Nonlinear Dyn (2011) 65:457–466 DOI 10.1007/s11071-010-9904-2 ORIGINAL PAPER Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci Caibin Zeng · Qigui Yang · Junwei Wang Received: 1 August 2010 / Accepted: 27 November 2010 / Published online: 16 December 2010 © Springer Science+Business Media B.V. 2010 Abstract This paper reports a new fractional-order Lorenz-like system with one saddle and two stable node-foci. First, some sufficient conditions for local stability of equilibria are given. Also, this system has a double-scroll chaotic attractor with effective dimen- sion being less than three. The minimum effective dimension for this system is estimated as 2.967. It should be emphasized that the linear differential equa- tion in fractional-order Lorenz-like system seems to be less “sensitive” to the damping, introduced by a frac- tional derivative, than two other nonlinear equations. Furthermore, mixed synchronization of this system is analyzed with the help of nonlinear feedback con- trol method. The first two pairs of state variables be- tween the interactive systems are anti-phase synchro- nous, while the third pair of state variables is complete synchronous. Numerical simulations are performed to verify the theoretical results. C. Zeng ( ) · Q. Yang School of Science, South China University of Technology, Guangzhou 510640, P.R. China e-mail: zeng.cb@mail.scut.edu.cn Q. Yang e-mail: qgyang@scut.edu.cn J. Wang School of Informatics, Guangdong University of Foreign Studies, Guangzhou 510006, P.R. China e-mail: wangjunweilj@yahoo.com.cn Keywords Fractional order system · Chaos · Lorenz-like system · Mixed synchronization · Minimum effective 1 Introduction Fractional calculus is a field of applied mathemat- ics that deals with derivatives and integrals of arbi- trary orders (including complex orders). The first ref- erence is popularly believed to be a question raised in the year 1695 by L’Hôspital to Leibniz, which sought the meaning of the derivative of order 1/2. Although it is a mathematical topic with more than 300 years old history, the applications of fractional calculus to physics and engineering are just a recent focus of interest [1, 2]. The list of applications of fractional calculus has been growing and includes viscoelastic materials and rheology, electrical engineering, elec- trochemistry, biology, biophysics and bioengineering, signal and image processing, mechanics, mechatron- ics, physics, and control theory [3]. According to the Poincaré–Bendixson theorem [4], a chaotic attractor can only arise in a continuous au- tonomous dynamical system (specified by differential equations) if it has three or more dimensions. How- ever, over the last decade or so, it has been found that chaos [5–12] and hyperchaos [13, 14] exist in fractional-order systems. In a seminal letter [6], the ef- fective dimension of such systems was denoted by .