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Advances and Applications in Chaotic Systems pp 189–210Cite as

Analysis, Adaptive Control and Synchronization of a Novel 3-D Highly Chaotic System

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Part of the book series: Studies in Computational Intelligence ((SCI,volume 636))

Abstract

In this work, we describe an eight-term novel highly chaotic system with three quadratic nonlinearities. The phase portraits of the novel highly chaotic system are illustrated and the dynamic properties of the highly chaotic system are discussed. The novel highly chaotic system has three unstable equilibrium points. We show that the equilibrium point at the origin is a saddle point, while the other two equilibrium points are a saddle-focus and a critical point. The novel highly chaotic system has rotation symmetry about the \(x_3\) axis. The Lyapunov exponents of the novel highly chaotic system are obtained as \(L_1 = 7.6557\), \(L_2 = 0\) and \(L_3 = -24.6796\), while the Kaplan–Yorke dimension of the novel chaotic system is obtained as \(D_{KY} = 2.3102\). Since the maximal Lyapunov exponent of the novel chaotic system has a high value, viz. \(L_1 = 7.6557\), the novel chaotic system is highly chaotic. Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Next, we derive new results for the global chaos control of the novel highly chaotic system with unknown parameters using adaptive control method. We also derive new results for the global chaos synchronization of the identical novel highly chaotic systems with unknown parameters using adaptive control method. The main control results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate the phase portraits of the novel highly chaotic system and also the adaptive control results derived in this work.

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  1. Research and Development Centre, Vel Tech University, Avadi, Chennai, 600062, Tamil Nadu, India

    Sundarapandian Vaidyanathan

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  1. Research and Development Centre, Vel Tech University, Chennai, India

    Sundarapandian Vaidyanathan

  2. Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

    Christos Volos

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Vaidyanathan, S. (2016). Analysis, Adaptive Control and Synchronization of a Novel 3-D Highly Chaotic System. In: Vaidyanathan, S., Volos, C. (eds) Advances and Applications in Chaotic Systems . Studies in Computational Intelligence, vol 636. Springer, Cham. https://doi.org/10.1007/978-3-319-30279-9_8

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