Experimental control of chaos in a non-linear electrical circuit

Abstract

It is a common misconception to regard chaos as an absence of order and is therefore unpredictable. In fact, it was observed that a chaotic attractor typically has imbedded in it an infinite number of periodic orbits. The sensitive dependence on initial conditions inherent in chaotic systems prevent the accurate measurement of future events. But if there is a way to stabilize such systems and clamp it down to a stable orbit, then accurate prediction of future events may be possible.
Recently, considerable efforts have been made toward the development of techniques that will effectively control a chaotic system. There are two practical considerations involved in the control design. The first consideration is to allow only minimal changes to be done to the system so as not to alter its original dynamics. Second, the method should be versatile enough to be applicable to a wide range of systems.
In 1990, Ott, Grebogi and Yorke (OGY) [Phys. Rev. Lett. 64, 1196 (1995)] proposed a general solution to achieve control using the feedback technique that utilizes small, time-dependent perturbations derived from one of the original system parameters. The OGY method was successfully implemented to a gravitationally buckled magnetoelastic ribbon by Ditto, Rauseo and Spano [Phys. Rev. Lett. 65, 3211 (1990)], converting its chaotic motion into period-1 and period-2 orbits. In 1991, a spin-off from this technique was developed by E. R. Hunt [Phys. Rev. Lett. 67, 1953 (1991)] using analog circuit operations to stabilize the waveforms generated by a diode resonator. The new method, an integration of the OGY method and linear feedback procedures, is called the Occasional Proportional Feedback (OPF) method.
In the OPF method, the system variable is sampled within a window of selected offset and width. If a peak of the signal falls within the adjustable range of the window comparator, a trigger is generated for the timing circuit. A stable oscillator is used to generate the synchronizing frequency for which the chaotic output is sampled. The window comparator is activated every time a waveform transits through the window. When the synchronizing input is coincident with this event, the sample-and-hold acquires the waveform voltage. The sampled signal is allowed to pass through the switch at time periods shorter compared to the period of the synchronizing input, the deviation of the sampled peaks from the reference level is then switched through an inverting amplifier to become the control feedback signal which amplitude modulates the source.
In this paper, the applicability of the OPF technique for controlling chaos was investigated as applied to the Piecewise-Linear Lorenz Circuit (PLLC), in an effort to gain further insights on the extent of applicability and limitations of the method.