Dual Penta-Compound Combination Anti-Synchronization with Analysis and Application to a Novel Fractional Chaotic System

This paper studies a fractional-order chaotic system with sine non-linearities and highlights its dynamics using the Lyapunov spectrum, bifurcation analysis, stagnation points, the solution of the system, the impact of the fractional order on the system, etc. The system considering uncertainties and disturbances was synchronized using dual penta-compound combination anti-synchronization among four master systems and twenty slave systems by non-linear control and the adaptive sliding mode technique. The estimates of the disturbances and uncertainties were also obtained using the sliding mode technique. The application of the achieved synchronization in secure communication is illustrated with the help of an example.


Introduction
The three-centuries-old theory of fractional calculus, which is the generalization of integer calculus, has found glory quite recently. Fractional calculus is being implemented to model various real-life systems in almost all disciplines. In fact, fractional calculus is seen as the future of mathematical modeling. Fractional calculus with its better hereditary properties is used to model various biological models such as the prey-predator model [1] and the human liver [2], engineering models such as harmonic oscillators [3] and thermostats [4], geophysical models such as Earth's dynamo [5], and so on. Fractional computations have attracted researchers worldwide in dynamical systems such as the Chua circuit, the Duffing system, and the logistic system. Chaotic systems being highly sensitivity to system values have contributed greatly in the development of information security, computers, communications [6], and encryption [7]. Chaotic systems help mask the information signal to avoid hacking by intruders. Chaos synchronization plays a major role in the decryption of the information signal from the masked signal. Chaotic systems with their positive Lyapunov spectrum component resist their synchronization with other where α is the fractional derivative, n − 1 < α < n, n ∈ N, and Γ(α) = ∞ 0 x α−1 e −x is the Gamma function.
Among the many fractional derivative [27] definitions, in this paper, Caputo's derivative of fractional order is considered. Caputo's definition uses the integer derivative for computation, whereas Riemann-Liouville's derivative uses the fractional derivative for computation. Therefore, Caputo's derivative has an edge over other derivatives.

New Fractional Chaotic System
Introduce the fractional chaotic system as: where Z ∈ R 3 are state variables for A, B ∈ R.
For A = 5, B = 5, and initial conditions (0.000001, 3, 0) for fractional order 0.987, System (1) is chaotic as displayed in Figures 1 and 2 in time series and phase plots, respectively.

Symmetry, Dissipativity, and Stagnation Points
The novel fractional-order chaotic system (1) does not show rotational symmetry about any axis, as the system does not remain invariant under the transformation Z i → −Z i , Z j → −Z j , Z k → Z k . However, the system shows symmetry about the origin as the system The matrix form of System (1) is: The divergence of vector field K is: i.e., ∇K = −1 < 0.

Solution of the Novel Fractional-Order Chaotic System
The novel fractional-order chaotic system is expressed as: where t ∈ (0, T] and Z(0) = Z o . Here: The solution is examined in the region ω × I, where I = (0, T] and ω = (Z i ) : max|Z i | ≤ P for i = 1, 2, 3, P > 0, where constant P designs a boundary in the phase space.

Lyapunov Dynamics and Bifurcation Analysis
With the idea of the rate of separation of closely lying trajectories in the phase space, the Lyapunov spectrum values are found. For A = B = 5 and initial conditions (0.000001, 3, 0), the Lyapunov dynamics for q = 0.987 are: The first component being positive confirms chaos. The dimension (K.Y.) is found by: where p is maximum number satisfying ∑ p s=1 L.E. s ≥ 0 and ∑ p+1 s=1 L.E. s < 0. Therefore, the K.Y. dimension is 2.34062.
Parameter values and initial conditions have a very sensitive dependence on the system dynamics. A slight change in these values of the system can lead to drastically different dynamics. Even in the smallest neighborhood of the parameter values, the dynamics may vary in the number of equilibrium points (none, finite, countably infinite, uncountably infinite), their stability, and from the regular nature to the periodic to the chaotic nature. In bifurcation analysis, only one parameter value is varied in a slight neighborhood, while others are kept constant. In Figure 3a, A is varied in (4.75, 5.25) and B = 5; in Figure 3b, B is varied in (4.75, 5.25) and A = 5. The effect of the fractional order is given in Figure 4.

Stability of the Trivial Equilibrium Point
Clearly, the origin is the equilibrium point of the system. We check its stability by finding the Jacobian matrix as: The eigenvalues of the above matrix are: Here, µ 1 , µ 2 are complex eigenvalues with a positive real part and µ 3 is a negative eigenvalue, which together imply that the system is unstable.
Since all eigenvalues have a non-zero real part, the equilibrium point is hyperbolic.

Dual Penta-Compound Combination Anti-Synchronization
Consider four master systems and twenty slave systems, the first ten slave systems corresponding to the first two master systems and the next ten slave systems corresponding to the next two master systems for dual penta-compound combination anti-synchronization.
Proof. Stability using Lyapunov's direct method [29] is proven considering positive definite function V and the negative derivative proving error convergence to zero. Let: where: Differentiating:V From (31), we have: Substituting the values in Equation (38): Now ∃ at ≥ 0 ∈ R such that: From the Lyapunov stability theory, errors converge to S i = 0.

Simulations and Proposed Application
The dual penta-compound combination anti-synchronized trajectories and error using the non-linear control method are displayed in Figure 7. The synchronized trajectories using the adaptive SMC technique are displayed in Figures 8a-c and 9a-c. Figures 8d and 9d show the anti-synchronization error using the SMC technique. The sliding surfaces tending to zero are shown in Figures 8e and 9e. The disturbances and uncertainties H 1 = 0, D 1 = 7sin(t), H 2 = 0, D 2 = sin(7t), H 3 = cos(Z 21 ), D 3 = 0, H 4 = 0, D 4 = 7sin(t), H 5 = 0, D 5 = sin(7t), H 6 = cos(Z 21 ), D 6 = 0 are estimated and shown in Figures 8f,g and 9f,g by estimating the parameters using I.C. asÂ i =B i = 0.1 and c 1 = 1, c 2 = 2, c 3 = 3, c 4 = 1, c 5 = 2, c 6 = 3, respectively.      The achieved anti-synchronization is illustrated with an example for application in secure communication. In this synchronization, we have two pairs of penta-compound combinations from which to choose. This adds to the diversity in the options for encrypting the original message. The sum of the chaotic signals from the master systems is added to the original signal; the encrypted signal is formed and transmitted. At the receiving end, upon performing synchronization, the controllers are applied and decrypted.
Suppose S(t) = sin(5t) + cos(6t) as the original signal. Mix S(t) with chaotic signals Z 11 + Z 21 to obtain S 1 (t). Apply controllers to recover S 2 (t) at the receiver, as illustrated in Figure 10.

Conclusions
In this paper, dual penta-compound combination anti-synchronization was performed on a chaotic system with two sine non-linearities. A thorough analysis of the newly introduced fractional-order system was performed. The achieved synchronization was performed between four master systems and twenty slave systems using two techniques. Uncertainties and disturbances were estimated. An application in secure communication was illustrated with the help of an example.
Studying the hidden attractors of the system and its electronic circuit implementation are the future scope.