Finite-Time Synchronization of the Rabinovich and Rabinovich-Fabrikant Chaotic Systems for Different Evolvable Parameters

ABSTRACT

control [16], sliding mode control, and active control [17], among others.Among the intriguing phenomena associated with chaotic systems is synchronization, where the dynamics of two chaotic systems, whether identical or non-identical, are coupled through appropriate control laws within finite time.Numerous synchronization schemes have been developed to facilitate this coupling, including adaptive control-based synchronization [18], adaptive hybrid synchronization [19], functional projective synchronization [20], and sliding mode control-based synchronization [21], [22], fuzzy synchronization [23], invariant ellipsoids method [24].Each scheme serves specific purposes in different contexts and applications.In this study, we delve into the synchronization of two particular chaotic systems: the Rabinovich chaotic system [25]- [27], and the Rabinovich-Fabrikant system [28].These systems, characterized by sets of three-coupled differential equations, possess unique properties that have attracted significant attention in diverse fields, such as image encryption, power system dynamics, and control.The Rabinovich-Fabrikant system, in particular, is renowned for its complexity owing to the presence of quadratic and cubic terms [29], making it a subject of extensive analysis and exploration [30]- [32].
In the following sections, we will focus on the finite-time synchronization of these intriguing chaotic systems, presenting our approach and numerical findings.This study aims to contribute to the understanding of chaos synchronization, with practical implications in various interdisciplinary domains.

Rabinovich System
The architecture of the Rabinovich system consists of a set of first-order, three-coupled ordinary differential equations given by where , , , , and  > 0 are positive parameters, whose values determines the type of attractor that is evolved, and  ∈ [ 1 ,  2 ,  3 ] are the state variables that gives the order of the system.When  = 4,  = 6.75,  = 6.75,  = 1,  = 1, the Rabinovich system evolves the attractors shown in Fig. 1.

Summarization of the Finite-Time Control Scheme
Conceptually, two identical or non-identical chaotic systems are said to synchronize in finite time, when for a given initial conditions, their coupling trajectories settle uniformly after a specified time.Finite-time synchronization is essential for time-critical applications to avoid overshoots and other undesirable consequences [33].Finite-time schemes are realized with the application of Lyapunov stability criteria in conjunction with definitions and lemmas.In recent years, finite-time control and synchronization schemes have attracted much attention they have been applied to the synchronization of chaotic systems with different dimensions and uncertainties [34]- [36], coronary artery chaotic systems [37], memristor chaotic systems [38] and hyperchaotic systems [33] amongst others.In this section, we summarized a body of definitions and lemmas suitable for the synchronization of the Rabinovich and Rabinovich-Fabrikant systems.

Architecture of the Finite-Time Synchronization Scheme
The architecture of the proposed Finite-time synchronization scheme is shown in Fig. 5.The master chaotic system is the Rabinovich system which is used to drive the slave system.The slave chaotic system is the Rabinovich Fabrikant system that is controlled to couple the dynamics of the two systems.The synchronization error system is used to track the dynamics of the master and slave systems to bring them into a state of synchrony in finite time.The parameter vector are the parameters whose values evolves the set of attractors.The finite time control laws are the set of functions based on the Lyapunov stability criteria that provides the sufficient conditions to couple the dynamics of the chaotic systems in finite time.The signum function and non-negative indices constitutes the weighted parameters that constrains overshoots of the asymptotically stable coupled dynamics to keep them within the finite time.The synchronization error dynamics of the systems (1) and (2) using the Definition 2 is given by Where   are vectors of the associate control laws of the synchronization error system, and are given by Vol.
We proposed a global finite-time control law that combines the associate control laws and control strength coefficients that comprise weighted parameters signum functions and non-negative indices and given by Where Π    are control strength coefficients,  is non-negative index and sgn is a Signum function and   has been defined in Definition 2. By replacing   in ( 7) with ( 9) and using (8), results in the following Theorem.The dynamics of the master chaotic system (1) and the controlled slave chaotic system (2) can be synchronized in finite time, if the proposed global finite-time controller given by ( 9) is applied.Proof: Consider the quadratic Lyapunov function candidate.
The time derivative of (10) along the trajectories of ( 7) is given by Using (10) in (12) results in the following By using the convention: Inserting ( 14) into (13) gives Eq. ( 15) is negative-definite in ℝ 3 .Thus Lemma 1 is satisfied, which implies that the synchronization error system (7) can be uniformly asymptotically stabilized in finite time while simultaneously synchronizing the dynamics of the master and slave chaotic systems.If the control strength coefficients are set to Π  1  = Π  2  = Π  3  = Π    , where "G" is a prefix for global control strength coefficient, then (15) reduces to Based on Assumption 1, By virtue of Assumption 1, According to Lemma 1, the two chaotic systems, the time of synchronization is estimated as Overall, the negative-definiteness of the partial derivative of the Lyapunov function along the trajectories of the coupling systems proves that the proposed control laws can synchronize the systems in finite time.

Result and Discussion
Few cases which comprise the coupling of the master and slave systems for different parameter sets were explored to test the robustness of the synchronization scheme.When the Master system:  = 4,  = 6.75,  = 6.75,  = 1,  = 1 and Slave system:  = 0.87,  = 0.87,  = 1.1 , the plots are shown in Fig. 6.In finite time control scheme, stabilized dynamics of the finite-time control laws and synchronization error functions are sufficient proof of the viability of a proposed control architecture.Thus, Fig. 7 and Fig. 8 confirmed the effectiveness of the proposed synchronization architecture.There was no significant difference in the synchronized dynamics of the master and slave systems for other parameters and initial conditions because all the varied parameters.Thus, the finite-time synchronization scheme is robust in the face of parametric disturbances.

Conclusion
A finite-time synchronization scheme was proposed to couple two non-identical chaotic systems.the dynamics of the two systems are highly sensitive to initial conditions and parameter variations.We derived a set of control laws that coupled the dynamics of the two systems, based on Lyapunov stability criteria.Robustness to parameter variations is an indication of effectiveness of the scheme.To examined the robustness of the synchronization scheme, the system parameters and initial conditions were varied.The results shows that the variations in system parameters had no noticeable impact on the synchronization time.It was observed that the signum function and nonnegative indices played a crucial role in accelerating the synchronization process and preventing overshoots.The higher the non-negative index, the shorter the synchronization time.

Fig. 5 .
Fig. 5. Architecture of the Finite Time Synchronization scheme

Fig. 6 .Fig. 7 .Fig. 8 .
Fig. 6.Synchronized dynamics of the state variables Fig. 6 (a), (b) and (c) depicts the synchronization of the dynamics of the master ad slave system.itcan be seen the trajectories coupled in less than 0.002s.these are relatively fast.The stabilized dynamics of the synchronization error functions and the finite time control law are shown in Fig.7and Fig.8respectively.