Summary: We numerically investigate the chaotic behaviors of the fractional-order Arneodo’s system [cf. A. Arneodo, P. H. Coullet, E. A. Spiegel
and C. Tresser
[Physica D 14, 327–347 (1985; Zbl 0595.58030
)]. We find that chaos exists in the fractional-order Arneodo’s system with order less than 3. The lowest order we find to have chaos is 2.1 in this fractional-order Arneodo’s system. Our results are validated by the existence of a positive Lyapunov exponent. The linear and nonlinear drive-response synchronization methods are also presented for synchronizing the fractional-order chaotic Arneodo’s systems only using a scalar drive signal. The two approaches, based on stability theory of fractional-order systems, are simple and theoretically rigorous. They do not require the computation of the conditional Lyapunov exponents. Simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization methods.