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A Novel 5-D Hyperchaotic System with a Line of Equilibrium Points and Its Adaptive Control

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Advances and Applications in Chaotic Systems

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

Abstract

Chaos theory describes the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems. Chaos theory has applications in several fields of science and engineering. A hyperchaotic system is generally defined as a chaotic system with at least two positive Lyapunov exponents. In this research work, we announce a novel 5-D hyperchaotic system with an infinite line of equilibrium points. The novel 5-D hyperchaotic system has fifteen terms on the right hand side with two quadratic nonlinearities. The phase portraits of the 5-D novel hyperchaotic system are depicted and the qualitative properties of the novel hyperchaotic system are discussed. All the equilibrium points of the novel 5-D hyperchaotic system are unstable. The Lyapunov exponents of the 5-D novel hyperchaotic system are obtained as \(L_1 = 1.2995\), \(L_2 = 0.2505\), \(L_3 = 0.0615\), \(L_4 = 0\) and \(L_5 = -17.5932\). The maximal Lyapunov exponent of the novel hyperchaotic system is \(L_1 = 1.2995\). Also, the Kaplan–Yorke dimension of the 5-D novel hyperchaotic system is obtained as \(D_{KY} = 4.0916\). Since the sum of the Lyapunov exponents is negative, the 5-D novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to globally stabilize the novel hyperchaotic system with unknown parameters. Finally, an adaptive controller is also designed to achieve global chaos synchronization of the identical hyperchaotic systems with unknown parameters. MATLAB simulations are depicted to illustrate all the main results derived in this work for the 5-D novel hyperchaotic system.

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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). A Novel 5-D Hyperchaotic System with a Line of Equilibrium Points and Its Adaptive Control. 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_20

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