Chua's circuit decomposition: a systematic design approach for chaotic oscillators
Introduction
Until recently, very few chaotic oscillators existed in the literature. This was a consequence of the fact that it was not clear how such strongly nonlinear circuits could be designed to produce chaos. In fact, many circuit designers had doubts about the possibility of designing robust chaotic oscillators. These doubts were supported by observing the reported chaotic oscillators which were either emulating a set of mathematical equations that are known to be chaotic, such as the Van der Pol, Lorenz and Mackey-Glass systems [1], [2], [3] or were intuitively discovered, such as Chua and Saito's oscillators [4], [5], [6]. Although detailed studies of the nonlinear dynamics governing these oscillators have been carried out, much less effort has been devoted to developing circuit-design methodologies [7]. A starting point for a design process did not seem to exist. Hence, more circuit-design-oriented studies have been carried out.
The objective of these studies is to establish methods for generating a chaotic signal with prescribed statistical properties from an electronic circuit that satisfies a set of design constraints. Such constraints may include the type of active devices used, the type and location of energy storage elements, active or passive nonlinearities, current or voltage signal processing, as well as frequency response, supply voltage and power dissipation requirements. Although statistical measures that can characterize a chaotic signal generated by an analog circuit as being suitable for a certain application are still lacking,1 this work aims to propose a solid basis for designing analog chaotic oscillators with prescribed circuit requirements.
We demonstrate here that Chua's circuit can be physically decomposed into a sinusoidal oscillator coupled to a voltage-controlled nonlinear resistor. In particular, the passive LC tank resonator can be replaced by a general second-order sinusoidal oscillator, such as the Wien-bridge oscillator, or even a third-order one, such as the Twin-T oscillator. In this way, inductorless realizations can be obtained without using RC emulations of the inductor, as was done in [8], [9], [10]. We have recently performed a similar decomposition [11] of Saito's double-screw hysteresis oscillator [6], which employs a current-controlled nonlinear resistor. Our work on decomposing Chua's and Saito's classical chaotic oscillators was motivated by the observation of chaos in the Colpitts oscillator [12] and the large number of chaotic oscillators that have been introduced thereafter [13], [14], [15], [16], [17], [18]. The common feature of these oscillators is that they all use a sinusoidal oscillator engine. However, the chaotic oscillators reported in [14], [15], [16], [17], [18] use a simple passive nonlinearity (diode or diode-connected transistor) instead of an active one, by contrast with Chua's and Saito's circuits. This should not be surprising, since there is no need for more than one energy source in the circuit. Therefore, an active nonlinear element is not necessary for chaos generation. The task of being able to design a chaotic oscillator with predefined circuit requirements is greatly simplified if only passive nonlinear devices are used since it is then sufficient to introduce any desired feature of the chaotic oscillator into its core sinusoidal (relaxation) oscillator [19]. We have proposed diode-inductor and FET-capacitor composites to permit classical sinusoidal oscillators to behave chaotically [20].
In this work, new realizations of Chua's circuit based on Wien-bridge and Twin-T sinusoidal oscillators are proposed. We present a conjecture and make clear that chaotic behavior is not associated with individual elements but rather with general circuit-independent functional blocks. PSpice simulations, numerical simulations of the derived models, and experimental results are shown.
Section snippets
Design concept
From a design-oriented point of view, understanding the behavior of a complicated circuit can be achieved by decomposing it into functional blocks. This decomposition process should continue until blocks with clearly defined functions and corresponding design procedures are obtained.
Estimating the frequency of oscillation
When the signal produced by a chaotic oscillator is nearly sinusoidal, the output power is concentrated in a single-frequency component centered near the operating frequency of the core sinusoidal oscillator (ωosc). As the circuit is perturbed towards its chaotic region of operation, the power is spread to more frequency components both higher and lower than the center frequency of oscillation. The linearized form of Chua's circuit (see Fig. 2(a)) is given by s3+a2s2+a1s+a0=0 where a2=1/RC1+(1+
Experimental results
The chaotic oscillator shown in Fig. 2(b) was constructed using the AD844 CFOA and the AD713 VOAs. Capacitors C1 and C2 were set to 39 nF, R is , RB is a pot. for tuning, and the rest of the components are as in the PSpice simulations. At , a simple limit cycle either in the lower or the upper scrolls is born. When RB is increased (the gain K is increased), a period-doubling sequence starts and a single scroll is formed. The upper and lower single scrolls are shown in Fig. 5
Conjecture and concluding remarks
We have demonstrated here that the chaotic behavior of Chua's circuit can be preserved when it is physically decomposed into two separate blocks: a sinusoidal oscillator and a voltage-controlled nonlinear resistor. This decomposition also applies to Saito's chaotic oscillator (which is simply a classical LC−R sinusoidal oscillator coupled to a hysteresis current-controlled nonlinear resistor [11]), the Colpitts oscillator [12], and all the chaotic oscillators reported in [13], [14], [15], [16],
Acknowledgements
The AD844 and AD713 amplifiers used in this work were provided by Analog Devices. This work has been sponsored by the Enterprise Ireland Basic Research Program under grant number SC/98/740.
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