Simulation 11(9): 671-677, 2010 ©Freund Publishing House Ltd., International Journal of Nonlinear Sciences & Numerical Coexistence of Multiple Strange Attractors Governed by Different Initial Conditions in a Deterministic System Shi-Jian Cang a,† , Zeng-Qiang Chen b, ‡ , Zeng-Hui Wang c and Yan-Xia Sun d a,† Department of Industry Design, Tianjin University of Science and Technology, Tianjin 300222, PR China(E-mail:sj.cang@gmail.com) b, ‡ Department of Automation, Nankai University, Tianjin 300071, PR China(E-mail: chenzq@nankai.edu.cn) c Department of Electrical and Mining Engineering, University of South Africa, Florida 1710, South Africa d Department of Electrical Engineering, Tshwane University of Technology, Pretoria 0001, South Africa Keywords: periodic attractor，chaotic attractor，coexistence，deterministic system Abstract This paper presents a new four-dimension autonomous system which shows extraordinary dynamical properties . Chaotic attractor and periodic attractor or hyper-chaotic attractor and quasi-periodic attractor, which are governed by different initial conditions instead of the system parameters, can coexist in the deterministic system. These interesting phenomena are verified through numerical simulations and analyses including time series, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents. 1. Introduction In general, many deterministic systems always display one of these forms such as static state, periodic or quasi-periodic motion, chaos or hyper-chaos, and so on. If a system is said to be chaotic, its behavior is so unpredictable as to show random dynamics owing to great sensitivity to the initial conditions. This property implies that two trajectories emerging from two different closely initial conditions separate exponentially with the passage of time. The necessary requirements for a deterministic smooth nonlinear system to be chaotic are that the system must be nonlinear, and be at least three dimensional. The fact that some dynamical systems showing the above necessary conditions possess such a critical dependence on the initial conditions has been known since the end of the last century. However, only in the last thirty years, experimental observations have pointed out that. Lorenz found the atmosphere dynamical model in 1963 [1]. Since then, the Lorenz system and other chaotic systems, such as the Rössler system [2] and the Chua circuit equation [3, 4], have been investigated profoundly and comprehensively. The theory of chaos, when being applied to the fields of control, synchronization and security communication, can help us to realize important value of chaos in practical engineering [5-8]. Moreover, chaotic systems are common in nature. Many natural phenomena can be characterized as being chaotic. They can be found in meteorology, solar system, heart and brain of living organisms and so on. Therefore, chaos has become an important research topic in the fields of nonlinearity in the past thirty years, and some new characteristics of chaos were also found. It is common that a nonlinear system with changeable system parameters shows different