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Abstract First, this paper announces a seven-term novel 3-D conservative chaotic system with four quadratic nonlinearities. The conservative chaotic systems are characterized by the important property that they are volume conserving. The phase portraits of the novel conservative chaotic system are displayed and the mathematical properties are discussed. An important property of the proposed novel chaotic system is that it has no equilibrium point. Hence, it displays hidden chaotic attractors. The Lyapunov exponents of the novel conservative chaotic system are obtained as L1 = 0.0395,L2 = 0 and L3 = −0.0395. The Kaplan-Yorke dimension of the novel conservative chaotic system is DKY =3. Next, an adaptive controller is designed to globally stabilize the novel conservative chaotic system with unknown parameters. Moreover, an adaptive controller is also designed to achieve global chaos synchronization of the identical conservative chaotic systems with unknown parameters. MATLAB simulations have been depicted to illustrate the phase portraits of the novel conservative chaotic system and also the adaptive control results.
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Abstrakt

This paper announces an eight-term novel 3-D jerk chaotic system with three quadratic nonlinearities. The phase portraits of the novel jerk chaotic system are displayed and the qualitative properties of the jerk system are described. The novel jerk chaotic system has two equilibrium points, which are saddle-foci and unstable. The Lyapunov exponents of the novel jerk chaotic system are obtained as L1 = 0.20572,L2 = 0 and L3 = −1.20824. Since the sum of the Lyapunov exponents of the jerk chaotic system is negative, we conclude that the chaotic system is dissipative. The Kaplan-Yorke dimension of the novel jerk chaotic system is derived as DKY = 2.17026. Next, an adaptive controller is designed via backstepping control method to globally stabilize the novel jerk chaotic system with unknown parameters. Moreover, an adaptive controller is also designed via backstepping control method to achieve global chaos synchronization of the identical jerk chaotic systems with unknown parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict feedback systems. MATLAB simulations have been depicted to illustrate the phase portraits of the novel jerk chaotic system and also the adaptive backstepping control results.
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