Forming Mechanizm of Bhalekar-Gejji Chaotic Dynamical System

Chaotic dynamical systems are used to model various natural phenomena. Bhalekar-Gejji chaotic dynamical system is a system of three ordinary differential equations containing only two nonlinear terms. Th is system shows two-scroll butterfly-shaped attractor for certain values of parameters . In this article we show that the two-scroll attractor in this system is formed from two one-scroll attractors. We have used a control parameter in the third equation of the system to study the forming procedure of the attractor.

Chaotic trajectories are very sensitive to initial conditions i.e. the trajectories starting nearby could have co mp letely different future. Though there is unpredictability, it is possible to make the behaviour of two (or many) nearby starting trajectories identical after so me t ime period. This process is done by applying a suitable control and is termed as a synchronization. Examp les of synchronization are abundant in nature. For the detailed discussion on this topic, readers are referred to [18][19][20]. Synchronization of chaotic systems have applications in secure communicat ion [21]. Due to unpredictability, the crypto-systems based on chaotic synchronization are difficult to decode. The review on this topic is available in [22].
In this art icle we show that the t wo-scro ll att ractor in Bh alekar-Gejji system is fo rmed fro m t wo o ne -scro ll attractors. We have used a control parameter in third equation of the proposed system to study the forming procedure of the attractor.

Bhalekar-Gejji System
A new chaotic system [23] proposed by Bhalekar and Daftardar-Gejji is g iven by the system of three ord inary differential equations.   An equilibriu m point of the system (2.1) is called a saddle point if the Jacobian matrix at has at least one eigenvalue with negative real part (stable) and one eigenvalue with non-negative real part (unstable). A saddle point is said to have index one (/two) if there is exact ly one (/two) unstable eigenvalue/s. It is established in the literature [24][25][26][27] that, scrolls are generated only around the saddle points of index two. Saddle points of index one are responsible only for connecting scrolls.

Forming Mechanism of Attractor
In order to study the compound structure of the new attractor, we add a constant gain to the third equation.
= − 2 , = ( − ), = a y − b z + x y + m, (3.1) Figure 2(a). Left attractor m=18. 5 We get one-scroll right-attractor for m=18.5 (cf. Fig. 2(a)) whereas m=-18.5 g ives the mirror image of the right-attractor i.e. the left-attractor as shown in Fig. 2(b). Thus, the new attractor is a co mpound structure obtained by merging together two simp le one-scroll attractors. Now we study the behavior of the controlled system (3.1) for different values of parameter m.
 |m|<3. 2 The system is chaotic and shows double-scroll complete attractor.
 |m|<4. 8 The system shows limit cycles for this range. In Fig. 3(a), the limit cycle is shown for m=3.2.

Conclusions
In this article, the forming mechanis m of Bhalekar-Gejji chaotic system is discussed. It is observed that the two-scroll attractor in the Bhalekar-Gejji system is formed fro m t wo one-scroll attractors. For this study, we have introduced a control parameter m in the third equation of the system. The complete double-scroll attractor observed for |m|<3.2 is transformed to a partial attractor in the range 11.4≤|m|<18.5. Limit-cycles are also observed for certain values of parameter m.