Analysis of a No Equilibrium Linear Resistive-Capacitive-Inductance Shunted Junction Model , Dynamics , Synchronization , and Application to Digital Cryptography in Its Fractional-Order Form

1Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon 2Department of Physics, Higher Teacher Training College, University of Bamenda, P.O. Box 39, Bamenda, Cameroon 3Research Group on Experimental and Applied Physics for Sustainable Development, Faculty of Science, Department of Physics, University of Dschang, P.O. Box 412, Dschang, Cameroon 4Laboratory of Electronics and Signal Processing Faculty of Science, Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon 5Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes (LaMSEBP) and TWAS Research Unit, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon


Introduction
A Josephson junction (JJ) is made up of two superconductors, separated by a nonsuperconducting layer so thin that electrons can cross through the insulating barrier [1].The flow of current between the superconductors in the absence of an applied voltage is called a Josephson current and the movement of electrons across the barrier is known as Josephson tunneling [2,3].The JJ finds practical use as a millimetre or submillimetre wave oscillator [4], in digital systems [5] and in superconducting quantum interference devices such as magnetometer applications [6].Based on the rapid development of superconducting devices in the fabrication technology different models of JJ have been reviewed in order to understand whether such superconducting junction can be used for ultrahigh-speed chaotic generators for applications of code generation in spread-spectrum communications and random key generation in secure communication and encryption [7].[14].(b) Current-voltage characteristics at a temperature  (in Kelvin) of the intrinsic junction shunt resistance () [3].(c) The schematic representation of LRCLSJ model.
In the literature, we can find different models of JJ, namely, linear resistive-capacitive shunted junction (RCSJ) model, nonlinear resistive-capacitive shunted junction model, and the NRCLSJ model [3,[8][9][10][11][12].The first two models show chaotic behaviors when driven by an external sinusoidal signal [11] while the NRCLSJ model generates chaotic oscillation with external dc bias only [12][13][14].Regular spiking, intrinsic bursting, and fast spiking which are usually seen in the mammalian neocortex have been found in the NRCLSJ model [15].For large inductance, the NRCLSJ model behaves as a relaxation oscillator [16].The NRCLSJ model [8,9] is used to simulate JJ, resulting in a fairly good agreement with experiment.The RCSJ model, however, fails to reproduce significant features on experimental - curves when the shunt of the JJ contains an inductance component [12,17].
In this paper, we use the NRCLSJ model where the nonlinear piecewise resistance is replaced by a linear resistance to make it the LRCLSJ model.In [18], Neumann and Pikovsky reported on the study of a nontrivial type of slow-fast dynamics in LRCLSJ model.However, to the best of our knowledge, the dynamical behavior of the LRCLSJ model without equilibrium and its fractional-order form remains unaddressed.It is important to note that, due to the absence of equilibrium, the LRCLSJ model belongs to a class of systems with hidden attractors [19][20][21][22].Systems with hidden attractors have three different families: systems with an infinite number of equilibrium points, systems with only stable equilibrium point, and systems without equilibrium points.A hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points, while a self-excited attractor has a basin of attraction that is associated with an unstable equilibrium.Almost all famous chaotic attractors are self-excited.Hidden attractors are of high interest in engineering applications because they can exhibit unexpected and potentially disastrous responses to perturbations in a structure like a bridge or an airplane wing [23][24][25][26].The paper is articulated around four sections presented as follows: in Section 2, the analytical and numerical analysis of the LRCLSJ model are investigated.In Section 3, we focus on the dynamical behavior, synchronization, and application to digital cryptography in the fractionalorder form of LRCLSJ model without equilibrium points.Finally the conclusion of the paper is drawn in Section 4.

Rate-Equations and Analysis of the Linear Resistive-Capacitive-Inductance Shunted Junction Model
In this work, we consider the NRCLSJ model where the nonlinear resistance () is replaced by a linear resistance  as shown in Figure 1.
The NRCLSJ model is presented in Figure 1(a).The intrinsic junction shunt nonlinear resistance () is modelled by a piecewise linear resistor as shown in Figure 1(b).In Figure 1(c), the shunted nonlinear resistance in Figure 1(a) is replaced by a linear resistor .The JJ is represented by the supercurrent channel   . is the bias current applied to the JJ and  is the junction capacitance.A current   flows through the shunt inductance   and its internal resistance   .The application of the Kirchhoff laws to the circuit of Figure 1(c) leads to the following differential equation: where  denotes the phase difference of JJ.Using the dimensionless variables, ( The dimensionless set of (1a)-(1c) can be rewritten as where   ,   ,   ,  are positive parameters which represent the dc bias current, the ratio of resistors, the capacitor, and the inductance, respectively.Therefore, the LRCLSJ model is described by the set of (3a)-(3c).

Dynamical Behaviors of the Linear Resistive-Capacitive-
Inductance Shunted Junction Model.Dynamics of LRCLSJ model is investigated by considering the effect of parameters on system's behavior in order to see if it can exhibit some of the dynamical behaviors of the NRCLSJ model.Our simulations show that for  < 1.0 the trajectories of system (3a)-(3c) converge to one of the equilibrium points  1,2 while, for  > 1.0, the trajectories of system (3a)-(3c) display periodic or complex behaviors as will be seen in the paragraphs below.It is interesting to note that, for  > 1.0, system (3a)-(3c) has no equilibrium point.We find regular spiking, intrinsic bursting, fast spiking, and periodic bursting in the junction as shown in Figure 2 when the capacitive parameter   is kept fixed at   = 0.007, while the dc bias , the inductive   , and resistive   parameters are varied.
For   = 1.5,   = 0.1, and  = 1.25, the junction exhibits regular spiking as shown in Figure 2(a).The regular spiking is sensitive to dc bias  and the parameter   .In Figure 2(b), for the same values of   and   parameters, intrinsic bursting is observed for larger dc bias current  = 1.75 when the spiking frequency is obviously larger.For   = 0.5, the fast spiking is found in Figure 2(c) when the dc bias current  and other parameters remain the same as for intrinsic bursting.For   = 0.01, the periodic bursting is observed in Figure 2(d) when the dc bias current  and other parameters are kept fixed as for regular spiking.Such spiking behaviors have been found in the NRCLSJ model [16] and are typical of the mammalian neocortex [27].As in the NRCLSJ model, the LRCLSJ model behaves as a relaxation oscillator for large value of parameter   [15].This is shown in Figure 3 which presents the time series of () and corresponding phase portrait for   = 26,   = 0.5,  = 1.25, and   = 0.1.
In the following, the dynamical behavior of system (3a)-(3c) is illustrated using bifurcation diagrams, Lyapunov exponent, time series, phase portraits, and basin of attraction.In Figure 4, we plot the bifurcation diagram depicting the local maxima of () as a function of dc bias current  for   = 2.6,   = 0.707, and   = 0.06.
When the dc bias current  increases from 1.05 to 2.1, the bifurcation diagram of Figure 4(a) shows period-2oscillations followed by an intermittency route to chaos interspersed with periodic windows.For  > 1.255, a reverse period-tripling bifurcation to period-2-oscillations at  ≈ 1.594 is observed.By increasing the dc bias current , the output () displays period-2-oscillations followed by period-1-oscillations for  ≥ 2.0085.The chaotic behavior is confirmed by the largest Lyapunov exponent shown in Figure 4(b).The chaotic behavior is illustrated in Figure 5 for a specific value of dc bias current .
For  = 1.25,   = 0.707, and   = 0.06, we plot the bifurcation diagram depicting the local maxima of () and the largest Lyapunov exponent of system (3a)-(3c) versus the parameter   as shown in Figure 6.
When the parameter   increases from 1.9 to 2.8 [see black dot in Figure 6(a)], the bifurcation diagram of the output () shows period-2-oscillations followed by an intermittency route to chaos interspersed with periodic windows.For   > 2.625, a reverse period-tripling bifurcation to period-3oscillations at   ≈ 2.725 is observed.Period-3-oscillations persists for   > 2.725.When performing the same analysis by ramping the parameter   [see red dot in Figure 6(a)], the output () displays the same dynamical behaviors as in Figure 6(a) (see black dot) in the ranges 1.9 ≤   < 2.014 and 2.0725 <   ≤ 2.8, while, in the range 2.014 ≤   ≤ 2.0725, the output () shows period-1-oscillations, perioddoubling bifurcation, and chaotic oscillations, respectively.By comparing the two sets of data [for increasing (black) and decreasing (red)] used to plot Figure 6(a), one can notice that system (3a)-(3c) displays coexistence of attractors in the range 2.014 ≤   ≤ 2.0725.For 2.014 ≤   < 2.0375, period-2-oscillations coexist with period-1-oscillations.Period-2oscillations coexist with period-doubling bifurcation in the range 2.0375 ≤   < 2.056.For 2.056 <   ≤ 2.0725, period-1-oscillations coexist with chaotic oscillations.The chaotic behavior is confirmed by the largest Lyapunov exponent shown in Figure 6(b).The coexistence of attractors found in Figure 6(a) is illustrated in Figure 7 which depicts the phase portraits and cross section of the basin of attraction of system (3a)-(3c) for specific value of parameter   .

Fractional-Order Form of Chaotic No Equilibrium Linear Resistive-Capacitive-Inductance Shunted Junction Model
In this section, we consider the commensurate fractionalorder form of the no equilibrium LRCLSJ model given by where  is the derivative order satisfying 0 <  ≤ 1.For numerical solutions of the above set of commensurate fractional-order differential equations, the Adams Bashforth-Moulton predictor-corrector scheme [38] is used.This method is based on the Caputo definition of the fractional-order derivative, given by [39,40] where  − 1 <  < ,  1 = ,  2 = ,  3 = , and Γ(⋅) is the Gamma function.In the coming subsections, we will focus on the effect of fractional derivation on the chaotic system (3a)-(3c) when  = 1.25,   = 2.6,   = 0.707, and   = 0.06 (see Figure 5).Chaos synchronization of unidirectional identical coupled commensurate fractionalorder system (7a)-(7c) and engineering applications shall also be investigated.

Effect of Commensurate Fractional Derivation on the Chaotic No Equilibrium Linear Resistive-Capacitive-
Inductance Shunted Junction Model.For the dc bias current  > 1, the commensurate fractional-order system (7a)-(7c) has no equilibrium points.Therefore we cannot determine analytically the lowest order of the commensurate fractionalorder system (7a)-(7c) to exhibit chaotic behavior.Rather, the effect of commensurate fractional derivation on chaotic system (3a)-(3c) for  = 1.25,   = 2.6,   = 0.707, and   = 0.06 can only be numerically investigated.Here accordingly, using the Adams Bashforth-Moulton predictorcorrector scheme presented in this subsection, we plot the bifurcation diagram showing the local maxima of the state variable  with respect to the commensurate fractionalorder  as shown in Figure 8 in order to find the lowest order of system (7a)-(7c) to remain chaotic.
The bifurcation diagram of Figure 8 indicates period-2-oscillations followed by chaotic behavior for  ≥ 0.978.Hence, the lowest order for the commensurate fractionalorder form of the no equilibrium LRCLSJ model to show chaos is 3 ≈ 2.934.In Figure 9, the time series of () and the phase portraits in the plane (, ) of significant results was obtained for specific values of commensurate fractionalorder .

Adaptive Finite-Time Synchronization with Parameter Estimation of Unidirectional Coupled Identical Fractional-Order Form Chaotic No Equilibrium Linear Resistive-
Capacitive-Inductance Shunted Junction Models.In many practical cases, it is difficult (or impossible) to accurately determine the values of the parameters of dynamical systems to be synchronized.As it was proven that accurate control of these parameters can significantly affect the synchronization process, however, this problem can be solved by adaptive synchronization with parameter estimation [41].The aim of this subsection is to provide an example of adaptive finitetime synchronization with parameter estimation applied to the chaotic commensurate fractional-order system (7a)-(7c).
The drive system can be written in the form And the response system is described by The controller signal is () = ( 2 −  1 ) + (), where () is the adaptive feedback coupling designed to achieve finite-time synchronization.We set the synchronization error between drive and response systems as Theorem 2. The response system (10a)-(10c) synchronizes with the drive system (8) in the finite-time If the adaptive feedback controller and the estimated parameter are set as follows: where , , and  are the positive parameters.The slave system can be also written as Proof.We consider the following Lyapunov candidate function defined by The fractional-order derivative of (15) gives Using the assumption, ( 16) becomes exists such that   ()/  ≤ −, then the update laws for the controller () and the estimated parameter β are designed through the following relations: In [42], the authors have shown that the finite-time synchronization of fractional-order system is shorter than the finite-time synchronization of integer-order systems.This implies that the finite-time of synchronization for integerorder systems is also finite-time of synchronization for the fractional-order version of the same systems with the same controller involved.Therefore the fractional-order response system (10a)-(10c) synchronizes with the drive system (8) in a finite-time of synchronization given by (12).Figures 10 and 11 are the graphical representation of the synchronization process obtained numerically by carrying out a numerical integration of drive system described by ( 8) and the response system (10a)-(10c) for their respective initial conditions chosen as ( 1 (0),  1 (0),  1 (0)) = (0, 0, 1) and ( 2 (0),  2 (0),  2 (0), β (0), (0)) = (0.1, 0.1, 1.2, −0.3, 0.1).To guarantee the chaotic synchronization, the other parameters are fixed at  = 0.98,   = 00.707, = 1.25,   = 2.6,   = 0.06,  = 0.01,  = 0.004, and  = 3.Based on these values, the theoretical settled time of synchronization is  1 syn = 45.375.
The dynamic of the synchronization error is present in Figure 11.
From Figure 11, it can be seen that the numerical time of synchronization is   syn ≈ 40.By comparing with analytical time of synchronization obtained above, it can be noted that   syn ≤  1  syn .This result respects the finite-time condition given in [42].Thus, these results can be used for chaos based communications.

Application to Digital Cryptography.
In this subsection, we propose a digital cryptography scheme based on the adaptive finite-time synchronization with parameters estimation of unidirectional coupled identical commensurate fractional-order form chaotic no equilibrium LRCLSJ models developed in the previous subsection.The technique of digital cryptography to be used in the present paper is an improved version of the method developed and implemented in [42][43][44] but we improve these methods.In [42], the authors used the adaptive finite-time synchronization with parameter estimation for message encryption; meanwhile in the present work we use the adaptive finite-time synchronization with parameters estimation which is recognized to be more robust than its counterpart [41].Cryptography is a way to encode a message by changing its original message form into another different message whose conversion rule is known only by the sender.In the digital cryptography, for instance, the letters can be replaced by numbers which can be assigned by the sender.In this work, the message (plaintext) will be replaced by their ASCII codes before the encryption.As the ASCII code table has 128 characters, we define the formula for the ciphertext and decrypted message corresponding to the assignment of numbers as follows: where   is the plaintext to be encrypted,   is the plaintext to be recovered,   is the secret keys of sender,   is the secret keys of the receiver, and  is the ciphertext.

Proposed Affine Cipher.
Consider  and , respectively, as the sender and the receiver in the cryptosystem.Also consider the drive system (8) as a sender's system and the response system (10a)-(10c) as a receiver's system. and  agree on a time  ≥   syn , fractional-order  = 0.98, and a number of characters of the ASCII codes table equal to 128.The public key  = Day/Month/year is only shared by  and .This public key allows building the two privates keys.The complete procedure between  and  is described as follows.

Key Generation. (1)
The plaintext can be presented into a form of integers using the ASCII code table.After this assignation that plaintext can be organized in blocks of four terms.If the number of letters constituting the message is a multiple of 4, we will have  1 ,  2 ,  3 ,  4 as a first bloc and  5 ,  6 ,  7 ,  8 as a second block and so on.On the other hand, if the number of letters of that message is not a multiple of 4 then blank spaces have to be insert at the end of the sequence to complete it to a multiple of 4.
(5) The ciphertext and recovered plaintext are obtained by applying  =   +   (mod 128) and   =  −   (mod 128), respectively.After the treatment of the first block of plaintext we restart the process in the next block and so on.
As an example, we want to send the message PASSWORD to a third party.By following the five steps mentioned above, we obtain the summarized results in Table 1.

Security Analysis of Proposed Affine
Cipher.The present proposed affine cipher consists of three keys: one public key  and two private keys  1 and  2 .The two private keys have been computed with the public key.The four receive keys and four sending keys have been computed with the private and sending keys as well as the Fibonacci numbers.The receiver cannot recover an original text without the knowledge of the receiver private key  2 .Compared to the existing work in this field to the best of our knowledge, our proposed technique is fully authenticated and more adapted for applications in telecommunications because we have used the adaptive finite-time synchronization with parameters estimation.In fact more recently in [42], the authors used the adaptive finite-time synchronization to improve the digital cryptography developed in [43,44].Yet, the adaptive finitetime synchronization is less robust than the adaptive finitetime synchronization with parameters estimation.The finitetime adaptive synchronization based on the identification of system parameters appears to be of great practical interest, especially when the system state is available to external measurements [45].Therefore, the proposed affine cipher is more efficient than the ordinary affine cipher method due to the adaptive finite-time synchronization with parameters estimation.

Conclusion
The present study dealt with the analysis of the linear resistive-capacitive-inductance shunted junction model and its fractional-order form.The linear resistive-capacitiveinductance junction shunted model has two or no equilibrium points depending on the dc bias current.By using the Routh-Hurwitz stability criteria, it has been found that the two equilibrium points are stable and saddle nodes, respectively.In spite of the absence of equilibrium, the linear resistive-capacitive-inductance shunted junction model can exhibit fast and regular spiking, intrinsic and periodic bursting, periodic and chaotic behaviors, and coexistence of attractors.Then, the commensurate fractional-order form of chaotic no equilibrium linear resistive-capacitiveinductance shunted junction model has been investigated.Chaos has been shown to exist in commensurate fractionalorder form of no equilibrium linear resistive-capacitiveinductance shunted junction model with orders less than 3.
Using adaptive finite-time synchronization with parameter estimation, chaos synchronization has been found between unidirectional coupled identical fractional-order forms of chaotic no equilibrium linear resistive-capacitive-inductance shunted junction models.Applying the synchronized commensurate fractional-order form of chaotic no equilibrium linear resistive-capacitive-inductance shunted junction models in digital cryptography, a well secured key system has been obtained.

Figure 10 :Figure 11 :
Figure 10: The time evolution of the adaptive feedback controller () (a) and the controller signal () (b) as well as the estimated parameter β (c).

Table 1 :
Summary of sending and receive message (PASSWORD).