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Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation

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Vaidyanathan S. , Volos C. , Pham V.-T. , Madhavan K.

Archives of Control Sciences

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2015

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tom Vol. 25, no. 1

135--158

EN

A hyperjerk system is a dynamical system, which is modelled by an nth order ordinary differential equation with n 4 describing the time evolution of a single scalar variable. Equivalently, using a chain of integrators, a hyperjerk system can be modelled as a system of n first order ordinary differential equations with n 4. In this research work, a 4-D novel hyperchaotic hyperjerk system has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel hyperjerk system are obtained as L1 = 0:1448;L2 = 0:0328;L3 = 0 and L4 = −1:1294. The Kaplan-Yorke dimension of the novel hyperjerk system is obtained as DKY = 3:1573. Next, an adaptive backstepping controller is designed to stabilize the novel hyperjerk chaotic system with three unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve global hyperchaos synchronization of the identical novel hyperjerk systems with three unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using SPICE is presented in detail to confirm the feasibility of the theoretical hyperjerk model.

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Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation

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Vaidyanathan S. , Volos C. , Pham V.-T.

Archives of Control Sciences

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2014

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tom Vol. 24, no. 4

409--446

EN

In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system (Jia, 2007) with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system (Jia, 2007) has the Lyapunov exponents L1 = 0.3684,L2 = 0.2174,L3 = 0 and L4 =−12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as DKY =3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L1 = 0.4195,L2 = 0.2430,L3 = 0.0145,L4 = 0 and L5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as DKY =4.0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE), which is greater than the maximal Lyapunov exponent (MLE) of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.

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A general unified approach to chaos synchronization in continuous-time systems (with or without equilibrium points) as well as in discrete-time systems

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Grassi G. , Ouannas A. , Pham V.-T.

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tom Vol. 28, No. 1

135--154

EN

By analyzing the issue of chaos synchronization, it can be noticed the lack of a general approach, which would enable any type of synchronization to be achieved. Similarly, there is the lack of a unified method for synchronizing both continuous-time and discrete-time systems via a scalar signal. This paper aims to bridge all these gaps by presenting a novel general unified framework to synchronize chaotic (hyperchaotic) systems via a scalar signal. By exploiting nonlinear observer design, the approach enables any type of synchronization defined to date to be achieved for both continuous-time and discrete-time systems. Referring to discrete-time systems, the method assures any type of dead beat synchronization (i.e., exact synchronization in finite time), thus providing additional value to the conceived framework. Finally, the topic of synchronizing special type of systems, such as those characterized by the absence of equilibrium points, is also discussed.

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A 4-D chaotic hyperjerk system with a hidden attractor, adaptive backstepping control and circuit design

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Vaidyanathan S. , Jafari S. , Pham V.-T. , Azar A. T. , Alsaadi F. E.

Archives of Control Sciences

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2018

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tom Vol. 28, No. 2

239--254

EN

A novel 4-D chaotic hyperjerk system with four quadratic nonlinearities is presented in this work. It is interesting that the hyperjerk system has no equilibrium. A chaotic attractor is said to be a hidden attractor when its basin of attraction has no intersection with small neighborhoods of equilibrium points of the system. Thus, our new non-equilibrium hyperjerk system possesses a hidden attractor. Chaos in the system has been observed in phase portraits and verified by positive Lyapunov exponents. Adaptive backstepping controller is designed for the global chaos control of the non-equilibrium hyperjerk system with a hidden attractor. An electronic circuit for realizing the non-equilibrium hyperjerk system is also introduced, which validates the theoretical chaotic model of the hyperjerk system with a hidden chaotic attractor.

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Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities

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Vaidyanathan S. , Volos C. , Pham V.-T. , Madhavan K. , Idowu B. A.

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tom Vol. 24, no. 3

375--403

EN

In this research work, a six-term 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel jerk system are obtained as L1 = 0.07765,L2 = 0, and L3 = −0.87912. The Kaplan-Yorke dimension of the novel jerk system is obtained as DKY = 2.08833. Next, an adaptive backstepping controller is designed to stabilize the novel jerk chaotic system with two unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve complete chaos synchronization of the identical novel jerk chaotic systems with two unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model.