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Application of triple compound combination anti-synchronization among parallel fractional snap systems & electronic circuit implementation

Abstract

In this article we examine the dynamical properties of the fractional version of the snap system by means of chaotic attractor, existence, and uniqueness of the solution, symmetry, dissipativity, stagnation point analysis, Lyapunov dynamics, K.Y. dimension, bifurcation diagram, etc. Also, parallel systems to this system are synchronized in presence of uncertainties and external disturbances using triple compound combination anti-synchronization by two ways. Synchronization time is compared with some other works. Also the utilization of achieved synchronization is illustrated in secure transmission. By constructing the snap system’s signal flow graph and its real electronic circuit, some of its additional invariants are investigated.

Introduction

Chaos is the common attribute of nonlinear dynamical systems. It has some interesting features such as irregular, complex, and unpredictable behavior. It was in 1960 while studying atmospheric turbulence that Lorenz [25, 26] realized that the 3-D differential equation system possessed a very irregular chaos like solution. After the Lorenz system, the Lu system and Chen system were proposed which too had complex dynamical behavior. In order to utilize chaos, many methods have been developed by researchers, such as chaos control, suppression, and synchronization, which led to the revolution in chaos theory. Because of these developments, chaotic systems found many applications across many disciplines such as secure communication, robotics, weather systems, encryption, neurons, circuits, oscillators, ecology, biology, and finance systems. Chaos synchronization, in which two or more chaotic systems are made to follow the same path on application of suitable controllers, was introduced in 1990. Synchronization of chaos involves synchronizing systems which are highly sensitive to initial conditions (I.C.) and parametric values. Many synchronization techniques [17, 1922, 27, 30, 31, 35, 54] have been developed since then using various control methods. Table 1 gives an idea of how new synchronization techniques have been developing every now and then. Many new chaotic dynamical systems [37, 56, 58] of finite dimension are being developed in chaos theory. Chaotic systems are also studied on the basis of attractors, hidden attractors, multi-scroll attractors, self-excited attractors, double-wing, etc. Designing of chaotic system with some special characteristics has become an attractive topic of research in recent years. For example, chaotic systems having only one stable equilibrium, surfaces or curves or planes of equilibria, no equilibria, non-hyperbolic equilibria, etc. As the fractional order adds an extra dimension in dynamics of chaotic systems, a fractional version of Lu system, Liu system, Rossler system etc. is being studied. Nowadays fractional discrete time chaotic systems [6164] are also being studied. Fractional calculus has been used recently to study nonlinear fractional B.V.P. [16], biological snap oscillator [44], harmonic oscillator, etc. [1, 2, 15, 34].

Table 1 Different Types of Synchronization Techniques

In chaos applications, a very important topic is the hardware realization of mathematical models [5].

The graph \(H(V,E)\) is defined as a pair of two sets: vertices set V and edges set E and written as H for simplicity. The eigenvalues (E.V.) of the adjacency matrix of H are said to be the E.V. of H and form its spectrum. A graph of order \(n= \vert V \vert \) has n eigenvalues; we denote these by \(\lambda _{1}, \ldots, \lambda _{n}\). The basics of graph spectrum can be found in [45]. The graph spectrum can be used to calculate a very useful graph invariant, namely the graph energy \(E(H)\) [7, 8] where

$$ E(H)=\sum_{i=1}^{n} \vert \lambda _{i} \vert . $$
(1)

Motivated by the above discussion, here the fractional version of the snap system with only one nonlinear term is dynamically analyzed using tools like Lyapunov exponents (L.E.) [4, 13, 59], Kaplan York (K.Y.) dimension, bifurcations, etc. The system is synchronized with its parallel systems considering uncertainties and external disturbances in triple compound combination anti-synchronization. The synchronization method is illustrated in secure communication [6, 12] by an example. Synchronization time is compared with some other works. Also the electronic circuit of the snap system is realized to assure its real life applications.

Summarizing the paper:

  1. 1.

    A fractional version of the previously studied integer order dimensional chaotic system is studied. The system is explored using stagnation points analysis, symmetric character, Lyapunov dynamics, bifurcation analysis, etc.

  2. 2.

    The effect of fractional order on system dynamics for varying fractional order in the 0.8 to 1 range is studied.

  3. 3.

    A novel technique—triple compound combination anti-synchronization—is developed to anti-synchronize two leader systems and six follower systems by two methods.

  4. 4.

    Bounds of uncertainties and disturbances are also estimated using the adaptive SMC technique.

  5. 5.

    Achieved results are compared with some earlier published literature.

  6. 6.

    Application of the achieved synchronization on the introduced fractional version of the snap system in secure transmission is illustrated.

  7. 7.

    Electronic circuit of the introduced snap system is also provided.

The fractional snap system and its dynamics

The integer order snap chaotic system was proposed in 2019 by Pham et al. [38] with only one quadratic nonlinear term given as follows:

$$ \begin{aligned} &\dot{z_{1}}=z_{2}, \\ &\dot{z_{2}}=z_{3}, \\ &\dot{z_{3}}=z_{4}, \\ &\dot{z_{4}}=-az_{1}-z_{2}-z_{4}+bz_{1}z_{3}, \end{aligned} $$
(2)

where \(z=(z_{1},z_{2},z_{3},z_{4})^{T}\in {R}^{4}\) are state variables.

Though many snap systems have been proposed by researchers over the time, developing new snap systems has always been of interest to researchers owing to their application in random number generations [11], cryptosystems [60], steganography [23], etc. Moreover, studying the fractional version of the snap system holds importance because fractional ordered systems have better memory properties and hence model real life situations more accurately than the integer counterparts. Also it is interesting to see their chaotic behavior vary for different fractional orders.

Among the many definitions of fractional derivatives, the following Caputo’s definition is used:

$$\begin{aligned} {{}_{t_{0}} D^{\alpha }_{t} f(t)}= \frac{1}{\Gamma (n-\alpha )} \int _{t_{0}}^{t} \frac{f^{(n)}(\tau )}{(t-\tau )^{\alpha -n+1}}\,d\tau , \quad t>t_{0}. \end{aligned}$$

We here examine the fractional version of this newly introduced system:

$$ \begin{aligned} &D^{q}{z_{1}}=z_{2}, \\ &D^{q}{z_{2}}=z_{3}, \\ &D^{q}{z_{3}}=z_{4}, \\ &D^{q}{z_{4}}=-az_{1}-z_{2}-z_{4}+bz_{1}z_{3}. \end{aligned} $$
(3)

Figures 1 and 2 show the chaotic attractors and state trajectories for \(a=0.2\), \(b=1\) with I.C. \((0, 0.1, 0.1, 0)\) and \(q=0.95\).

Figure 1
figure1

Chaotic attractor of (3)

Figure 2
figure2

Trajectories of (3)

Note: For numerical simulations we have used Grünwald–Letnikov approximation function in MATLAB software.

Solution of a novel system

Theorem

The I.V.P. of (3):

$$\begin{aligned}& D^{q}z(t)=B_{1}z(t)+z_{1}(t)B_{2}z(t), \\& z(0)=z_{o}, \end{aligned}$$

where \(0< t\leq \tau \), \(z(t)=\) \((z_{1}(t), z_{2}(t), z_{3}(t), z_{4}(t))^{T}\in \mathbb{R}^{4}\), \(z_{o}=(z_{1o}, z_{2o}, z_{3o}, z_{4o})^{T}\), \(q=(q_{1}, q_{2}, q_{3}, q_{4})^{T}\), \(0< q_{i}<1 \), for \(i=1, 2, 3, 4\).

B 1 = [ 0 1 0 0 0 0 1 0 0 0 0 1 0.2 1 0 1 ] & B 2 = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ]

for some \(\tau >0\), then a unique solution is guaranteed.

Proof

As \(G(z)=B_{1} z+z_{1}B_{2} z\) is continuous on \(\mathbb{R}^{4}\) and hence bounded on any compact interval \([z_{o}-\epsilon ,z_{o}+\epsilon ]\) for \(\epsilon >0\), therefore the Lipschitz continuity on this interval is sufficient to apply the existence and uniqueness theorem in [9].

We have \(|G(z)-G(w)|=|B_{1} z +z_{1} B_{2} z-B_{1} w -w_{1}B_{2} w|\).

As

$$ \vert z_{1}-w_{1} \vert \leq \vert z-w \vert , $$

we get

$$\begin{aligned} \vert z_{1}B_{2}z-w_{1}B_{2}w \vert &= \bigl\vert B_{2}[z_{1}-w_{1}]z+B_{2} \bigl[w_{1}[z-w]\bigr] \bigr\vert \\ &\leq \Vert B_{2} \Vert \cdot\bigl[ \vert z \vert + \vert w_{1} \vert \bigr]\bigl( \vert z-w \vert \bigr), \end{aligned}$$

which implies that

$$\begin{aligned} \bigl\vert G(z)-G(w) \bigr\vert &\leq \Vert B_{1} \Vert \cdot \vert z-w \vert + \Vert B_{2} \Vert \bigl[2 \vert z_{o} \vert +2\epsilon \bigr]\bigl( \vert z-w \vert \bigr) \\ &\leq \bigl( \Vert B_{1} \Vert + \Vert B_{2} \Vert \bigl[2 \vert z_{o} \vert +2 \epsilon \bigr]\bigr) \vert z-w \vert \\ &\leq \psi \vert z-w \vert , \end{aligned}$$

where \(\psi =\|B_{1}\|+\|B_{2}\|[2|z_{o}|+2 \epsilon ]\).

This implies that G(z) is Lipschitz continuous, and hence the system has a unique solution. □

Stagnation points, dissipativity, and symmetry

We evaluate the stagnation points of (3) by equating \(D^{q}z_{i}\) to zeroi.e.

$$\begin{aligned}& z_{2}=0, \\& z_{3}=0, \\& z_{4}=0, \\& -az_{1}-z_{2}-z_{4}+bz_{1}z_{3}=0. \end{aligned}$$

For \(a=0.2\), \(b=1\), we obtain a stagnation point \(\textit{E}=(0,0,0,0)\).

Theorem

The stagnation point E is an unstable saddle focus type stagnation point.

Proof

Jacobian matrix J of (3) is

J= [ 0 1 0 0 0 0 1 0 0 0 0 1 0.2 + z 3 1 b z 1 1 ] .

At E, J is

[ 0 1 0 0 0 0 1 0 0 0 0 1 0.2 1 0 1 ] .

The eigenvalues of J at E are

$$\begin{aligned}& \mu _{1}=-1.42393,\quad \mu _{2}=-0.194106,\quad \mu _{3}=0.309017+0.792537i, \\& \mu _{4}=0.309017-0.792537i \end{aligned}$$

implying E is an unstable saddle node type stagnation point.

Consider system (3) as

[ D q z 1 D q z 2 D q z 3 D q z 4 ] = [ z 2 z 3 z 4 a z 1 z 2 z 4 + b z 1 z 3 ] .

We find the divergence of this vector field by

$$\begin{aligned} \nabla G&=\frac{\partial z_{2}}{\partial z_{1}}+ \frac{\partial z_{3}}{\partial z_{2}}+\frac{\partial z_{4}}{\partial z_{3}}+ \frac{\partial (-az_{1}-z_{2}-z_{4}+bz_{1}z_{3})}{\partial z_{4}} \\ &=-1 \end{aligned}$$

i.e. \(\nabla G <0\) implying that the system under consideration is dissipative for any values of the parameters. □

Symmetric behavior about the \(z_{3} \) axis is observed as the system remains invariant under the transformation \(z_{1} \rightarrow -z_{1}\), \(z_{2} \rightarrow -z_{2}\), \(z_{3} \rightarrow z_{3}\), \(z_{4} \rightarrow -z_{4}\). This implies rotational symmetry about the \(z_{3}\) axis. However, the system does not remain invariant under such transformations about the \(z_{1}\), \(z_{2}\), \(z_{4}\) axis respectively.

Lyapunov exponents

L.E. is the rate at which trajectories which are very very close to each other diverge away. The positive L.E. confirms chaos. For \(a = 0.2\), \(b = 1\), and I.C. \((0, 0.1, 0.1, 0)\), (3) has the following L.E.:

$$ 0.107, -0.0002\approx 0, -0.9791, -0.3707 $$

i.e. (3) is a chaotic dynamical system.

The chaotic attractor of (3) possesses a fractional dimension defined as follows:

$$ D_{YK}=p+\frac{\sum_{j=1}^{p}L.E._{j}}{ \vert L.E._{p+1} \vert + \vert L.E._{p+2} \vert }. $$

Here, p is the integer such that \(\sum_{j=1}^{p}L.E._{j}\geq 0\) and \(\sum_{j=1}^{p+1}L.E._{j}<0\).

The K.Y. dimension for system (3) is 2.079122 which is a noninteger value.

Bifurcation analysis

The system dynamics is significantly dependent on the parameters. Different parameters have different influence on equilibrium points and system’s stability, hence on the dynamics of the system. Therefore observing the change in the system dynamics by varying a parameter in a range brings insight into the chaotic behavior over the specified range. The bifurcation diagram is obtained by plotting local maxima for changing parameter values in range.

For \(a=0.2\), \(b=1\). and I.C. \((0,0.1,0.1,0)\), Fig. 3(a) gives the bifurcation diagram for \(a\in (0,1)\) and Fig. 3(b) gives for \(b\in (0.5,1.5) \). The increase in the density of points in Fig. 3(a) on crossing a particular value highlights its route to chaos, whereas the same density of points in Fig. 3(b) shows chaotic behavior throughout the range.

Figure 3
figure3

Bifurcations of (3) for (a) \(0\leq a\leq 0.5\) and \(b=1\) (b) \(a=0.2\) and \(0.5 \leq b \leq 1.5\)

System dynamics for variable fractional order

In fractional order systems, the system shows much diverseness with respect to variable as shown in Fig. 5(a) and 5(b). The fractional order has significance in the system dynamics.

By [52], we have a necessary condition on fractional order to have chaos is \(q>\frac{2}{\pi }\arctan (\frac{|\operatorname{Im}(\mu )|}{\operatorname{Re}(\mu )})\). Therefore the smallest q for (3) to have chaos is 0.76. Different q plots of (3) are in Fig. 4.

Figure 4
figure4

Attractor of (3) for q (a) 0.6 (b) 0.7 (c) 0.76 (d) 0.8 (e) 0.9 (f) 0.95

In Fig. 5(a) the Lyapunov dynamics of (3) for q between 0.8 to 1 is shown. From the figure we observe more chaos in the system for fractional order 0.86 and above. In Fig. 5(b) the bifurcations of (3) for order between the 0.8 to 1 range are shown. On crossing a value, the increase in the density of points shows the increasing bifurcations undergoing in the system.

Figure 5
figure5

Lyapunov dynamics and bifurcations for varying q

Triple compound combination anti-synchronization

We first consider the six parallel systems of (3) defined as

$$ \begin{aligned} &D^{q}{v_{11}}=v_{12}, \\ &D^{q}{v_{12}}=v_{13}, \\ &D^{q}{v_{13}}=v_{14}, \\ &D^{q}{v_{14}}=-av_{11}-v_{12}-v_{14}+Pbv_{11}v_{13}, \end{aligned} $$
(4)

summarized in the form of a table as follows: triple compound combination anti-synchronization is achieved here between system (3) and six systems listed in Table 2 [18]. The phase portraits of such systems are parallel shown in Fig. 6.

Figure 6
figure6

Parallel phase portraits

Table 2 Parallel systems for \(a= 0.2\), \(b= 1\) and \(q= 0.95\)

In presence of uncertainty and disturbances consider two leader systems and six follower systems. Figure 7 shows the plots of a leader system.

Figure 7
figure7

Chaotic attractor of the master systems

Leader I:

$$ \begin{aligned} &D^{q}{z_{11}}=z_{12}+ \bigtriangleup g_{1}+\psi _{1}, \\ &D^{q}{z_{12}}=z_{13}+\bigtriangleup g_{2}+\psi _{2}, \\ &D^{q}{z_{13}}=z_{14}+\bigtriangleup g_{3}+\psi _{3}, \\ &D^{q}{z_{14}}=-az_{11}-z_{12}-z_{14}+bz_{11}z_{13}+ \bigtriangleup g_{4}+ \psi _{4}. \end{aligned} $$
(5)

Here, \(a=0.2\), \(b=1\), \(\bigtriangleup g_{1}=0\), \(\bigtriangleup g_{2}=0.7\sin (t)\), \(\bigtriangleup g_{3}=0\), \(\bigtriangleup g_{4}=0\), and I.C. \((0,0.1,0.1,0)\), \(q=0.95\) and \(\psi _{1}\), \(\psi _{2}\), \(\psi _{3}\), \(\psi _{4}\) are controllers.

Leader II:

$$ \begin{aligned} &D^{q}{z_{21}}=z_{22}+ \bigtriangleup h_{1}+\phi _{1}, \\ &D^{q}{z_{22}}=z_{23}+\bigtriangleup h_{2}+\phi _{2}, \\ &D^{q}{z_{23}}=z_{24}+\bigtriangleup h_{3}+\phi _{3}, \\ &D^{q}{z_{24}}=-az_{21}-z_{22}-z_{24}+bz_{21}z_{23}+ \bigtriangleup h_{4}+ \phi _{4}. \end{aligned} $$
(6)

Here, \(a=0.2\), \(b=1\), \(\bigtriangleup h_{1}=0\), \(\bigtriangleup h_{2}=\sin (\pi z_{23})\), \(\bigtriangleup g_{3}=0.2t\), \(\bigtriangleup g_{4}=0\), and I.C. \((0,0.1,0.1,0)\), \(q=0.95\) and \(\phi _{1}\), \(\phi _{2}\), \(\phi _{3}\), \(\phi _{4}\) are controllers.

Follower I:

$$ \begin{aligned} &D^{q}{w_{11}}=w_{12}, \\ &D^{q}{w_{12}}=w_{13}, \\ &D^{q}{w_{13}}=w_{14}, \\ &D^{q}{w_{14}}=-aw_{11}-w_{12}-w_{14}+bw_{11}w_{13}. \end{aligned} $$
(7)

Here, \(a=0.2\), \(b=1\), and I.C. \((w_{11}(0),w_{12}(0),w_{13}(0),w_{14}(0))=(0,0.1,0.1,0)\) with \(q=0.95\).

Follower II:

$$ \begin{aligned} &D^{q}{w_{21}}=w_{22}, \\ &D^{q}{w_{22}}=w_{23}, \\ &D^{q}{w_{23}}=w_{24}, \\ &D^{q}{w_{24}}=-aw_{21}-w_{22}-w_{24}+0.5bw_{21}w_{23}. \end{aligned} $$
(8)

Here, \(a=0.2\), \(b=1\), and I.C. \((0,0.05,0.05,0)\) with \(q=0.95\).

Follower III:

$$ \begin{aligned} &D^{q}{w_{31}}=w_{32}, \\ &D^{q}{w_{32}}=w_{33}, \\ &D^{q}{w_{33}}=w_{34}, \\ &D^{q}{w_{34}}=-aw_{31}-w_{32}-w_{34}+2bw_{31}w_{33}. \end{aligned} $$
(9)

Here, \(a=0.2\), \(b=1\), and I.C. \((0,0.2,0.2,0)\) with \(q=0.95\).

Follower IV:

$$ \begin{aligned} &D^{q}{w_{41}}=w_{42}, \\ &D^{q}{w_{42}}=w_{43}, \\ &D^{q}{w_{43}}=w_{44}, \\ &D^{q}{w_{44}}=-aw_{41}-w_{42}-w_{44}+1.5bw_{41}w_{43}. \end{aligned} $$
(10)

Here, \(a=0.2\), \(b=1\), and I.C. \((0,0.15,0.15,0)\) with \(q=0.95\).

Follower V:

$$ \begin{aligned} &D^{q}{w_{51}}=w_{52}, \\ &D^{q}{w_{52}}=w_{53}, \\ &D^{q}{w_{53}}=w_{54}, \\ &D^{q}{w_{54}}=-aw_{51}-w_{52}-w_{54}+0.33bw_{51}w_{53}. \end{aligned} $$
(11)

Here, \(a=0.2\), \(b=1\), and I.C. \((0,0.033,0.033,0)\) with \(q=0.95\).

Follower VI:

$$ \begin{aligned} &D^{q}{w_{61}}=w_{62}, \\ &D^{q}{w_{62}}=w_{63}, \\ &D^{q}{w_{63}}=w_{64}, \\ &D^{q}{w_{64}}=-aw_{61}-w_{62}-w_{64}+3bw_{61}w_{63}. \end{aligned} $$
(12)

Here, \(a=0.2\), \(b=1\), and I.C. \((0,0.3,0.3,0)\) with \(q=0.95\).

The anti-synchronization error is as follows:

$$ \begin{aligned} &e_{1}=(z_{11}+z_{21})+(w_{11}+w_{21}) (w_{31}+w_{41}) (w_{51}+w_{61}), \\ &e_{2}=(z_{12}+z_{22})+(w_{12}+w_{22}) (w_{32}+w_{42}) (w_{52}+w_{62}), \\ &e_{3}=(z_{13}+z_{23})+(w_{13}+w_{23}) (w_{33}+w_{43}) (w_{53}+w_{63}), \\ &e_{4}=(z_{14}+z_{24})+(w_{14}+w_{24}) (w_{34}+w_{44}) (w_{54}+w_{64}). \end{aligned} $$
(13)

Differentiating (13), from (5)–(12) we have:

$$\begin{aligned}& D^{q}{e_{1}}=(z_{12}+z_{22}+ \psi _{1}+\phi _{1})+(w_{12}+w_{22}) (w_{31}+w_{41}) (w_{51}+w_{61}) \\& \hphantom{D^{q}{e_{1}}=}{}+(w_{11}+w_{21}) (w_{32}+w_{42}) \\& \hphantom{D^{q}{e_{1}}=}{}\times (w_{51}+w_{61})+(w_{11}+w_{21}) (w_{31}+w_{41}) (w_{52}+w_{62}), \\& \begin{aligned}&D^{q}{e_{2}}=\bigl(z_{13}+0.7\sin (t)+z_{23}+\sin (\pi z_{23})+\psi _{2}+ \phi _{2}\bigr) \\ &\hphantom{D^{q}{e_{2}}=} {}+(w_{13}+w_{23}) (w_{32}+w_{42}) (w_{52}+w_{62}) \\ &\hphantom{D^{q}{e_{2}}=} {}+(w_{12}+w_{22}) (w_{33}+w_{43}) (w_{52}+w_{62})+(w_{12}+w_{22}) (w_{32}+w_{42}) (w_{53}+w_{63}), \\ &D^{q}{e_{3}}=(z_{14}+0.2t+z_{24}+ \psi _{3}+\phi _{3})+(w_{14}+w_{24}) (w_{33}+w_{43}) (w_{53}+w_{63}) \\ &\hphantom{D^{q}{e_{3}}=}{}+(w_{13}+w_{23}) (w_{34}+ w_{44}) (w_{53}+w_{63})+(w_{13}+w_{23}) (w_{33}+w_{43}) (w_{54}+w_{64}), \\ &D^{q}{e_{4}}=(-az_{11}-z_{12}-z_{14}+2bz_{11}z_{13}-az_{21}-z_{22}-z_{24}+3bz_{21}z_{23}+ \psi _{4}+\phi _{4}) \\ &\hphantom{D^{q}{e_{4}}=}{}+(-aw_{11}-w_{12}- w_{14}+bw_{11}w_{13}-aw_{21}-w_{22}-w_{24}+0.5bw_{21}w_{23}) \\ &\hphantom{D^{q}{e_{4}}=}{}\times (w_{34}+w_{44}) (w_{54}+w_{64}) \\ &\hphantom{D^{q}{e_{4}}=}{}+(w_{14}+w_{24})\end{aligned} \\& \hphantom{D^{q}{e_{4}}=}{}\times (-aw_{31}- w_{32}-w_{34}+2bw_{31}w_{33}-aw_{41}-w_{42}-w_{44} +1.5bw_{41}w_{43}) \\& \hphantom{D^{q}{e_{4}}=}{}\times (w_{54}+w_{64})+(w_{14}+w_{24}) (w_{34}+w_{44}) \\& \hphantom{D^{q}{e_{4}}=}{}\times (-aw_{51}-w_{52}-w_{54}+0.33bw_{51}w_{53} -aw_{61}-w_{62}-w_{64}+3bw_{61}w_{63}). \end{aligned}$$
(14)

Via nonlinear control

Choosing controllers as follows:

$$ \begin{aligned} &\psi _{1}=-z_{12}-(w_{12}+w_{22}) (w_{31}+w_{41}) (w_{51}+w_{61}) \\ &\hphantom{\psi _{1}=}{}-(w_{11}+w_{21}) (w_{31}+w_{41}) (w_{52}+w_{62}), \\ &\psi _{2}=-z_{13}-0.7\sin (t)-(w_{13}+w_{23}) (w_{32}+w_{42}) (w_{52}+w_{62}) \\ &\hphantom{\psi _{2}=}{}-(w_{12}+w_{22}) (w_{32}+w_{42}) (w_{53}+w_{63}), \\ &\psi _{3}=-z_{14}-(w_{14}+w_{24}) (w_{33}+w_{43}) (w_{53}+w_{63}) \\ &\hphantom{\psi _{3}=}{}-(w_{13}+w_{23}) (w_{33}+w_{43}) (w_{54}+w_{64}) \\ &\psi _{4}=(az_{11}+z_{12}+z_{14}-2bz_{11}z_{13}) \\ &\hphantom{\psi _{4}=}{}-(-aw_{11}-w_{12}-w_{14}+bw_{11}w_{13}-aw_{21}-w_{22}-w_{24} \\ &\hphantom{\psi _{4}=}{}+0.5bw_{21}w_{23}) (w_{34}+w_{44}) (w_{54}+w_{64})-4e_{4}, \\ &\phi _{1}=-z_{22}-(w_{11}+w_{21}) (w_{32}+w_{42}) (w_{51}+w_{61})-e_{1}, \\ &\phi _{2}=-z_{23}-\sin (\pi z_{23})-(w_{12}+w_{22}) (w_{33}+w_{43}) (w_{52}+w_{62})-2e_{2}, \\ &\phi _{3}=-z_{24}-0.2t-(w_{13}+w_{23}) (w_{34}+w_{44}) (w_{53}+w_{63})-3e_{3}, \\ &\phi _{4}=(az_{21}+z_{22}+z_{24}-3bz_{21}z_{23}) \\ &\hphantom{\phi _{4}=}{}-(w_{14}+w_{24}) (-aw_{31}-w_{32}-w_{34}+2bw_{31}w_{33}-aw_{41}-w_{42} \\ &\hphantom{\phi _{4}=}{}-w_{44}+1.5bw_{41}w_{43}) (w_{54}+w_{64})-(w_{14}+w_{24}) (w_{34}+w_{44}) (-aw_{51}-w_{52}-w_{54} \\ &\hphantom{\phi _{4}=}{}+0.33bw_{51}w_{53}-aw_{61}-w_{62}-w_{64}+3bw_{61}w_{63}). \end{aligned} $$
(15)

Substituting (15) into (14), we get

$$\begin{aligned}& D^{q}{e_{1}}=-e_{1}, \\& D^{q}{e_{2}}=-2e_{2}, \\& D^{q}{e_{3}}=-3e_{3}, \\& D^{q}{e_{4}}=-4e_{4}. \end{aligned}$$

Define the Lyapunov function

$$\begin{aligned}& V\bigl(e(t)\bigr)=\frac{1}{2}\bigl({e_{1}}^{2}+{e_{2}}^{2}+{e_{3}}^{2}+{e_{4}}^{2} \bigr), \\& D^{q}{V\bigl(e(t)\bigr)}\leq -{e_{1}}^{2}-2{e_{2}}^{2}-3{e_{3}}^{2}-4{e_{4}}^{2} \\& \hphantom{D^{q}{V(e(t))}}\Rightarrow D^{q}{V\bigl(e(t)\bigr)}\quad \mbox{is negative definite.} \end{aligned}$$

Lyapunov stability theory proves \(\Vert e_{i}\Vert \rightarrow 0 \) as \(t\rightarrow \infty \), implying synchronization.

Via adaptive SMC

Assuming bounded uncertainties and disturbances, we have \(| \bigtriangleup g_{i}| \leqslant a_{i} \) and \(| \bigtriangleup h_{i}| \leqslant b_{i} \) for positive real numbers \(a_{i} \) and \(b_{i} \), and \(\hat{a}_{i} \) and \(\hat{b}_{i} \) are estimated values of \(a_{i} \) and \(b_{i} \).

Choosing the sliding surface so as to minimize the error:

$$ s_{i}(t)=D^{q-1}e_{i}(t)+k_{i} \int _{0}^{t}e_{i}(\xi )\,d\xi . $$
(16)

For (14) to be at (16), we must have

$$ s_{i}(t)=0,\qquad \dot{s_{i}}(t)=0. $$
(17)

Taking Caputo’s derivative of (17)

$$ \dot{s_{i}}(t)=D^{q}e_{i}(t)+k_{i}e_{i}(t), $$
(18)

then from (17) we have

$$ D^{q}e_{i}(t)=-k_{i}e_{i}(t). $$
(19)

The error dynamical system is stable by the Matignon theorem [33]. Using SMC theory, we design controllers:

$$ \begin{aligned} &\psi _{1} =-z_{12}-(w_{12}+w_{22}) (w_{31}+w_{41}) (w_{51}+w_{61}) \\ &\hphantom{\psi _{1} =}{}-(w_{11}+w_{21}) (w_{31}+w_{41}) (w_{52}+w_{62}), \\ &\psi _{2} =-z_{13}-(w_{13}+w_{23}) (w_{32}+w_{42}) (w_{52}+w_{62}) \\ &\hphantom{\psi _{2} =}{}-(w_{12}+w_{22}) (w_{32}+w_{42}) (w_{53}+w_{63}), \\ &\psi _{3}=-z_{14}-(w_{14}+w_{24}) (w_{33}+w_{43}) (w_{53}+w_{63}) \\ &\hphantom{\psi _{3}=}{}-(w_{13}+w_{23}) (w_{33}+w_{43}) (w_{54}+w_{64}), \\ &\psi _{4}=(az_{11}+z_{12}+z_{14}-2bz_{11}z_{13}) \\ &\hphantom{\psi _{4}=}{}-(-aw_{11}-w_{12}-w_{14}+bw_{11}w_{13}-aw_{21}-w_{22}-w_{24} \\ & \hphantom{\psi _{4}=}{}+0.5bw_{21}w_{23}) (w_{34}+w_{44}) (w_{54}+w_{64}) \\ &\hphantom{\psi _{4}=}{}-(w_{14}+w_{24}) (w_{34}+w_{44}) (-aw_{51}-w_{52}-w_{54} \\ &\hphantom{\psi _{4}=}{}+0.33bw_{51}w_{53}-aw_{61}-w_{62}-w_{64}+3bw_{61}w_{63}), \\ &\phi _{1}=-z_{22}-(w_{11}+w_{21}) (w_{32}+w_{42}) (w_{51}+w_{61})-e_{1}-( \hat{a_{1}}+\hat{b_{1}}+\gamma _{1}) \operatorname{sign}s_{1}), \\ &\phi _{2} =-z_{23}-(w_{12}+w_{22}) (w_{33}+w_{43}) (w_{52}+w_{62})-2e_{2}-( \hat{a_{2}}+\hat{b_{2}}+\gamma _{2}) \operatorname{sign}s_{2}), \\ &\phi _{3}=-z_{24}-(w_{13}+w_{23}) (w_{34}+w_{44}) (w_{53}+w_{63})-3e_{3}-( \hat{a_{3}}+\hat{b_{3}}+\gamma _{3}) \operatorname{sign}s_{3}), \\ &\phi _{4}=(az_{21}+z_{22}+z_{24}-3bz_{21}z_{23}) \\ &\hphantom{\phi _{4}=}{}-(w_{14}+w_{24}) (-aw_{31}-w_{32}-w_{34}+2bw_{31}w_{33}-aw_{41} \\ &\hphantom{\phi _{4}=}{}-w_{42}-w_{44}+1.5bw_{41}w_{43}) (w_{54}+w_{64})-4e_{4}-(\hat{a_{4}}+ \hat{b_{4}}+\gamma _{4})\operatorname{sign}s_{4}) \end{aligned} $$
(20)

for signum function \(\operatorname{sign}(\cdot)\) and positive real numbers \(\gamma _{i}\).

Adaptive parameter update conditions are as follows:

$$ \begin{aligned} \dot{\hat{a}}_{i}=p_{i} \vert s_{i} \vert , \qquad \dot{\hat{b}}_{i}=q_{i} \vert s_{i} \vert , \end{aligned} $$
(21)

where \(p_{i}\) and \(q_{i}\) are positive constants.

Theorem 8.1

Stability and triple compound combination anti-synchronization is achieved in presence of uncertainties and disturbances between leader systems (10)(11) and follower systems (12)(17) with controllers and update conditions as (25) and (26).

Proof

Lyapunov’s direct method [57] is used to achieve stability between leader and follower systems. Considering V to be a positive definite function with negative definite derivative, asymptotic vanishing of error is proved. □

Let

$$ V=V_{1}+V_{2}+V_{3}+V_{4}, $$
(22)

where

$$ \begin{aligned} &V_{1}= \frac{1}{2}s_{1}^{2}+\frac{1}{2p_{1}}( \hat{a}_{1}-a_{1})^{2}+\frac{1}{2q_{1}}(\hat{b}_{1}-b_{1})^{2}, \\ &V_{2}=\frac{1}{2}s_{2}^{2}+ \frac{1}{2p_{2}}(\hat{a}_{2}-a_{2})^{2}+\frac{1}{2q_{2}}(\hat{b}_{2}-b_{2})^{2}, \\ &V_{3}=\frac{1}{2}s_{3}^{2}+ \frac{1}{2p_{3}}(\hat{a}_{3}-a_{3})^{2}+\frac{1}{2q_{3}}(\hat{b}_{3}-b_{3})^{2}, \\ &V_{4}=\frac{1}{2}s_{4}^{2}+ \frac{1}{2p_{4}}(\hat{a}_{4}-a_{4})^{2}+\frac{1}{2q_{4}}(\hat{b}_{4}-b_{4})^{2}. \end{aligned} $$
(23)

Differentiating (23):

$$ \begin{aligned} &\dot{V}_{1}=s_{1} \dot{s_{1}}+\frac{1}{p_{1}}(\hat{a}_{1}-a_{1}) \dot{\hat{a}}_{1}+\frac{1}{q_{1}}( \hat{b}_{1}-b_{1})\dot{\hat{b}}_{1}, \\ &\dot{V}_{2}=s_{2}\dot{s_{2}}+ \frac{1}{p_{2}}(\hat{a}_{2}-a_{2}) \dot{\hat{a}}_{2}+\frac{1}{q_{2}}( \hat{b}_{2}-b_{2})\dot{\hat{b}}_{2}, \\ &\dot{V}_{3}=s_{3}\dot{s_{3}}+ \frac{1}{p_{3}}(\hat{a}_{3}-a_{3}) \dot{\hat{a}}_{3}+\frac{1}{q_{3}}( \hat{b}_{3}-b_{3})\dot{\hat{b}}_{3}, \\ &\dot{V}_{4}=s_{4}\dot{s_{4}}+ \frac{1}{p_{4}}(\hat{a}_{4}-a_{4}) \dot{\hat{a}}_{4}+\frac{1}{q_{4}}(\hat{b}_{4}-b_{4}) \dot{\hat{b}}_{4}. \end{aligned} $$
(24)

From (18), we have

$$\begin{aligned}& \dot{V}_{1}=s_{1} \bigl(D^{q}e_{1}+k_{1}e_{1} \bigr)+\frac{1}{p_{1}}(\hat{a}_{1}-a_{1})\dot{\hat{a}}_{1}+\frac{1}{q_{1}}(\hat{b}_{1}-b_{1}) \dot{\hat{b}}_{1}, \\& \begin{aligned}&\dot{V}_{2}=s_{2}\bigl(D^{q}e_{2}+k_{2}e_{2} \bigr)+\frac{1}{p_{2}}(\hat{a}_{2}-a_{2})\dot{\hat{a}}_{2}+\frac{1}{q_{2}}(\hat{b}_{2}-b_{2}) \dot{\hat{b}}_{2}, \\ &\dot{V}_{3}=s_{3}\bigl(D^{q}e_{3}+k_{3}e_{3} \bigr)+\frac{1}{p_{3}}(\hat{a}_{3}-a_{3})\dot{\hat{a}}_{3}+\frac{1}{q_{3}}(\hat{b}_{3}-b_{3}) \dot{\hat{b}}_{3},\end{aligned} \\& \dot{V}_{4}=s_{4}\bigl(D^{q}e_{4}+k_{4}e_{4} \bigr)+\frac{1}{p_{4}}(\hat{a}_{4}-a_{4})\dot{\hat{a}}_{4}+\frac{1}{q_{4}}(\hat{b}_{4}-b_{4}) \dot{\hat{b}}_{4}. \end{aligned}$$
(25)

Substituting \(D^{q}e_{i}\), \(\dot{\hat{a}}_{i}\), and \(\dot{\hat{b}}_{i} \) in (25), we obtain

$$ \begin{aligned} \dot{V}_{i} & =s_{1}\bigl[ \bigtriangleup g_{i}- \bigtriangleup h_{i} -( \hat{a}_{i}+\hat{b}_{i}+\gamma _{i})\operatorname{sign}s_{i}\bigr]+( \hat{a}_{i}-a_{i}) \vert s_{i} \vert +( \hat{b}_{i}-b_{i}) \vert s_{i} \vert \\ & \leqslant \bigl( \vert \bigtriangleup g_{i} \vert + \vert \bigtriangleup h_{i} \vert \bigr) \vert s_{i} \vert -(\hat{a}_{i}+\hat{b}_{i}+\gamma _{i}) \vert \operatorname{sign}s_{i} \vert +(\hat{a}_{i}-a_{i}) \vert s_{i} \vert +(\hat{b}_{i}-b_{i}) \vert s_{i} \vert \\ & < (a_{i}+b_{i}) \vert s_{i} \vert -(\hat{a}_{i}+\hat{b}_{i}+\gamma _{i}) \vert \operatorname{sign}s_{i} \vert +(\hat{a}_{i}-a_{i}) \vert s_{i} \vert +(\hat{b}_{i}-b_{i}) \vert s_{i} \vert \\ & =-R_{i} \vert s_{i} \vert . \end{aligned}$$
(26)

Finally,

$$ \begin{aligned} \dot{V}&=\sum _{i=1}^{4}\dot{V}_{i} \\ & < -\sum_{i=1}^{4}\bigl(R_{i} \vert s_{i} \vert \bigr). \end{aligned}$$
(26)

Thus \(\exists R \geq 0 \in R\)

$$ \sum_{i=1}^{4}R_{i} \vert s_{i} \vert > R, $$

then we have

$$ \begin{aligned} \dot{V}&< -R\sqrt{s_{1}^{2}+s_{2}^{2}+s_{3}^{2}+s_{4}^{2}+s_{5}^{2}} \\ &< 0. \end{aligned}$$
(27)

Lyapunov stability theory proves \(\Vert s_{i}\Vert \rightarrow 0 \) as \(t\rightarrow \infty \), implying stability and synchronization.

Simulation results

The efficacy of our synchronization methods is verified using MATLAB software for simulations. The trajectories of the triple compound combination anti-synchronized leader system and follower system using nonlinear control are in Fig. 8. The anti-synchronization error vanishing has been displayed in Fig. 9 for the I.C. \((0, 0.5175, 0.4825, 0)\).

Figure 8
figure8

Nonlinear control synchronized trajectories

Figure 9
figure9

Nonlinear control synchronization error

Figure 10 gives the anti-synchronized trajectories using the adaptive SMC method with the error vanishing in Fig. 11. The sliding surfaces tend to zero in Fig. 12 with Fig. 13 giving estimates of bounds of disturbances of leader systems. The I.C. for estimates is \(\hat{a_{i}}=\hat{b_{i}}=0.1\) and \(\gamma _{1}=2\), \(\gamma _{2}=3\), \(\gamma _{3}=3\), \(\gamma _{4}=4\) respectively. Other parameters and conditions are as in Sect. 8.

Figure 10
figure10

Adaptive sliding mode control synchronized trajectories

Figure 11
figure11

Adaptive sliding mode control synchronization error

Figure 12
figure12

Sliding surfaces tending to zero

Figure 13
figure13

Bound estimates of uncertainties and disturbances

Comparison of synchronization results

The increasing usage of synchronization methods in various fields leads to discovery of new chaotic systems and new synchronization schemes. The time taken to achieve the synchronization is a very key aspect in this regard. Clearly the lesser the time of synchronization, the more is the efficacy of the designed controllers. Table 3 shows the time taken to attain different types of synchronization by different authors and presents a comparative study of combination and compound synchronizations.

Table 3 Error convergence time

In our paper we have performed triple compound combination synchronization among parallel systems of novel fractional order snap systems. In nonlinear control, error converges to zero at three units of time, and in the adaptive SMC method error approaches zero before two units but oscillates about it. These findings display the efficacy of our designed controllers.

Application in secure communication

Chaos synchronization, having found application in many different fields such as secure communication [53, 68] and image encryption [46, 65], has been growing rapidly [47, 48, 55]. The highly sensitive dependence on I.C. and parameter values makes the information secure from getting hacked by intruders.

Here we use the above anti-synchronization technique to explain its usage in information security. The original signal is mixed with chaotic signals and transmitted from the sending end. Only if prior knowledge of the type of synchronization, I.C., and parameter values are known, the original message can be recovered at the receiving end.

Illustration:

The original message \(x(t)=\sin (7t)\) is mixed with a combination of chaotic signals \(z_{14}+z_{24}\) produced from master systems to obtain the encrypted message as \({x}_{1}(t)\). On attaining the right synchronization after applying controllers at the receiver, the original message is recovered viz \(x_{2}(t)\). Simulation results are shown in Fig. 14.

Figure 14
figure14

Illustrative example in secure communication

Snap system realization

Snap system signal flow graph

Every state of the snap system (3) is represented as a vertex of the proposed signal flow graph \(\overrightarrow{H}\) (see Fig. 15, the arrow means its edges have directions). The graph \(\overrightarrow{H}\) is a weighted graph since each edge has a weight. The weight of the edge \((Z_{i}, Z_{j})\) is the impact by which the state variable \(Z_{i}\) affects the state variable \(Z_{j}\). Then the graph \(\overrightarrow{H}\) can be represented by the following weighted adjacency matrix \(A(\overrightarrow{H}) \) (28):

A( H )= Z 1 Z 2 Z 3 Z 4 Z 1 Z 3 [ 0 0 0 a 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 b 0 ] .
(28)
Figure 15
figure15

Snap system signal flow graph \(\overrightarrow{H}\)

For \(a=0.2\), \(b=1\), the eigenvalues of \(A(\overrightarrow{H})\) are \(\lambda _{1}=0.822 80\), \(\lambda _{2}=0.184 18-0.883 3i\), \(\lambda _{3}=0.184 18+0.883 3i\), \(\lambda _{4}=-1.095 6+0.540 83i\), \(\lambda _{5}=-1.095 6-0.540 83i\). Then energy (1) of the snap system (3) can calculated as

$$ E(H)=\sum_{i=1}^{5} \vert \lambda _{i} \vert =5.071 . $$

If the directions of the arcs and loops in \(\overrightarrow{H}\) are omitted, we can get the underlined graph H whose adjacency matrix \(A(H)\) (29) can be formed as follows (Note that \(A(i, j)=1\) if vertex i is linked with vertex j in H regardless of the direction and \(A(i, j)=0\) otherwise.):

A(H)= Z 1 Z 2 Z 3 Z 4 Z 1 Z 3 [ 0 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 0 1 1 0 1 1 0 ] .
(29)

\(A(H)\) represents the structure of the snap system and can help in calculating the energy of the system from the graph theory point of view as follows.

\(A(H)\) eigenvalues are \(\lambda _{1}=\frac{1}{2}\sqrt{17}+\frac{3}{2}\), \(\lambda _{2}=\frac{3}{2}-\frac{1}{2}\sqrt{17}\), \(\lambda _{3}=-2\), \(\lambda _{4}, \lambda _{5}=0\), then the energy of the snap system can calculated as follows:

$$ E(H)=\sum_{i=1}^{5} \vert \lambda _{i} \vert =2\sqrt{5}+2 . $$

Snap system electronic circuit realization

In chaos applications, a very important topic is the hardware realization of mathematical models [28]. We implement an electronic circuit for realizing the studied snap system (3) in this section.

Figure 16 shows the schematic of the designed electronic circuit for the studied snap system (3), which is based on four integrators \((Z1,\ldots, Z4)\). All integrators and inverters are constructed using the 741 operational amplifier with Package type DIP-8 and Package manufacturer IPC-2221A/2222.

Figure 16
figure16

The electronic circuit of system

Equations of the designed electronic circuit of Fig. 16 are as follows:

$$\begin{aligned}& \begin{aligned}&Z_{1} =\frac{1}{R_{1}C_{1}} \int \frac{dZ_{1}}{dt}\,dt, \\ &Z_{2} =\frac{1}{R_{3}C_{2}} \int \frac{dZ_{z}}{dt}\,dt, \\ &Z_{3} =\frac{1}{R_{5}C_{3}} \int \frac{dZ_{3}}{dt}\,dt, \\ &Z_{4} =\frac{1}{R_{11}C_{4}} \int \frac{dZ_{1}}{dt}\,dt, \end{aligned} \end{aligned}$$
(30)

where the symbols Z1, Z2, Z3, and Z4 are the voltages of the integrators.

The circuit has been realized via NI MULTISIM 14.0 for \(R_{1}=R_{3}=R_{5}=R_{11}=1\) kΩ, \(C_{1}=C_{2}=C_{3}=C_{4}=1\) μF. We use five electronic summers \(S_{1},\ldots, S_{5}\), one electronic multiplier \(Z1Z3\), and two inverters Z5 and Z6. An oscilloscope has been used for displaying the measurements and capturing the experimental results.

As a result, the experimental phase portraits of circuit’s outputs, for the values of a, b as in the corresponding plots of Fig. 1, are given as shown in Figs. 17, 18, and 19. As predicted, the acquired experimental outputs assure the feasibleness of the studied snap system (3).

Figure 17
figure17

The plot of \(Z_{2}\) versus \(Z_{1}\) from the electronic circuit

Figure 18
figure18

The plot of \(Z_{3}\) versus \(Z_{1}\) from the electronic circuit

Figure 19
figure19

The plot of \(Z_{4}\) versus \(Z_{1}\) from the electronic circuit

Conclusion

In this paper, the fractional snap system with only one nonlinear term has been analyzed using various tools such as L.E., equilibrium point analysis, bifurcation diagram, and dissipativity. The system is synchronized in triple compound combination anti-synchronization considering uncertainties and disturbances among parallel systems using two control methods. Comparison of the obtained results with previous synchronization literature is made. The achieved synchronization has been illustrated in security of information transmission. Also the circuit of the system has been implemented.

The future scope of this work lies in studying the hidden attractors of the introduced fractional snap system. The achieved synchronization can be made applicable in the field of image and voice encryption as well.

Availability of data and materials

Not applicable.

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Acknowledgements

Taif University Researchers Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.

Funding

This paper is funded by Taif University Researchers Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.

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Correspondence to Emad E. Mahmoud.

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Mahmoud, E.E., Trikha, P., Jahanzaib, L.S. et al. Application of triple compound combination anti-synchronization among parallel fractional snap systems & electronic circuit implementation. Adv Differ Equ 2021, 211 (2021). https://doi.org/10.1186/s13662-021-03362-4

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Keywords

  • Dynamical properties
  • Electronic circuit
  • Parallel systems
  • Triple compound combination anti-synchronization