# Circuit Implementation of a Chaotic Lorenz System

Posted on Feb 5, 2014

The Lorenz system is one of a few standard oscillators commonly used to explore chaos [1]. An accessible description of its mathematical features and chaotic dynamics is presented by Thompson and Stewart [2]. This important system was originally developed as a simplified mathematical model of atmospheric instabilities and does not derive from a ba

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sic circuit topology. As such, realizing an electronic Lorenz oscillator requires an explicit analog computer implementation of the oscillator equations. A few designs for realizing a Lorenz attractor have been published in the literature [3 ‘14]. Here, we present an original circuit design that is derived directly from the Lorenz system equations. where X, Y, and Z are the states of the system and s, R, and b are parameters. Typical parameter values that yields chaotic dynamics are s=10, R=30, and b=8/3. For these values, the attractor generated by numerically integrating (1) is shown in Figure1. Typical waveforms for the three dynamical states are shown in Figure2. To obtain a practical circuit design for the Lorenz system, it is necessary to scale the system equations. To this end, we define the new system states For a practical circuit, it is desirable to use ±10 ‘V signals to represent the system states. Based on the magnitude of the oscillations shown in Figure1, choosing a=1/3 yields states that map directly to the desired voltage range. As such, the scaled system is Figure3. Numerical Lorenz attractor for the scaled system with a=1/3. A circuit schematic that closely realizes the scaled Lorenz system is shown in Figure4. The voltages at the nodes labeled x, y, and z correspond to the states of the scaled system (4). The multipliers are each AD633, for which the additional summing node is grounded. The operational...

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