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A new chaotic system with fractional order and its projective synchronization

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Abstract

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.

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  1. Computing Center, Henan University, Kaifeng, 475004, People’s Republic of China

    Xiangjun Wu & Hui Wang

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  1. Xiangjun Wu
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  2. Hui Wang
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Correspondence to Xiangjun Wu.

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Wu, X., Wang, H. A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn 61, 407–417 (2010). https://doi.org/10.1007/s11071-010-9658-x

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  • DOI: https://doi.org/10.1007/s11071-010-9658-x

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