**1. Introduction**

Chaos behavior is one of the most studied topics in nonlinear dynamics in recent years. Such interest relies mainly on its extreme sensitivity to the initial conditions. From a real-world application point of view, the random-like patterns generated by chaotic oscillators are currently pointed out as the core for obtaining significant engineering applications, for instance, secure-communications schemes [1–7]; radars [8–10]; sonars [11,12]; liquid mixing [13,14]; adaptive logic gates [15,16]; true random number generators (TRNGs) [17,18]; collective phenomena in physics and biology [19,20]; navigation and control of autonomous mobile robots [21–23]; Internet of Things [24–29]; and so forth. Thereupon, the cutting edge chaos-based applications may need reliable, robust, compact, and faster chaos oscillators.

A remarkable solution to obtain chaotic behavior consists of exploiting the delay paths in autonomous Boolean networks (ABNs) [30–34]. Kauffman proposed the Boolean networks in 1969 as a mathematical framework for studying gene regulatory networks. The mathematics describing ABNs has shown that they could display aperiodic patterns if the Boolean functions have instantaneous response times, the link time-delays are incommensurate, and their nodes perform asymmetric Boolean operations, such as the combination of logic exclusive-OR (XOR) and XNOR functions.

In the context of ABNs, deterministic chaos, also known as Boolean chaos, was initially demonstrated by using Boolean functions implemented with electronic logic circuits (logic gates

and field-programmable gate arrays FPGAs) [10,35–38]. At circuit level, the basic principle for obtaining Boolean chaos depends on three main characteristics; the asymmetry between the logic states, the short-pulse rejection phenomenon, and, most importantly, the degradation effect [30,31].

As is well-known, the incommensurate delay between two different nodes of an ABN is the critical parameter to obtain chaotic behaviors since it induces the degradation effect [30–34]. Rosin et al. analyzed two ABNs, one composed of a logic XOR function and two delays *τnk* and *τnl* , and the other one with a logic XNOR function and three delays *τnk* , *τnl* , and *τnm* [38]. They showed that Boolean chaos arises in an FPGA-based implementation when the delays for each of the three delay paths are *τnk* ≥ 2.8 ns, *τnl* ≥ 1.7 ns, and *τnm* ≥ 0.56 ns, respectively. To attain the time-delays, they required 18 extra logic NOT gates to connect the nodes of ABN. Besides, if those time-delays reduce below the minimum, the ABN does not show chaos and evolves to periodic oscillations only, as was analyzed in Ref. [35]. Park et al. presented an ABN composed of a logic XOR gate and ring oscillator [39]. The proposed logic circuit synthesized on an application-specific IC (ASIC), but the design is not straightforward because it also demands specific incommensurate delays in the feedback path to observe Boolean chaos.

Based on the discussion mentioned above, we note a possible benefit of using ABNs can be to get Boolean chaos oscillators with relatively high oscillation frequencies and small form factors since they depend on logic functions only. However, we also found that the proposed ABNs have high sensitivity to the time-delay among network nodes for generating chaos behavior. From a practical point of view, that condition is very complicated to satisfy since the time-delays are heavily related to the electronics technology chosen for implementation. Due to the electronic logic gates being heterogeneous, they do not have the same intrinsic time-delay. As a consequence, the dynamical behaviors of the ABN can be affected by placing the oscillator on a different area into an integrated circuit or FPGA. In conclusion, the previously reported Boolean chaos oscillators may not be suitable for physical realizations with multiple hardware approaches.

In this paper, we propose two ABNs with three and two nodes, respectively. The nodes perform the logic XOR and XNOR operations. This asymmetric approach avoids fixed points in the ABNs, and therefore, their dynamics can converge to chaotic oscillations. By applying the Lyapunov exponent method, we experimentally demonstrate that the Boolean chaos oscillators do not require specific incommensurate time-delays to show chaotic behaviors. Indeed, the Boolean chaos was observed under a wide range of the time-delays for the ABNs nodes. We prove our findings by implementing the proposed ABNs using various logic electronic circuits without any modification neither of the proposed networks nor adding additional path delays. Three discrete physical realizations using commercial logic gates, a Generic Array Logic (GAL), and FPGA are presented. Besides, we design an integrated circuit realization at 180nm fabrication technology.

The structure of the manuscript is as follows. Section 2 introduces the two ABNs and gives the mathematical demonstrations of their equilibrium points. Section 3 shows the analysis based on the Lyapunov exponents to determine the insensibility to time-delays. Section 4 presents the Lyapunov exponents for three different discrete implementations to prove that the ABNs are not affected by the technology. Section 5 introduces a straightforward methodology to design an integrated circuit of the Boolean chaos oscillators. Time-series, phase space reconstruction, Lyapunov exponents, and Poincare maps validate the observed chaos behavior. Finally, the last section concludes the paper.

### **2. Mathematical Preliminaries**

A Boolean network consists of a number of logical nodes interconnected through direct or indirect links. These are nonlinear networks requiring a mathematical base for analysis. Among the present models there are: the Kauffman (N-K) networks, Boolean differential equations, and piecewise-linear differential equations [33,34]. This work uses the Boolean differential equations to develop important mathematical considerations.
