A simple multi-stable chaotic jerk system with two saddle-foci equilibrium points: Analysis, synchronization via backstepping technique and MultiSim circuit design

Aceng Sambas, Sundarapandian Vaidyanathan, Irene M. Moroz, Babatunde Idowu, Mohamad Afendee Mohamed, Mustafa Mamat, W. S. Mada Sanjaya Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Indonesia Research and Development Centre, Vel Tech University, Chennai, Tamil Nadu, India Mathematical Institute, University of Oxford, Andrew Wiles Building, ROQ, Oxford, United Kingdom Department of Physics, Lagos State University, Ojo, Nigeria Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, KualaTerengganu, Malaysia Department of Physics, Universitas Islam Negeri Sunan Gunung Djati Bandung, Indonesia

The presence of 1 0   in the LE spectrum (5) establishes the chaos nature of the jerk model (2). A comparison of the Genesio-Tesi jerk model (2) and the modified jerk model (4) can be encapsulated as follows. The Genesio-Tesi jerk model (2) has five linear terms and a single quadratic nonlinearity, while the modified model (4) has five linear terms and 2 quadratic nonlinearities. The maximal values of the Lyapunov exponents of the models (2) and (4) are easily seen as 1 = 0.1052 and 1 = 0.1080 respectively. Furthermore, the model (4) has complex behavior such as multi-stability with coexistence of chaos attractors. For finding the balance points of the model (4), we seek the roots of the system given as (6a), (6b), (6c): By finding the roots of the system (6), we arrive at two balance points of the model (4) as (7). 0 (0,0,0)  and 1 ( ,0,0). a  (7) In the chaos situation, ( , , , ) ( This shows that 0 (0,0,0)  and 1 (1.3,0,0)  are saddle-foci, unstable balance points of the modified model (4). Figure 1 represents various signal plots of the jerk model (4) in the 2-D planes.  It is worthwhile to observe that the jerk model (4) exhibits multistability phenomenon, which is the coexistence of chaos attractors when choosing different initial phase vectors [22,23]

 
and 0 ( 0.5,0, 0.5),     and the corresponding signal plots of the jerk model (4) are depicted in blue and red colors, respectively. Figure 2 illustrates the multistability of the jerk model (4).
A steady state bifurcation   0   can occur only when 0. a  For a Hopf bifurcation, we set i   in (10), which yields: Simplififying (11) and rearranging terms, it is deduced that: By equating the imaginary and real parts of both sides of (12), it is seen that: A steady state bifurcation occurs when 0. a  Thus, we have an exchange of stabilities beween 0  and 1 .
 However, for a Hopf bifurcation, we set i    in (14), which yields: Simplifying (15) and rearranging terms, it is deduced that: By equating the imaginary and real parts of both sides of (12), we deduce as (17): Since all four parameters are required to be positive, the first condition in (17)

GLOBAL CHAOS SYNCHRONIZATION OF THE NEW JERK SYSTEMS VIA ACTIVE BACKSTEPPING CONTROL
With the use of backstepping technique, new results are encapsulated in this section for the synchronising design of the new chaos models envisioned as leader and follower systems. The leader system is specified by the new chaos model dynamics given in (18). 12 23 Furthermore, the follower system is specified by the controlled chaos model dynamics given in (19). 12 23 In the systems (18) and (19)      are the states. Also, u is a backstepping controller that is to be determined in this section. The action of u is to enable synchronization of the respective phases of the jerk models (18) and (19). For this purpose, we shall define synchronizing error between the respective phases of the jerk models (18) and (19) as being (20). 1 1 1 The error phases satisfy the system of differential equations as given in (21).
Next, we shall outline a main backstepping control result providing a compact formula for the feedback control u which will achieve the desired synchronization between the leader and follower models (18) and (19). Theorem 1. The backstepping feedback control law mentioned by where 0 L  is a controller gain and 3 between the states of the leader and follower chaos models (18) and (19).
Proof. The assertion of Theorem 1 shall be established with an application of Lyapunov stability theory [24][25][26][27]. We start the proof by defining a scalar Lyapunov function as (23): where 11 .

 
The time-derivative of 1 W along the error dynamics (21) can be easily found as being (24) We can express (24) using (25) as (26): Next, we propose the Lyapunov function The time-derivative of 2 W along the error dynamics (21) can be found as (28): Next, we define We can express (28) using (29) as (30): Finally, we propose the Lyapunov function It is a straightforward calculation to verify that W is a positive definite function on 3 .
R The timederivative of W along the error dynamics (21) can be found as (32): A simple calculation yields the result (34): When we substitute the active feedback law (22) 6 0.4 -1.3). The initial state of the follower system (19) is picked as β(0)=(2.8 -0.7 1.2). Figure 5 pinpoints the complete synchronization of the new jerk models represented by (18) and (19). Furthermore, Figure 6 depicts the time-history of the synchronization error between the new chaos jerk models (18) and (19).

MULTISIM CIRCUIT SIMULATION OF THE NEW CHAOTIC JERK MODEL
A MultiSim circuit model of the new chaos jerk model (4) is implemented by using off-the-shelf components such as operational amplifiers, capacitors, resistors, and analog multipliers. The MultiSim circuit in Figure 7 has been designed by applying the general approach with operational amplifiers. Thus, the variables 1 ,  2  3  1  2  3  1 2  1  3 3  3 4  3 5  3 6  3 7   1   1   1  1  1  1  1  10 10 The power supplies of all active devices are ±15 VDC and the TL082CD operational amplifiers are used in this work. The values of components in Figure 8 are chosen to match the parameters of system (37) as: R1=R2=R6=R7=R8=R9=R10=R11=R12=R13=100 kΩ, R3=R4=76.92 kΩ, R5=200 kΩ, R7=10 kΩ and C1=C2=C3=1nF. The designed circuit has been examined by MultiSim software and results are described in Figure 7. It is easy to see a good agreement between the MultiSim simulation as shown in Figure 8 and MATLAB simulation as shown in Figure 1.  -plane 6. CONCLUSION In this paper, by modifying the Genesio-Tesi jerk dynamics (1992), a new jerk system model was derived and its dynamical properties were analyzed in detail. It was shown that the new jerk model has with multistability and dissipative chaos with two saddle-foci balance points. By invoking backstepping technique, new results for synchronizing chaos between the proposed jerk models were successfully yielded and proved using Lyapunov stability theory. Furthermore, MultiSim software was used in order to implement a circuit model for the new jerk dynamics. The control application and circuit implementation of the new jerk model have useful applications in engineering such as secure communications, and crypto-devices.