*5.2. Experimental Results of the Integrated BCO-1 and BCO-2*

Now, we present the continuous-time behavior of both BCO-1 and BCO-2 on the integrated circuit. Figure 9 shows the real-time obtained waveforms and the respective dynamical analysis considering:

(i) time-series of the output voltage; (ii) frequency spectra; (iii) time-lag reconstructions of the attractors; (iv) Poincaré mapping set; (v) and largest Lyapunov exponent.

**Figure 9.** Chaotic dynamics measured experimentally from the integrated circuit of 180 nm at distinct settings for both Boolean chaos oscillators. Top to bottom: Time-series, time-lag embedded attractor, frequency spectrum, Poincaré map, the divergence ln*d*(*s*) to determine the largest Lyapunov exponent *λmax* of the attractor. (**a**) Experimental results for BCO-1 @ *VDD* = 3.3 V with *λmax* = 0.4496; (**b**) Experimental results for BCO-1 @ *VDD* = 2.8 V with *λmax* = 0.4243; (**c**) Experimental results for BCO-2 @ *VDD* = 3.3 V with *λmax* = 0.2492.

Figure 9a presents the BCO-1 features for *VDD* = 3.3 V. The time-series shows a random evolution since it has variable cycle amplitudes regarding maxima and minima, and the frequency content is characterized for a predominant broad distribution and with strong content up to 200 MHz, therefore suggesting chaotic oscillations. Besides, the time-lag embedded attractor (lag equal to the first minimum of the time-lag mutual information function) exhibits a chaotic behavior in phase space, whose underlying complexity can be more properly appreciated on the corresponding Poincaré map for amplitudes of successive local maxima. In this manner, the arbitrarily chosen plane sections the attractor in two and thereby enables the visualization of its complex geometry. We found that the Poincaré map has a dense set of points, which has been identified as characteristic dynamics of the chaotic behavior. To quantify these observations, we determine the largest Lyapunov exponent (*λmax*) of the attractor. The result shows the time evolution of the ln*d*(*s*). This divergence presents an almost constant slope for the first part of the curve and then it saturates at a maximum value, corresponding to the uncorrelated signals *x*(*s* + *T* + *ta*) and *x*(*s* + *T* + *tb*). Next, we estimate the value of *λmax*, assuming that the divergence of the initially similar segments is exponential in the region of constant slope. As a result, the average of all pairs of similar segments is our estimate of the largest Lyapunov exponent for the BCO-1, giving *λmax* = 0.4496, which demonstrates that the CMOS Boolean oscillator integrated at 180 nm is chaotic.

Figure 9b shows the dynamical analysis for the same BCO-1 but now with *VDD* = 2.8 V. This BCO-1 displays a clear chaotic attractor in the 0–4 V range and is validated with the Poincaré set. The results for time-series, frequency spectrum up to 150 MHz, Poincaré map, and *λmax* = 0.4243 have a similar response to the previous case. Therefore, we can conclude the BCO-1 is robust against bias voltage variations.

The same test is included for the BCO-2 biased to VDD = 3.3 V. The circuit presents a chaotic oscillation but the on-chip pad originates an explicit limitation of the voltage swing. This prototype uses internal pad connections and the reduced swing is a consequence of the extended bonding and absence of I/O cells. Therefore, the on-die probes represent an important load impedance and limit output swing to 200 mV. Figure 9c shows the time evolution, spectral content up to 160 MHz, chaotic attractor, and Poincaré map showing the expected results. Finally, Figure 9c also shows the largest Lyapunov exponent, which has a slope less abrupt but still presents a positive exponent (*λmax* = 0.2492) in spite of the small values of the continuous-time sequence.

#### *5.3. Comparison with Similar Implementations*

For the sake of reference, Table 4 highlights the principal features of the recent True Random Number Generator (TRNG), systems based on chaotic circuits. The two new boolean chaotic oscillators exhibit competitive numbers compared to the references [20,42]. The comparison includes the most recent works with attempts to fully integrate the system-on-chip. The work in [20] presents a set of inverted-based chaotic oscillators. The area and power consumption are affordable, but the system uses additional off-chip biasing circuits. The chaotic circuit in [42] is fully integrated with the disadvantage of increasing the circuit resources. This is the result of the multiattractor analog system requiring a large bandwidth but the band limitation or frequency centroid is not reported. The two new boolean chaotic oscillators in this work present a reduced circuit size and power dissipation, not considering the chip input-output cells. The proposed boolean chaotic circuits present the most extended frequency span content compared to the recent on-chip implementations.


**Table 4.** Comparison of the principal features of recent TRNG based on chaotic systems.
