Chaotic Oscillations in a Fractional-Order Circuit with a Josephson Junction Resonator and Its Synchronization Using Fuzzy Sliding Mode Control

Center for Nonlinear Systems, Chennai Institute of Technology, Chennai, India Sakarya University of Applied Sciences, Faculty of Technology, Department of Electrical and Electronics Engineering, Sakarya 54050, Turkey Hitit University, Faculty of Engineering, Department of Computer Engineering, Corum 19030, Turkey Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET), Nanjing University of Information Science & Technology, Nanjing 210044, China Jiangsu Key Laboratory of Meteorological Observation and Information Processing, Nanjing University of Information Science & Technology, Nanjing, China


Introduction
e complex nature of nonlinear dynamical systems has long been a problem for science, but as computers have evolved, they have become much better at handling it. Particularly, in some nonlinear systems, chaotic behaviour is unveiled for interesting parameter values. e presence of chaotic behavior is often regarded as undesirable and problematic. e investigation about the sensitivity of the nonlinear dynamical system becomes mandatory for understanding how systems behave during working range of parameters. Even though there are lots of studies conducted on this topic, study of irregular behaviours of the nonlinear dynamical system is a potential research area. Fractional calculus is an e ective tool for exploring the unexplored region of system characteristics [1,2]. Chaotic systems are identi ed with more intricate response for not only parameter changes but also initial conditions. ere are many studies found in analysing the chaotic systems with fractional-order treatment, and really useful results were obtained. On the contrary some literatures proved that not all the chaotic systems need to study with fractional-order. So, it is obvious a question in our mind "which chaotic systems actually needs fractional-order treatment?" During the course of finding the answer for the question, we come up with a better understanding of fractional-order theory. e fractionalorder approach is particularly suited for analyzing chaotic systems that exclusively depend on the instant of time but they are affected by the history of the preceding stage [1][2][3][4]. Bifurcation theory is a mathematical tool that describes effectively the transformation of behaviour from one state to another [5]. Using this mathematical tool and fractional-order treatment will provide an insight analysis of nonlinear dynamical systems. It is proved that the fractional-order form holds unlimited memory and provides more degrees of freedom [1,3,6]. Hence, investigation of chaotic behaviour in nonlinear dynamical systems with fractional-order circuits is more complex and unearthed interesting characteristics.
Fractional systems can be used in chemical processes [7], biological systems [8], electrochemical systems [9], viscoelastic systems [7,10], and propagation of electromagnetic waves [7,11]. In the literature, it is encountered in the control of such systems or synchronization applications, communication [12], control of power systems, or control of chemical processes [13,14]. Many methods such as PID [15], sliding mode control [7], backstepping control [16,17]¸fuzzy sliding mode control [18], and adaptive sliding mode control [19] have been used in the control of such systems. e major significance of SMC is that it has a robust structure against varying parameters [7,20].
anks to that, the controller keeps the system on the designed surface under varying conditions. Moreover, the SMC method is a powerful method of controlling high-order systems against disturbances and parameter uncertainties [21,22]. On the other hand, sliding mode control suffers from high amplitude chattering problem. In this study, fuzzy logic is used to determine the chattering amplitude. In [23], a set of linguistic rules is determined and adaptive SMC control law can be applied for chaotic systems. It has also been studied in the control and synchronization of time-invariant/varying and SMC chaotic systems [23,24]. In addition, studies have been made on determining the surface of fractional chaotic systems with fuzzy logic [25,26]. In the study, rather than determining the surface, the chattering amplitude was determined by fuzzy logic and synchronization was achieved with the control rule.
Current-voltage relation can be described with resistor; similarly, the relation between voltage and charge can be described with capacitor; finally, the current-magnetic flux is defined with inductor. Memristor is a novel component, which defines the inter-connection of charge and magnet flux. Our interest in memristor is for its memory effect. From the literature, we identified that the memristor coupling can effectively describe the effects of memory and show the relation between output voltage and magnetic flux by generating induction current [2]. e nonlinearity of electric circuit is engendered, and the dynamical behaviour becomes more complex when memristor is used in circuits because the memductance is dependent on the inputs current [15]. Introduction of memristor in a circuit engendered the nonlinearity and exhibit intricate behaviours, it is because, the dependency of the memductance on the input current. e application of Josephson junction in superconductors [27][28][29] is considered as a milestone since its discovery in the 1960s and attracts attention due to its significance and suitability for different circuits. ese devices are very high speed and sensitivity [30]. e prominence is now extended to various applications such as analog devices [31]. Many studies [32][33][34][35][36] discussed the usage of Josephson junction for high-frequency oscillators with high critical current density, but most of them are deliberately designed to avoid chaotic regions. Initial investigations show the need of rigorous studies to establish this application. An asymmetric memristive diode-bridge-based jerk circuit is proposed in [37] and studied the asymmetric coexisting bifurcations, and it has a potential future works to new applications.
Motivated from the above discussion, we propose a novel chaotic circuit consisting both memristor and Josephson junction, and dynamical behaviours are analysed using the fractional-order approach. Section 2 deals the formulation of circuit and its mathematical model. Section 3 provides the numerical simulations. Section 4 describes synchronization with fuzzy sliding mode. Section 5 presents the simulation results. Finally, we highlight the significance and effectiveness of the proposed system in conclusion.

Mathematical Model
A Josephson junction with topologically nontrivial barrier can be defined with a linear resistive capacitive inductive shunted model discussed in [3,38,39]. We include a memristor parallel to the Josephson junction to derive the new proposed chaotic oscillator consisting of both memristor and Josephson junction as discussed in [40], where the authors considered a flux-controlled memristor with a Josephson junction [3]. is memristor is included to model the feedback flux effects in a Josephson junction device. We propose the new chaotic oscillator as shown in Figure 1 wherein we used a fractional-order memristor whose mathematical model is used from [2]. e circuit of Figure 1 is made of a flux-controlled fractional-order memristor W, a Josephson junction with topologically nontrivial barrier J, a capacitor C, and a shunt inductor L S whose internal resistance is R S . By applying an external DC current I, the current passing through and the voltage across the circuit elements can be derived using the KVL and KCL as where h is the Planck constant, e is the charge of the electron, and q is the fractional order of the memristor. e memductance function can be derived with Faraday's law through which we can define the electromagnetic induction current as whose memductance is given by Using (3) in (2), By using equations (3) and (4) in equation (1) and by substituting dimensionless state variables and parameters, we can derive the mathematical model as where the dimensionless state variables are defined as v � V/R S I C , i S � I S /I C , t � τω 0 and the dimensionless parame- To numerically simulate system (5), we considered the modified Adams-Bashforth method [41] for the Caputo-Fabrizio (CF) fractional operator [6]. By definition, the general form of a CF fractional operator can be described in the form e definition in (6) can be modified as By the definition from [41], (6) can be numerically expanded to the form e solution of (8) can be derived as Using (7) in (5), we could derive the solution for the fractional-order system as By proper selection of the system parameters and commensurate fractional orders q, the circuit in Figure 1 exhibits chaotic oscillations.
e system parameters are defined as e initial conditions of the state variables are [0, 0, 1, 0], and the phase portraits are shown in Figure 2 for the fractional order q � 0.95. Mathematical Problems in Engineering 3

Numerical Analysis and Discussion
It can be easily noted that system (5)  e characteristic polynomial of system (5) for the equilibrium points [0, 0, ϕ * , 0] is given by By using the theorem, we could say that the condition for stability is that the fractional order q < (2/π)|arg(λ)| for all λ. We have shown the stable and unstable regions in Figure 2 where the stable regions are shown by red colour and unstable regions are shown by pale green.
To investigate the sensitivity of the nonlinear dynamical system with respect to parameter changes, bifurcation diagram is an effective mathematical tool. e influence of parameter changes on the proposed system is analysed in this section. Firstly, we showed the bifurcation diagram for the parameter k 0 variation. e range considered for the investigation is 0 ≤ k 0 ≤ 2. It is observed that system (5) under consideration shows very rich bifurcation structures when slowly tuning the control parameter k 0 . From Figure 3(a), we can easily identify some striking bifurcation events including period doubling scenario to chaos, period halving exit to chaos, symmetry boundary, and interior crises.
A small change in parameter value ends up with entirely different system behaviours. e oscillation can be periodic, period doubling, or chaotic. In order to reveal fine changes of oscillations, a tiny window is zoomed with the range of 1.02 to 1.16 and presented in Figure 3. Within this piece of range, we could observe periodic, period doubling, and chaotic regions. Multistability is identified as a significant property in nonlinear dynamics [14][15][16][17][18][19]. With any abrupt change in the states or parameters, the said multistable system may enter into a new stable situation, which may be entirely different from the desired state. Realizing such properties in dynamical systems and investigating the parameter ranges where multistability occurs is essential and interesting. Dark blue dots are obtained by increasing value of control parameter k 0 from 0 to 2 and red dots are obtained by decreasing the value from 2 to 0 in Figure 3(a). e end value of the states is considered as the initial condition for the consecutive iteration. In Figure 3(b), during parameter range 1.146 ≤ k 0 ≤ 1.151, we can recognize multiperiodic oscillation with blue dots and chaotic oscillation with red dots. Hence, the existence of multistability property is highlighted using the bifurcation plot.
Secondly, we considered the bifurcation parameter as m and other parameters are taken as discussed in the previous section. From Figure 4, periodic oscillations (0.2 to 0.31), period doubling (0.376 to 0.425, 0.62 to 0.65, 0.79 to 0.81, 0.90 to 0.91, and more tiny ranges), period halving (0.322 to 0.375), and cascades of chaotic regions (0.31 to 0.321, 0.376 to 0.39, 0.45 to 0.60, and many more small windows) can be observed. e parameter range considered for the investigation is 0.2 ≤ m ≤ 1.2; with this small variation, the system shows variety of oscillations, and very rapidly, it changes the behaviours. e increment of the parameter shows the increase of amplitude of the oscillations, and the frequency of the occurrence of chaotic regions is reduced. We could also observe multistability property, and it is clearly shown in Figure 4(a).
We increased (or decreased) in tiny steps of parameter m and plotted the local maxima of v. e final state at each iteration of the parameter is considered as the initial state for the next iteration. is strategy is identified as forward (blue plot) and backward (red plot) continuation, and it signifies a simple way to localize the window in which the system advances to multistability. In Figure 4(a), the multistability region is identified and highlighted during 0.322 ≤ m ≤ 0.35 parameter range.
Investigating the influence of order while analysing the fractional-order system is more important. In Figure 4(b), we showed how the system behaves when it is treated with different orders. As we mentioned before, the system is very sensitive in nature. We can observe it even a small change it enters from multiperiodical state to chaotic state. For example, for q � 0.98, the system shows chaos, for q � 0.99, it is with multiperiodic oscillation, and for q � 1 (integer order), it oscillates in chaotic behaviour.

Fractional Chaotic System Synchronization with Fuzzy Sliding Mode
In this study, although the two systems to be synchronized are the same in structure, the one that is to be synchronized parametrically is different. Fractional chaotic system is given in (12) which is parametrically equivalent to (5).
Fractional chaotic system (12) is given in equation (13) which is to be synchronized to Error dynamics of difference of master and slave system are determined as in (14) to accomplish synchronization.

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4.1. Sliding Mode Control. In order to control a fractionalorder system on the sliding surface, it is necessary to define the surface. For this, a surface definition is made as Also, this surface function is a derivative as in the following equation: In order to provide stability, the Lyapunov function is determined on the surface defined as in equation (16) and its derivative is taken. Here, the control signal is chosen to ensure _ V ≤ 0'i in equality (18).
In this section, the process that is the basis of the sliding mode control will be repeated. In other words, equation (19) must be provided to satisfy _ V � s _ s ≤ 0. w is a constant value and a required amplitude to satisfy the condition _ V � s _ s ≤ 0. e corresponding -sign(S) function will be used.
us, _ V ≤ s(−sign(s)) ≤ 0 is obtained. is is given more clearly in Figure 5.
_ s(t) � k 1 D q e 1 + c 1 e 1 + c 2 e 2 + c 2 e 2 + c 2 e 2 � −sign(s). (21) In equation (21), variables are substituted and expanded as in equation (22). When the equation is arranged to determine the control law to be applied, it is obtained as in the following equation.

Fuzzy
System. Although fuzzy logic applications are quite easy and have a broad mathematical background, they have an uncomplicated and easy-to-understand structure. us, an easier and more durable control law can be produced. Fuzzy logic can be defined with membership functions and rules, which are designed as expertbased and usually expressed linguistically, as well as structures that can learn by data (ANFIS). However, in this study, membership functions and rules, which will be expressed linguistically, have been established. It was then used to determine the amplitude of the control signal using this structure. e general structure of fuzzy logic is given in Figure 6. e basic structure used in the fuzzy logic system is the membership function. Membership functions are used both for fuzzification and for defuzzification. Figure 7 shows the structures of the 6 Mathematical Problems in Engineering   Table 1. e rule table allowing the association of membership functions for input and output is given in Table 2. A total of 13 rules have been created and these rules are associated with inputs and outputs.
is structure used is generally called the Takagi-Sugeno (TS) model in the literature.
Fuzzy logic is used in the modelling and control of systems in decision making and many other situations. In this section, it will be used to adjust the amplitude of the control signal to be applied with the sliding mode control. Fuzzy-based control and sliding mode have been used to prevent rapid and high amplitude changes, which will be applied in the actuator especially in a stationary situation where the error is reduced. e general structure of the system synchronized with fuzzy sliding is given in Figure 8. Especially in the application, when the system approaches the desired surface, the integral value of this error can remain constant. erefore, a feedback gain (H) is added to reduce the effect of the integral over time, so that this integral value does not remain constant. In order for this structure, which acts as a stable low filter, to be close to the pure integrator, the feedback gain is low.

Simulation Studies
e states of the system synchronized in the simulation studies, the amplitude of the control signal, and the amplitude of k s are given in Figure 9. As can be seen, the system was started to be controlled after 15 seconds and the amplitude of s (t) surface increased until the system was controlled. After the system started to be controlled, s(t) approached 0 and when it reached this value, rapid changes occurred in the control signal at a high frequency due to the amplitude of k s . However, according to the fuzzy logic decision according to the situation of s(t), the value of k s was reduced after a certain time and the desired performance was tried to be achieved.
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Conclusion
In this study, we have proposed a chaotic oscillator with a Josephson junction device whose feedback flux effects are modelled using a fractional-order memristor. We have derived the dimensionless model of the proposed oscillator and the dynamical properties are investigated using eigenvalues, Lyapunov spectrum, and bifurcation plots. To show the application prospective of the proposed oscillator, we have derived the synchronization between master and slave systems with different parameter sets. e control laws required for the synchronization of the chaotic systems are determined by the sliding mode control technique. en, fuzzy logic was used to determine the chattering amplitude in the sliding mode control. With the determination of this amplitude, high amplitude changes in the control signal are prevented. Subsequently, this approach has been performed to synchronize fractional chaotic systems. As a future direction, the discussed model can be formulated using the Abu-Shady-Kaabar fractional derivative [42] to obtain analytical solutions and can widen the application in many fields.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 9