1. Chaotic Time Series
2. Power Spectrum
3. Bifurcation Diagram
4. Largest Lyapunov Exponent
5. Propagation Delay

1. Chaotic Time Series

The figure below shows the voltage time series at the output of node 2, using VCC = 2.75 V. The Boolean logic states are clearly defined for most of the pulses. The series was acquired with a high-impedance active probe on a 8 GHz bandwidth oscilloscope at 40 Gsamples/s.

Fig.1: Time series of the voltage measured at node 2.


2. Power Spectrum

We acquire long time series of the chaotic signal and then calculate the magnitude of the Fourier transform from them. Integrating the Fourier magnitude over a frequency window gives the power spectral density (PSD). A typical unit used in communication engineering is the dBm, decibel relative to one milliwatt of power. Figure 2, below, show the PSD for the same signal of the time series in figure 1, above (the actual time series used is much longer than the segment displayed in figure 1), using a integration window of 1 MHz. The Federal Communications Commission (FCC) defines a ultra-wideband (UWB) frequency range when the bandwidth of a signal is greater than 500 MHz or the fractional bandwidth is greater than 20%. Ultra-wideband devices operating inside a specified power mask may broadcast without licensing in the United States [1]. Our device is a good candidate to be used as a UWB source, if attenuated to fit inside the power mask.

Fig.2: Power spectrum of the voltage measured at node 2.


3. Bifurcation Diagram

Changing the value of VCC affects the delay times, rise times and the memory effects of the circuit. Thus, one can use this parameter to tune the circuit into different regimes of chaos or periodic oscillation. Figure 3, below, shows a bifurcation diagram, where we plot the values of the time intervals between two sucessive transitions from 0 to 1 of the voltage in node 2, as we scan the supply voltage. For a given value of VCC, the abcissa in the figure, we can see a set of points corresponding to the intervals between rises that occur at this voltage. If the set appear as a discrete number of points it is an evidence that the oscillation is periodic. If the points appear to cover intervals on the ordinates axis, it is an evidence that the trajectory does not repeat exactly. the number of points we can plot is finite, though, and a very long periodic orbit could be confused with a nonperiodic oscillation. Furthermore, the bifurcation diagram does not tell the difference between a quasi-periodic orbit (composed of incommensurate frequencies of oscillation) or chaos.

Fig.3: Bifurcation Diagram.


4. Largest Lyapunov Exponent

To verify the existence of chaos we need to check whether nearby trajectories go apart exponentially fast, at least while the distance between them is small. The hypothesis we want to check is whether the average distance between trajectories obeys the equation:
(1) d(t)  =  d0 exp(λt),
when both the initial distance d0 and the final distance d(t) are small. The characteristic exponent λ is the largest Lyapunov exponent of the system. The calculation of the largest Lyapunov exponent has been extensively discussed in the literature [2], but problems may arise in systems where the dynamics are determined by time delayed terms [3]. Also, numerical problems may arise in Boolean systems or systems with dynamics determined by approximatelly Boolean values, because an appropriate definition of distance should weigh the differences that do influence the evolution and not the differences that do not contribute to the dynamics. In the figure below we plot the evolution of the distance in a m-dimensional space
(2) dn(m)  =  [∑mk = 1 (tn-k+1 -tn-k)2 ]1/2,
formed by m sucessive intervals between rise transitions, averaged over many sequences of transitions that start with a small distance. The slope of the natural logarithm of the distance defined in this space gives the average value of λ ≈ 0.6, from equation (1), above, for m ≥ 6.

Fig.4: Evolution of the distance between close trajectories and Lyapunov exponent.


5. Propagation Delay

Simulations of systems with constant propagation delay do not yield chaotic behavior. To understand how chaos is possible in the experimental systems we need to study the propagation delays. We measure simultaneously the input and output of gate, using high-impedance active probes and calculate the propagation delay for each transition. The objective is to find what is the delay function g(p) in the experimental system. We define a new variable

(3) Pn  =  tnij -tkn-1,
for the pulse width between the current transition in the input tn and the previous transition in the output ij (tkn-1), where τkij is a constant that depends on the specific link of the network, indicated by the indexes ij and on whether the transition tn is a rise (k = r) or a fall (k = f). We fit the experimental observations to an empirical function gij(p) for each one of the links ij in the network.
(4) gij(Pn)  =  τkij +A exp(-B Pn) cos(ΩPn+φ).
Figure 5 shows the measured delays for the link 33 in our circuit and a fit with equation (4).
Fig.5: Propagation delay as function of the pulse width along the link 33 for transitions from 1 to 0.


We find that, with delays given by equation (4) above and parameters measured from the experiment, our simulations give chaotic solutions with Lyapunov exponents similar to the ones observed in the experiment.



[1] FCC Report and order with revision of Part 15 of the Commision's rules. Released April 22, 2002.

[2] Nonlinear Time Series Analysis, H. Kantz and T. Schreiber (Cambridge Univ. Press, Cambridge, UK, 1997).

[3] Chaotic attractors of an infinite-dimensional dynamical system, J. D. Farmer, Physica 4D, 366 (1982).

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