International Journal of Bifurcation and Chaos, Vol. 18, No. 3 (2008) 841–844 c  World Scientific Publishing Company A SYSTEM AND CIRCUIT FOR GENERATING “MULTI-BUTTERFLIES” A. S. ELWAKIL Department of Electrical and Computer Engineering, University of Sharjah, P. O. Box 27272, Sharjah, United Arab Emirates elwakil@ieee.org elwakil@sharjah.ac S. ¨ OZOGUZ Istanbul Technical University, Faculty of Electrical-Electronics Engineering, 34469 Maslak, Istanbul, Turkey Received December 20, 2006; Revised February 20, 2007 A system for generating a multi-butterfly chaotic attractor using the multi-level-logic pulse- excitation technique is proposed. Two-butterfly, three-butterfly and four-butterfly attractors are demonstrated. Results from an experimental setup are also shown. Keywords : Lorenz system; butterfly chaos; chaotic oscillators. 1. Introduction The Lorenz system which produces the well-known butterfly chaotic attractor has served as a proto- type for studying chaotic behavior for a long time [Sparrow, 1982]. New systems of equations, which are not topologically equivalent to the original sys- tem but can maintain the butterfly attractor were proposed in [Chen & Ueta, 1999] and [Ueta & Chen, 2000]. The composite nature of the butterfly attrac- tor was first explained in [Elwakil & Kennedy, 2001] and experimentally verified in [ ¨ Ozoguz et al., 2002]. This enabled more complex butterfly architectures to be designed [Elwakil et al., 2002; Elwakil et al., 2003; Qi & Chen, 2006]. On the other hand, developing various tech- niques to generate multi-scroll chaotic attractors has received considerable interest [Lu et al., 2004]. The objective is to generate more scrolls to form 1D, 2D or 3D scroll-grids, instead of the con- ventional double-scroll attractor, which has only two scrolls in a 1D structure. A technique was recently developed in [Elwakil & ¨ Ozoguz, 2006] which is capable of generating multi-“attractors” of any type. For example, not only can a multi-“scroll” chaotic attractor be obtained, but also multi-“Rossler”, multi-“Colpitts” or multi- “Duffing” chaotic attractors can be obtained where the individual “Rossler-type”, “Colpitts-type” and “Duffing-type” attractors are readily famous and well-known. The aim of this work is to show how a multi- “butterfly” attractor can be obtained by using the technique proposed in [Elwakil & ¨ Ozoguz, 2006], which is basically a nonautonomous technique cen- tered around utilizing multi-level-logic pulse excit- ing sources. 2. Proposed System of Equations Consider the following novel system of differential equations ˙ x = x - y · sgn[f r (t)+ f p (t)+ x] (1a) ˙ y = a · abs[f p (t)+ x] - by - c (1b) 841