed under scalar multiplication∥Proof: For any arbitrary cɛR, the scalar multiplication of cf(x) can be expressed by: ${cf}_{1} (x) = \frac{\sum_{l = 1}^{M_{1}} {\bar{y_{1}}}^{l} [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l} v'_{1_{i}}^{l} k_{1_{i}}^{l}) - {\bar{x_{1_{i}}}}^{l})}^{2} / - (2 σ_{1_{i}}^{l_{1}})^{2}])]}{\sum_{l = 1}^{M_{1}} [(\prod_{i = 1}^{n} \exp [- {((u'_{1_{i}}^{l} v'_{1_{i}}^{l} k_{1_{i}}^{l}) - {\bar{x_{1_{i}}}}^{l})}^{2} / - (2 σ_{1_{i}}^{l})^{2}])]},$ (26) ∥Which is again in the form of Equation (21) (the main equation of chaotic fuzzy system). Hence, Cf₁ɛY, and therefore, Y is closed under scalar multiplication. As a result, based on the lemma 1, 2, and 3, Y is an algebra. Q. E. D.∥B. Proof of Part 2 of the Stone-Weierstrass theorem (Y is a point separator on U)∥Lemma 4. This part (Y distinguishes the points on U) is proven by constructing the required fuzzy system f(x), i.e., the number of rules, the parameters of chaotic membership function, and the number of fuzzy sets in U are specified, such that the achieved f in the form of Equation (21), has the feature that f(x⁰) ≠ g(z⁰) for each x⁰, z⁰ɛU and x⁰ ≠ z⁰. Let x⁰, z⁰ɛU are two arbitrary points on U and, x⁰ ≠ z⁰. The f(x) parameters are again selected as follows: $\begin{matrix} M = 2, {\bar{y}}^{1} = 0, {\bar{y}}^{2} = 1, σ_{i}^{l} = 1, {\bar{x}}_{i}^{1} = x_{i}^{0}, {\bar{x}}_{i}^{2} = z_{i}^{0}, \\ l = 1, 2, \\ i = 1, 2, \dots ., n . \end{matrix}$ (27) ∥Also, considering the interactions between the excitatory and inhibitory neurons, we have: ${\begin{matrix} \overset{=}{u} a_{1} u - a_{2} v + a_{3} z + a_{4} I \\ \overset{=}{v} b_{3} z - b_{1} u - b_{2} v . \end{matrix},$ (28) ∥Considering the parameters a_i and b_i as follows, we will have: ${\begin{matrix} a_{1} = 1 \\ a_{2} = 1 \\ a_{3} = 1 \\ a_{4} = 1 \end{matrix}}, {\begin{matrix} b_{1} = - 1 \\ b_{2} = - 1 \\ b_{3} = - 1 \end{matrix}} \overset{So we have}{\Rightarrow} {\begin{matrix} u^{'} = u - v + z + x_{i}, \\ v^{'} = - z + u + v . \end{matrix}}$ (29) ∥Let ${\begin{matrix} u = 1 \\ v = 1 \\ z = 1 \end{matrix}}$ and k = 1, hence: ${\begin{matrix} u^{'} = 1 + x_{i} \\ v^{'} = 1 \end{matrix}}$ , so the achieved system is as follows: $f (x) = \frac{\sum_{l = 1}^{M} {\bar{y}}^{l} [(\prod_{i = 1}^{n} \exp [- {((1 + x_{i}) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}])]}{\sum_{l = 1}^{M} [(\prod_{i = 1}^{n} \exp [- {((1 + x_{i}) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}])]} .$ (30) ∥Inserting the selected parameters of Equation (27) in Equation (30), we will have: $f (x) = \frac{\sum_{l = 1}^{2} [(\prod_{i = 1}^{n} \exp [- {((1 + x) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}])]}{\sum_{l = 1}^{M} [(\prod_{i = 1}^{n} \exp [- {((1 + x) - {\bar{x_{i}}}^{l})}^{2} / - (2 σ_{i}^{l})^{2}])]} .$ (31) By inserting ${\bar{y}}^{1} = 0, {\bar{y}}^{2} = 1, σ_{i}^{l} = 1, {\bar{x}}_{i}^{1} = x_{i}^{0} and {\bar{x}}_{i}^{2} = z_{i}^{0}$ , we will have: $f (x) = \frac{\exp [- {(1 + x - z^{0})}^{2} / - 2]}{\exp [- {(1 + x - x^{0})}^{2} / - 2] + \exp [- {(1 + x - z^{0})}^{2} / - 2]} .$ (32) ∥Therefore: $f (x^{0}) = \frac{\exp [- {(1 + x^{0} - z^{0})}^{2} / - 2]}{\exp [- {(1 + x^{0} - x^{0})}^{2} / - 2] + \exp [- {(1 + x^{0} - z^{0})}^{2} / - 2]}$ (33) $= \frac{\exp [- {(1 + x^{0} - z^{0})}^{2} / - 2]}{e^{- 1 / 2} + \exp [- {(1 + x^{0} - z^{0})}^{2} / - 2]} .$ (34) Similarly: $f (z^{0}) = \frac{\exp [- {(1 + z^{0} - z^{0})}^{2} / - 2]}{\exp [- {(1 + z^{0} - x^{0})}^{2} / - 2] + \exp [- {(1 + z^{0} - z^{0})}^{2} / - 2]}$ (35) $= \frac{e^{- 1 / 2}}{\exp [- {(1 + z^{0} - x^{0})}^{2} / - 2] + e^{- 1 / 2}} .$ (36) By reversing the Equation (34), we will have: $\begin{matrix} \frac{1}{f (x^{0})} = \frac{e^{- 1 / 2} + \exp [- {(1 + x^{0} - z^{0})}^{2} / - 2]}{\exp [- {(1 + x^{0} - z^{0})}^{2} / - 2]} = 1 + \frac{e^{- 1 / 2}}{\exp [- {(1 + x^{0} - z^{0})}^{2} / - 2]} \\ = 1 + \exp [- \frac{1}{2} + \frac{{(1 + x^{0} - z^{0})}^{2}}{2}] = 1 + \exp [\frac{{(1 + x^{0} - z^{0})}^{2} - 1}{2}] = \\ 1 + \exp [\frac{1 + x^{0^{2}} + 2 x^{0} + z^{0} - 2 z^{0} - 2 x^{0} z^{0} - 1}{2}] = \exp [\frac{x^{0^{2}} + 2 x^{0} - z^{0} - 2 x^{0} z^{0}}{2}] . \end{matrix}$ (37) Similarly, by reversing the Equation (36), we will have: $\begin{matrix} \frac{1}{f (z^{0})} = \frac{e^{- 1 / 2} + \exp [- {(1 + z^{0} - x^{0})}^{2} / - 2]}{e^{- 1 / 2}} = \\ 1 + \frac{\exp [- {(1 + z^{0} - x^{0})}^{2} / - 2]}{e^{- 1 / 2}} = 1 + \exp [\frac{1}{2} - \frac{{(1 + z^{0} - x^{0})}^{2}}{2}] = \\ 1 + \exp [\frac{1 - {(1 + z^{0} - x^{0})}^{2}}{2}] = 1 + \exp [\frac{1 - (1 + z^{0^{2}} + 2 z^{0} + x^{0^{2}} - 2 x^{0} - 2 x^{0} z^{0})}{2}] \\ = 1 + \exp [\frac{- z^{0^{2}} - 2 z^{0} - x^{0^{2}} + 2 x^{0} + 2 x^{0} z^{0}}{2}] . \end{matrix}$ (38) ∥By comparing equations (37) and (38) and removing similar sentences, we will have: $\overset{Removing similar sentences}{\Rightarrow} x^{0^{2}} + z^{0^{2}} - 2 x^{0} z^{0} = {(x^{0} - z^{0})}^{2} .$ (39) $\overset{Removing similar sentences}{\Rightarrow} - z^{0^{2}} - x^{0^{2}} + 2 x^{0} z^{0} = - {(x^{0} - z^{0})}^{2} .$ (40) ∥By comparing Equations (39) and (40), it is observed that these two equations are not equal and therefore, $\frac{1}{f (x^{0})} \neq \frac{1}{f (z^{0})}$ . As a result, their inverse are not equal, too and f(x⁰) ≠ f(z⁰); hence: Y distinguishes the points on U. Q. E. D.

C. Proof of Part 3 of the Stone-Weierstrass theorem (Y vanishes at no point of U)

Lemma 5. For each xɛU, there exists a fɛY such that f(x) ≠ 0. Therefore, Y vanishes at no point of U. considering Equation (21) (the main equation of CCFS), it is observed that by choosing ${\bar{y}}^{l} > 0, (l = 1, 2, 3, \dots . M$ ), then: f(x) > 0, ∀ xɛU. It means that any fɛY with ${\bar{y}}^{l} > 0$ , acts as the required f. so, Y vanishes at no point of U. Q. E. D.

Therefore, using lemma 1, 2, and 3, Y is proved to be an algebra. Lemma 4 is demonstrated that Y is a point separator, and finally, using lemma 5, it is depicted that Y vanishes at no point of U.

5.4 Proof of GFA Theorem

From equations (17) and (18), it is clear that Y is a set of real continuous functions on U. applying the Stone-Weierstrass theorem along with lemma 1–5 lead to proof of function approximation theory for the proposed system. In fact, the proposed chaotic fuzzy system (CCFS) encompass the general function approximation capability. Q. E. D.

6 System implementation and experimental results

This section evaluates the proposed structure. Cascade chaotic fuzzy system (CCFS) is verified through forecasting the chaotic Mackey-Glass time series. According to extended usage of T1FS and T2FS in time-series forecasting [1], [40] and [41], these frameworks are selected to make comparisons with the proposed model in this paper. The studies mentioned above reported that the T2FS results in better forecasting accuracy than the T1FS. In the systems implementation, Due to more degrees of freedom in the CCFS than in T2FS, better performance of the proposed model is most likely, compared to classic fuzzy systems.

6.1 The Mackey-Glass chaotic time series

Forecasting of time series has several real-world applications in areas such as signal processing, traffic prediction [42] and industrial modeling [43]. The Mackey-Glass (MG) time series is one of the most referred benchmarks in nonlinear time-delay modeling. These nonlinear differential-delay equations are initially used for describing physiological control systems. In particular, the MG time series is a model of white blood cell generation in leukemia patients. The MG time series is a good examination for learning programs and verifying forecaster’s performance [44]. This capability is related to the straightforward definition and hard prediction of MG elements due to the chaotic nature of data. Another noteworthy property of the MG time series is that real-valued outputs are needed in place of the discrete output values found in most fuzzy test benches. The forecasting of future MG time series values is a benchmark problem that has been regarded by several connectionist scholars [45, 46].

MG equation is a nonlinear time-delay system that generates a chaotic time series under certain conditions. A broad versatility of dynamical behavior, including limit cycle fluctuation, along with the diversity of waveforms and seemingly aperiodic or “chaotic” solutions, are exhibited by this equation. The MG equation is as follows [47]: $\frac{dx (t)}{dt} = β \frac{x (t - τ)}{1 + x^{n} (t - τ)} - γ x (t) .$ (41)

Equation 41 displays different qualitative dynamics. Increasing the value of τ converting from an initially stable equilibrium to an unstable one and a stable solution. Moreover, if τ is increased further, a sequence of bifurcations in the dynamics is revealed. These behaviors seem to be utterly similar to bifurcations found in first- and second-order finite difference equations. Let β=0.2, γ=0.1, and n = 10, using these values for the parameters, Equation (41) can be rewritten as follows: $\frac{dx (t)}{dt} = 0.2 \frac{x (t - τ)}{1 + x^{10} (t - τ)} - 0.1 x (t) .$ (42)

If τ >16.8, time-series exhibit chaotic behavior. This equation is simulated by the fourth-order Runge-Kutta procedure to calculate the discrete-time equation: $f (x, n) = 0.2 \frac{x (n - τ)}{1 + x^{10} (n - τ)} - 0.1 x (n),$ (43)

where, $x (n + 1) = x (n) + hf (x, n),$

where h is a time step that is supposed to be one and for n ⩽ τ, the initial values of time series are randomly set.

6.2 Forecasting chaotic MG time series using CCFS

MG time series is simulated for τ=17 and 0 ⩽ n ⩽ 1175. The typical method for time series forecasting is to generate a map from D sample time series points, sampled every Δ units in time, (x (t-(D-1) Δ) . . . , x (t-Δ), x (t)) to a forecasted future point x (t + p). Following the common parameter setting for Mackey-Glass time series forecasting [44], D and Δ are considered 4 and 6, respectively. For each t, the input pattern for network training is a four-dimensional vector as follows:

Input: W (t)=[x (t-18) x (t-12) x (t-6) x (t)]

Then, trajectory forecasting is done via output training data:

Output: S (t)=x (t + 6)

For each t, the input/output training data is a matrix whose first element is the four-dimensional input w, and the second element is the output s.

6.2.1 Forecasters specification

As mentioned above, T1FS and T2FS are selected to be compared with the proposed chaotic fuzzy system. The specification of these forecasters is as follows:

Four antecedent, x(t-3), x(t-2), x(t-1) and x(t) is used to forecast consequent x(t + 1).

The fuzzy rules extraction in T2FS is done via the Wang-Mendel process [48].

Three membership functions are used for each antecedent. Therefore, 81 rules are generated.

The type of implication is the product.

The Defuzzification operator for T1FS and type-reduction operator in T2FS is the center of sets.

Gaussian MFs and triangular MFs are selected, considering a better understanding of the system. Although, there is no limitation in this case. In T2FS, the MFs are used with uncertain center and mean, respectively.

The initial position of the antecedent MFs in T1FS is set based on the mean and variance of the training dataset. For all antecedents, 4σ_t and m_t ± 2σ_t are the initial values of spread and centers, respectively. Also, a random number between 0 and 1 is selected for the initial value of the consequent.

$[m_{1}^{Lrn} - 2 c_{1} \sqrt{3} σ_{n}, m_{1}^{Lrn}]$ , $[m_{2}^{Lrn} - \sqrt{3} σ_{n}, m_{2}^{Lrn} + 2 c_{1} \sqrt{3} σ_{n},]$ and $[m_{3}^{Lrn}, m_{3}^{Lrn} + 2 c_{1} \sqrt{3} σ_{n},]$ , are the span of uncertainty in the centers of antecedent membership functions. $m_{1}^{Lrn}$ , $m_{2}^{Lrn}$ , and $m_{3}^{Lrn}$ are the centers of type-1 membership function in a learned T1FS, and the spreads of antecedent membership functions are Δ^Lrn, similar to antecedent membership function, Δ^Lrn is obtained from the spreads of learned T1FS and C₁ = 0.24. $[{\bar{y}}^{Lrn}, {\bar{y}}^{Lrn} + 2 c_{2} \sqrt{3} σ_{n}]$ is the span of uncertainty for the rules consequent. ${\bar{y}}^{Lrn}$ is the center of consequent in a learned T1FS and C₁ = 0.06.

In the CCFS forecaster, the initial values of membership function parameters are achieved from the learned T1FS. The Lee oscillator parameters is as follows similar to [31]:

a1 = 0.7856; a2 = 0.6681; a₃ = 0.36; a₄ = 0.11

b1=–0.2043; b2 = 0.5078; b₃=–0.989; b₄ = 0.091; K = 0.4188;

It should be noted that the values for a_i and b_i are not so inclusive, and these values should put the oscillator in the chaotic mode. This range is not vast. After finding the values, they are regarded as constants and not parameters. θ_u = 0; θ_v = 0; There is no limitation in the selection of f. One example can be exponential function (f(x)=exp(x)).

The backpropagation algorithm is used for training T1FS with an error tolerance of 0.001. As mentioned earlier, T2FS and the proposed model use the parameters of learned T1FS as the initial values of parameters.

In the experiments, there are 2500 pairs of input/output data. Dividing the generated data to train, validation and test dataset is done randomly according to the percentage mentioned below:

The percentage of train/validation/test data is 60/20/20. Experiments are conducted 20 times, and the results are obtained. The stated errors are based on the mean absolute error (MAE) and the root mean square error (RMSE). MAE computes the average value of errors in a forecasting set, while RMSE is a quadratic scoring rule that computes the average value of the error [49]. These two parameters can be applied together to distinguish the error variations in a forecasting set. The greater difference among them, the greater the variance in the particular errors in the instances. In the experiments, 20 realizations of the MG chaotic time series are achieved with different values of τ, and the average values of RMSEs and MAEs are calculated for each FIS. The results for train and test datasets are displayed in Table 1. As seen in Table 1, as the value of τ increases, the accuracy of forecasters is decreased. This is concluded via both train and test data. After presenting the detailed results related to the proposed model, to compare the method presented in this paper with popular methods, two outstanding models, including T1FS and T2FS, are implemented according to conditions described earlier in this section. These two classic methods are compared with the two chaotic models, including the cascading and non-cascading chaotic models, namely CFS and CCFS, to analyze the impact of each instrument separately. All four models are implemented in the same experimental condition, and results are achieved. Specifications of these forecasters are mentioned in section 6.2.1. Simulation results for these models and the proposed models are shown in Tables 2 and 3 for train and test data set, for different values of τ. As seen from the tables, the difference between RMSE and MAE is decreased from T1FS to T2FS and from T2FS to CFS and CCFS. The most negligible differences between these two parameters are obtained in the CFS and CCFS forecasters. This reduction reflects lower variances in resulting errors, and this means more prediction stability and uniformity in the performance of the CFS and CCFS models. Figures 7 and 8 depict the results of all FIS compared to each other, based on the RMSE parameter. These figures compare the results of four forecasters for train and test datasets, respectively, with different values of τ. As can be seen from the figures, the CCFS outperforms the other forecasters and performs better for all values of τ, except for a few limited cases. In τ=80 for training data, the values of RMSE for the CFS and CCFS are almost the same, both better than T1FS and T2FS. Also, in τ=50 for training data, the value of RMSE for the T1FS is better than T2FS. Similarly, considering the test dataset and Fig. 8, in τ=73, the value of RMSE for the CFS is better than CCFS, both better than T1FS and T2FS. Again, the accuracy of forecasters, in general, is decreased by increasing the value of τ. This is concluded via both train and test data. Another observation is that the increasing uncertainty in the FIS forecasters from fixed membership to interval and from interval to chaotic, and finally from chaotic to cascade chaotic enhances the efficiency of forecasters. Increasing the variety and hence the nonlinearity in this sequence improves the capability to understand the uncertainties of the considered system. Similarly, figures 9 and 10 show the results related to four forecasters based on the MAE parameter. Similar outcomes are obtained from these figures, but with a few exceptions. For instance, in τ=17, the MAE value of T2FS is lower than that of the CFS. However, for a general understanding of the forecasters’ efficiency, these two parameters (RMSE and MAE) must be considered together, as mentioned earlier in this section. As can be seen from Tables 2 and 3 and figures 7 to 10, the CCFS presents a lower RMSE, MAE, and a lower difference between RMSE and MAE. These values exhibit better efficiency, stability, and unified performance compared to other forecasters. In τ=17 in the training dataset, where the difference between the two mentioned parameters in the CCFS model (0.012) is greater than the CFS model (0.005), the values of both parameters are lower in the proposed model. Therefore, outstanding performance of the proposed model is demonstrated.

Fig. 7

Comparison of T1FS, T2FS, CFS and the CCFS in forecasting of Mackey-Glass time series with different values of τ for train dataset.

Fig. 8

Comparison of T1FS, T2FS, CFS and the CCFS in forecasting of Mackey-Glass time series with different values of τ for test dataset.

Table 1

Performance of the CCFS with different values of τ

τ	Train Data		Test Data
	RMSE	MAE	RMSE	MAE
17	0.05421	0.04153	0.05248	0.04175
25	0.06312	0.04898	0.0607	0.0532
50	0.0819	0.06411	0.0798	0.0548
80	0.0940	0.07167	0.0911	0.0622

Table 2

Performance of T1FS, T2FS, CFS and CCFS for train data in MG time series forecasting with different values of τ

T1FS		T2FS		CFS		CCFS
τ	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE
17	0.1072	0.07481	0.0742	0.0561	0.0639	0.0585	0.05421	0.04153
25	0.1123	0.08476	0.09052	0.0698	0.0709	0.0599	0.06312	0.04987
50	0.1363	0.1002	0.1385	0.0989	0.1003	0.0741	0.0819	0.0662
80	0.1687	0.1252	0.1529	0.1132	0.1121	0.0879	0.1107	0.0915

Table 3

Performance of T1FS, T2FS, CFS and CCFS for test data in MG time series forecasting with different values of τ

T1FS		T2FS		CFS		CCFS
τ	RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE
17	0.0915	0.07843	0.0754	0.0602	0.071	0.0543	0.0602	0.04175
25	0.1267	0.10341	0.0921	0.0698	0.0777	0.0631	0.0679	0.06002
50	0.1465	0.11789	0.148	0.1183	0.0994	0.0705	0.0835	0.07
80	0.1755	0.1525	0.1586	0.1254	0.1132	0.0901	0.1188	0.0911

Fig. 9

Comparison of T1FS, T2FS, CFS and the CCFS in forecasting of Mackey-Glass time series with different values of τ for train dataset.

Fig. 10

Comparison of T1FS, T2FS, CFS and the CCFS in forecasting of Mackey-Glass time series with different values of τ for test dataset.

7 Conclusion

This paper has proposed a new chaotic fuzzy system, namely a cascade chaotic fuzzy system (CCFS) based on the Lee oscillator. The function approximation capability of the proposed model has been proven by applying the GFA and Stone-Weierstrass Theorem. The paper’s significant contribution is to narrow the gap between the conventional fuzzy system, cascade network, and chaotic systems. From the structure point of view, this paper embeds the CCS structure into the membership layer of the classic fuzzy systems. The resulting framework has the characteristics of the chaotic fuzzy system in modeling uncertainties and also characteristics of cascade chaotic systems in enhancing the chaos capabilities. To dominate complexities in the design of T2FS, CCFS presents an innovative general chaotic structure to incorporate fuzzy reasoning, self-adaptation of the neural networks, and chaotic signal generation. Also, the cascade framework enhances the diversities of the chaotic behavior originating from the lee oscillator.

From the fuzzy operation point of view, the proposed system is neither fuzzy type-1 nor fuzzy type-2. Unlike the T2FS, which continuously covers a Min to Max interval and considers all Memberships, the CCFS scans all space and provides samples of the entire space. Then picks the output and sends it to FIS. The extraction process of membership values is dynamically followed. Inherent features of chaotic systems such as ergodicity and maximum entropy help the system search the membership values more efficiently and in the shortest time without searching the whole domain as in T2FS. Practically, the T2FS is implemented by a chaotic type of T1FS, making the overall system more robust. Behavioral and fundamental analyses are done to approve the performance of the proposed structure. The bifurcation diagram of the resulting chaotic system confirms the enhanced chaotic behavior of the CCS compared to conventional chaotic maps and networks. Moreover, according to the lack of studies related to investigating the GFA properties of chaotic fuzzy systems, the proposed CCFS has been proven to be a general function approximator. Besides, evaluation and comparison results have shown the outstanding results of the new CCFS compared to other fuzzy systems such as CFS, T1FS, and T2FS when predicting the chaotic time series. The nature of data is similar to the structure of the model, both of which are chaotic.

For future works, studying the impact of chaos type on the overall performance of the fuzzy system will be followed. Besides, setting the parameters of the chaotic network to increase or decrease the chaotic degree of the model will be studied. Also, although the range of oscillator parameters is restricted for chaos generation, presenting a training algorithm for the oscillator will be followed.

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Multiple Input Single Output

Membership Function

Type-1 Fuzzy Neural Network

Interval Type-2 Fuzzy Neural Network

Activation Function

Chaotic Fuzzy System

Membership Function

General Function Approximation

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