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Analysis, Control and Circuit Design of a Novel Chaotic System with Line Equilibrium

In this paper, we describe a new 3-D chaotic system with line equilibrium. A detailed analysis of the proposed chaotic system is provided with MATLAB phase portraits, equilibrium points, symmetry, invariance, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and a Poincarè map. For specific values of the parameters, the proposed system displays chaotic behaviour. We apply adaptive control method for the global chaos synchronization of the new chaotic systems with unknown system parameters. In addition, a new circuit implementation of the new chaotic system is reported and examined in MultiSIM. A good qualitative agreement is shown between the simulations and the MultiSIM results.


Introduction
The first experimental model of a chaotic system was obtained by Lorenz [1], when he was modelling the atmospheric convection with a 3-D model in 1963. Lorenz system [1] is a classical chaotic system with a two-scroll chaotic attractor. Since this monumental work by Lorenz, chaos has been studied extensively and many chaotic systems have been developed in the literature. Both theoretical and experimental chaotic systems have been reported and their applications have been discussed as well.
In [2], Rössler constructed a seven-term chaotic system with just one quadratic nonlinearity. In [3][4], Rikitake discovered a new chaotic system to explain the irregular polarity switching of the earth's geomagnetic field. A model to explain the irregular variability in the luminosity of stars was presented in 1976 [5] and this system is known as the Moore-Spiegel system. In [6], Rucklidge discovered a double convection chaotic model in a horizontal layer of Boussinesq fluid with lateral constraints.
Synchronization of chaotic systems [34] deals with a pair of chaotic systems called the master and slave systems and the design goal is to synchronize the respective state trajectories of the two systems asymptotically with the help of a controller attached to the slave system.
The synchronization of chaotic systems is a challenging research problem as the state trajectories of chaotic systems are very sensitive to small changes in initial conditions [35]. Several methods have been proposed in the literature for the synchronization of chaotic systems [36][37][38].
In this paper, we describe a new 3-D chaotic system with line equilibrium. Our novel chaotic system has six terms with four quadratic nonlinearities. In Section 2, we first provide the dynamics and phase portraits of the new chaotic system. We also present a qualitative analysis of the new chaotic system with properties such as dissipativity, equilibrium points, symmetry, invariance, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and a Poincarè map. In Section 3, we apply adaptive control method to derive new results for the global chaos synchronization of the new chaotic systems with unknown system parameters. In Section 4, we, a new circuit implementation of the new chaotic system is reported and examined in MultiSIM. A good qualitative agreement is shown between the simulations and the MultiSIM results. In Section 5, we give final conclusions of this work.

A new 3-D chaotic system
In this work, inspired by the method and structure proposed in [39][40], we announce a new 3-D chaotic system described by the nonlinear dynamics x y z are state variables and , a bare positive system parameters.
We shall show that the system (1) is chaotic when the system parameters take the values 2.8, 0.6 a b = = (2) For numerical simulations, we take the initial conditions as  (1) is chaotic. Also, the sum of the Lyapunov exponents of the system (1) is found as Hence, the system (1) is a dissipative chaotic system with a strange chaotic attractor. The phase portraits of the new chaotic system (1) are shown in Figure 1. The Lyapunov exponents of the new chaotic system (1) are shown in Figure 2.
The Kaplan-Yorke dimension of the new chaotic system (1) is calculated as We take the parameter values as in the chaotic case (2). A simple calculation shows that the new chaotic system (1) has a line of equilibrium points It is easy to see that the new chaotic system (1) is invariant under the change of coordinates ( , , ) ( , , ) x y z x y z -- (9) This shows that the system (1) has rotation symmetry about the zaxis. Also, we find that zaxis is invariant under the flow of the new chaotic system (1). This invariant motion is characterized by the 1-D dynamics Next, to have a detailed view of the new chaotic system (1), the behaviour of the system with respect to the bifurcation parameter b is studied. The new chaotic system (1) displays the expected chaotic behavior for 0.5 ≤ b ≤ 0.84 and for 0.85 ≤ b ≤ 0.86 periodic behavior is noted in the system. Also, the new chaotic system has chaotic behavior for b > 0.86 as presented in Figure 3. Poincarè map of the system (1) is also illustrated in Figure 4, which exhibits that the new chaotic system (1) has complex dynamics.

Adaptive synchronization of the new chaotic system
In this section, we apply adaptive control method [42][43][44][45] for the global chaos synchronization of identical new chaotic systems with unknown system parameters.
As the master system, we consider the new chaotic system dynamics x y x z y z y ax z z y bz where 1 1 1 , , x y z are the states and , a bare unknown system parameters.
As the slave system, we consider the controlled new chaotic system dynamics where 2 2 2 , , x y z are the states and , , x y z u u u are the adaptive controls to be designed.
We define the synchronization error as follows: We consider the adaptive control defined by ( )  (15) where , , x y z k k k are positive constants.
By substitution of (15) into (14), the following closed-loop system is obtained: ( ) Differentiating V along the trajectories of (18) and (19), we obtain In view of (21), we consider the following update law for parameter estimates: ( ) Proof. We establish this result by using Lyapunov stability theory [46]. First, we note that the quadratic Lyapunov function V defined by (20) is positive definite in 5 . R After substituting the parameter update law (22) into (21), we obtain the time-derivative of V as which is negative semi-definite in 5 . R Hence, by Barbarat's lemma [46], we conclude that the closed-loop synchronization error dynamics (18) is globally exponentially stable.
This completes the proof. < For numerical simulations, we take the parameters as in the chaotic case (2), i.e.
2.8 a = and 0.6. b = We take the initial values of the parameter estimates as ˆ(0) 6.4 a = and ˆ( 0) 9.5. b = Also, we take the gain constants as 10.
x y z k k k = = = As the initial state of the master system (11), we take

Circuit realization of the new chaotic system
In this section, an electronic circuit which emulates the new chaotic system (1) is described to show its feasibility. The circuit design in Figure 9 has been described following an approach based on operational amplifiers [47][48][49][50][51][52], where the state variables , , x y z of the new system (1) are associated with the voltages across the capacitors 1 , C 2 C and 3 C respectively.
By applying Kirchhoff's circuit laws, the corresponding circuital equations of the designed circuit can be written as

Conclusions
In this paper, we derived a new chaotic system with a line of equilibrium points. The dynamical behaviors of the modified Rucklidge chaotic system are analyzed, both analytically and numerically, including some basic dynamical properties, symmetry, invariance, phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map. The designed circuit has been implemented and examined using the MultiSIM and numerical results using MATLAB. Comparison of the MultiSIM result and MATLAB simulations showed good qualitative agreement between the chaotic system and its circuitry realization.