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

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.

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 = |V | has n eigenvalues; we denote these by λ 1 , . . . , λ 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 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. 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. The effect of fractional order on system dynamics for varying fractional order in the 0.8 to 1 range is studied.
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. Bounds of uncertainties and disturbances are also estimated using the adaptive SMC technique. 5. Achieved results are compared with some earlier published literature. 6. Application of the achieved synchronization on the introduced fractional version of the snap system in secure transmission is illustrated. 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: where z = (z 1 , z 2 , z 3 , z 4 ) T ∈ 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: We here examine the fractional version of this newly introduced system:

Solution of a novel system
Theorem The I. V.P. of (3): for some τ > 0, then a unique solution is guaranteed.
Proof As G(z) = B 1 z + z 1 B 2 z is continuous on R 4 and hence bounded on any compact interval [z o -, z o + ] for > 0, therefore the Lipschitz continuity on this interval is sufficient to apply the existence and uniqueness theorem in [9]. As we get This implies that G(z) is Lipschitz continuous, and hence the system has a unique solution.
Theorem The stagnation point E is an unstable saddle focus type stagnation point.
We find the divergence of this vector field by i.e. ∇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 → -z 1 , z 2 → -z 2 , z 3 → z 3 , z 4 → -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. The chaotic attractor of (3) possesses a fractional dimension defined as follows: Here, p is the integer such that p j=1 L.E. j ≥ 0 and p+1 j=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 ∈ (0, 1) and Fig. 3(b) gives for b ∈ (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.

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 > 2 π arctan( | Im(μ)| Re(μ) ). Therefore the smallest q for (3) to have chaos is 0.76. Different q plots of (3) are in Fig. 4.
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.

Triple compound combination anti-synchronization
We first consider the six parallel systems of (3) defined as summarized in the form of a table as follows: triple compound combination antisynchronization 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.
In presence of uncertainty and disturbances consider two leader systems and six follower systems. Figure 7 shows the plots of a leader system.  Leader I: D q z 13 = z 14 + g 3 + ψ 3 , D q z 14 = -az 11z 12z 14 + bz 11 z 13 + g 4 + ψ 4 .

Via nonlinear control
Choosing controllers as follows:  Lyapunov stability theory proves e i → 0 as t → ∞, implying synchronization.

Via adaptive SMC
Assuming bounded uncertainties and disturbances, we have | g i | ≤ a i and | h i | ≤ b i for positive real numbers a i and b i , andâ i andb i are estimated values of a i and b i . Choosing the sliding surface so as to minimize the error: For (14) to be at (16), we must have Taking Caputo's derivative of (17) then from (17) we have The error dynamical system is stable by the Matignon theorem [33]. Using SMC theory, we design controllers: for signum function sign(·) and positive real numbers γ i .
Adaptive parameter update conditions are as follows: where p i and q i are positive constants. (10)- (11) and follower systems (12)- (17) with controllers and update conditions as (25) and (26).

Theorem 8.1 Stability and triple compound combination anti-synchronization is achieved in presence of uncertainties and disturbances between leader systems
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. where Differentiating (23): From (18), we havė Substituting D q e i ,ȧ i , andḃ i in (25), we obtaiṅ Finally, Thus then we havė V < -R s 2 1 + s 2 2 + s 2 3 + s 2 4 + s 2 Lyapunov stability theory proves s i → 0 as t → ∞, 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 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 isâ i =b i = 0.1 and γ 1 = 2, γ 2 = 3, γ 3 = 3, γ 4 = 4 respectively. Other parameters and conditions are as in Sect. 8.

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.  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,  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  Combination [24] B o , X i a b i n g , Y u n 3 Compound [51] S u n , Y i , S h e n 5 Multi-switching compound [39] Prajapati, Khan, Khattar 4.5 Double compound [67] Zhang, Deng 5 Modified compound [49] S u n , L i , W a n g , W a n g 3 Double compound combination [29] Mahmoud, Abed-Elhameed 4 Triple compound [66] Yadav, Prasad, Srivastava 5 Dual compound compound [14] Ibraheem 4 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.

Snap system signal flow graph
Every state of the snap system (3) is represented as a vertex of the proposed signal flow graph − → H (see Fig. 15, the arrow means its edges have directions). The graph − → 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 − → H can be For a = 0.2, b = 1, the eigenvalues of A( − → H ) are λ 1 = 0.82280, λ 2 = 0.18418 -0.8833i, λ 3 = 0.18418 + 0.8833i, λ 4 = -1.0956 + 0.54083i, λ 5 = -1.0956 -0.54083i. Then energy (1) of the snap system (3) can calculated as If the directions of the arcs and loops in − → 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) 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.

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, . . . , Z4). All integrators and inverters are constructed using the 741 operational amplifier with Package type DIP-8 and Package manufacturer IPC-2221A/2222.
Equations of the designed electronic circuit of Fig. 16 are as follows: 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 k , C 1 = C 2 = C 3 = C 4 = 1 μF. We use five electronic summers S 1 , . . . , S 5 , one electronic multiplier Figure 16 The electronic circuit of system Figure 17 The plot of Z 2 versus Z 1 from the electronic circuit 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 18
The plot of Z 3 versus Z 1 from the electronic circuit Figure 19 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 antisynchronization 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.