In this paper, the complex simplified Lorenz system is proposed. It is the complex extension of the simplified Lorenz system. Dynamics of the proposed system are investigated by theoretical analysis as well as numerical simulation, including bifurcation diagram, Lyapunov exponent spectrum, phase portraits, Poincaré section, and basins of attraction. The results show that the complex simplified Lorenz system has non-trivial circular equilibria and displays abundant and complicated dynamical behaviors. Particularly, the coexistence of infinitely many attractors, i.e., extreme multistability, is discovered in the proposed system. Furthermore, the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is achieved. Based on Lyapunov stability theory, the corresponding adaptive controllers and parameter update law are designed. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed synchronization scheme. It provides a theoretical and experimental basis for the applications of the complex simplified Lorenz system. KeywordsComplex chaos • Simplified Lorenz system • Coexisting attractors • Extreme multistability • Adaptive control • Generalized function projective synchronization 1 Introduction Since Lorenz discovered the first chaotic attractor in 1963 [1], chaos has been extensively investigated over

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In this paper, a novel fractional-order hyperchaotic complex system is proposed by introducing the Caputo fractional-order derivative operator and a constant term to the complex simplified Lorenz system. The proposed system has different numbers of equilibria for different ranges of parameters. The dynamics of the proposed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The research results show that it displays abundant dynamical characteristics. Particularly, the phenomenon of extreme multistability as well as hidden attractors are discovered. In addition, the complex generalized projective synchronization is implemented between two fractional-order hyperchaotic complex systems with different fractional orders. Based on the fractional Lyapunov stability theorem, the synchronization controllers are designed, and the theoretical results are verified and demonstrated by the numerical simulations. It lays the foundation for the practical applications of the proposed system.

In this paper, the complex simplified Lorenz system is proposed. It is the complex extension of the simplified Lorenz system. Dynamics of the proposed system are investigated by theoretical analysis as well as numerical simulation, including bifurcation diagram, Lyapunov exponent spectrum, phase portraits, Poincaré section, and basins of attraction. The results show that the complex simplified Lorenz system has non-trivial circular equilibria and displays abundant and complicated dynamical behaviors. Particularly, the coexistence of infinitely many attractors, i.e., extreme multistability, is discovered in the proposed system. Furthermore, the adaptive complex generalized function projective synchronization between two complex simplified Lorenz systems with unknown parameter is achieved. Based on Lyapunov stability theory, the corresponding adaptive controllers and parameter update law are designed. The numerical simulation results demonstrate the effectiveness and feasibility of the proposed synchronization scheme. It provides a theoretical and experimental basis for the applications of the complex simplified Lorenz system.