OPTOELECTRONICS AND ADVANCED MATERIALS – RAPID COMMUNICATIONS Vol. 6, No. 7-8, July - August 2012, p. 742 - 745 Chaos, coexisting attractors, and circuit design of the generalized sprott C system with only two stable equilibria ZHOUCHAO WEI * , IHSAN PEHLİVAN a School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, PR China a Electronics and Computer Education Department, Sakarya University, TR-54187, Esentepe Campus, Sakarya, Turkey This paper carries out the coexistence of chaotic attractors and stable equilibria in a generalized Sprott C system with only two stable equilibria. The discovery of this result is striking, because one typically would anticipate non-chaotic and even asymptotically converging behaviors. The simulation results are verified with a circuit implementation. (Received May 27, 2012; accepted July 19, 2012) Keywords: Chaotic attractors, Sil'nikov theorem, Lyapunov exponent, Coexisting attractors, Circuit implementation 1. Introduction Since Lorenz found the first classical chaotic attractor in 1963 [1], chaos, as a very interesting nonlinear phenomenon, has been intensively studied in the last decades [2-6]. It is found to be either useful or has great potential in many fields, such as in engineering, biology and economics. For a generic three-dimensional smooth quadratic autonomous system, Sprott found by exhaustive computer searching about 19 simple chaotic systems with no more than three equilibria[1-3]. It is very important to note that some 3D autonomous chaotic systems have three particular fixed points: one saddle and two unstable saddle-foci(for example, Lorenz system[1], Chen system [4], Lü system [5], the conjugate Lorenz-type systemet. et al [6]). The other 3D chaotic system, such as the original Rölsser system [2], DLS [7] and Burke-Show system [8], have two unstable saddle-foci. Yang and Chen found another 3D chaotic system with three fixed points: one saddle and two stable fixed points [9]. Recently, Yang, Wei and Chen [10] introduced and analyzed a new 3-D chaotic system in a form very similar to the Lorenz, Chen, Lü and Yang-Chen system [9], but it has only two stable node-foci. In 2011, Wang and Chen discovered a simple three-dimensional autonomous quadratic system that has only one stable equilibrium [11], revealing some new mysterious features of chaos. That is to say, the analytic criterion that the system has at least an unstable equilibrium for emergence of chaos is certainly not necessary. Moreover, many theoretical analysis and numerical simulation results about these systems were obtained [12-16]. It should be noted that one commonly used analytic criterion for generating and proving chaos in autonomous systems is based on the fundamental work of Sil'nikov [17, 18] and its subsequent embellishment and slight extension [19]. However, Shi'linikov criteria is sufficient but certainly not necessary for emergence of chaos. Another form of complexity arises when two or more asymptotically stable equilibria and other attracting sets coexist as the system parameters are being varied. The trajectories of the kinds of system selectively converges on either of the attracting sets depending on the initial state of the system. Another form of complexity arises when two or more asymptotically stable equilibria or attracting sets co-exist as the system parameters are being varied. There has been increasing interest in exploiting chaotic dynamics in engineering applications, where some attention has been focused on effectively creating chaos via simple physical systems, such as some electronic circuits [20-26]. In this paper, by using linear feedback, we introduce a generalized Sprott C system with six terms. When all of equilibria of generalized Sprott C system are stable, the system generates a double-scroll chaotic attractor, which can coexist with period attractors and stable equilibria.This is usually referred to as co-existing attractors and when this occurs, the trajectories of the system selectively converges on either of the attracting sets depending on the initial state of the system. When co-existing attractors occur in a system, engineers and scientists are usually interested in obtaining the basins of attraction of the different attracting sets, defined as the set of initial points whose trajectories converge on the given attractor. Trajectories selectively converge on either of the attracting sets depending on the initial condition of the system. It