Eur. Phys. J. Plus (2014) 129: 76
https://doi.org/10.1140/epjp/i2014-14076-4

Regular Article

Three-dimensional chaotic autonomous system with only one stable equilibrium: Analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form

¹ Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes (LaMSEBP), Department of Physics, Faculty of Science, University of Yaoundé I, Po. Box 812, Yaoundé, Cameroon
² Department of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., 15875-4413, Tehran, Iran

^* e-mail: stkingni@gmail.com

Received: 15 February 2014
Accepted: 6 March 2014
Published online: 2 May 2014

Abstract

This paper proposes a three-dimensional chaotic autonomous system with only one stable equilibrium. This system belongs to a newly introduced category of chaotic systems with hidden attractors. The nonlinear dynamics of the proposed chaotic system is described through numerical simulations which include phase portraits, bifurcation diagrams and new cost function for parameter estimation of chaotic flows. The coexistence of a stable equilibrium point with a strange attractor is found in the proposed system for specific parameters values. The physical existence of the chaotic behavior found in the proposed system is verified by using the Orcard-PSpice software. A good qualitative agreement is shown between the simulations and the experimental results. Based on the Routh-Hurwitz conditions and for a specific choice of linear controllers, it is shown that the proposed chaotic system is controlled to its equilibrium point. Chaos synchronization of an identical proposed system is achieved by using the unidirectional linear and nonlinear error feedback coupling. Finally, the fractional-order form of the proposed system is studied by using the stability theory of fractional-order systems and numerical simulations. A necessary condition for the commensurate fractional order of this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than three.