IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 3, NO. 2, APRIL 2016 157 Chaos and Combination Synchronization of a New Fractional-order System with Two Stable Node-foci Zeeshan Alam, Liguo Yuan, and Qigui Yang Abstract—A new fractional-order Lorenz-like system with two stable node-foci has been thoroughly studied in this paper. Some sufficient conditions for the local stability of equilibria considering both commensurate and incommensurate cases are given. In addition, with the effective dimension less than three, the minimum effective dimension of the system is approximated as 2.8485 and is verified numerically. It should be affirmed that the linear differential equation in fractional-order Lorenz- like system appears to be less sensitive to the damping, rep- resented by a fractional derivative, than the two other non- linear equations. Furthermore, combination synchronization of this system is analyzed with the help of nonlinear feedback control method. Theoretical results are verified by performing numerical simulations. Index Terms—Fractional order system, chaos, Lorenz-like sys- tem, combination synchronization, minimum effective dimension. I. I NTRODUCTION T HE very first notion of fractional calculus, that deals with derivatives and integrals of non-integer orders (including complex orders), is believed to be a question raised in the year 1695 by L’Hˆ ospital in a letter addressed to Leibniz, which was seeking the meaning of the derivative of half order. The appli- cations of fractional calculus to physics and engineering, just recently attracted interest [1-2] . Although, it is developed as a field of applied mathematics with a history of more than three centuries old. There is an increase in the list of applications of fractional calculus and comprises of viscoelastic materials and rheology, electrical engineering, electrochemistry, biology, biophysics and bioengineering, signal and image processing, mechatronics, and control theory [3] . Poincar´ e-Bendixson theorem [4] states that, chaos can only arise in a continuous autonomous dynamical system (repre- sented by a differential equation or set of differential equa- tions) if it has three or more dimensions. However, over the last few years, it has been revealed that chaos [5-9] and hyperchaos [10-11] exist in fractional-order systems. In a sem- inal letter [6] , the effective dimension of such systems was Manuscript received September 6, 2015; accepted December 21. This work was supported by National Natural Science Foundation of China (11271139), Guangdong Natural Science Foundation (2014A030313256, S2013040016144), Science and Technology Projects of Guangdong Province (2013B010101009), and Tianhe Science and Technology Foundation of Guangzhou (201301YG027). Recommended by Associate Editor YangQuan Chen. Citation: Zeeshan Alam, Liguo Yuan, Qigui Yang. Chaos and combination synchronization of a new fractional-order system with two stable node-foci. IEEE/CAA Journal of Automatica Sinica, 2016, 3(2): 157-164 Zeeshan Alam and Qigui Yang are with the School of Mathematics, South China University of Technology, Guangzhou 510640, China (e-mail: zeeshan alam84@hotmail.com; qgyang@scut.edu.cn). Liguo Yuan is with the School of Mathematics and Information, South China Agricultural University, Guangzhou 510640, China (e-mail: liguoy@scau.edu.cn). denoted by Σ. Moreover, the minimum effective dimension Σ cr was defined as a certain critical value, under which the system exhibits periodic dynamics. Thus one gratifying work is to determine the minimum effective dimension Σ cr for a given fractional-order system. Note that a chaotic system is referred to be extremely sensitive to its initial conditions, and the “memory” time of the system can be estimated as t mem = λ -1 max , where λ max is the largest Lyapunov exponent, which is obtained through numerical computations. Generally, a positive maximal Lyapunov exponent is commonly taken as an indication that the system is chaotic. On the other hand, consider a 3-D nonlinear dynamical system ˙ x = f (x),x ∈ R 3 , and x e is an equilibrium point, then x e is called a saddle- focus if the eigenvalues of the Jacobian A = Df evaluated at x e are λ 1 = μ, λ 2,3 = σ ± iω, and satisfy μσ < 0,ω =0, where μ, σ, ω ∈ R. Also, x e is called a node-focus if the eigenvalues of the Jacobian A = Df evaluated at x e are λ 1 = μ, λ 2,3 = σ ± iω, and satisfy μσ > 0,ω =0. Note that many fractional-order systems [6-10] and their references display chaotic dynamics with one saddle and two saddle- foci, while in [12] a Lorenz-like fractional order chaotic system is introduced with one saddle and two stable node-foci. However, this paper proposes a new fractional-order system with only two stable node-foci. It is crystal clear that the system will be topologically non-equivalent to the original fractional-order Lorenz and all Lorenz-like systems. Therefore, what is interesting is to further find out what kind of new dynamics this system has. The paper is organized as follows. In Section II, basic defi- nitions of fractional calculus as well as some useful theorems of stability of fractional-order systems are briefly introduced. Section III is dedicated to the local stability and chaotic dynamics of the new fractional-order Lorenz-like system. In Section IV, the combination synchronization of this system with two other systems is analyzed. Conclusions are drawn in Section V. II. PRELIMINARIES A. Basic Definitions There are several definitions of fractional derivatives [13] . Two most commonly used definitions are Riemann-Liouville and Caputo definitions. Definition 1. Let m − 1 <δ<m, m ∈ N, the Riemann- Liouville fractional derivative of order δ of any function f (t) is defined as follows D δ t f (t)= 1 Γ(m − δ) d m dt m  t 0 (t − τ ) m-δ-1 f (τ )dτ. (1)