Control and Synchronization of Chaos in RCL-Shunted Josephson Junction with Noise Disturbance Using Only One Controller Term

Abstract

This paper investigates the control and synchronization of the shunted nonlinear resistive-capacitive-inductance junction (RCLSJ) model under the condition of noise disturbance with only one single controller. Based on the sliding mode control method, the controller is designed to eliminate the chaotic behavior of Josephson junctions and realize the achievement of global asymptotic synchronization of coupled system. Numerical simulation results are presented to demonstrate the validity of the proposed method. The approach is simple and easy to implement and provides reference for chaos control and synchronization in relevant systems.

1. Introduction

Josephson junction, a strongly nonlinear device, has attracted considerable attention due to the advantage of ultra low noise, low power consumption, and high working frequency [1]. And thus different models were proposed as follows: the shunted linear resistive-capacitive junction (RCSJ) [2, 3], the shunted nonlinear resistive-capacitive junction (SNRCJ) [4], the shunted nonlinear resistive-capacitive-inductance junction (RCLSJ) [5] and the periodically modulated Josephson junction (PMJJ) [6, 7].

The chaotic behavior of Josephson junctions has been widely investigated. Take vortex dynamics, for example, Josephson vortices in intrinsic Josephson junctions made of single crystalline Bi₂Sr₂CaCu₂O_8+δ [8], vortex dynamics in S-shaped Josephson junctions [9], vortex dynamics in Josephson junction arrays with magnetic flux noise measurements [10], dynamics of vortices in disordered Josephson junction arrays [11], vortex dynamics in Josephson junction arrays with percolative disorder [12]. At the same time, intrinsic Josephson junctions, formed in cuprate high-Tc superconductors, have won great attention. MachHida et al. [13] put forward microscopic theory for intrinsic Josephson junctions and established proper model equation describing the dynamic in intrinsic Josephson junctions. Kashiwaya et al. [14] studied the switching dynamics of Bi₂Sr_1.6La_0.4CuO_6+δ intrinsic Josephson junctions to show the effectiveness of the strong coupling between stacked junctions. Hayashi et al. [15] described the dynamics in a differential equation of intrinsic Josephson junctions under applied current.

However, it is essential to avoid all types of noise, chaos, and so forth, hence the need for chaos control and synchronization to make good use of the system. Motivated by important applications, researchers have put forward kinds of approaches to chaos control and synchronization, such as fuzzy control method [16], backstepping control method [17], feedback control method [18, 19], adaptive control method [20], OGY method [21], and sliding mode control method [22, 23]. As to Josephson junctions, in applications of parametric amplifiers [24], voltage standard [25], and superconducting quantum interference devices (SQUID) magnetometer [26], and so forth, chaos control and synchronization is of great help. Zhao and Wang [27] investigated the SNRCJ model, which contains two state variables and showed chaotic behavior with an external sinusoidal signal, and synchronized the model in two types of active siding control method as follows: strong restricting of the controller form and no superabundant restrictive assumption imposed for the controller. The study of the RCLSJ model which includes three state variables and exhibits chaotic dynamics with external DC bias only has been popular over years, and the model has been found to be more appropriate in high-frequency applications. In paper [28] a nonlinear active controller was proposed to synchronize the coupled RCLSJ model, and in paper [29] the RCLSJ model was synchronized via a variable structure controller with regard to uncertainties. Synchronization of superconducting junction arrays is also important. Gaifullin et al. [30] studied the synchronization of the stacked Josephson junctions array, fabricated by high-quality Bi₂Sr₂CaCu₂O_8+δ whiskers and crystals, making it possible to be applied in the fields as voltage standard or HTSC Josephson spectral detector. Filatrella and Pedersen [31] set up an intrinsic Josephson junction model of the highly anisotropic BSCCO type, and a cavity is used to enhance fluxon synchronization in a stacked junction, providing application for enhancing microwave radiation from BSCCO single crystals. In addition to the classic method of synchronization, in paper [32], cluster synchronization was introduced to realize the synchronization of coupled Josephson junction arrays. Recently, the chaos control and synchronization of the RCLSJ model has been achieved in paper [33, 34]. In both papers, three control inputs were introduced to realize the achievement of the model control, and although the synchronization of the model was achieved with one control term, the control inputs did not consider the external disturbance in practical systems.

Consequently, the aim of this paper lies in the achievement of the control and synchronization of the RCLSJ model by means of sliding mode method with only one control term under the condition of noise disturbance. It is worth noting the control and synchronization aim considered in this paper can be both achieved with only a single control input. Furthermore, the single controller proposed in this paper regardless of the system’s noise disturbance, which is of great significance in reducing the cost and complexity for controller implementation. This paper is organized as follows. Section 2 describes the model and discusses the dynamics behavior of the system, and Section 3 presents the sliding mode designs, together with numerical simulations. In Section 4, we summarize this work.

2. The Model of Josephson Junction System and Dynamics Analysis

A standard form of the RCLSJ model [35] is proposed as ℎ4𝜋𝑒𝑑𝛾𝐶𝑑𝑡=𝑉,𝑑𝑉+𝑉𝑑𝑡𝑅_𝑉+𝐼_𝑐sin𝛾+𝐼_𝑠𝐿=𝐼,𝑑𝐼_𝑠𝑑𝑡+𝐼_𝑠𝑅_𝑠=𝑉, (2.1) where 𝑅_𝑠 and 𝐿 are the shunt resistance and inductance, 𝐼_𝑠 the shunt current, 𝐼 the input DC bias. 𝐼_𝑐, 𝐶, 𝑅_𝑉 are critical current, capacitance, and resistance of the junction, respectively. The junction resistance 𝑅_𝑉 is nonlinear, expressed as follows: 𝑅_𝑉=𝑅_𝑁,|𝑉|>𝑉_𝑔𝑅_sg,|𝑉|≤𝑉_𝑔,with𝑉_𝑔=2Δ𝑒, (2.2) where 𝑅_𝑁 and 𝑅_sg are the normal state resistance and energy gap resistance, 𝑉_𝑔 the gap voltage. 𝛾 and 𝑉 are the superconducting order parameter phase difference and the junction voltage.

For numerical simulation and analysis, the standard form of nonlinear nondimensional differential equation is got as 𝑑𝑥₁𝑑𝜏=𝑥₂,𝑑𝑥₂=1𝑑𝜏𝛽_𝑐𝑖−𝑔𝑥₂−sin𝑥₁−𝑥₃,𝑑𝑥₃=1𝑑𝜏𝛽_𝐿𝑥₂−𝑥₃, (2.3) with 𝑥₁=𝛾,𝑥₂=𝑉/𝐼_𝑐𝑅_𝑠,𝑥₃=𝐼_𝑠/𝐼_𝑐. Normalized time: 𝜏=𝜔_𝑐𝑡,𝜔_𝑐=4𝜋𝐼_𝑐𝑅_𝑠/ℎ. Normalized current: 𝑖=𝐼/𝐼_𝑐. Normalized voltage: 𝑣=𝑉/𝐼_𝑐𝑅_𝑠.

Other dimensionless parameters: 𝛽_𝑐=4𝜋𝐼_𝑐𝑅²_𝑠𝐶/ℎ, 𝛽_𝐿=4𝜋𝑒𝐼_𝑐𝐿/ℎ, 𝑔=𝑅_𝑠/𝑅_𝑉, where 𝛽_𝑐 and 𝛽_𝐿 are simplified capacitance and inductance, respectively. Further, the equation of dynamic system [33] can be obtained as 𝑥̇₁=𝑥₂,𝑥̇₂=𝑎𝑑−𝑐𝑥₂−sin𝑥₁−𝑥₃,𝑥̇₃=𝑏𝑥₂−𝑥₃. (2.4)

Let 𝛽_𝑐=0.707, 𝛽_𝐿=2.68, 𝑔=0.0478, 𝑖=1.2, that is, 𝑎=1.4144, 𝑏=0.3731, 𝑐=0.0478, 𝑑=1.2 and the initial condition (0, 0, 0). The phase diagram is shown in Figure 1.

Figure 1 

The 𝑥2-𝑥3 two-dimensional phase diagram.

Bifurcation diagrams can be applied to discover the transitions between periodic motion and chaotic motion of the system with the system parameter varying. Figure 2 displays the bifurcation of the system (2.4) with respect to parameter d.

Figure 2 

The bifurcation diagrams varying with d.

3. Design of the Controller with Only One Control Term

3.1. Chaos Control in RCLSJ Model

The controlled system is defined as follows: 𝑥̇₁=𝑥₂,𝑥̇₂=𝑎𝑑−𝑐𝑥₂−sin𝑥₁−𝑥₃+𝛿(𝑡)+𝑢,𝑥̇₃=𝑏𝑥₂−𝑥₃, (3.1) where  𝑢 is the control input and 𝛿(𝑡) is the external disturbance of the system. Assume disturbance term 𝛿(𝑡) is bounded, that is, |𝛿(𝑡)|≤𝛼 where 𝛼 is positive constant.

The goal is to design a sliding controller and stabilize the system for any given initial condition. First of all, an adaptive switching surface is defined as 𝑆(𝑡)=𝑥₂(𝑡)+𝜑(𝑡), (3.2) where 𝜑(𝑡) is an adaptive function given by 𝜑̇=𝑥₁+𝑏𝑥₃+𝑎𝑐𝑥₂+𝜌𝑥₂,𝜌>0. (3.3) When the system operates in sliding mode, it satisfies the following equation: 𝑆(𝑡)=𝑆̇(𝑡)=0. (3.4) Therefore, 𝑆̇(𝑡)=𝑥̇₂(𝑡)+𝜑̇(𝑡)=0⟹𝑥̇₂(𝑡)=−𝜑̇(𝑡)=−𝑥₁−𝑏𝑥₃−𝑎𝑐𝑥₂−𝜌𝑥₂. (3.5) From (3.5) the following sliding mode dynamics can be obtained as 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑥₁−𝑏𝑥₃−𝑎𝑐𝑥₂−𝜌𝑥₂,𝑥̇₃=𝑏𝑥₂−𝑥₃. (3.6) In the following, the Lyapunov stability theory is used to analyze the stability of the sliding mode dynamics (3.6). The Lyapunov function is selected as 𝑣(𝑡)=0.5(𝑥₁²+𝑥₂²+𝑥₃²), which leads to 𝑣̇=𝑥₁𝑥̇₁+𝑥₂𝑥̇₂+𝑥₃𝑥̇₃=𝑥₁𝑥₂+𝑥₂−𝑥₁−𝑏𝑥₃−𝑎𝑐𝑥₂−𝜌𝑥₂+𝑥₃𝑏𝑥₂−𝑥₃=−𝑎𝑐+𝜌𝑥₂²−𝑏𝑥₃²≤0. (3.7) According to Lyapunov stability theory, it appears that the sliding motion on the sliding manifold is stable and ensures lim_𝑡→∞‖[𝑥₁,𝑥₂,𝑥₃]‖=0 where ‖⋅‖ is the Euclidean norm of a vector.

The next step is to design a control scheme to drive the system trajectories onto the sliding mode 𝑠=0. The equivalent control law is obtained: 𝑢_eq=−𝑥₁−𝑏𝑥₃−𝜌𝑥₂−𝑎𝑑−sin𝑥₁−𝑥₃−𝛿(𝑡). (3.8) In general, the overall control signal has the following form: 𝑢=𝑢_eq+𝑘_𝑠sgn(𝑆), (3.9) where 𝑘_𝑠 is the switching gain.

In practice, the system uncertainty, 𝛿(𝑡), is unknown. To overcome this, the equivalent control input is therefore modified to 𝑢=𝑢_eq+𝑘_𝑠sgn(𝑆)=−𝑥₁−𝑏𝑥₃−𝜌𝑥₂−𝑎𝑑−sin𝑥₁−𝑥₃+𝑘_𝑠sgn(𝑆). (3.10)

Theorem 3.1. When 𝑘_𝑠<−𝛼, the controller (3.10) can make the system (2.4) reach sliding mode 𝑆=0 and the trajectory of the system converge to the sliding surface 𝑆(𝑡)=0 in a finite time.

Proof. The Lyapunov function of the system is constructed as 𝑉=0.5𝑆², and then its first derivative with respect to time is 𝑉̇=𝑆𝑆̇=𝑆𝑎𝑑−𝑐𝑥₂sin𝑥₁−𝑥₃+𝑥₁+𝑏𝑥₃+𝑎𝑐𝑥₂+𝜌𝑥₂(𝑡)𝛿(𝑡)+𝛿+𝑢=𝑆+𝑘_𝑠sgn(𝑆)≤𝛼|𝑆|+𝑘_𝑠|𝑆|=𝑘_𝑠+𝛼|𝑆|<0. (3.11) Thus the proof is achieved completely.

Without loss of generality, we choose the uncertainty term 𝛿(𝑡)=0.1cos(𝑡), where |𝛿(𝑡)|≤𝛼=0.1 and the initial conditions of the system [0.5, 0.8, 0.4], 𝑘_𝑠=−7 and 𝜌=7. The system parameters 𝑎=1.4144, 𝑏=0.3731, 𝑐=0.0478, 𝑑=1.2 are specified for simulation.

The simulation results are modeled in MATLAB software by using fourth-order Runge-Kutta method and shown in the following figures. Figure 3 shows the time responses of the state variables of the uncontrolled system. Figure 4 shows the time domain charts of state variables of system when the control is active. Figure 5 shows the time-varying graph of sliding surface. Obviously, the simulation results presented confirm the validity of the proposed control.

(a) 𝑥1-t

(b) 𝑥2-t

(c) 𝑥3-t

Figure 3 

The time domain charts of state variables without controller.

(a) 𝑥1-t

(b) 𝑥2-t

(c) 𝑥3-t

Figure 4 

The time domain charts of state variables when the control is applied.

Figure 5 

The time domain chart of sliding surface.

3.2. Synchronization of the Coupled RCLSJ Model

For the advantage of ultralow noise, low power consumption and high working frequency for RCL, we were led to ask whether it would be possible to synchronize two different RCLSJ systems together.

The controlled system with noise perturbation is described as follows: 𝑦₁̇=𝑦₂,𝑦₂̇=𝑎𝑑−𝑐𝑦₂−sin𝑦₁−𝑦₃+𝛿(𝑡)+𝑈,𝑦₃̇=𝑏𝑦₂−𝑦₃, (3.12) where 𝑈 is the control input and 𝛿(𝑡) is the external disturbance of the system. Assume disturbance term 𝛿(𝑡) is bounded, that is, |𝛿(𝑡)|≤𝛿 where 𝛿 is positive constant.

Define the error states of system as follows: 𝑒₁=𝑦₁−𝑥₁,𝑒₂=𝑦₂−𝑥₂,𝑒₃=𝑦₃−𝑥₃. (3.13) The error dynamics system is obtained: 𝑒₁̇=𝑒₂,𝑒₂̇=−𝑎𝑐𝑒₂+𝑒₃+𝑑(𝑡)+𝑈₁,𝑒₃̇=𝑏𝑒₂−𝑒₃, (3.14) with 𝑈₁=𝑈−𝑎(sin𝑦₁−sin𝑥₁).

We establish an extended system as follows: 𝑒₁̇=𝑒₂,𝑒₂̇=𝑒₄,𝑒₃̇=𝑏𝑒₂−𝑒₃,𝑒₄̇=−𝑎𝑏𝑒₂−𝑒₃−𝑎𝑐𝑒₄+𝑑̇(𝑡)+𝑈̇₁, (3.15) and choose a sliding mode surface 𝑠=0, where 𝑠(𝑡)=𝑒₄+^𝑡₀𝑐₁𝑒₁+𝑐₂𝑒₂+𝑐₄𝑒₄𝑑𝜏 (3.16) and 𝑐₁, 𝑐₂, 𝑐₄ are constants waiting for designing.

Theorem 3.2. If the controller 𝑈₁ is selected as 𝑈₁=^𝑡₀𝑎𝑐𝑒₄+𝑎𝑏𝑒₂−𝑒₃−𝑐₁𝑒₁−𝑐₂𝑒₂−𝑐₄𝑒₄−𝑘sgn(𝑠(𝑡))𝑑𝜏, (3.17) where 𝑠(𝑡) is as in (3.16) and 𝑘 is constant satisfying 𝑘>𝛿+1, then the states of the error system (3.13) will approach the sliding mode surface 𝑠=0 in a finite time.

Proof. The Lyapunov function of the system is selected as 𝑉=𝑠²; with the section of the sliding mode surface (3.16) and the controller (3.17), we have 𝑉̇=2𝑠𝑒₄̇+𝑐₁𝑒₁+𝑐₂𝑒₂+𝑐₄𝑒₄=2𝑠−𝑎𝑐𝑒₄−𝑎𝑏𝑒₂−𝑒₃+𝑐₁𝑒₁+𝑐₂𝑒₂+𝑐₄𝑒₄−𝛿(𝑡)+𝑢̇₁=2𝑠−𝑑̇(𝑡)−𝑘sgn(𝑠)≤2−𝑑̇(𝑡)−𝑘|𝑠|≤−2|𝑠|=−2𝑉^1/2 (3.18) which implies that, 𝑉^1/2(𝑡)≤𝑉^1/2(0)−𝑡, 𝑡∈[0,𝑡_𝑠) and 𝑉(𝑡)=0 when 𝑡≥𝑡_𝑠 where 𝑡_𝑠=𝑉^1/2(0). The proof is finished as desired.

Next we analyze the dynamics of the error system on the sliding manifold. On the sliding manifold 𝑠=0 the error system (3.15) reads 𝑒₁̇=𝑒₂,𝑒₂̇=𝑒₄,𝑒₃̇=𝑏𝑒₂−𝑒₃,𝑒₄̇=−𝑐₁𝑒₁−𝑐₂𝑒₂−𝑐₄𝑒₄+𝛿(𝑡). (3.19) Thus, 𝑒₁𝑒₂𝑒₄=𝑒^𝐴𝑡𝑒₁(0)𝑒₂(0)𝑒₄(0)+^𝑡₀𝑒^−𝐴𝑠00𝑑̇(𝑡)𝑑𝑠, (3.20) where 𝐴=010001−𝑐₁−𝑐₂−𝑐₄. (3.21) Its characteristic polynomial is 𝑓(𝜆)=|𝜆𝐼−𝐴|=𝜆³+𝑐₄𝜆²+𝑐₂𝜆+𝑐₁. (3.22) According to Routh-Hurwitz theorem, we know that the real parts of its all characteristic roots are negative if and only if Δ₁=𝑐₄>0,Δ₂=𝑐₄𝑐₂−𝑐₁>0,Δ₃=𝑐₁𝑐₄𝑐₂−𝑐₁>0. (3.23) Therefore, there exists positive constants 𝑎₁ and 𝑏₁ such that |𝑒^𝐴𝑡𝑥|≤𝑎₁𝑒^−𝑏1𝑡|𝑥| for every 𝑥∈𝑅³ and 𝑡≥0. Thus, 𝑒_𝑖≤𝑎₁𝑒^−𝑏1𝑡max_𝑖=1,2,4𝑒_𝑖(0+𝑎)¹𝛿𝑏₁,𝑖=1,2,4. (3.24) Additionally, (3.19) implies that 𝑒₃=𝑒^−𝑏𝑡𝑒₃(0)+^𝑡₀𝑒^𝑏𝑡𝑏𝑒₂𝑑𝑡. (3.25) Therefore, 𝑒₃≤𝑒^−𝑏𝑡𝑒₃(0)+^𝑡₀𝑒^𝑏𝑡𝑏𝑒₂𝑑𝑡. (3.26) Obviously, we can get the following theorem.

Theorem 3.3. If the controller U is selected as 𝑈=𝑈₁+𝑎(sin𝑦₁−sin𝑥₁), where 𝑈₁=∫^𝑡₀[𝑎𝑐𝑒₄+𝑎𝑏(𝑒₂−𝑒₃)−𝑐₁𝑒₁−𝑐₂𝑒₂−𝑐₄𝑒₄−𝑘sgn(𝑠(𝑡))]𝑑𝜏, 𝑐₁, 𝑐₂, 𝑐₄ are constants satisfying (3.23) and 𝑘>𝛿+1, there exists a constant M such that the controlled chaotic system with noise perturbation (3.12) is synchronous with the system (2.4) with ultimate error bound 𝑀𝛿, that is: lim_𝑡→∞𝑦₁−𝑥₁≤𝑀𝛿,lim_𝑡→∞𝑦₂−𝑥₂≤𝑀𝛿,lim_𝑡→∞𝑦₃−𝑥₃≤𝑀𝛿. (3.27)

Without loss of generality, we choose the uncertainty term 𝛿(𝑡)=0.1cos(𝑡), where |𝛿(𝑡)|≤𝛿=1. In the following numerical simulations, the parameters of the system are 𝑎=1.4144, 𝑏=0.3731, 𝑐=0.0478, 𝑑=1.2; the initial conditions of the system (2.4) and the system (3.12) are [0.40.50.7]^𝑇 and [0.50.60.8]^𝑇, respectively. The constants in the sliding mode are selected 𝑐₁=1, 𝑐₂=2, 𝑐₄=2. And the constant in the sliding mode controller is selected as 𝑘=10.

Numerical simulations are presented to demonstrate the effectiveness of the proposed method. Figure 6 displays the time response of the synchronization errors defined in (3.13) and sliding mode surface defined in (3.16) when the control signal has been activated. It is very clear that the error dynamics converge asymptotically to zero as soon as the control is activated. The numerical simulation verifies the theoretical analysis.

(a) 𝑒1-t

(b) 𝑒2-t

(c) 𝑒3-t

(d) 𝑠-t

Figure 6 

The time response of the error system and sliding mode surface.

4. Conclusions and Discussion

We have investigated the control and synchronization of the RCLSJ model via sliding mode method. In the sliding mode design, the single controller is constructed in the case of noise disturbance for the chaos control of the junction, and numerical simulation results are employed to verify the effectiveness of the control scheme. And it is same with the chaos synchronization of the junction. In practical system, with noise disturbance considered, the control scheme is of significant importance due to its simple and effective execution as well as good robustness.

Here, another interesting remark is that chaos control and chaos synchronization could be realized by using the same method. Comparing the chaos control and synchronization, we can get the conclusion that chaos control and chaos synchronization are the same with each other in essence. In other words, chaos control is special case of chaos synchronization, and it is considered to achieve synchronization with O (0, 0, 0). Moreover, applying this control method to multiscroll chaotic systems [36–39] is our future work.

Acknowledgments

This wok was supported by the scientific research foundation of National Natural Science Foundation (51109180) and the “111” Project from the Ministry of Education of P.R. of China and the State Administration of Foreign Experts Affairs of P.R. of China (B12007) and the National Science and Technology Supporting Plan from the Ministry of Science and Technology of P.R. of China (2011BAD29B08).