CONTROLLING AND SYNCHRONIZATION IN CHAOTIC SYSTEMS USING PARAMETER IDENTIFICATION METHOD

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this manuscript, we investigate projective synchronization (PS) between identical chaotic new Hamiltonian systems based on Hénon-Heiles Model using parameter identification method (PIM). Initially, Lyapunov’s theory of stability is used to design the proper adaptive controllers in view of master-slave configuration to achieve the global asymptotic stability. Also, the proposed technique establishes identification of parameter simultaneously via PS scheme. Additionally, numerical simulations are performed using MATLAB software for visualizing the efficiency and feasibility of the proposed methodology. Furthermore, the discussed approach has numerous significant applications in encryption and secure communication.


INTRODUCTION
Chaos theory is undoubtedly an important and intriguing field of applied mathematics that deals with highly complex nonlinear dynamical systems. This theory has played a vital role in several disciplines, including physics [1], biomedical engineering [2], chemistry [3], ecology as OGY method to control chaotic systems.
Synchronization is a procedure to adjust two or more (identical or non-identical) chaotic systems in a manner that both (all) exhibit the similar behavior owing to pairing to gain stability.
Synchronization among chaotic systems using parameter identification method (PIM) or adaptive control method was introduced in 1989 by Hubler [30]. In [16,25,31,32,33,34,35,36,37,38], many control techniques are studied for controlling and synchronizing the chaotic/ hyperchaotic systems. Vaidyanathan et al. [39] defined and studied a new chaotic Hamiltonian system based on Hénon-Heiles system, describing the nonlinear complex motion of a star around a galactic centre with the motion restricted to a plane, which was modeled by Hénon and Heiles [40] in 1964.
Considering the above literature review and discussion, the main objective of this manuscript is to carry out an investigation of projective synchronization (PS) scheme in a new Hamiltonian chaotic systems using PIM. The underlying idea here is to use Lyapunov's theory of stability to design proper adaptive controllers to achieve global asymptotic stability.
The remainder of this manuscript is organized as: Sect. 2 comprising of few mathematical preliminaries having essential notations and terminology to be utilised within the manuscript. In Sect. 3, basic structured properties of Hamiltonian system to be synchronized, are mentioned.
Sect. 4 investigates the active control method to stabilize the given chaotic system asymptotically. Sect. 5 deals with the numerical simulations illustrating the effectiveness and feasibility of the discussed complete synchronization approach using active control method. Finally, Sect. 6 concludes the paper.

MODEL ANALYSIS
Reported by Sundarapandian Vaidyanathan et al. [39], the discussed chaotic system can be written as: where is the state vector and p and q are parameters. When p = −1.95 and q = 1.48, the system (4) displays chaos. Also, the Lyapunov exponents of system (4) are Fig. 1(a-d) display the phase graphs of (4). Moreover, the detailed analytic study and simulation results for system (4) may be found in [39].

STABILITY ANALYSIS
In this section, we study PS scheme for the new chaotic Hamiltonian system to design proper adaptive controllers using PIM in such a way that each state variable v m1 , v m2 , v m3 and y m4 approaching to equilibrium points as t tending to infinity.
The system (4) is chosen as master system and corresponding slave system is defined as: where η 1 , η 2 , η 3 and η 4 are adaptive nonlinear controllers to be constructed so that PS scheme between two identical Hamiltonian chaotic systems will be attained.
The resulting error dynamics would be written as: Next, we describe the adaptive controllers by where M 1 > 0, M 2 > 0, M 3 > 0, M 4 > 0 are gain constants.
By substituting the controllers as defined in eq. (8) in error dynamics eq. (7), one finds that wherep,q are estimated quantities of unknown parameter p, q respectively.
Defining the parameter estimation error as: Using eq. (10), the error dynamics eq. (9) is written as: On differentiating eq. (12), one finds thaṫ Lyapunov function is defined as: which implying that V is positive definite.
Derivative of V may be written as: Keeping eq. (14) in view, we formulate the parameter estimates laws as: where M 5 > 0 and M 6 > 0 are gain constants. Proof. The Lyapunov functional V as defined in eq. (13) is a positive definite function. By solving eq. (11), eq. (14) and eq. (15), we havė confirming thatV is negative definite.

NUMERICAL SIMULATION AND DISCUSSION
This section presents some simulation experiments to illustrate the efficiency of the proposed PS scheme via PIM. Here, we use the fourth order Runge-Kutta methodology to solve system of differential equations. The initial states of master(4) and slave systems (5) are we have achieved complete PS in master(4) and slave (5) systems. The control gains are selected as K i = 6 for i = 1, 2, 3, 4. Simulation results are displayed in Fig. 2(a-d) which depict state trajectories of master(4) and slave systems(5) and Fig. 3(a) displays that synchronization 0.2, 0.6, 0) tends to zero as t tends to infinity. Further, Fig. 3(b) displays the estimated values (p,q) asymptotically with time. Therefore, the investigated PS scheme among master and slave systems has been attained computationally. In addition, if scaling matrix A is selected as ξ = −5, we have achieved anti-PS scheme in master(4) and slave (5) systems.The Fig. 4(a-e) shows that anti-PS scheme in systems (4) and (5) is achieved numerically. Also, Fig. 5(a) shows that synchronization error (E 1 , E 2 , E 3 , E 4 ) = (1.2, 0.2, −0.8, 0) tends to zero as t tends to infinity. Moreover, Fig. 5

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.