MOS realization of the modiﬁed Lorenz chaotic system A.G. Radwan * , A.M. Soliman, A. El-Sedeek Faculty of Engineering, Cairo university, Cairo, Egypt Accepted 24 February 2003 Abstract A new chaotic oscillator circuit that realizes three attractors, the modiﬁed Lorenz system, Lorenz attractor ‘‘But- terﬂy attractor’’ and unsymmetrical modiﬁed Lorenz system [IEEE Trans. Circ. Syst. I 48 (2001) 289] is given in the paper. The general block diagram of this circuit is introduced based on g m –C integrators. The overall circuit realization using MOS transistors and using low supply voltage is given. The proposed circuit depends on the use of grounded capacitors which provides the freedom to be oﬀ chip. A new block diagram called voltage controlled current direction is also introduced and its realization using MOS transistors is given. Numerical and PSpice simulations are also provided to conﬁrm its functionality. Ó 2004 Published by Elsevier Ltd. 1. Introduction Recently, during the last few decades, chaos as a very interesting nonlinear phenomenon has been extensively studied within the scientiﬁc, engineering and mathematical communities. Chaos has great potential to be useful in many dis- ciplines which include information processing, encryption, modulation, demodulation and biomedical engineering applications such as research of human brain and heart [2–4]. Realization of chaotic equations has been a powerful area of research due to the previously mentioned applications. The basic element of any chaotic application is the chaotic oscillator. Realization of the chaotic oscillator passed many stages, from the use of diode as in Chua’s circuit [5], large block diagrams such as operational ampliﬁer (op-amp) or current feedback op-amp (CFOA) as in [6,7] till reaching transistor level [8]. Chaotic equations are a very great area of research, many mathematicians tried to simplify the chaotic equations as much as they can in order to analyze the chaotic behavior and to study how these equations govern chaos or try to answer the basic question of what is the necessary and suﬃcient conditions for the diﬀerential equations to become chaotic? [9,10] Lorenz from few decades proved that there is a chaotic behavior in our life from his study of the system of 12 diﬀerential equations that model a miniature atmosphere. He was able to transform them into only three diﬀerential equations which govern the same characteristics and same attractor which is called ‘‘Butterﬂy attractor’’ [11]. The main disadvantage of these equations appeared in its realization due to the presence of two multipliers which made it diﬃcult. Modiﬁed Lorenz system was introduced in [1] which is represented by three diﬀerential equations with no multipliers. This system can capture the essential behavior of Lorenz attractor and can produce the ‘‘Butterﬂy eﬀect’’, modiﬁed Lorenz and unsymmetrical Lorenz systems. The chaotic oscillator circuit proposed in this paper is designed using MOS transistor level based on Mietec 0.5 micron technology and operates on low supply voltage 1.5 V and small die area in order to satisfy the required conditions for portable devices. This circuit can realize three chaotic attractors which are the modiﬁed Lorenz system, Lorenz attractor ‘‘Butterﬂy eﬀect’’ and unsymmetrical Lorenz system with very small modiﬁcations. * Corresponding author. 0960-0779/$ - see front matter Ó 2004 Published by Elsevier Ltd. doi:10.1016/S0960-0779(03)00077-8 Chaos, Solitons and Fractals 21 (2004) 553–561 www.elsevier.com/locate/chaos