In this note, we describe a simple electronic oscillator and controller for demonstrating chaos control. The oscillator is built using a LC tank circuit tuned near 1 kHz, and the controller is an active limiter. Both circuits can be constructed using common, inexpensive parts. As such, this system is well suited for laboratory demonstrations of chaos control, thereby providing students with valuable exposure to the emerging field of chaos engineering. One of the more intriguing aspects of

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... research in nonlinear dynamics has been the ability to control chaos by applying tiny, but cleverly placed perturbations. The key to this ability lies in the exponential sensitivity to fluctuations that is inherent in all chaotic systems. This so-called butterfly effect is often illustrated as a butterfly flapping its wings in one part of the world and causing a hurricane in another. Though anathema to the meteorologist, this sensitivity creates for us a fine scalpel with which to dissect chaos in the laboratory. Standard tools of linear analysis, such as least-squares fitting, autocorrelation, and Fourier transforms, are of limited use for chaotic dynamics. This is due to the very broad spectral content of chaos and its tendency to mix inseparably random microscopic fluctuations together with deterministic dynamics. Instead one characterizes chaos by dimensionality, global rates of divergence, and the topology or shape of the attractor. The "bones" of a chaotic system are its unstable steady states and unstable periodic orbits ͑UPOs͒. The set of these unstable states constitute the skeleton of a chaotic attractor, and from them one can deduce a great deal about the system at hand. 1 A typical chaotic attractor has an infinite number of UPOs, each corresponding to a unique solution of the equations of motion. Amazingly, the number of periodic orbits is dwarfed by the set of unstable aperiodic solutions. All of the periodic and aperiodic trajectories can be stabilized and viewed real-time using chaos control. The topic of control strikes right into the heart of the nature of chaos. A physical chaotic system exhibits random behavior over long time scales, this in spite of the fact that its macroscopic dynamics are deterministic. Determinism means that the dynamics are predictable in principle and thus controllable. Randomness implies quite the opposite. This apparent contradiction is due to the fact that chaotic systems are continuously unstable. Ever-present microscopic fluctuations are magnified by this instability, resulting in dynamics that are truly unpredictable over time scales exceeding the time scale of the instability. But for time scales much less than that of the instability, the dynamics proceed in a very predictable way. Chaos control uses this window of determinism to override the fluctuations while they are still small and to replace them with control perturbations of our choosing, thereby selecting a particular trajectory for the system dynamics to follow. For the most part, chaos control can be thought of as a special case of the well-known control theory taught in engineering curricula; however, there is a key difference in approach. Conventional control theory targets an operating point based on the application at hand, whereas chaos control selectively targets a natural state of the system. Chaotic systems have a large number of such states, making it possible to design applications around them. The advantage of doing so appears in the efficiency of the controller and the extremely low control power needed to achieve stable operation. Although there were earlier results describing control of chaotic dynamical systems using arbitrary perturbations or system modifications, the original description of efficient chaos control as described above is due to Ott, Grebogi, and Yorke ͑OGY͒. 2 Their original treatment is highly mathematical, involving relatively complex vector operations and essentially requiring a computer in the feedback loop to calculate the required control perturbations in experimental implementations. 3 A simplification of OGY, called occasional proportional feedback control, was demonstrated for electronic systems using calculations performed in analog circuitry. 4 In addition, delay feedback control techniques have been developed for controlling periodic orbits. 5 Although simpler than OGY, these techniques are still relatively complicated and require a significant cost in time and materials to implement. Recently, our group has pursued further simplification of chaos control to target potential applications in remote sensing and communications using microwave and optical chaos. 6 For this need, we have developed several chaos control techniques that are simple to implement, including limiter control, which uses a simple limiting device to stabilize various unstable states of a chaotic system. 7 In this note, we describe an electronic circuit for a simple chaotic oscillator that is amenable to limiter control. We also show an active limiter that can stabilize low-order periodic waveforms in the oscillator. Both circuits can be constructed using commonly available, inexpensive parts. This system is well suited for laboratory demonstrations of chaos control, thereby providing students with insight into the nature of chaotic motion. We also discuss how limiter control can be extended to handle higher-order periods and arbitrary trajectories. Finally, we illustrate how these ideas form a basis for chaos engineering, which is a field that explores the use of chaos for practical applications. The schematic for our chaotic electronic oscillator is shown in Fig. 1 . This low-frequency circuit can be constructed using any customary prototyping technique, including a solderless breadboard. This circuit uses an unstable LC harmonic oscillator coupled to a nonlinear folding circuit to generate chaos. The circuit is modeled as where i L is the current through the tank inductor, v C is the voltage across the tank capacitor, v F is the voltage in the folding circuit, V D Ӎ0.7 V is the voltage drop for a conducting diode, Cϭ0.1 F is the tank capacitance, and R is a variable resistance for tuning the chaotic behavior of the circuit. The tank inductor is implemented using a general impedance converter, with equivalent inductance of Lϭ0.2 H. In operation, the inductor current i L is derived using and v C is sampled directly using a high-impedance probe. The system of Eq. ͑1͒ also includes i D , which models the control current due to the limiter that is introduced later. For the uncontrolled circuit shown in Fig. 1 , i D ϭ0. Fig. 1 . Chaotic electronic LC oscillator. The inductor is realized using a generalized impedance converter, the negative resistor provides gain for the LC harmonic oscillator, and the folding circuit introduces a nonlinear element that folds the instability and results in chaos. All op amps are type TL082 or equivalent, all resistors are 5% tolerance, and the circuit is supplied with Ϯ15 V. All resistances are in ohms. Fig. 2. Simply folded band attractor exhibited by the uncontrolled chaotic LC oscillator. Trajectories in this phase-space projection proceed counterclockwise. 273 273 Am. For large R, the circuit generates a periodic, nearly sinusoidal waveform around 1 kHz. As R is decreased, the oscillator gain increases and the uncontrolled circuit exhibits a period-doubling route to chaos. Throughout this bifurcation sequence, the oscillator exhibits a sinusoidal-like waveform, although with cycle-to-cycle amplitude fluctuations. At R Ӎ800⍀, the circuit is fully chaotic, exhibiting the attractor shown in Fig. 2 . This attractor is a simply folded band and is topologically equivalent to that generated by Rossler's oscillator. 8 An ideal diode limiter may be used to control a chaotic LC oscillator as shown in Fig. 3 . The limiter acts on the peak voltages of the chaotic waveform generated by the oscillator. When the tank voltage v C gets too large, the diode turns on and conducts excess charge out of the oscillator LC tank, thereby reducing the oscillation amplitude. Since the limiter conducts only at the waveform peak, the effective control perturbation due to the limiter is a short current pulse. The adjustable bias voltage V LIM controls the threshold at which the diode turns on, which can be set precisely at the maximum of a targeted state, such as an unstable periodic orbit. Since the limiter turns on only when the tank voltage exceeds the threshold, the target state is not modified by the presence of the limiter; however, deviations from this state that exceed the threshold will trigger the limiter and be suppressed. For the present circuit, we use the active limiter shown in Fig. 4 . The limiter is modeled as a piecewise-linear switch, where R D is the series resistance of the limiter. This limiter functions much as an ideal diode limiter for low-frequency Fig. 3. Limiter configuration using an ideal diode with series resistance R D for controlling chaos in a LC oscillator. The box N represents the negative resistor and folding circuit elements in the chaotic oscillator. V LIM sets the limiter threshold level. Fig. 4. Active circuit for realizing an ideal diode limiter with series resistance R D ϭ470 ⍀. The output voltage v D is proportional to the limiter current i D and is used to monitor the magnitude of the control perturbation. All resistances are in ohms. Fig. 5. Controlled ͑a͒ period-1 and ͑b͒ period-2 orbits stabilized using limiter control of chaos. The controlled orbits overlay the uncontrolled attractor, shown in gray. 274 274 Am. J. Phys., Vol. 72, No. 2, February 2004 Apparatus and Demonstration Notes applications. The effective diode limiter current is given by where v D is the output of the limiter circuit. For the examples presented here, we use R D ϭ470 ⍀. As the limiter level V LIM is varied, the circuit exhibits a variety of states, including periodic and chaotic states. In general, the effect of the limiter is significant and modifies the natural behavior of the chaotic oscillator. However, at special levels, the effects of the limiter nearly vanish and the observed periodic state is consistent with the uncontrolled oscillator. At these levels, we recognize that efficient chaos control has been achieved. In Fig. 5 we show two such periodic orbits stabilized using limiter control. These orbits are shown superimposed on the uncontrolled attractor, which is shown in gray. For both orbits, the controlled state is consistent with the uncontrolled dynamics. Further evidence is shown in Fig. 6 , where the average value of the peak current for each control pulse through the limiter is shown as a function of the limiter level. In this plot, distinct local minima are evident at the levels of optimal control for period-1 and period-2 control. At these minima, the effects of the limiter nearly vanish, thereby indicating that the system evolves with dynamics of the oscillator without the limiter. In Fig. 7 we show in detail the oscillator waveform and corresponding control signal at and near the optimal control for period-1 stabilization. In Fig. 7͑b͒ we show optimal period-1 control, noting the nearly complete absence of control perturbations. In Figs. 7͑a͒ and 7͑c͒ the limiter level is below and above the optimal level, respectively. In both cases, we now see significant control perturbations that distort the oscillator waveforms. As a result, the oscillator exhibits waveforms contrary to the natural dynamics of the uncontrolled system. The results shown here illustrate chaos control of the two lowest-order UPOs available in this oscillator. Since the circuit is chaotic, we know that an infinite number of UPOs are available for control; however, significantly longer, higherorder UPOs cannot be controlled using a single static limiter. For a period-n orbit, the time between intersections with the limiter goes as n, where is the mean orbital period. When n becomes longer than the time scale set by the instabilities in the system, single-point control is no longer effective. For this reason, a single static limiter level is only effective for controlling low-period UPOs. To enhance the capabilities of limiter control, we extended the control concept to dynamic limiting. 9 In this process, the limiter is set to a different level for each cycle of the chaotic waveform; consequently, the chaotic system can be controlled to virtually any natural state, including aperiodic states, using only small limiting perturbations. As part of our research we are considering the use of chaotic oscillators as waveform generators in communication and remote sensing systems. The primary advantage lies in Fig. 6. Average value of the peak control perturbation i D as a function of the limiter level V LIM . The pronounced minimum at V LIM ϭ3.47 V corresponds to the period-1 unstable periodic orbit ͑UPO͒. The local minimum at V LIM ϭ4.31 V corresponds to the period-2 UPO. The vanishing control perturbation for V LIM Ͼ4.5 V results as the limiter leaves the attractor and the circuit reverts to uncontrolled chaos. Fig. 7. Waveform, limiter level, and control current for controlled oscillator at ͑a͒ V LIM ϭ3.34 V, ͑b͒ V LIM ϭ3.47 V, and ͑c͒ V LIM ϭ3.51 V. The limiter levels in ͑a͒ and ͑c͒ are below and above, respectively, the period-1 UPO optimal control level shown in ͑b͒, which exhibits the minimal control perturbation i D . 275 275 Am. J. Phys., Vol. 72, No. 2, February 2004 Apparatus and Demonstration Notes utilizing the inherent ability of simple chaotic oscillators to generate very wide bandwidth waveforms directly and efficiently. With chaos control, we can produce a vast array of waveforms that are ideal for spread-spectrum communications and for highly accurate ranging. We believe a system designed around a chaotic source can perform comparably to conventional systems while requiring far fewer components. The idea of exploiting the complexity of chaos in order to simplify hardware is the crux of chaos engineering. For these applications, the oscillators and controllers must operate at frequencies at and above several hundred megahertz. To control chaos at these speeds, we certainly cannot employ digital processors or complex analog circuitry to calculate the requisite control perturbations: the extremely short latency demanded of the feedback loop necessitates a simple controller. Besides such practical concerns, the true efficiency of chaos control is suspect if the controller is considerably more complex than the system under control. Hence, we believe the development of simple control techniques-such as limiter control and dynamic limiting-is essential for the development of practicable chaos engineering. In summary, we have described a circuit for generating chaos along with an active limiter that can be used to stabilize periodic states using efficient chaos control. We have used this limiter to stabilize period-1 and period-2 orbits in the circuit, and have shown that the magnitudes of the control perturbations are minimal when controlling these natural states of the system. We then indicated how this simple control scheme can be extended to control higher order UPOs and arbitrary chaotic waveforms using a dynamic limiter. Due to its simple construction and operation, this system is well suited for laboratory demonstrations of chaos control that can provide students with exposure to the emerging field of chaos engineering. a͒