# Simultaneous amplitude and frequency noise analysis in Chua's circuit

Jean-Michel Friedt, Daniel Gillet, Michel Planat

► To cite this version:

Jean-Michel Friedt, Daniel Gillet, Michel Planat. Simultaneous amplitude and frequency noise analysis in Chua's circuit. International journal of bifurcation and chaos in applied sciences and engineering, 2003, 13, pp.2301-2308. 10.1142/S0218127403007953 . hal-00078147

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# SIMULTANEOUS AMPLITUDE AND FREQUENCY NOISE ANALYSIS IN CHUA'S CIRCUIT

J.-M. FRIEDT

IMEC, MCP/BIO, Kapeldreef 75, 3001 Leuven, Belgium

D. GILLET and M. PLANAT

Laboratoire de Physique et Métrologie des Oscillateurs (LPMO, CNRS) 32,
avenue de l'observatoire, 25044 Besançon Cedex, France

A large number of simultaneous frequency and amplitude data from an electronic chaotic circuit (Chua's circuit) have been obtained. These acquisitions are validated by plotting the bifurcation diagrams of the experimental data versus the bifurcation parameter. We introduce a topological parallel between the Colpitts oscillator and Chua's circuit, and look for similar behavior of the frequency fluctuations using the Allan deviation.

Keywords: Chua's circuit; frequency fluctuations; Colpitts oscillator; amplitude; Allan deviation.

## 1. Introduction

Chua's circuit is a nonlinear electronic circuit which exhibits chaotic behavior for high enough values of one of its parameters, the inductor's value in our case. We have studied the characteristics of the frequency and amplitude fluctuations of the oscillator in the periodic and chaotic regimes (while the signal is still pseudo-periodic, i.e. before the bifurcation towards the double-scroll regime). The amplitude and period of successive oscillations were measured and analyzed by plotting the return maps, the transfer functions and the bifurcation diagrams. The type of fluctuations were characterized using the Allan deviation, and compared to the classical results obtained for a quartz resonator based Colpitts oscillator.

## 2. Motivation of This Study

Our aim is to be able to characterize frequency instabilities in oscillators [Eckert *et al.*, 1996]. We have thus chosen to analyze Chua's chaotic circuit as an example of a highly unstable oscillator. The

conclusions can be tentatively extended to more classical oscillators such as the Colpitts oscillator used in conjunction with a quartz resonator.

Quartz oscillators show a well-known phase (and hence frequency) fluctuation behavior observed to follow a series of power law functions. The power spectral density of relative frequency fluctuations is thus $S_y(F) = h_\alpha F^\alpha$, where F is the Fourier frequency, $y = \Delta\nu/\nu_0$ the relative frequency shift with respect to the carrier of mean frequency $\nu_0$ and the integer $\alpha$ varies from -2 to +2. The power spectral density of phase fluctuations is $S_\varphi(F) = (\nu_0/F^2)S_y(F)$. Hence a slope equal to -3 in $S_\varphi(F)$ corresponds to $1/F$ noise in $S_y(F)$. This type of noise is still not well understood. This is one reason for attempting to detect it in a Chua's oscillator. Instead of power spectra we will use the characterization of time dynamics in terms of Allan variance which is defined as $\sigma_y^2(\tau) = (1/2)\langle(y_{k+1}(\tau) - y_k(\tau))^2\rangle$, with $y_k(\tau)$ the $k$th average of samples $y$ over the integration time $\tau$. The coefficient $y$ is obtained by computing
<figure><img src="image_2.png" /><figcaption>Fig. 1. Comparison of the Colpitts oscillator often used for using a quartz crystal resonator in an oscillator configuration, and the canonical Chua circuit.</figcaption></figure>

the average $y_{\tau} = \langle x_{n=[t;t+\tau]} \rangle$, $x_t$ being our time series indexed by $t$ and $\langle \cdot \rangle$ an average over the data sets. For a stationary fluctuation the power spectrum and Allan variance are related. The graph of the variance against the sampling time also obeys power laws $\sigma_{\tau}^2(y) = \tau^{-m}$, with $m = \alpha + 1$ (if $-2 \le \alpha \le 1$) and $m = 2$ (if $\alpha \ge 1$). Therefore the $1/F$ noise in $S_y(F)$ corresponds to a flicker floor $m = 0$. Allan variance is often used for describing the stability of oscillators since it converges for the kinds of noises observed experimentally, as opposed to the classical variance which does not always converge to a finite value [Allan et al., 1997; Audoin et al., 2001].

A parallel between the schematics of the Colpitts oscillator and the canonical Chua’s circuit [Sarafian & Kaplan, 1995; Kennedy, 1995] can be displayed and analyzed in order to justify our extension of the results to quartz oscillators (Fig. 1). The circuit we actually studied is the usual Chua’s circuit in which $L$ and $R_2$ are exchanged [Chua & Lin, 1990] as shown in Fig. 3. The nonlinear differential equations used to predict the behavior of the circuit lead to chaotic oscillations after a series of bifurcations. Here we have studied simultaneously the evolution of the instantaneous amplitude and instantaneous frequency with the value $L$ of the inductor parameter.

### 3. Experimental Setup

A preliminary study shows a large variation in the shape of the power spectra (Fig. 2) observed on Chua’s circuit for various kinds of attractors (from periodic to chaotic behaviors of the circuit with varying values of the bifurcation parameter). We will here attempt to precisely characterize the various fluctuation regimes depending on the inductor value by means of signal processing tools including the Allan deviation and a display of return maps.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Evolution of the power spectra around the oscillation frequency of Chua’s circuit. A precise analysis is needed to describe the slope of the spectra depending on the oscillation regime.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3. Schematic of Chua's circuit we used for our data acquisitions. The parameter is the inductor's value L. Other components value are C₂ = 1847 nF, C₁ = 20.3 nF, inductor's internal resistance 3.3 Ω. R₂ is a 10 kΩ variable resistor. The variables measured are the frequency and amplitude of the voltage at C₂.</figcaption></figure>

For the realization of Chua’s circuit, we used a simulated inductor described by Weldon [1990]: it is less bulky and its value is defined by a variable resistor and thus can be computer controlled thanks to the availability of computer controlled variable resistors. The nonlinear negative resistor is a negative impedance converter (NIC) [Horowitz & Hill, 1989] based on an operational-amplifier [Weldon, 1990]. The details of the electronic circuits we built are given in Fig. 4.

An external circuit displayed in Fig. 5 is used for detecting single periods and measuring their duration thanks to a comparator and a frequency counter with an internal clock at 14.31818 MHz. The amplitude is measured simultaneously by using a peak holder circuit and a 12 bit analog to digital converter. The trigger signal for the frequency

measurement and the amplitude signal can be seen on Fig. 6. All consecutive periods can be measured for signals of frequency less than 1 kHz.

4. Statistical Analysis and Bifurcation Diagrams

All our statistical analysis have been limited to regions of the *L* parameter for which the signal is observed to be periodic or pseudo-periodic with a constant mean value (as opposed to smaller oscillations overlapping large scale fluctuations such as the signals observed in the double scroll oscillator). This limitation is required by our data acquisition electronic circuit as well as by our intuitive definition of frequency and amplitude of pseudo-periodic signals.

We plotted the values taken by the amplitude and frequency time series against the value of the *L* parameter in Figs. 7 and 8. One can observe the first bifurcation, followed by the chaotic behavior [Feigenbaum, 1978, 1979].

We also display the Allan deviation $\sigma_y(\tau)$: a -0.5 slope in the Allan deviation indicates a white frequency noise, while a -1 slope indicates $\alpha = +1$ (flicker phase fluctuations) or $\alpha = +2$ (white phase noise) frequency fluctuation spectra (Fig. 8).

We then display the evolution of various parameters which are characteristic of the dynamic of Chua's circuit for several chosen values of the *L* parameter: the frequency return maps (Fig. 9), the transfer functions (Fig. 10) and the amplitude return maps (Fig. 11). The return map is defined by displaying $f_{n+1}$ versus $f_n$ when given a time series ($f_{n,n \in [1..N]}$). The amplitude and frequency measured are similar to their usual definition when the

<figure><img src="image_5.png" /><figcaption>Fig. 4. Details of the electronic circuits used in these experiments. The nonlinear element is described in the left schematic, the simulated inductance is depicted in the middle schematic and includes a computer controlled variable resistance, *P*, whose detailed schematic is presented on the right. The DS1669 computer controlled linear potentiometers are powered between -1.44 V and 5.28 V. The value of the simulated inductance is $L = 0.1 \times 10^{-6} \times 3.3 \times P$ H.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5. Experimental setup used for measuring the frequency and amplitude of successive oscillations of Chua's circuit. A buffer was included so as to minimize the disturbances on the chaotic circuit due to the measurement apparatus. The diode used for the amplitude detection is a germanium diode in order to minimize the voltage drop. The frequency counter and the analog to digital converter (12 bits) are custom built PC-104 cards: this allows fast transfer time and precise control of the timings, allowing all successive periods to be measured during a given data acquisition time lapse.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Trigger signal for the frequency measurement (left, top curve) and amplitude measurement analog signal (right, top curve) output from the electronic circuit connected to Chua's circuit. In both images, the bottom curve is the signal observed at the output of Chua's circuit (chaotic regime). The amplitude signal (right) is read by a 12 bit analog to digital converter while a custom made frequency counter samples the pulse (left). Both values are read in less than 500 µs, half of the duration of a period of Chua's circuit, and allows the measurement of all successive periods.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Bifurcation diagram for the amplitude data.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 8. Slope of Allan deviation (top) and bifurcation diagram (bottom) for the frequency data.</figcaption></figure>

signal is periodic, and is intuitively extended to re-
spectively the maximum of the voltage over each
period and the inverse of the duration between two
crossings of a reference level, chosen here to be 0 V
(ground level being the average value of the volt-
ages in the periodic and quasi-periodic regimes).
Figure 6 illustrates these definitions by displaying
both the signal being analyzed (bottom signal) and

the frequency (left) and amplitude (right) measure-
ments (top waveforms). We have selected values of
the parameter for which we can see a periodic sig-
nal followed by signals characteristic of each fur-
ther bifurcation, until obtaining a fully developed
(Rössler like) chaotic attractor. The chosen values
of L are the following: 13.167 mH for a periodic
signal, 13.497 mH and 13.926 mH for signals after
<figure><img src="image_10.png" /><figcaption>Fig. 9. Evolution of the return maps ($a_n$ versus $f_n$) for the period data with the $L$ parameter.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10. Evolution of the transfer functions (the abscissa are not here the frequency as usually used when displaying transfer functions, but the period, which is more relevant in this study) with the parameter $L$.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 11. Evolution of the return maps for the amplitude data with the $L$ parameter. Notice that the relation between $a_n$ and $a_{n+1}$ ($a_n$ being the amplitude data) is no longer a function for a value large enough of $L$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. 3D plots of the experimental amplitude return map at 1 and 2 steps in the chaotic regime of the circuit. $L = 14.415$ mH. These plots can be compared with Fig. 11: adding a third dimension unwraps the return map.</figcaption></figure>
the first bifurcation, 14.207 mH for a signal after
the second bifurcation, and finally 14.058 mH and
14.322 mH for fully developed chaos (Rössler-like
attractor).

Figure 10 displays the evolution of the transfer
functions with the parameter *L*. Absicissa are here
given in duration units, rather than in the usual
frequency units as classically used. The splitting of
the fundamental spot (top-left) is clearly identified.
A major difference between the return maps of the
frequency data (Fig. 9) and the return maps of the
amplitude data (Fig. 11) can be observed: the re-
turn maps of the frequency data are always func-
tions (*f*<sub>n</sub> defines one *f*<sub>n+1</sub> only) while return maps
of the amplitude data are no longer functions for
large enough values of the parameter *L*. By adding
additional dimensions, i.e. plotting *a*<sub>n</sub> as a func-
tion of *a*<sub>n+1</sub> and *a*<sub>n+2</sub> (*a*<sub>n,n∈[1..N]</sub> being the ampli-
tude data), the relation becomes a function again
by unwrapping the return map, and the prediction
is again possible (Fig. 12).

5. Conclusion

After validating our data acquisition electronics
and software by plotting the bifurcation diagrams,
we have shown the experimental return maps and
transfer functions of Chua's circuit. We have then
analyzed the frequency fluctuations using the Allan
deviation.

The slope of the Allan deviation was shown to
be equal to -0.5 over most of the bifurcation pa-
rameter range, meaning the frequency fluctuations

are mainly characterized by white frequency noise.
The same kind of behavior is observed in quartz
crystal based oscillators far enough from the carrier
(100 kHz to 1 MHz from the 11.0592 MHz carrier
for a Colpitts oscillator).

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