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

Authors: Sifeu Takougang Kingni; Gaetan Fautso Kuiate; Romanic Kengne; Robert Tchitnga; Paul Woafo
Journal: Complexity
 (2017)

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## Abstract

A linear resistive-capacitive-inductance shunted junction (LRCLSJ) model obtained by replacing the nonlinear piecewise resistance of a nonlinear resistive-capacitive-inductance shunted junction (NRCLSJ) model by a linear resistance is analyzed in this paper. The LRCLSJ model has two or no equilibrium points depending on the dc bias current. For a suitable choice of the parameters, the LRCLSJ model without equilibrium point can exhibit regular and fast spiking, intrinsic and periodic bursting, and periodic and chaotic behaviors. We show that the LRCLSJ model displays similar dynamical behaviors as the NRCLSJ model. Moreover the coexistence between periodic and chaotic attractors is found in the LRCLSJ model for specific parameters. The lowest order of the commensurate form of the no equilibrium LRCLSJ model to exhibit chaotic behavior is found to be 2.934. Moreover, adaptive finite-time synchronization with parameter estimation is applied to achieve synchronization of unidirectional coupled identical fractional-order form of chaotic no equilibrium LRCLSJ models. Finally, a cryptographic encryption scheme with the help of the finite-time synchronization of fractional-order chaotic no equilibrium LRCLSJ models is illustrated through a numerical example, showing that a high level security device can be produced using this system.


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## Body

## 1. 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]. 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, 8–12]. 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 [12–14]. 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 I-V 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 [19–22]. 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 [23–26]. 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 fractional-order form of LRCLSJ model without equilibrium points. Finally the conclusion of the paper is drawn in Section 4.


## 2. Rate-Equations and Analysis of the Linear Resistive-Capacitive-Inductance Shunted Junction Model

In this work, we consider the NRCLSJ model where the nonlinear resistanceR(V) is replaced by a linear resistance R as shown in Figure 1.Figure 1
(a) The schematic representation of NRCLSJ model [14]. (b) Current-voltage characteristics at a temperature T (in Kelvin) of the intrinsic junction shunt resistance R(V) [3]. (c) The schematic representation of LRCLSJ model.
(a)
(b)
(c)The NRCLSJ model is presented in Figure1(a). The intrinsic junction shunt nonlinear resistance R(V) 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 R. The JJ is represented by the supercurrent channel IC. I is the bias current applied to the JJ and C is the junction capacitance. A current IS flows through the shunt inductance LS and its internal resistance RS. The application of the Kirchhoff laws to the circuit of Figure 1(c) leads to the following differential equation:(1a)I=CdVdt′+VR+ICsin⁡ϕ+IS(1b)V=LSdISdt′+RSIS(1c)V=ħ2edϕdt′,where ϕ denotes the phase difference of JJ. Using the dimensionless variables,(2)x=VRSIC,y=ISIC,z=ϕ,t′=tω0,ω0=2πeRSICh,βR=RSR,βC=2πeCICRS2h,βL=2πeLSICh,i=IIC.The dimensionless set of (1a)–(1c) can be rewritten as(3a)dxdt=1βCi-y-βRx-sin⁡z(3b)dydt=1βLx-y(3c)dzdt=x,where βR,βC,βL,i 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).


### 2.1. Analytical Analysis of the Linear Resistive-Capacitive-Inductance Shunted Junction Model

System (3a)–(3c) is dissipative because ∇V=∂x˙/∂x+∂y˙/∂y+∂z˙/∂z=-βR/βC+1/βL<0. For i>1, it does not have an equilibrium point, while, for i≤1, that same system has two equilibrium points E1=(0,0,arcsin⁡i) and E2=(0,0,π-arcsin⁡i). The characteristic equation associated with the equilibrium point E=(x∗,y∗,z∗) is (4)λ3+βRβC+1βLλ2+1βCβL+βRβCβL+1βCcos⁡z∗λ+1βCβLcos⁡z∗=0,which, for E1, gives (5)λ3+βRβC+1βLλ2+1βCβL+βRβCβL+1βCcos⁡arcsin⁡iλ+1βCβLcos⁡arcsin⁡i=0and, for E2, the characteristic equation becomes (6)λ3+βRβC+1βLλ2+1βCβL+βRβCβL-1βCcos⁡arcsin⁡iλ-1βCβLcos⁡arcsin⁡i=0.According to the Routh–Hurwitz criteria, the equilibrium point E1 is a stable node and the equilibrium point E2 is a saddle node.


### 2.2. 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 fori<1.0the trajectories of system (3a)–(3c) converge to one of the equilibrium points E1,2 while, for i>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 i>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 βC is kept fixed at βC=0.007, while the dc bias i, the inductive βL, and resistive βR parameters are varied.Figure 2
Time series ofxt and corresponding phase portraits of system (3a)–(3c) in plane x,y for specific values of (a) βL=1.5, i=1.25, and βR=0.1, (b) βL=1.5, i=1.75, and βR=0.1, (c) βL=0.5, i=1.75, and βR=0.1, and (d) βL=1.5, i=1.25, and βR=0.01. The other parameter is βC=0.007.
(a)
(b)
(c)
(d)ForβL=1.5, βR=0.1, and i=1.25, the junction exhibits regular spiking as shown in Figure 2(a). The regular spiking is sensitive to dc bias i and the parameter βR. In Figure 2(b), for the same values of βL and βR parameters, intrinsic bursting is observed for larger dc bias current i=1.75 when the spiking frequency is obviously larger. For βL=0.5, the fast spiking is found in Figure 2(c) when the dc bias currentiand other parameters remain the same as for intrinsic bursting. For βR=0.01, the periodic bursting is observed in Figure 2(d) when the dc bias current i 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 βL [15]. This is shown in Figure 3 which presents the time series of xt and corresponding phase portrait for βL=26, βC=0.5, i=1.25, and βR=0.1.Figure 3
Time series ofxt and corresponding phase portrait of system (3a)–(3c) in plane x,y for βL=26, βC=0.5, i=1.25, and βR=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 xt as a function of dc bias current i for βL=2.6, βC=0.707, and βR=0.06.Figure 4
The bifurcation diagram depicting the local maxima ofxt (a) and the largest Lyapunov exponent (b) of system (3a)–(3c) versus the dc bias current i for βL=2.6, βC=0.707, and βR=0.06.
(a)
(b)When the dc bias currenti increases from 1.05 to 2.1, the bifurcation diagram of Figure 4(a) shows period-2-oscillations followed by an intermittency route to chaos interspersed with periodic windows. For i>1.255, a reverse period-tripling bifurcation to period-2-oscillations at i≈1.594 is observed. By increasing the dc bias current i, the output xt displays period-2-oscillations followed by period-1-oscillations for i≥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 i.Figure 5
Time series ofxt, yt, and zt and corresponding phase portrait of system (3a)–(3c) in plane x,y for i=1.25. The other parameters are βL=2.6, βC=0.707, and βR=0.06. The initial conditions are (x(0),y(0),z(0))=(0.0,0.0,1.0).From Figure5, we observe that system (3a)–(3c) exhibits chaotic attractor similar to the one found in the NRCLSJ model [12–16].Fori=1.25, βC=0.707, and βR=0.06, we plot the bifurcation diagram depicting the local maxima of xt and the largest Lyapunov exponent of system (3a)–(3c) versus the parameter βL as shown in Figure 6.Figure 6
The bifurcation diagrams depicting the local maxima ofxt (a) and the largest Lyapunov exponent (b) of system (3a)–(3c) versus the parameter βL for i=1.25, βC=0.707, and βR=0.06. Bifurcation diagrams are obtained by scanning the parameter βL upwards (black) and downwards (red).
(a)
(b)When the parameterβL increases from 1.9 to 2.8 [see black dot in Figure 6(a)], the bifurcation diagram of the output xt shows period-2-oscillations followed by an intermittency route to chaos interspersed with periodic windows. For βL>2.625, a reverse period-tripling bifurcation to period-3-oscillations at βL≈2.725 is observed. Period-3-oscillations persists for βL>2.725. When performing the same analysis by ramping the parameter βL [see red dot in Figure 6(a)], the output xt displays the same dynamical behaviors as in Figure 6(a) (see black dot) in the ranges 1.9≤βL<2.014 and 2.0725<βL≤2.8, while, in the range 2.014≤βL≤2.0725, the output xt shows period-1-oscillations, period-doubling 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≤βL≤2.0725. For 2.014≤βL<2.0375, period-2-oscillations coexist with period-1-oscillations. Period-2-oscillations coexist with period-doubling bifurcation in the range 2.0375≤βL<2.056. For 2.056<βL≤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 βL.Figure 7
(a) Phase portraits in the planex,y for specific initial conditions: (blue line) (x(0),y(0),z(0))=(0.0,0.0,1.0) and (black line) (x(0),y(0),z(0))=(3.3,0.0,1.0). (b) Cross section of the basin of attraction of system (3a)–(3c) in the xz-plane at y = 0. In subplot (b), periodic attractors are in red color and chaotic attractors are in black color. The parameters values are βL=2.06, i=1.25, βC=0.707, and βR=0.06.
(a)
(b)System (3a)–(3c) exhibits period-1-oscillations and chaotic attractor for βL=2.06and depending on the initial conditions as shown in Figure 7(a). One can see from Figure 7(a) that system (3a)–(3c) can exhibit either chaotic or periodic attractor depending on the initial conditions. It is worth noting that coexistence of attractors has been observed in various nonlinear systems including laser [28, 29], biological system [30], chemical reactions [31], Lorenz systems [32–35], and electrical circuits [36, 37], just to name a few.


## 2.1. Analytical Analysis of the Linear Resistive-Capacitive-Inductance Shunted Junction Model

System (3a)–(3c) is dissipative because ∇V=∂x˙/∂x+∂y˙/∂y+∂z˙/∂z=-βR/βC+1/βL<0. For i>1, it does not have an equilibrium point, while, for i≤1, that same system has two equilibrium points E1=(0,0,arcsin⁡i) and E2=(0,0,π-arcsin⁡i). The characteristic equation associated with the equilibrium point E=(x∗,y∗,z∗) is (4)λ3+βRβC+1βLλ2+1βCβL+βRβCβL+1βCcos⁡z∗λ+1βCβLcos⁡z∗=0,which, for E1, gives (5)λ3+βRβC+1βLλ2+1βCβL+βRβCβL+1βCcos⁡arcsin⁡iλ+1βCβLcos⁡arcsin⁡i=0and, for E2, the characteristic equation becomes (6)λ3+βRβC+1βLλ2+1βCβL+βRβCβL-1βCcos⁡arcsin⁡iλ-1βCβLcos⁡arcsin⁡i=0.According to the Routh–Hurwitz criteria, the equilibrium point E1 is a stable node and the equilibrium point E2 is a saddle node.


## 2.2. 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 fori<1.0the trajectories of system (3a)–(3c) converge to one of the equilibrium points E1,2 while, for i>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 i>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 βC is kept fixed at βC=0.007, while the dc bias i, the inductive βL, and resistive βR parameters are varied.Figure 2
Time series ofxt and corresponding phase portraits of system (3a)–(3c) in plane x,y for specific values of (a) βL=1.5, i=1.25, and βR=0.1, (b) βL=1.5, i=1.75, and βR=0.1, (c) βL=0.5, i=1.75, and βR=0.1, and (d) βL=1.5, i=1.25, and βR=0.01. The other parameter is βC=0.007.
(a)
(b)
(c)
(d)ForβL=1.5, βR=0.1, and i=1.25, the junction exhibits regular spiking as shown in Figure 2(a). The regular spiking is sensitive to dc bias i and the parameter βR. In Figure 2(b), for the same values of βL and βR parameters, intrinsic bursting is observed for larger dc bias current i=1.75 when the spiking frequency is obviously larger. For βL=0.5, the fast spiking is found in Figure 2(c) when the dc bias currentiand other parameters remain the same as for intrinsic bursting. For βR=0.01, the periodic bursting is observed in Figure 2(d) when the dc bias current i 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 βL [15]. This is shown in Figure 3 which presents the time series of xt and corresponding phase portrait for βL=26, βC=0.5, i=1.25, and βR=0.1.Figure 3
Time series ofxt and corresponding phase portrait of system (3a)–(3c) in plane x,y for βL=26, βC=0.5, i=1.25, and βR=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 xt as a function of dc bias current i for βL=2.6, βC=0.707, and βR=0.06.Figure 4
The bifurcation diagram depicting the local maxima ofxt (a) and the largest Lyapunov exponent (b) of system (3a)–(3c) versus the dc bias current i for βL=2.6, βC=0.707, and βR=0.06.
(a)
(b)When the dc bias currenti increases from 1.05 to 2.1, the bifurcation diagram of Figure 4(a) shows period-2-oscillations followed by an intermittency route to chaos interspersed with periodic windows. For i>1.255, a reverse period-tripling bifurcation to period-2-oscillations at i≈1.594 is observed. By increasing the dc bias current i, the output xt displays period-2-oscillations followed by period-1-oscillations for i≥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 i.Figure 5
Time series ofxt, yt, and zt and corresponding phase portrait of system (3a)–(3c) in plane x,y for i=1.25. The other parameters are βL=2.6, βC=0.707, and βR=0.06. The initial conditions are (x(0),y(0),z(0))=(0.0,0.0,1.0).From Figure5, we observe that system (3a)–(3c) exhibits chaotic attractor similar to the one found in the NRCLSJ model [12–16].Fori=1.25, βC=0.707, and βR=0.06, we plot the bifurcation diagram depicting the local maxima of xt and the largest Lyapunov exponent of system (3a)–(3c) versus the parameter βL as shown in Figure 6.Figure 6
The bifurcation diagrams depicting the local maxima ofxt (a) and the largest Lyapunov exponent (b) of system (3a)–(3c) versus the parameter βL for i=1.25, βC=0.707, and βR=0.06. Bifurcation diagrams are obtained by scanning the parameter βL upwards (black) and downwards (red).
(a)
(b)When the parameterβL increases from 1.9 to 2.8 [see black dot in Figure 6(a)], the bifurcation diagram of the output xt shows period-2-oscillations followed by an intermittency route to chaos interspersed with periodic windows. For βL>2.625, a reverse period-tripling bifurcation to period-3-oscillations at βL≈2.725 is observed. Period-3-oscillations persists for βL>2.725. When performing the same analysis by ramping the parameter βL [see red dot in Figure 6(a)], the output xt displays the same dynamical behaviors as in Figure 6(a) (see black dot) in the ranges 1.9≤βL<2.014 and 2.0725<βL≤2.8, while, in the range 2.014≤βL≤2.0725, the output xt shows period-1-oscillations, period-doubling 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≤βL≤2.0725. For 2.014≤βL<2.0375, period-2-oscillations coexist with period-1-oscillations. Period-2-oscillations coexist with period-doubling bifurcation in the range 2.0375≤βL<2.056. For 2.056<βL≤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 βL.Figure 7
(a) Phase portraits in the planex,y for specific initial conditions: (blue line) (x(0),y(0),z(0))=(0.0,0.0,1.0) and (black line) (x(0),y(0),z(0))=(3.3,0.0,1.0). (b) Cross section of the basin of attraction of system (3a)–(3c) in the xz-plane at y = 0. In subplot (b), periodic attractors are in red color and chaotic attractors are in black color. The parameters values are βL=2.06, i=1.25, βC=0.707, and βR=0.06.
(a)
(b)System (3a)–(3c) exhibits period-1-oscillations and chaotic attractor for βL=2.06and depending on the initial conditions as shown in Figure 7(a). One can see from Figure 7(a) that system (3a)–(3c) can exhibit either chaotic or periodic attractor depending on the initial conditions. It is worth noting that coexistence of attractors has been observed in various nonlinear systems including laser [28, 29], biological system [30], chemical reactions [31], Lorenz systems [32–35], and electrical circuits [36, 37], just to name a few.


## 3. Fractional-Order Form of Chaotic No Equilibrium Linear Resistive-Capacitive-Inductance Shunted Junction Model

In this section, we consider the commensurate fractional-order form of the no equilibrium LRCLSJ model given by(7a)dqxdtq=1βCi-y-βRx-sin⁡z(7b)dqydtq=1βLx-y(7c)dqzdtq=x,where q is the derivative order satisfying 0<q≤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](8)dqXidtq=1Γq-n∫0tXint′t-t′q-n+1dt′,where n-1<q<n, X1=x, X2=y, X3=z, 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 i=1.25, βL=2.6, βC=0.707, and βR=0.06 (see Figure 5). Chaos synchronization of unidirectional identical coupled commensurate fractional-order system (7a)–(7c) and engineering applications shall also be investigated.


### 3.1. Effect of Commensurate Fractional Derivation on the Chaotic No Equilibrium Linear Resistive-Capacitive-Inductance Shunted Junction Model

For the dc bias currenti>1, the commensurate fractional-order system (7a)–(7c) has no equilibrium points. Therefore we cannot determine analytically the lowest order of the commensurate fractional-order system (7a)–(7c) to exhibit chaotic behavior. Rather, the effect of commensurate fractional derivation on chaotic system (3a)–(3c) for i=1.25, βL=2.6, βC=0.707, and βR=0.06 can only be numerically investigated. Here accordingly, using the Adams Bashforth–Moulton predictor-corrector scheme presented in this subsection, we plot the bifurcation diagram showing the local maxima of the state variablexwith respect to the commensurate fractional-order q as shown in Figure 8 in order to find the lowest order of system (7a)–(7c) to remain chaotic.Figure 8
Bifurcation diagram showing the local maxima of the state variablex with respect to the commensurate fractional-order q. The parameters are set as i=1.25, βL=2.6, βC=0.707, and βR=0.06.The bifurcation diagram of Figure8 indicates period-2-oscillations followed by chaotic behavior for q≥0.978. Hence, the lowest order for the commensurate fractional-order form of the no equilibrium LRCLSJ model to show chaos is 3q≈2.934. In Figure 9, the time series of xt and the phase portraits in the plane x,y of significant results was obtained for specific values of commensurate fractional-order q.Figure 9
Time series ofxt and corresponding phase portraits in plane x,y of the fractional-order system (7a)–(7c) with commensurate fractional-orders: (a) q=0.98, (b) q=0.96, (c) q=0.8, and (d) q=0.7. The other parameters are βL=2.6, βC=0.707, i=1.25, and βR=0.06. The initial conditions are x0,y0,z0=0,0,1.
(a)
(b)
(c)
(d)Forq=0.98, Figure 9(a) shows chaotic attractor. Period-2-oscillations (see Figure 9(b)), regular spiking (see Figure 9(c)), and fast spiking (see Figure 9(d)) are shown at q=0.96, q=0.8, and q=0.7, respectively.


### 3.2. 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 finite-time synchronization with parameter estimation applied to the chaotic commensurate fractional-order system (7a)–(7c). The drive system can be written in the form(9a)dqx1dtq=1βCi-y1-βRx1-sin⁡z1(9b)dqy1dtq=1βLx1-y1(9c)dqz1dtq=x1.And the response system is described by(10a)dqx2dtq=1βCi-y2-β^Rx2-sin⁡z2-st(10b)dqy2dtq=1βLx2-y2(10c)dqz2dtq=x2.The controller signal is s(t)=z2-z1+kut, where utis the adaptive feedback coupling designed to achieve finite-time synchronization. We set the synchronization error between drive and response systems as ex=x2-x1, ey=y2-y1, ez=z2-z1, and eβ=β^R-βR. The errors system is obtained as follows:(11a)dqexdtq=1βC-ey-x2eβ-βRex-fz1,z2-ez-kut(11b)dqeydtq=1βLex-ey(11c)dqezdtq=eywith the nonlinear function fz1,z2 defined by fz1,z2=sin⁡z2-sin⁡z1. The response system (10a)–(10c) synchronizes with the drive system (8) at a finite-time and final time tsyn exists such that limt→tsyn⁡eit=0.Assumption 1.
The nonlinear functionfz1,z2 with f0,0=0 respects the Lipschitz condition, if and only if scalar numbers η1 and η2 exist such that fz1,z2≤η1ez≤η2ex.Theorem 2.
The response system (10a)–(10c) synchronizes with the drive system (8) in the finite-time(12)tsynq≤tsyn1=12χβCex20+βLey20+ez20+eβ20+u0χ.If the adaptive feedback controller and the estimated parameter are set as follows:(13a)dqβ^Rdtq=x2ex(13b)dqudtq=kuex-ηex2-χsign⁡u,where χ, η, and k are the positive parameters. The slave system can be also written as(14a)dqx2dtq=1βCi-y2-β^Rx2-sin⁡z2-z2-z1-ku(14b)dqy2dtq=1βLx2-y2(14c)dqz2dtq=x2(14d)dqβ^Rdtq=x2ex(14e)dqudtq=kuex-ηex2-χsign⁡u.Proof.
We consider the following Lyapunov candidate function defined by(15)Vt=12βCex2+βLey2+ez2+eβ2+ut.The fractional-order derivative of (15) gives(16)dqVtdtq=-x2exeβ-βRex2-fz1,z2ex-kuex-ey2+eβdqβ^Rdtq+sign⁡udqudtq.Using the assumption, (16) becomes dqVt/dtq≤-x2exeβ+ηex2-kuex+eβDqβ^R+sign⁡(u)dqu/dtq. If a positive constant χ exists such that dqVt/dtq≤-χ, then the update laws for the controller ut and the estimated parameter β^R are designed through the following relations:(17a)dqβ^Rdtq=x2ex(17b)dqudtq=kuex-ηex2-χsign⁡u.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 integer-order 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 x10,y10,z10=0,0,1 and x20,y20,z20,β^R0,u0=0.1,0.1,1.2,-0.3,0.1. To guarantee the chaotic synchronization, the other parameters are fixed at q=0.98, βC=00.707, i=1.25, βL=2.6, βR=0.06, η=0.01, χ=0.004, and k=3. Based on these values, the theoretical settled time of synchronization is tTsyn1=45.375.Figure 10
The time evolution of the adaptive feedback controlleru(t) (a) and the controller signal S(t) (b) as well as the estimated parameter β^R (c).
(a)
(b)
(c)Figure 11
Synchronization errors between the drive and response systems (8) and (10a)–(10c). The parameters are given in the text.The dynamic of the synchronization error is present in Figure11.From Figure11, it can be seen that the numerical time of synchronization is tNsynq≈40. By comparing with analytical time of synchronization obtained above, it can be noted that tNsynq≤tTsyn1. This result respects the finite-time condition given in [42]. Thus, these results can be used for chaos based communications.


### 3.3. 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 [42–44] 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:(18a)c=ps+ksmod⁡128(18b)pr=c-krmod⁡128,where ps is the plaintext to be encrypted, pr is the plaintext to be recovered, ks is the secret keys of sender, kr is the secret keys of the receiver, and c is the ciphertext.


#### 3.3.1. Proposed Affine Cipher

ConsiderS and R, 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. S and R agree on a time t≥tsynq, fractional-order q=0.98, and a number of characters of the ASCII codes table equal to 128. The public key D=Day/Month/year is only shared by S and R. This public key allows building the two privates keys. The complete procedure between S and R 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 p1, p2, p3, p4 as a first bloc and p5, p6, p7, p8 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.(
2
) For t≥tsynq and q≥0.98, the sender S has x1,y1,z1 as solution of commensurate fractional-order chaotic system (8), and the receive R has x2,y2,z2 as solution of system (10a)–(10c). The private keys of the sender and receiver are, respectively, defined as follows:(19a)k1=Day∗z1+Month∗y1+year∗x1mod⁡128(19b)k2=Day∗z2+Month∗y2+year∗x2mod⁡128,where a is the integer part of a.(
3
) The first four sending keys k1s, k2s, k3s, k4s are determined by(20)Qk1=Fk1+1Fk1Fk1Fk1-1=k~1k~2k~3k~4,where k1s=k~1 (mod 128), k2s=k~2 (mod 128), k3s=k~3 (mod 128), k4s=k~4 (mod 128), and Fn is the Fibonacci number; Fn+1=Fn+Fn-1 with n≥3 taking into account the fact that the initial terms are F0=0 and F1=F2=0.(
4
) The first four receiving keys k1r, k2r, k3r, k4r are determined by (21)Qk2=Fk2+1Fk2Fk2Fk2-1=k~1k~2k~3k~4,where k1r=k~1 (mod 128), k2r=k~2 (mod 128), k3r=k~3 (mod 128), and k4r=k~4 (mod 128).(
5
) The ciphertext and recovered plaintext are obtained by applying c=ps+ks (mod 128) and pr=c-kr (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 Table1.Table 1
Summary of sending and receive message (PASSWORD).
Times
k
1
Plaintext
Keysks
Ciphertext
k
2
Keyskr
Plaintext recovered
t
=
80
55
80 (P)
23
103
55
23
80 (P)
65 (A)
34
99
34
65 (A)
83 (S)
34
117
34
83 (S)
83 (S)
117
200
117
83 (S)
t
=
100
56
87 (W)
57
144
56
57
87 (W)
79 (O)
23
102
23
79 (O)
82 (R)
23
105
23
82 (R)
68 (D)
34
102
34
68 (D)


#### 3.3.2. Security Analysis of Proposed Affine Cipher

The present proposed affine cipher consists of three keys: one public keyD and two private keys k1 and k2. 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 k2. 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 finite-time synchronization is less robust than the adaptive finite-time synchronization with parameters estimation. The finite-time 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.


## 3.1. Effect of Commensurate Fractional Derivation on the Chaotic No Equilibrium Linear Resistive-Capacitive-Inductance Shunted Junction Model

For the dc bias currenti>1, the commensurate fractional-order system (7a)–(7c) has no equilibrium points. Therefore we cannot determine analytically the lowest order of the commensurate fractional-order system (7a)–(7c) to exhibit chaotic behavior. Rather, the effect of commensurate fractional derivation on chaotic system (3a)–(3c) for i=1.25, βL=2.6, βC=0.707, and βR=0.06 can only be numerically investigated. Here accordingly, using the Adams Bashforth–Moulton predictor-corrector scheme presented in this subsection, we plot the bifurcation diagram showing the local maxima of the state variablexwith respect to the commensurate fractional-order q as shown in Figure 8 in order to find the lowest order of system (7a)–(7c) to remain chaotic.Figure 8
Bifurcation diagram showing the local maxima of the state variablex with respect to the commensurate fractional-order q. The parameters are set as i=1.25, βL=2.6, βC=0.707, and βR=0.06.The bifurcation diagram of Figure8 indicates period-2-oscillations followed by chaotic behavior for q≥0.978. Hence, the lowest order for the commensurate fractional-order form of the no equilibrium LRCLSJ model to show chaos is 3q≈2.934. In Figure 9, the time series of xt and the phase portraits in the plane x,y of significant results was obtained for specific values of commensurate fractional-order q.Figure 9
Time series ofxt and corresponding phase portraits in plane x,y of the fractional-order system (7a)–(7c) with commensurate fractional-orders: (a) q=0.98, (b) q=0.96, (c) q=0.8, and (d) q=0.7. The other parameters are βL=2.6, βC=0.707, i=1.25, and βR=0.06. The initial conditions are x0,y0,z0=0,0,1.
(a)
(b)
(c)
(d)Forq=0.98, Figure 9(a) shows chaotic attractor. Period-2-oscillations (see Figure 9(b)), regular spiking (see Figure 9(c)), and fast spiking (see Figure 9(d)) are shown at q=0.96, q=0.8, and q=0.7, respectively.


## 3.2. 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 finite-time synchronization with parameter estimation applied to the chaotic commensurate fractional-order system (7a)–(7c). The drive system can be written in the form(9a)dqx1dtq=1βCi-y1-βRx1-sin⁡z1(9b)dqy1dtq=1βLx1-y1(9c)dqz1dtq=x1.And the response system is described by(10a)dqx2dtq=1βCi-y2-β^Rx2-sin⁡z2-st(10b)dqy2dtq=1βLx2-y2(10c)dqz2dtq=x2.The controller signal is s(t)=z2-z1+kut, where utis the adaptive feedback coupling designed to achieve finite-time synchronization. We set the synchronization error between drive and response systems as ex=x2-x1, ey=y2-y1, ez=z2-z1, and eβ=β^R-βR. The errors system is obtained as follows:(11a)dqexdtq=1βC-ey-x2eβ-βRex-fz1,z2-ez-kut(11b)dqeydtq=1βLex-ey(11c)dqezdtq=eywith the nonlinear function fz1,z2 defined by fz1,z2=sin⁡z2-sin⁡z1. The response system (10a)–(10c) synchronizes with the drive system (8) at a finite-time and final time tsyn exists such that limt→tsyn⁡eit=0.Assumption 1.
The nonlinear functionfz1,z2 with f0,0=0 respects the Lipschitz condition, if and only if scalar numbers η1 and η2 exist such that fz1,z2≤η1ez≤η2ex.Theorem 2.
The response system (10a)–(10c) synchronizes with the drive system (8) in the finite-time(12)tsynq≤tsyn1=12χβCex20+βLey20+ez20+eβ20+u0χ.If the adaptive feedback controller and the estimated parameter are set as follows:(13a)dqβ^Rdtq=x2ex(13b)dqudtq=kuex-ηex2-χsign⁡u,where χ, η, and k are the positive parameters. The slave system can be also written as(14a)dqx2dtq=1βCi-y2-β^Rx2-sin⁡z2-z2-z1-ku(14b)dqy2dtq=1βLx2-y2(14c)dqz2dtq=x2(14d)dqβ^Rdtq=x2ex(14e)dqudtq=kuex-ηex2-χsign⁡u.Proof.
We consider the following Lyapunov candidate function defined by(15)Vt=12βCex2+βLey2+ez2+eβ2+ut.The fractional-order derivative of (15) gives(16)dqVtdtq=-x2exeβ-βRex2-fz1,z2ex-kuex-ey2+eβdqβ^Rdtq+sign⁡udqudtq.Using the assumption, (16) becomes dqVt/dtq≤-x2exeβ+ηex2-kuex+eβDqβ^R+sign⁡(u)dqu/dtq. If a positive constant χ exists such that dqVt/dtq≤-χ, then the update laws for the controller ut and the estimated parameter β^R are designed through the following relations:(17a)dqβ^Rdtq=x2ex(17b)dqudtq=kuex-ηex2-χsign⁡u.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 integer-order 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 x10,y10,z10=0,0,1 and x20,y20,z20,β^R0,u0=0.1,0.1,1.2,-0.3,0.1. To guarantee the chaotic synchronization, the other parameters are fixed at q=0.98, βC=00.707, i=1.25, βL=2.6, βR=0.06, η=0.01, χ=0.004, and k=3. Based on these values, the theoretical settled time of synchronization is tTsyn1=45.375.Figure 10
The time evolution of the adaptive feedback controlleru(t) (a) and the controller signal S(t) (b) as well as the estimated parameter β^R (c).
(a)
(b)
(c)Figure 11
Synchronization errors between the drive and response systems (8) and (10a)–(10c). The parameters are given in the text.The dynamic of the synchronization error is present in Figure11.From Figure11, it can be seen that the numerical time of synchronization is tNsynq≈40. By comparing with analytical time of synchronization obtained above, it can be noted that tNsynq≤tTsyn1. This result respects the finite-time condition given in [42]. Thus, these results can be used for chaos based communications.


## 3.3. 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 [42–44] 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:(18a)c=ps+ksmod⁡128(18b)pr=c-krmod⁡128,where ps is the plaintext to be encrypted, pr is the plaintext to be recovered, ks is the secret keys of sender, kr is the secret keys of the receiver, and c is the ciphertext.


### 3.3.1. Proposed Affine Cipher

ConsiderS and R, 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. S and R agree on a time t≥tsynq, fractional-order q=0.98, and a number of characters of the ASCII codes table equal to 128. The public key D=Day/Month/year is only shared by S and R. This public key allows building the two privates keys. The complete procedure between S and R 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 p1, p2, p3, p4 as a first bloc and p5, p6, p7, p8 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.(
2
) For t≥tsynq and q≥0.98, the sender S has x1,y1,z1 as solution of commensurate fractional-order chaotic system (8), and the receive R has x2,y2,z2 as solution of system (10a)–(10c). The private keys of the sender and receiver are, respectively, defined as follows:(19a)k1=Day∗z1+Month∗y1+year∗x1mod⁡128(19b)k2=Day∗z2+Month∗y2+year∗x2mod⁡128,where a is the integer part of a.(
3
) The first four sending keys k1s, k2s, k3s, k4s are determined by(20)Qk1=Fk1+1Fk1Fk1Fk1-1=k~1k~2k~3k~4,where k1s=k~1 (mod 128), k2s=k~2 (mod 128), k3s=k~3 (mod 128), k4s=k~4 (mod 128), and Fn is the Fibonacci number; Fn+1=Fn+Fn-1 with n≥3 taking into account the fact that the initial terms are F0=0 and F1=F2=0.(
4
) The first four receiving keys k1r, k2r, k3r, k4r are determined by (21)Qk2=Fk2+1Fk2Fk2Fk2-1=k~1k~2k~3k~4,where k1r=k~1 (mod 128), k2r=k~2 (mod 128), k3r=k~3 (mod 128), and k4r=k~4 (mod 128).(
5
) The ciphertext and recovered plaintext are obtained by applying c=ps+ks (mod 128) and pr=c-kr (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 Table1.Table 1
Summary of sending and receive message (PASSWORD).
Times
k
1
Plaintext
Keysks
Ciphertext
k
2
Keyskr
Plaintext recovered
t
=
80
55
80 (P)
23
103
55
23
80 (P)
65 (A)
34
99
34
65 (A)
83 (S)
34
117
34
83 (S)
83 (S)
117
200
117
83 (S)
t
=
100
56
87 (W)
57
144
56
57
87 (W)
79 (O)
23
102
23
79 (O)
82 (R)
23
105
23
82 (R)
68 (D)
34
102
34
68 (D)


### 3.3.2. Security Analysis of Proposed Affine Cipher

The present proposed affine cipher consists of three keys: one public keyD and two private keys k1 and k2. 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 k2. 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 finite-time synchronization is less robust than the adaptive finite-time synchronization with parameters estimation. The finite-time 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.


## 3.3.1. Proposed Affine Cipher

ConsiderS and R, 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. S and R agree on a time t≥tsynq, fractional-order q=0.98, and a number of characters of the ASCII codes table equal to 128. The public key D=Day/Month/year is only shared by S and R. This public key allows building the two privates keys. The complete procedure between S and R 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 p1, p2, p3, p4 as a first bloc and p5, p6, p7, p8 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.(
2
) For t≥tsynq and q≥0.98, the sender S has x1,y1,z1 as solution of commensurate fractional-order chaotic system (8), and the receive R has x2,y2,z2 as solution of system (10a)–(10c). The private keys of the sender and receiver are, respectively, defined as follows:(19a)k1=Day∗z1+Month∗y1+year∗x1mod⁡128(19b)k2=Day∗z2+Month∗y2+year∗x2mod⁡128,where a is the integer part of a.(
3
) The first four sending keys k1s, k2s, k3s, k4s are determined by(20)Qk1=Fk1+1Fk1Fk1Fk1-1=k~1k~2k~3k~4,where k1s=k~1 (mod 128), k2s=k~2 (mod 128), k3s=k~3 (mod 128), k4s=k~4 (mod 128), and Fn is the Fibonacci number; Fn+1=Fn+Fn-1 with n≥3 taking into account the fact that the initial terms are F0=0 and F1=F2=0.(
4
) The first four receiving keys k1r, k2r, k3r, k4r are determined by (21)Qk2=Fk2+1Fk2Fk2Fk2-1=k~1k~2k~3k~4,where k1r=k~1 (mod 128), k2r=k~2 (mod 128), k3r=k~3 (mod 128), and k4r=k~4 (mod 128).(
5
) The ciphertext and recovered plaintext are obtained by applying c=ps+ks (mod 128) and pr=c-kr (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 Table1.Table 1
Summary of sending and receive message (PASSWORD).
Times
k
1
Plaintext
Keysks
Ciphertext
k
2
Keyskr
Plaintext recovered
t
=
80
55
80 (P)
23
103
55
23
80 (P)
65 (A)
34
99
34
65 (A)
83 (S)
34
117
34
83 (S)
83 (S)
117
200
117
83 (S)
t
=
100
56
87 (W)
57
144
56
57
87 (W)
79 (O)
23
102
23
79 (O)
82 (R)
23
105
23
82 (R)
68 (D)
34
102
34
68 (D)


## 3.3.2. Security Analysis of Proposed Affine Cipher

The present proposed affine cipher consists of three keys: one public keyD and two private keys k1 and k2. 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 k2. 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 finite-time synchronization is less robust than the adaptive finite-time synchronization with parameters estimation. The finite-time 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.


## 4. 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-capacitive-inductance 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-capacitive-inductance shunted junction model has been investigated. Chaos has been shown to exist in commensurate fractional-order form of no equilibrium linear resistive-capacitive-inductance 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.



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*Source: 4107358-2017-10-12.xml*