*3.1. BCO-1*

Figure 1a shows the first Boolean chaos oscillator (BCO). It consists of three nodes where each node has three inputs and one output that propagates to three different nodes. Nodes *A* and *B* perform the XOR logic operation while node *C* executes the XNOR. Expressing BCO-1 in the form of Equation (1), we obtain the following system of Boolean delay equations:

$$\begin{array}{lll} X\_{\mathfrak{a}}(t) &=& X\_{\mathfrak{a}}(t - \tau\_{\mathfrak{a}\mathfrak{a}}) \oplus X\_{\mathfrak{b}}(t - \tau\_{\mathfrak{a}\mathfrak{b}}) \oplus X\_{\mathfrak{c}}(t - \tau\_{\mathfrak{b}\mathfrak{c}}), \\ X\_{\mathfrak{b}}(t) &=& X\_{\mathfrak{a}}(t - \tau\_{\mathfrak{b}\mathfrak{a}}) \oplus X\_{\mathfrak{b}}(t - \tau\_{\mathfrak{b}\mathfrak{b}}) \oplus X\_{\mathfrak{c}}(t - \tau\_{\mathfrak{b}\mathfrak{c}}), \\ X\_{\mathfrak{c}}(t) &=& X\_{\mathfrak{a}}(t - \tau\_{\mathfrak{c}\mathfrak{a}}) \oplus X\_{\mathfrak{b}}(t - \tau\_{\mathfrak{c}\mathfrak{b}}) \oplus X\_{\mathfrak{c}}(t - \tau\_{\mathfrak{c}\mathfrak{c}}) \oplus 1, \end{array} \tag{2}$$

with Boolean functions *fi* : <sup>B</sup><sup>3</sup> <sup>→</sup> <sup>B</sup>, *<sup>i</sup>* <sup>=</sup> 1, ... , 3, and <sup>⊕</sup> the logic XOR operation. The signal propagation time from node *j* to node *i* is *τij* for *i*, *j* = *a*, *b*, *c*.

**Figure 1.** (**a**) Autonomous Boolean networks (ABN) for the proposed Boolean chaos oscillator (BCO-1). (**b**) An implementation of BCO-1 using electronic logic gates and its look-up table.

**Theorem 3.** *For an autonomous Boolean network given by the system of Equation* (2)*, the orbits are always oscillating [36].*

**Proof.** A Boolean fixed point provokes nonoscillating dynamics due to some orbits eventually collapsing into the fixed point. To demonstrate the proposed Boolean chaos oscillator converges to sustained oscillations indefinitely, we must prove that there is not a fixed point. By contradiction, we demonstrate this theorem. Let us assume that the BCO-1 has a fixed point (*X*∗ *<sup>a</sup>* , *X*<sup>∗</sup> *<sup>b</sup>* , *X*<sup>∗</sup> *<sup>c</sup>* ), such that:

$$\begin{aligned} X\_a^\* &= X\_a(t - \tau), \\ X\_b^\* &= X\_b(t - \tau), \\ X\_c^\* &= X\_c(t - \tau), \end{aligned}$$

for *t* >> *τ* = max{*τaa*, *τab*, *τac*, *τba*, *τbb*, *τbc*, *τca*, *τcb*, *τcc*}. In this manner, the system of Equation (2) recast as:

$$X\_d(t) \quad = \quad X\_d(t) \oplus X\_b(t) \oplus X\_c(t), \tag{3}$$

$$X\_b(t) \quad = \quad X\_a(t) \oplus X\_b(t) \oplus X\_c(t), \tag{4}$$

$$X\_{\mathfrak{c}}(t) \quad = \quad X\_{\mathfrak{a}}(t) \oplus X\_{\mathfrak{b}}(t) \oplus X\_{\mathfrak{c}}(t) \oplus \mathbf{1}. \tag{5}$$

By substituting Equations (3) and (4) into (5), we obtain:

$$\begin{aligned} X\_{\mathfrak{c}}(t) &= \quad X\_{\mathfrak{a}}(t) \oplus X\_{\mathfrak{b}}(t) \oplus X\_{\mathfrak{c}}(t) \oplus X\_{\mathfrak{a}}(t) \oplus X\_{\mathfrak{b}}(t) \oplus \\ X\_{\mathfrak{c}}(t) &\oplus X\_{\mathfrak{c}}(t) \oplus 1. \end{aligned} \tag{6}$$

Equation (6) implies *Xc*(*t*) = *Xc*(*t*). Since the Boolean space is 2*n*, the possible states for *Xc*(*t*) are {1, 0}. Thus, "1" = "0", or vice-versa indicates a contradiction which leads us to conclude that Boolean network (2) does not have fixed points, and therefore, always oscillates.

*3.2. BCO-2*

Figure 2a shows the second Boolean chaos oscillator introduced in this work. The proposed topology consists of two nodes, where each node has three inputs and one output connecting to two different nodes. While node *A* performs the XOR logic operation, node *B* executes the XNOR. Additionally, the ABN includes two logic NOT operations to obtain the opposed Boolean states for both nodes. The set of Boolean delay equations for the BCO-2 are:

$$\begin{aligned} X\_d(t) &= -X\_d(t - \tau\_{ad}) \oplus X\_b(t - \tau\_{db}) \oplus \neg X\_b(t - \tau\_{ad}), \\ X\_b(t) &= -X\_d(t - \tau\_{ba}) \oplus X\_b(t - \tau\_{bb}) \oplus \\ &- X\_b(t - \tau\_{bb}) \oplus 1, \end{aligned} \tag{7}$$

with ⊕ and ¬ indicating the logic XOR and NOT operations, respectively. In addition, the Boolean functions *fi* : <sup>B</sup><sup>2</sup> <sup>→</sup> <sup>B</sup>, *<sup>i</sup>* <sup>=</sup> 1, ... , 2. Similarly to the previous case, it is necessary to prove that the system (7) does not have a Boolean fixed point.

**Figure 2.** (**a**) ABN for the second Boolean chaos oscillator (BCO-2). (**b**) An implementation of BCO-2 using electronic logic gates and its look-up table.

**Theorem 4.** *For an autonomous Boolean network given by the system of Equation* (7)*, the orbits are always oscillating [36].*

**Proof.** We assume that the BCO-2 has the fixed point (*X*∗ *<sup>a</sup>* , *X*<sup>∗</sup> *<sup>b</sup>* ), such that *X*<sup>∗</sup> *<sup>a</sup>* = *Xa*(*t* − *τ*) and *X*<sup>∗</sup> *<sup>b</sup>* = *Xb*(*t* − *τ*), for *t* >> *τ* = max{*τaa*, *τab*, *τaa*˜, *τba*, *τbb*, *τb*˜ *<sup>b</sup>*}. Therefore, system (7) is rewritten as:

$$X\_{\mathfrak{a}}(t) \quad = \quad X\_{\mathfrak{a}}(t) \oplus X\_{\mathfrak{b}}(t) \oplus \neg X\_{\mathfrak{a}}(t), \tag{8}$$

$$X\_b(t) \quad = \quad X\_a(t) \oplus X\_b(t) \oplus \neg X\_b(t) \oplus \mathbf{1},\tag{9}$$

By inserting Equation (8) into (9), we obtain:

$$X\_b(t) = \begin{aligned} X\_a(t) \oplus X\_b(t) \oplus \neg X\_a(t) \oplus X\_b(t) \oplus \neg X\_b(t) \oplus \neg X\_b(t), \end{aligned} \tag{10}$$
 
$$\neg X\_b(t) \oplus 1. \tag{11}$$

Equation (10) means *Xb*(*t*) = *Xb*(*t*). This again implies a contradiction and it is possible to claim that the autonomous Boolean network (7) does not have a fixed point and it will oscillate permanently.
