862 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 9, SEPTEMBER 2006 Multiscroll Chaotic Oscillators: The Nonautonomous Approach A. S. Elwakil, Senior Member, IEEE, and S. Özo˘ guz, Member, IEEE Abstract—A novel technique for designing chaotic oscillators capable of producing multiscrolls is presented. The technique is based on multilevel-logic pulse-excitation. A modified forced Van der Pol oscillator, a forced active tank resonator, and a forced cross-coupled oscillator are given as design examples. I. INTRODUCTION T HE generation of multiscroll chaotic attractors has been a topic of both theoretical and practical interest [1]–[8]. The first multiscroll oscillators [1], [2] were derived from the original Chua’s circuit by introducing a nonlinear resistor with multiple breakpoints. The number of breakpoints, and hence the number of possible scrolls, was practically limited by the dc characteristics of the nonlinear resistor. In [9], a canonical model, which generated a double-scroll-like attractor using a comparator nonlinearity, was introduced. This model was later modified by inserting one, two, or three multistep comparator nonlinearities to generate one-dimensional (1-D), two-dimen- sional (2-D), or three-dimensional (3-D) scroll-grid attractors [3]. The key point is that each extra break-point of the folding nonlinearity sets an extra equilibrium point in the state space around which chaotic trajectories may evolve. Different forms of this nonlinearity were used in different chaotic oscillators to obtain 1-D [4]–[6] and higher-D [7], [8] multiscrolls. However, the technique remains basically the same: a static dc approach centered around redesigning the nonlinearity so as to increase the number of possible 1-D, 2-D, or 3-D scrolls. This approach always has an upper practical limit imposed by the value of the used dc supply. Recently, in [10]–[12], novel nonautonomous pulse-excited chaotic oscillators were introduced. It was particularly shown that equilibrium points with fixed positions in space can be gen- erated by using a binary pulse-exciting source. In this study, we show that multiscrolls can hence be generated by suitably intro- ducing composite multilevel-logic pulse-exciting sources into a chaotic oscillator structure. The number of generated equilib- rium points, and hence possible scrolls, depends on the number of available logic levels, which in turn depends not only on the difference between the amplitudes of these sources but also on the difference between their respective oscillation frequencies . It is thus possible to control the number of scrolls either through or through or through both. This Manuscript received June 19, 2005; revised October 23, 2005. This paper was recommended by Associate Editor J. L. Moiola. A. S. Elwakil is with the Department of Electrical Engineering, University of Sharjah, United Arab Emirates (e-mail: elwakil@sharjah.ac.ae). S. Özo˘ guz is with the Faculty of Electrical-Electronics Engineering, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey. Digital Object Identifier 10.1109/TCSII.2006.880032 technique transforms the multiscroll generation problem from a static problem into a dynamic problem and does not require any redesign of the existing nonlinearity in the chaotic oscillator. In what follows, three design examples are given to validate this nonautonomous approach. II. MULTILEVEL-LOGIC EXCITATION:FORCED VAN DER POL EXAMPLE Here, the generation of multiscrolls from a forced Van der Pol equation, by using a multilogic-level exciting source, is ex- plained. The classical Van der Pol equation is given by (1) where is usually a sinusoidal periodic force of the form and is a quadratic nonlinearity of the form . Evidently, the location of the equilibrium points of this system are time varying in space. Now, consider using a periodic binary pulse-exciting force of the form . In this case, the system has two fixed time-in- variant equilibrium points located at . It is also possible to modify (1) to include a binary-switching non- linearity of the form , instead of the quadratic one. Equation (1) in this case may be rewritten in the form (2) where (3) The two equilibrium points are still located at . Note that the engine driving oscillations in this system is a quadrature oscillator with oscillation condition and oscillation frequency [9]. The role of the external excitation is only to force the system to alternate between the two equilibrium points with alternation frequency without affecting the state transition matrix. Now, consider using a three-level-logic source obtained by combining the outputs of two binary sources with different fre- quencies. In particular, the signal has three possible values namely 0, 1, and 1. Using this in the above system with , , and , a 1-D three-scroll attractor is observed, as shown in Fig. 1(a). The intersource frequency and amplitude separa- tion are . A four-scroll can be generated 1057-7130/$20.00 © 2006 IEEE