Journal Details
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Open Access

# Electronic Information Security Model of Nonlinear Differential Equations

###### Accepted: 28 Apr 2022
Journal Details
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

The nonlinear calculus model can more accurately describe and characterize many physical phenomena in the engineering field than the traditional calculus model. The research and application fields of nonlinear calculus are deepening and expanding. The study of accurate and reliable solutions for nonlinear differential equations has always been favored by researchers [1]. The existing methods for solving nonlinear differential equations mainly include analytical and numerical approximation solutions. But they each have their pros and cons. The circuit simulation solution method uses the fractional reactance approximation circuit to construct the calculus operator sμ(−1 < μ < 0), and then build the system simulation block diagram on the Simulink simulation platform to solve the nonlinear differential equation. Compared with the traditional nonlinear differential equation solving method, this method has the advantages of stable and accurate solution results, fast operation speed, and high real-time performance.

Nonlinear Bagley-Torvik equation and its mathematical solution

The Bagley-Torvik equation is a typical nonlinear differential equation with two derivative terms. This method aims to demonstrate the shear stress-induced at any point in the fluid [2]. The model can be directly represented by the nonlinear time derivative of the fluid velocity distribution.

Bagley-Torvik equation

The initial value problem of the Bagley-Torvik equation with inhomogeneous constant coefficients is: $Ay″(t)=B0Dtvy(t)+Cy(t)=f(t)(t>0,1 Ay^{''}\left( t \right) = {B_0}D_t^vy\left( t \right) + Cy\left( t \right) = f\left( t \right)\left( {t > 0,1 < v < 2} \right) $y(0)=0,y′(0)=0$ y\left( 0 \right) = 0,\,y'\left( 0 \right) = 0

In formula (1), A, B, C is a constant. f(t) is a known function (or excitation signal). y(t) is the unknown function to be sought. The values in the original problem considered by the Bagley-Tor-vik equation are as follows: $v=3/2,A=1,B=1/2,C=1/2$ v = 3/2,\,A = 1,\,B = 1/2,\,C = 1/2 $f(t)={8,(0≤t≤1)0,(t>1)$ f\left( t \right) = \left\{ {\matrix{ {8,} & {\left( {0 \le t \le 1} \right)} \cr {0,} & {\left( {t > 1} \right)} \cr } } \right.

The classical solutions of nonlinear differential equations include Grünwald - Letnikov numerical approximate solution method and Green's function solution method (analytical solution).

Grünwald - Letnikov numerical approximate solution method

We take the fractional v derivative of the function (or signal) f(t) using the Grünwald - Letnikov definition $aDtbf(t)=limh→0Δh,pvf(t),(0 _aD_t^bf\left( t \right) = \mathop {\lim }\limits_{h \to 0} \,\Delta _{h,p}^v\,f\left( t \right),\,\left( {0 < v < \infty } \right) $Δh,pvf(t)=h−v∑j=0∞gp(v)[j]f(t−jh)$ \Delta _{h,p}^v\,f\left( t \right) = {h^{ - v}}\,\sum\limits_{j = 0}^\infty {g_p^{\left( v \right)}\left[ j \right]f\left( {t - jh} \right)}

Where h is the time step (also called the sampling interval). The weighting coefficient $gp(v)$ g_p^{\left( v \right)} is the p order approximation Grünwald system number. The first-order approximation Grünwald coefficient $g1(v)[j]$ g_1^{\left( v \right)}\left[ j \right] is generated by the generating function $φ1v(z)−(1−z−1)v$ \varphi _1^v\left( z \right) - {\left( {1 - {z^{ - 1}}} \right)^v} . ${φ1v(z)=(1−z−1)v=∑j=0∞g1(v)(j)z−jg1(v)[j]=(−1)j(vj)$ \left\{ {\matrix{ {\varphi _1^v\left( z \right) = {{\left( {1 - {z^{ - 1}}} \right)}^v} = \sum\limits_{j = 0}^\infty {g_1^{\left( v \right)}\left( j \right){z^{ - j}}} } \hfill \cr {g_1^{\left( v \right)}\left[ j \right] = {{\left( { - 1} \right)}^j}\left( {\matrix{ v \cr j \cr } } \right)} \hfill \cr } } \right.

The second-order approximation Grünwald coefficient $g2(v)[j]$ g_2^{\left( v \right)}\left[ j \right] is generated by the generating function $φ2v(z)$ \varphi _2^v\left( z \right) . $φ2v(z)=(32−2z−1+12z−2)v=∑j=0∞g2(v)[j]z−j$ \varphi _2^v\left( z \right) = {\left( {{3 \over 2} - 2{z^{ - 1}} + {1 \over 2}{z^{ - 2}}} \right)^v} = \sum\limits_{j = 0}^\infty {g_2^{\left( v \right)}\left[ j \right]{z^{ - j}}}

And obtained by $φ2v(z)=(32)v(1−z−1)v(1−13z−1)v$ \varphi _2^v\left( z \right) = {\left( {{3 \over 2}} \right)^v}{\left( {1 - {z^{ - 1}}} \right)^v}{\left( {1 - {1 \over 3}{z^{ - 1}}} \right)^v} $g2(v)[j]=(32)v(13)jg1(v)[j]*g1(v)[j],(j=N)$ g_2^{\left( v \right)}\left[ j \right] = {\left( {{3 \over 2}} \right)^v}{\left( {{1 \over 3}} \right)^j}g_1^{\left( v \right)}\left[ j \right]*g_1^{\left( v \right)}\left[ j \right],\,\left( {j = N} \right)

In formula (9), * is the convolution operator. We use the generalized binomial method to calculate the first-order approximation Grünwald coefficients. Calculated as follows. $g1(v)[j]=(−1)jΓ(v+1)Γ(v−j+1)Γ(j+1)$ g_1^{\left( v \right)}\left[ j \right] = {\left( { - 1} \right)^j}{{\Gamma \left( {v + 1} \right)} \over {\Gamma \left( {v - j + 1} \right)\Gamma \left( {j + 1} \right)}}

The first-order approximation numerical solution uses Grünwald - Letnikov equations (5) and (6) to approximate the second-order derivatives y″(t) and v order derivatives $0Dtvy(t)$ _0D_t^vy\left( t \right) . The first-order approximation numerical difference equation of equation (1) is: ${Ay˙m−2y˙m−1+y˙m−2h2+Bhv∑j=0mg1(v)[j]y˙m−j+Cy˙m=fmy˙m=y˙(mh),f(m)=f(mh) (m≥2)y˙0=0,y˙1=0$ \left\{ {\matrix{ {A{{{{\dot y}_m} - 2{{\dot y}_{m - 1}} + {{\dot y}_{m - 2}}} \over {{h^2}}} + {B \over {{h^v}}}\sum\limits_{j = 0}^m {g_1^{\left( v \right)}\left[ j \right]{{\dot y}_{m - j}} + C{{\dot y}_m} = {f_m}} } \hfill \cr {{{\dot y}_m} = \dot y\left( {mh} \right),\,f\left( m \right) = f\left( {mh} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {m \ge 2} \right)} \hfill \cr {{{\dot y}_0} = 0,\,{{\dot y}_1} = 0} \hfill \cr } } \right.

From this, the first-order approximation numerical solution algorithm is derived as follows. ${y˙0=0,y˙1=0y˙m=h2fm+A(2y˙m−1−y˙m−2)−Bh2−v∑j=1mg1(v)[j]y˙m−jA+Bh2−v+Ch2$ \left\{ {\matrix{ {{{\dot y}_0} = 0,\,{{\dot y}_1} = 0} \hfill \cr {{{\dot y}_m} = {{{h^2}\,{f_m} + A\left( {2{{\dot y}_{m - 1}} - {{\dot y}_{m - 2}}} \right) - B{h^{2 - v}}\,\sum\limits_{j = 1}^m {g_1^{\left( v \right)}\left[ j \right]{{\dot y}_{m - j}}} } \over {A + B{h^{2 - v}} + C{h^2}}}} \hfill \cr } } \right.

Thus, the first-order approximation numerical solution $y˙m$ {\dot y_m} of equation (1) is obtained and has $y˙m=ym+O(h),ym=y(mh),m∈N$ {\dot y_m} = {y_m} + O\left( h \right),\,{y_m} = y\left( {mh} \right),\,m \in N

In Eq. (13), ym is the true solution. Second-order approximation numerical solution equation (1) The second-order approximation numerical difference equation is: ${Ah2∑j=0mg2(2)[j]y¨m−j+Bhv∑j=0mg2(v)[j]y¨m−j+Cy¨m=fmy¨m=y¨(mh),f(m)=(mh),y¨0=0,y¨1=0,(m≥2)$ \left\{ {\matrix{ {{A \over {{h^2}}}\sum\limits_{j = 0}^m {g_2^{\left( 2 \right)}\left[ j \right]{{\ddot y}_{m - j}} + {B \over {{h^v}}}\sum\limits_{j = 0}^m {g_2^{\left( v \right)}\left[ j \right]} \,{{\ddot y}_{m - j}} + C{{\ddot y}_m} = {f_m}} } \hfill \cr {{{\ddot y}_m} = \ddot y\left( {mh} \right),\,f\left( m \right) = \left( {mh} \right),\,{{\ddot y}_0} = 0,\,{{\ddot y}_1} = 0,\,\,\left( {m \ge 2} \right)} \hfill \cr } } \right.

From this, we deduce the second-order approximation numerical solution algorithm ${y¨0=0,y¨1=0y¨m=h2fm−A∑j=1mg2(2)[j]y¨m−j−Bh2−v∑j=1mg2(v)[j]y¨m−jA(3/2)2+B(3/2)vh2−v+Ch2,(m≥2)$ \left\{ {\matrix{ {{{\ddot y}_0} = 0,\,{{\ddot y}_1} = 0} \hfill \cr {{{\ddot y}_m} = {{{h^2}\,{f_m} - A\sum\limits_{j = 1}^m {g_2^{\left( 2 \right)}\left[ j \right]{{\ddot y}_{m - j}} - B{h^{2 - v}}\sum\limits_{j = 1}^m {g_2^{\left( v \right)}\left[ j \right]{{\ddot y}_{m - j}}} } } \over {A{{\left( {3/2} \right)}^2} + B{{\left( {3/2} \right)}^v}{h^{2 - v}} + C{h^2}}},\,\left( {m \ge 2} \right)} \hfill \cr } } \right.

Thus we obtain the second-order approximation numerical solution ÿm of equation (1) and have $y¨m=ym+O(h2),m∈N$ {\ddot y_m} = {y_m} + O\left( {{h^2}} \right),\,m \in N

Green's function solution method - analytical solution

For equation (1), the reference describes and proves Green's function solution method in detail. Using the Green's function solution method, the analytical solution of equation (1) is obtained as $y(t)=∫0tG3(t−τ)f(τ)dτ$ y\left( t \right) = \int\limits_0^t {{G_3}\left( {t - \tau } \right)f\left( \tau \right)d\tau }

Where Green's function $G3(t)=1A∑j=0∞(−BA)j∑k=0∞(−CA)k(j+kk)t2(j+k+1)−vj−1Γ2(j+k+1)−vj)$ {G_3}\left( t \right) = {1 \over A}\sum\limits_{j = 0}^\infty {{{\left( {{{ - B} \over A}} \right)}^j}} \sum\limits_{k = 0}^\infty {{{\left( {{{ - C} \over A}} \right)}^k}\left( {\matrix{ {j + k} \cr k \cr } } \right){{{t^{2\left( {j + k + 1} \right) - vj - 1}}} \over {\left. {\Gamma 2\left( {j + k + 1} \right) - vj} \right)}}}

Do a double infinite series define a special function. In the specific numerical calculation, only the truncated approximate form can be taken as follows $G˜3(t)=1A∑j=0J(−BA)j∑k=0k(−CA)k(j+kk)t2(j+k+1)−vj−1Γ2(j+k+1)−vj)(J,K∈Z)$ {\tilde G_3}\left( t \right) = {1 \over A}\sum\limits_{j = 0}^J {{{\left( { - {B \over A}} \right)}^j}} \sum\limits_{k = 0}^k {{{\left( { - {C \over A}} \right)}^k}\left( {\matrix{ {j + k} \cr k \cr } } \right){{{t^{2\left( {j + k + 1} \right) - vj - 1}}} \over {\left. {\Gamma 2\left( {j + k + 1} \right) - vj} \right)}}\left( {J,\,K \in Z} \right)}

Then the corresponding approximate analytical solution $y˜(t)$ \tilde y\left( t \right) is $y˜(t)=∫0tG˜3(t−τ)f(τ)dτ$ \tilde y\left( t \right) = \int\limits_0^t {{{\tilde G}_3}\left( {t - \tau } \right)f\left( \tau \right)d\tau }

According to the algorithm formulas (12), (15), and (20) and using Matlab programming to solve the first-order approximation numerical solution, the second-order approximation numerical solution, and the approximate analytical solution. The result is shown in Figure 1.

By comparing these two classical nonlinear differential equation solving methods, it can be concluded that: (1) Green's function method is theoretically convergent. But the result of its operation will be related to the computer system [3]. This results in a non-convergence condition for a tail As2Y(s) + BsvY(s) + CY(s) = F(s), (1 < v < 2). (2) The operation result of the numerical approximation method is related to the time step. We need to perform approximate substitution operations on the differential equation to be solved.

Circuit simulation solution method

The circuit simulation method uses the fractional reactance approximation circuit to construct the nonlinear calculus operator sμ instead of the fractional derivative $aDtv$ _aD_t^v in the nonlinear differential equation [4]. Then we build the system simulation block diagram of nonlinear differential equations on the simulation platform to assist us in solving the nonlinear equations. Compared with the numerical approximation method, the circuit simulation method has the advantages of stable and accurate solution results, fast operation speed, and real-time performance.

Mathematical principle of circuit simulation solution method

The formula for the Laplace transform of the Grünwald - Letnikov fractional derivative is as follows. $L{0Dtvf(t);s}=∑k=0nf(k)(0)L{t−v+kΓ(−v+n+1);s}+L{tn−vΓ(−v+n+1)f(n+1)(t);s}=∑k=0nf(k)(0)sv−k−1+sv−n−1(sn+1F(s)−∑k=0nf(k)(0)sn−k)=svF(s)$ \matrix{ {L\left\{ {_0D_t^vf\left( t \right);s} \right\} = \sum\limits_{k = 0}^n {{f^{\left( k \right)}}\left( 0 \right)L\left\{ {{{{t^{ - v + k}}} \over {\Gamma \left( { - v + n + 1} \right)}};s} \right\} + } } \hfill \cr {L\left\{ {{{{t^{n - v}}} \over {\Gamma \left( { - v + n + 1} \right)}}\,{f^{\left( {n + 1} \right)}}\left( t \right);s} \right\} = \sum\limits_{k = 0}^n {{f^{\left( k \right)}}\left( 0 \right){s^{v - k - 1}} + {s^{v - n - 1}}} } \hfill \cr {\left( {{s^{n + 1}}F\left( s \right) - \sum\limits_{k = 0}^n {{f^{\left( k \right)}}\left( 0 \right){s^{n - k}}} } \right) = {s^v}F\left( s \right)} \hfill \cr }

Where n < v < n + 1, $F(s)=L{f(t);s}=∫0∞e−stf(t)dt$ F\left( s \right) = L\left\{ {f\left( t \right);s} \right\} = \int\limits_0^\infty {{e^{ - st}}f\left( t \right)dt}

We take the one-sided Laplace transform of equation (1) to get the differential operator sv in the equation can be decomposed into $sv=sμ s2,−1<μ<0$ {s^v} = {s^\mu }\,{s^2},\, - 1 < \mu < 0

Operator s2 corresponds to the second-order differential operation in the time domain. The μ order operator sμ can be realized by the μ order fractional anti-approximation circuit [5]. This corresponds to the operation in the time domain as follows: $0Dtvy(t)=0Dtμ{y″(t)},−1<μ<0$ _0D_t^vy\left( t \right) ={ _0}D_t^\mu \left\{ {y^{''}\left( t \right)} \right\},\, - 1 < \mu < 0

So equation (22) can be transformed into $s2Y(s)=1A{F(s)−CY(s)−Bsμ[s2Y(s)]},−1<μ<0$ {s^2}Y\left( s \right) = {1 \over A}\left\{ {F\left( s \right) - CY\left( s \right) - B{s^\mu }\left[ {{s^2}Y\left( s \right)} \right]} \right\},\, - 1 < \mu < 0

Its corresponding time-domain expression is $y″(t)−1A{f(t)−Cy(t)−B0Dtμ[y″(t)],−1<μ<0$ y^{''}\left( t \right) - {1 \over A}\left\{ {f\left( t \right) - Cy\left( t \right) - {B_0}D_t^\mu \left[ {y^{''}\left( t \right)} \right]} \right.,\, - 1 < \mu < 0

Draw a block diagram of the circuit system according to formula (25) (Fig. 2).

It can be seen from Figure 2 that the circuit simulation system is a negative feedback system. The feedback system has the characteristics of high stability and small error. This ensures the stability and reliability of solving the Bagley-Torvik equation by the circuit simulation method [6]. The actual physical circuit system built according to this schematic diagram can solve nonlinear differential equations in real-time.

Circuit realization of calculus operator sμ scale fractal lattice fractional anti-approximation circuit

The circuit element that realizes the operation function of the calculus operator sμ(−1 < μ < 0) is the fractional reactant [7]. Under certain conditions, the fractional reactance approximation circuit can realize the ideal fractional reactance element. The impedance function Zk(s) of the fractional reactance approximation circuit is rational, and its rational impedance function sequence can be expressed as follows. $Zk(s)=Nk(s)Dk(s)=∑i=0nkβksi∑i=0dkαksi→k→∞I(μ)(s)=F(μ)sμ$ {Z_k}\left( s \right) = {{{N_k}\left( s \right)} \over {{D_k}\left( s \right)}} = {{\sum\limits_{i = 0}^{nk} {{\beta _k}{s^i}} } \over {\sum\limits_{i = 0}^{dk} {{\alpha _k}{s^i}} }}\buildrel {k \to \infty } \over \longrightarrow {I^{\left( \mu \right)}}\left( s \right) = {F^{\left( \mu \right)}}{s^\mu }

In the formula, μ(0 < | μ | < 1) is the operation order of the fractional anti-approximation circuit [8]. k is the number of sections of the circuit. s is a Laplace complex variable (or operational variable). As the fractional reactance approaches, the circuit's section number k tends to infinity. The circuit realizes the ideal fractional reactance I(μ) (s).

The Carlson scale fractal lattice anti-approximation circuit is shown in Figure 3.

According to the iterative circuit in the figure, the input impedance function Zk(s) can be obtained, as shown in the following formula. $Zk(s)=Fcexp(Zk−1(s))=2R+(1+RCs)λZk−1(σs)1+RCs+2CsλZk−1(σs),(k∈Z+)$ {Z_k}\left( s \right) = {F_{c\,\exp }}\left( {{Z_{k - 1}}\left( s \right)} \right) = {{2R + \left( {1 + RCs} \right)\lambda {Z_{k - 1}}\left( {\sigma s} \right)} \over {1 + RCs + 2Cs\lambda {Z_{k - 1}}\left( {\sigma s} \right)}},\,\left( {k \in {Z^ + }} \right)

In the formula, k is the section times of the split-reaction circuit. α is the resistance approximation ratio (αR+, a ≠ 1). β is the capacitance approximation ratio (βR+, β ≠ 1); scale factor σ = a β (σ > 0, σ ≠ 1). where $Fcexp(x)=2ab+(a+b)axa+b+2ax(x∈R+,a=R,b=1Cs)$ {F_{c\exp }}\left( x \right) = {{2ab + \left( {a + b} \right)ax} \over {a + b + 2ax}}\left( {x \in {R^ + },\,\,a = R,\,\,b = {1 \over {Cs}}} \right) . We call it the iterative function of the Carlson scale fractal lattice fractional resistance [9]. Carlson scale fractal lattice fractional anti-approximation circuit Ｌｉｕ operation order μLiu = −lgα/lgσ, μLiu ∈ (−1, 0).

Simulink environment setting and system block diagram construction

Matlab Simulink platform has powerful functions. SimulinkLibrary Browser Browser contains common toolboxes, such as DSPSystem Toolbox, Control System Toolbox, Communications System Toolbox, etc. Various arithmetic modules are provided in each toolbox. We can use these encapsulated modules to build simulation block diagrams to solve differential equations [10]. The parameters of these modules need to be set before calculating the solution.

First, set the parameters of the Simulink simulation platform. Set the simulation start time to 0s in the Simulations-Mode Configuration Parameter Parameter. The end time is the 30s. The maximum step size is 0.1. Solver selects ode45. ode45 indicates that the fourth-order to fifth-order Runge-Kutta algorithm is used. It uses a 4th-order method to provide candidate solutions and a 5th-order method to control the error [11]. It is a numerical solution method of ordinary differential equations with adaptive step size.

Next, we set the input and output. The Step function block connected to the input generates a step signal. We set its initial step time to 1s. The initial value of the simulation is 8. The simulation end value is 0. In this way, a function similar to Eq. (2) can be generated. We let the image displayed by the oscilloscope follow. The image drawn by the m file is drawn in the same figure. We need to set the parameters of the oscilloscope. The article will set the history of the parameter column of the oscilloscope [12]. At the same time we save the data to the workspace, set the variable name, and save format to Array. In this way, the output result y(t) can be called in the m file, and its image can be drawn on the figure. The transfer function module TransferFcn is used to characterize the transfer function. The specific transfer function expression is: $H(s)=Y(s)X(s)=bmsm+bm−1sm−1+…+b1s+b0ansn+bn−1sn−1+…+a1s+a0$ H\left( s \right) = {{Y\left( s \right)} \over {X\left( s \right)}} = {{{b_m}{s^m} + {b_{m - 1}}{s^{m - 1}} + \ldots + {b_1}s + {b_0}} \over {{a_n}{s^n} + {b_{n - 1}}{s^{n - 1}} + \ldots + {a_1}s + {a_0}}}

Y(s), X(s) is the Laplace transform of the system output and input y(s), x(s), respectively. We extract the numerator and denominator coefficients of the impedance function sequence of the fractional reactance approximation circuit and substitute them into the transfer function module to realize the nonlinear calculus operation [13]. The block diagram of the nonlinear differential equation system built according to formula (26) is shown in Figure 4. The gain parameters in Figure 4 are expressed as follows: k1 = 1 / A, k2 = C / A, k3 = B / A.

Simulation results and analysis

Set up the circuit parameters of the Carlson scale fractal lattice fractional reactance circuit according to the simulation environment built in Figure 4: k = 20, μ = −1/2, σ = 9, λ = 3. We substitute the calculated coefficient vector into the transfer function module [14]. Fig. 5 shows the comparison between Green's function method, Grünwald numerical approximation method, and circuit simulation results.

Figure 5 compares the solution results of three numerical solutions to the Bagley-Torvik equation [15]. The circuit simulation method can accurately solve nonlinear differential equations with constant coefficients.

Solving of Differential Equations of Arbitrary Order

There is a calculus order v = 3/2 in equation (1). We input the function f(t) as a square wave. Assuming the differential order v ∈ (1, 2), the input function f(t) is three cases of the sine wave, periodic triangular wave, and periodic square wave [16]. We explore the accuracy of the circuit simulation method for solving arbitrary-order differential equations. The simulation results are compared as shown in Figure 6. Carlson scale fractal lattice fractional anti-approximation circuit scale factor σ = 9, Circuit Section k = 20.

From the above results, we can draw the following conclusions: 1) The circuit simulation method can solve any nonlinear differential equation. 2) The frequency of the input function has a great influence on the solution result of the circuit simulation method. 3) In Different operation orders μ, the frequency range of the input function that can be accurately calculated by the circuit simulation method is different.

Conclusion

In this paper, the Grünwald numerical approximation method and the Green's function method are introduced first. Then we elaborate the circuit simulation method and solve the Bagley-Torvik equation. Compared with the traditional numerical method, the solution results have higher accuracy. After changing the input function and calculus order of the Bagley Torvik equation, the solution results show that the circuit simulation method can solve any nonlinear differential equation.

Yuan, H., Bi, Y., Fu, H. C., & Lam, A. Stability analysis of supply chain in evolutionary game based on stability theory of nonlinear differential equation. Alexandria Engineering Journal., 2020; 59(4): 2331–2337 YuanH. BiY. FuH. C. LamA. Stability analysis of supply chain in evolutionary game based on stability theory of nonlinear differential equation Alexandria Engineering Journal 2020 59 4 2331 2337 10.1016/j.aej.2020.02.025 Search in Google Scholar

Javeed, A., & Shah, T. Design of an S-box using Rabinovich-Fabrikant system of differential equations perceiving third order nonlinearity. Multimedia Tools and Applications., 2020; 79(9): 6649–6660 JaveedA. ShahT. Design of an S-box using Rabinovich-Fabrikant system of differential equations perceiving third order nonlinearity Multimedia Tools and Applications 2020 79 9 6649 6660 10.1007/s11042-019-08393-4 Search in Google Scholar

Aghili, A. Complete Solution For The Time Fractional Diffusion Problem With Mixed Boundary Conditions by Operational Method. Applied Mathematics and Nonlinear Sciences., 2021; 6(1): 9–20 AghiliA. Complete Solution For The Time Fractional Diffusion Problem With Mixed Boundary Conditions by Operational Method Applied Mathematics and Nonlinear Sciences 2021 6 1 9 20 10.2478/amns.2020.2.00002 Search in Google Scholar

de Assis, R., Pazim, R., Malavazi, M., Petry, P., de Assis, L. & Venturino, E. A Mathematical Model to describe the herd behaviour considering group defense. Applied Mathematics and Nonlinear Sciences., 2020; 5(1): 11–24 de AssisR. PazimR. MalavaziM. PetryP. de AssisL. VenturinoE. A Mathematical Model to describe the herd behaviour considering group defense Applied Mathematics and Nonlinear Sciences 2020 5 1 11 24 10.2478/amns.2020.1.00002 Search in Google Scholar

Tariq, S., Elmoasry, A., Batool, S. I., & Khan, M. Quantum harmonic oscillator and schrodinger paradox based nonlinear confusion component. International Journal of Theoretical Physics., 2020;59(11): 3558–3573 TariqS. ElmoasryA. BatoolS. I. KhanM. Quantum harmonic oscillator and schrodinger paradox based nonlinear confusion component International Journal of Theoretical Physics 2020 59 11 3558 3573 10.1007/s10773-020-04616-9 Search in Google Scholar

Rusyn, V. Modeling, analysis and control of chaotic Rucklidge system. Journal of Telecommunication, Electronic and Computer Engineering (JTEC)., 2019;11(1): 43–47 RusynV. Modeling, analysis and control of chaotic Rucklidge system Journal of Telecommunication, Electronic and Computer Engineering (JTEC) 2019 11 1 43 47 Search in Google Scholar

Alghafis, A., Munir, N., & Khan, M. An encryption scheme based on chaotic Rabinovich-Fabrikant system and S 8 confusion component. Multimedia Tools and Applications., 2021; 80(5): 7967–7985 AlghafisA. MunirN. KhanM. An encryption scheme based on chaotic Rabinovich-Fabrikant system and S 8 confusion component Multimedia Tools and Applications 2021 80 5 7967 7985 10.1007/s11042-020-10142-x Search in Google Scholar

Arshad, U., Batool, S. I., & Amin, M. A novel image encryption scheme based on Walsh compressed quantum spinning chaotic Lorenz system. International Journal of Theoretical Physics., 2019; 58(10): 3565–3588 ArshadU. BatoolS. I. AminM. A novel image encryption scheme based on Walsh compressed quantum spinning chaotic Lorenz system International Journal of Theoretical Physics 2019 58 10 3565 3588 10.1007/s10773-019-04221-5 Search in Google Scholar

Khan, M., & Masood, F. A novel chaotic image encryption technique based on multiple discrete dynamical maps. Multimedia Tools and Applications., 2019; 78(18): 26203–26222 KhanM. MasoodF. A novel chaotic image encryption technique based on multiple discrete dynamical maps Multimedia Tools and Applications 2019 78 18 26203 26222 10.1007/s11042-019-07818-4 Search in Google Scholar

Tariq, S., Khan, M., Alghafis, A., & Amin, M. (2020). A novel hybrid encryption scheme based on chaotic Lorenz system and logarithmic key generation. Multimedia Tools and Applications, 79(31): 23507–23529. TariqS. KhanM. AlghafisA. AminM. 2020 A novel hybrid encryption scheme based on chaotic Lorenz system and logarithmic key generation Multimedia Tools and Applications 79 31 23507 23529 10.1007/s11042-020-09134-8 Search in Google Scholar

Musanna, F., & Kumar, S. A novel fractional order chaos-based image encryption using Fisher Yates algorithm and 3-D cat map. Multimedia Tools and Applications., 2019; 78(11): 14867–14895 MusannaF. KumarS. A novel fractional order chaos-based image encryption using Fisher Yates algorithm and 3-D cat map Multimedia Tools and Applications 2019 78 11 14867 14895 10.1007/s11042-018-6827-2 Search in Google Scholar

Khan, M., Jamal, S. S., & Waqas, U. A. A novel combination of information hiding and confidentiality scheme. Multimedia Tools and Applications., 2020;79(41): 30983–31005 KhanM. JamalS. S. WaqasU. A. A novel combination of information hiding and confidentiality scheme Multimedia Tools and Applications 2020 79 41 30983 31005 10.1007/s11042-020-09610-1 Search in Google Scholar

Cang, S., Li, Y., Xue, W., Wang, Z., & Chen, Z. Conservative chaos and invariant tori in the modified Sprott A system. Nonlinear Dynamics., 2020; 99(2): 1699–1708 CangS. LiY. XueW. WangZ. ChenZ. Conservative chaos and invariant tori in the modified Sprott A system Nonlinear Dynamics 2020 99 2 1699 1708 10.1007/s11071-019-05385-9 Search in Google Scholar

Li, C., Luo, G., & Li, C. An Image Encryption Scheme Based on The Three-dimensional Chaotic Logistic Map. Int. J. Netw. Secur., 2019; 21(1): 22–29 LiC. LuoG. LiC. An Image Encryption Scheme Based on The Three-dimensional Chaotic Logistic Map Int. J. Netw. Secur 2019 21 1 22 29 Search in Google Scholar

Sun, Y., Shi, Y., & Zhang, Z. Finance big data: Management, analysis, and applications. International Journal of Electronic Commerce., 2019; 23(1): 9–11 SunY. ShiY. ZhangZ. Finance big data: Management, analysis, and applications International Journal of Electronic Commerce 2019 23 1 9 11 10.1080/10864415.2018.1512270 Search in Google Scholar

Alghafis, A., Munir, N., Khan, M., & Hussain, I. An encryption scheme based on discrete quantum map and continuous chaotic system. International Journal of theoretical physics., 2020; 59(4): 1227–1240 AlghafisA. MunirN. KhanM. HussainI. An encryption scheme based on discrete quantum map and continuous chaotic system International Journal of theoretical physics 2020 59 4 1227 1240 10.1007/s10773-020-04402-7 Search in Google Scholar

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