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

and Applied Analysis 3 2. The Model of Josephson Junction System and Dynamics Analysis A standard form of the RCLSJ model 35 is proposed as


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 Bi 2 Sr 2 CaCu 2 O 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

The Model of Josephson Junction System and Dynamics Analysis
A standard form of the RCLSJ model 35 is proposed as 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: 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 Other dimensionless parameters: , where β c and β L are simplified capacitance and inductance, respectively.Further, the equation of dynamic system 33 can be obtained as   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.
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.

Chaos Control in RCLSJ Model
The controlled system is defined as follows: 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 where ϕ t is an adaptive function given by φ x 1 bx 3 acx 2 ρx 2 , ρ > 0. 3.3 When the system operates in sliding mode, it satisfies the following equation: From 3.5 the following sliding mode dynamics can be obtained as

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 According to Lyapunov stability theory, it appears that the sliding motion on the sliding manifold is stable and ensures lim t → ∞ x 1 , x 2 , x 3 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: 3.8 In general, the overall control signal has the following form: u u eq k s sgn 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

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 0.5S 2 , and then its first derivative with respect to time is

3.11
Thus the proof is achieved completely.
Without loss of generality, we choose the uncertainty term δ t 0.1 cos 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.

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: 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:

3.13
The error dynamics system is obtained:  We establish an extended system as follows: 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 s 2 ; with the section of the sliding mode surface 3.16 and the controller 3.17 Next we analyze the dynamics of the error system on the sliding manifold.On the sliding manifold s 0 the error system 3.  where Its characteristic polynomial is

3.22
According to Routh-Hurwitz theorem, we know that the real parts of its all characteristic roots are negative if and only if Therefore, there exists positive constants a 1 and b 1 such that |e At x| ≤ a 1 e −b 1 t |x| for every x ∈ R 3 and t ≥ 0. Thus, 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: Without loss of generality, we choose the uncertainty term δ t 0.1 cos 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 0.4 0.5 0.7 T and 0.5 0.6 0.8 T , respectively.The constants in the sliding mode are selected c 1 1, c 2 2, c 4 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.

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.

Figure 3 :
Figure 3: The time domain charts of state variables without controller.

ė1 e 2 ,ė2 −a ce 2 e 3 d t U 1 , ė3 b e 2 − e 3 , 3 . 14 with U 1 UFigure 4 :
Figure 4: The time domain charts of state variables when the control is applied.

2 sFigure 5 :
Figure 5: The time domain chart of sliding surface.

Figure 6 :
Figure 6: The time response of the error system and sliding mode surface.