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The double scroll

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3 Author(s)

A detailed analysis is given of the geometric structure of a chaotic attractor observed from an extremely simple autonomous electrical circuit. It is third order, reciprocal, and has only one nonlinear element: a 3-segment piecewise-linear resistor. Extensive laboratory measurements from this circuit and a detailed geometrical analysis and computer simulation reveal the following rather intricate "anatomy" of the associated strange attractor. In addition to a microscopically infinite sheet-like composition the attractor has a macroscopic "double-scroll" structure, i.e., two sheetlike objects are curled up together into spiral forms with infinitely many rotations. (See frontispiece.) The chaotic nature of this circuit is further confirmed by calculating its associated Lyapunov exponents and Lyapunov dimension. The double-scroll attractor has one positive, one zero and one negative Lyapunov exponent. The Lyapunov dimension turns out to be a fractal between 2 and 3 which agrees with the observed structures. The power spectra of the three associated state variables obtained by both measurement and computer simulation show a continuous broad spectrum typical of chaotic systems.

Published in:

IEEE Transactions on Circuits and Systems  (Volume:32 ,  Issue: 8 )