SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\boldsymbol {z} _ {t} = F _ {\boldsymbol {\theta}} \left(\boldsymbol {z} _ {t - 1}, \boldsymbol {s} _ {t}\right) = \boldsymbol {A} \boldsymbol {z} _ {t - 1} + \boldsymbol {W} _ {1} \phi \left(\boldsymbol {W} _ {2} \boldsymbol {z} _ {t - 1} + \boldsymbol {h} _ {2}\right) + \boldsymbol {C} \boldsymbol {s} _ {t} + \boldsymbol {h} _ {1}, \tag {6}
$$
with  $\pmb{W}_1 \in \mathbb{R}^{M \times L}$  and  $\pmb{W}_2 \in \mathbb{R}^{L \times M}$ ,  $L \geq M$ , connectivity matrices,  $\pmb{h}_1 \in \mathbb{R}^M$ ,  $\pmb{h}_2 \in \mathbb{R}^L$  bias terms, and all other parameters and variables as in eq. (1). While this formulation is beneficial for training, the shPLRNN can essentially be rewritten in standard PLRNN form (see [24]).
Assume that  $D_{\Omega(t)} \coloneqq \mathrm{diag}(d_{\Omega(t)})$  is a diagonal matrix with an indicator vector  $d_{\Omega(t)} \coloneqq (d_1, d_2, \dots, d_M)$  such that  $d_m(z_{m,t}) \eqqcolon d_m = 1$  whenever  $z_{m,t} > 0$ , and zero otherwise. Then eq. (1) can be rewritten as
$$
\boldsymbol {z} _ {t} = \left(\boldsymbol {A} + \boldsymbol {W D} _ {\Omega (t - 1)}\right) \boldsymbol {z} _ {t - 1} + \boldsymbol {C} \boldsymbol {s} _ {t} + \boldsymbol {h} =: \boldsymbol {W} _ {\Omega (t - 1)} \boldsymbol {z} _ {t - 1} + \boldsymbol {C} \boldsymbol {s} _ {t} + \boldsymbol {h}.
$$
Let us ignore the inputs for simplicity. There are  $2^{M}$  different configurations for matrix  $\pmb{D}_{\Omega (t - 1)}$  and so  $2^{M}$  different forms for matrix  $\pmb{W}_{\Omega (t - 1)}$  in the system
$$
\boldsymbol {z} _ {t} = F _ {\boldsymbol {\theta}} \left(\boldsymbol {z} _ {t - 1}\right) = \boldsymbol {W} _ {\Omega (t - 1)} \boldsymbol {z} _ {t - 1} + \boldsymbol {h}. \tag {7}
$$
Thus, the phase space of the system is divided into  $2^{M}$  sub-regions corresponding to the indexed matrices
$$
\boldsymbol {W} _ {\Omega^ {k}} := \boldsymbol {A} + \boldsymbol {W} \boldsymbol {D} _ {\Omega^ {k}}, \quad k = 1, 2, \dots , 2 ^ {M}, \tag {8}
$$
see [38, 39] for more details. For  $M = 2$ , assuming
$$
\boldsymbol {W} = \left( \begin{array}{l l} w _ {1 1} & 0 \\ w _ {2 1} & 0 \end{array} \right), \tag {9}
$$
in (8), we have
$$
\boldsymbol {W} _ {\Omega^ {1}} = \boldsymbol {W} _ {\Omega^ {3}} = \left( \begin{array}{c c} a _ {1 1} & 0 \\ 0 & a _ {2 2} \end{array} \right) = \boldsymbol {A},
$$
$$
\boldsymbol {W} _ {\Omega^ {2}} = \boldsymbol {W} _ {\Omega^ {4}} = \left( \begin{array}{c c} a _ {1 1} + w _ {1 1} & 0 \\ w _ {2 1} & a _ {2 2} \end{array} \right). \tag {10}
$$
Hence, for this parameter constellation, the map simplifies as there exists only one border which divides the phase space into two distinct sub-regions, such that (7) can be rewritten as a map of the form
$$
\begin{array}{l} \left( \begin{array}{c} z _ {1, t} \\ z _ {2, t} \end{array} \right) = T (z _ {1, t - 1}, z _ {2, t - 1}) \\ = \left\{ \begin{array}{l} T _ {\mathcal {L}} \left(z _ {1, t - 1}, z _ {2, t - 1}\right) = \underbrace {\left( \begin{array}{c c} a _ {l} & c \\ b _ {l} & d \end{array} \right)} _ {\boldsymbol {A} _ {\mathcal {L}}} \binom {z _ {1, t - 1}} {z _ {2, t - 1}} + \binom {h _ {1}} {h _ {2}}; \quad z _ {1, t - 1} \leq 0 \\ T _ {\mathcal {R}} \left(z _ {1, t - 1}, z _ {2, t - 1}\right) = \underbrace {\left( \begin{array}{c c} a _ {r} & c \\ b _ {r} & d \end{array} \right)} _ {\boldsymbol {A} _ {\mathcal {R}}} \binom {z _ {1, t - 1}} {z _ {2, t - 1}} + \binom {h _ {1}} {h _ {2}}; \quad z _ {1, t - 1} \geq 0 \end{array} , \right. \tag {11} \\ \end{array}
$$
with  $a_{l} = a_{11}$ ,  $a_{r} = a_{11} + w_{11}$ ,  $b_{r} = w_{21}$ ,  $d = a_{22}$ ,  $b_{l} = c = 0$ . The map (11) is a PWL dynamical system whose phase space is split into left and right half-planes (sub-regions) by the borderline  $\Sigma$  ( $z_{2}$ -axis). Note that bifurcation curves of the  $2d$  PLRNN (7) in the  $(a_{11}, a_{11} + w_{11})$ -parameter space can be determined analogous to those of the PWL map (11) in the  $(a_{l}, a_{r})$ -parameter space.
Another way to simplify the PLRNN to a  $2d$  ( $M = 2$ ) PWL map with just a single border is to remove one of the ReLU nonlinearities and define  $\phi(\pmb{z}_{t-1}) = (\phi_1(z_{1,t-1}), \beta z_{2,t-1})^{\mathrm{T}}$ , where  $\beta \in \mathbf{R}$  and  $\phi_1$  is some variant of the ReLU function such as the leaky or parametric ReLU given by
$$
\phi_ {1} (z) = \left\{ \begin{array}{l l} z; & z > 0 \\ \alpha z; & z \leq 0 \end{array} \quad (\alpha \in \mathbf {R}). \right. \tag {12}
$$
Then  $D_{\Omega (t)}\coloneqq \mathrm{diag}(d_1,\beta)$  such that
$$
d _ {1} \left(z _ {1, t}\right) =: d _ {1} = \left\{ \begin{array}{l l} 1; & z _ {1, t} > 0 \\ \alpha ; & z _ {1, t} \leq 0 \end{array} , \right. \tag {13}
$$
$$
\boldsymbol {W} _ {\Omega^ {1}} = \boldsymbol {W} _ {\Omega^ {3}} = \left( \begin{array}{c c} a _ {1 1} + \alpha w _ {1 1} & \beta w _ {1 2} \\ \alpha w _ {2 1} & a _ {2 2} + \beta w _ {2 2} \end{array} \right) =: \left( \begin{array}{c c} a _ {l} & c \\ b _ {l} & d \end{array} \right),
$$
$$
\boldsymbol {W} _ {\Omega^ {2}} = \boldsymbol {W} _ {\Omega^ {4}} = \left( \begin{array}{c c} a _ {1 1} + w _ {1 1} & \beta w _ {1 2} \\ w _ {2 1} & a _ {2 2} + \beta w _ {2 2} \end{array} \right) =: \left( \begin{array}{c c} a _ {r} & c \\ b _ {r} & d \end{array} \right). \tag {14}
$$
This gives another example of a representative of  $2d$  PWL maps with only one border defined in eq. (11). We are pointing this out because eq. (11) is a generic system considered more widely in the discrete dynamical systems literature [5, 6], and also was the basis for the analyses below and in Fig. 1.
A.1.2 Fixed points of the map (11) and their bifurcations
For  $a_{l}, a_{r}, b_{l}, b_{r}, c, d, h_{1}, h_{2} \in \mathbb{R}$ , the map (11) has the following two fixed points
$$
\mathcal {O} _ {\mathcal {L} / \mathcal {R}} = \left(z _ {1} ^ {\mathcal {L} / \mathcal {R}}, z _ {2} ^ {\mathcal {L} / \mathcal {R}}\right) ^ {\top} = \left(\frac {(1 - d) h _ {1} + c h _ {2}}{(1 - d) (1 - a _ {l / r}) - b _ {l / r} c}, \frac {b _ {l / r} h _ {1} + (1 - a _ {l / r}) h _ {2}}{(1 - d) (1 - a _ {l / r}) - b _ {l / r} c}\right) ^ {\top}. \tag {15}
$$
The fixed points  $\mathcal{O}_{\mathcal{L}}$  and  $\mathcal{O}_{\mathcal{R}}$  exist iff  $z_1^{\mathcal{L}} < 0$  and  $z_1^{\mathcal{R}} > 0$  respectively; otherwise they are virtual. Hence, the existence regions of admissible fixed points are
$$
E _ {\mathcal {O} _ {\mathcal {L}}} = \left\{(h _ {1}, h _ {2}, a _ {l}, b _ {l}, c, d) \mid \begin{array}{c} (1 - d) h _ {1} + c h _ {2} \\ \hline (1 - d) (1 - a _ {l}) - b _ {l} c \end{array} <   0 \right\},
$$
$$
E _ {\mathcal {O} _ {\mathcal {R}}} = \left\{\left(h _ {1}, h _ {2}, a _ {r}, b _ {r}, c, d\right) \mid \frac {(1 - d) h _ {1} + c h _ {2}}{(1 - d) (1 - a _ {r}) - b _ {r} c} > 0 \right\}. \tag {16}
$$
Let  $\mathcal{D}_{\mathcal{L} / \mathcal{R}}$  be the determinant and  $\mathcal{T}_{\mathcal{L} / \mathcal{R}}$  the trace of  $A_{\mathcal{L} / \mathcal{R}}$ , and
$$
\mathcal {P} _ {\mathcal {L} / \mathcal {R}} (\lambda) = \lambda^ {2} - \left(a _ {l / r} + d\right) \lambda + a _ {l / r} d - b _ {l / r} c = \lambda^ {2} - \mathcal {T} _ {\mathcal {L} / \mathcal {R}} \lambda + \mathcal {D} _ {\mathcal {L} / \mathcal {R}}, \tag {17}
$$
its characteristic polynomial. The corresponding eigenvalues are given by
$$
\lambda_ {1, 2} \left(\mathcal {O} _ {\mathcal {L} / \mathcal {R}}\right) = \frac {a _ {l / r} + d}{2} \pm \frac {\sqrt {\left(a _ {l / r} - d\right) ^ {2} + 4 b _ {l / r} c}}{2} = \frac {\mathcal {T} _ {\mathcal {L} / \mathcal {R}}}{2} \pm \frac {\sqrt {\mathcal {T} _ {\mathcal {L} / \mathcal {R}} ^ {2} - 4 \mathcal {D} _ {\mathcal {L} / \mathcal {R}}}}{2}, \tag {18}
$$
which are always real for  $b_{l / r}c \geq 0$ , while for  $b_{l / r}c < 0$  they are real provided that  $|a_{l / r} - d| > 2\sqrt{-b_{l / r}} c$ . For complex conjugate eigenvalues of  $A_{\mathcal{L} / \mathcal{R}}$  obviously  $|\lambda|^2 = \mathcal{D}_{\mathcal{L} / \mathcal{R}}$ . Thus computing the real eigenvalues, the stability condition for the fixed points is determined as
$$
- \left(1 + \mathcal {D} _ {\mathcal {L} / \mathcal {R}}\right) <   \mathcal {T} _ {\mathcal {L} / \mathcal {R}} <   1 + \mathcal {D} _ {\mathcal {L} / \mathcal {R}}. \tag {19}
$$
Accordingly, the stability region of the fixed points  $\mathcal{O}_{\mathcal{L}}$  and  $\mathcal{O}_{\mathcal{R}}$  can be obtained by  $\mathcal{P}_{\mathcal{L} / \mathcal{R}}(\pm 1) = 1 \mp (a_{l / r} + d) + a_{l / r}d - b_{l / r}c > 0$  and  $\mathcal{D}_{\mathcal{L} / \mathcal{R}} < 1$  as
$$
\mathcal {S} _ {\mathcal {L} / \mathcal {R}} = \left\{(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d) \in E _ {\mathcal {O} _ {\mathcal {L} / \mathcal {R}}} \mid a _ {l / r} d - b _ {l / r} c <   1, \right.
$$
$$
\left. 1 \pm \left(a _ {l / r} + d\right) + a _ {l / r} d - b _ {l / r} c > 0 \right\}. \tag {20}
$$
Note that when  $\mathcal{D}_{\mathcal{L} / \mathcal{R}} < 0$ , all the eigenvalues are real and so there cannot be any spiralling orbit.
Remark 1. Consider the PLRNN (2) with  $M = 2$ . For the parameter setting (3), i.e.
$$
\boldsymbol {W} _ {\Omega^ {1}} = \boldsymbol {W} _ {\Omega^ {3}} = \left( \begin{array}{c c} a _ {1 1} & 0 \\ 0 & a _ {2 2} \end{array} \right) =: \boldsymbol {A} _ {\mathcal {L}}, \quad \boldsymbol {W} _ {\Omega^ {2}} = \boldsymbol {W} _ {\Omega^ {4}} = \left( \begin{array}{c c} a _ {1 1} + w _ {1 1} & 0 \\ w _ {2 1} & a _ {2 2} \end{array} \right) =: \boldsymbol {A} _ {\mathcal {R}}, \tag {21}
$$
the two fixed points  $\mathcal{O}_{\mathcal{L} / \mathcal{R}} = \left(z_1^{\mathcal{L} / \mathcal{R}},z_2^{\mathcal{L} / \mathcal{R}}\right)^{\top}$  are given by
$$
\mathcal {O} _ {\mathcal {L}} = \left(\frac {h _ {1}}{1 - a _ {1 1}}, \frac {h _ {2}}{1 - a _ {2 2}}\right) ^ {\top}, \quad \mathcal {O} _ {\mathcal {R}} = \left(\frac {h _ {1}}{1 - a _ {1 1} - w _ {1 1}}, \frac {w _ {2 1} h _ {1} + (1 - a _ {1 1} - w _ {1 1}) h _ {2}}{(1 - a _ {2 2}) (1 - a _ {1 1} - w _ {1 1})}\right) ^ {\top}. \tag {22}
$$
Hence, the existence regions of admissible fixed points are
$$
E _ {\mathcal {O} _ {\mathcal {L}}} = \left\{(h _ {1}, a _ {1 1}, a _ {2 2}) \big | \frac {h _ {1}}{1 - a _ {1 1}} <   0 \right\}, \qquad E _ {\mathcal {O} _ {\mathcal {R}}} = \left\{(h _ {1}, a _ {1 1}, a _ {2 2}, w _ {1 1}) \big | \frac {h _ {1}}{1 - a _ {1 1} - w _ {1 1}} > 0 \right\},
$$
and their stability regions can be obtained as
$$
\mathcal {S} _ {\mathcal {L}} = \left\{\left(h _ {1}, a _ {1 1}, a _ {2 2}\right) \in E _ {\mathcal {O} _ {\mathcal {L}}} \mid a _ {1 1} a _ {2 2} <   1, 1 \pm \left(a _ {1 1} + a _ {2 2}\right) + a _ {1 1} a _ {2 2} > 0 \right\}, \tag {23}
$$
$$
\mathcal {S} _ {\mathcal {R}} = \Bigg \{\left(h _ {1}, a _ {1 1}, a _ {2 2}, w _ {1 1}\right) \in E _ {\mathcal {O} _ {\mathcal {R}}} \big | (a _ {1 1} + w _ {1 1}) a _ {2 2} <   1, 1 \pm \left(a _ {1 1} + w _ {1 1} + a _ {2 2}\right) + \left(a _ {1 1} + w _ {1 1}\right) a _ {2 2} > 0 \Bigg \}.
$$
Remark 2. If  $b_{l / r}c = 0$ , then  $\lambda_{1,2}(\mathcal{O}_{\mathcal{L} / \mathcal{R}})$  are real and the stability regions  $S_{\mathcal{L} / \mathcal{R}}$  become
$$
\mathcal {S} _ {\mathcal {L} / \mathcal {R}} = \left\{\left(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d\right) \in E _ {\mathcal {O} _ {\mathcal {L} / \mathcal {R}}} \mid b _ {l / r} c = 0, - 1 \leq a _ {l / r} \leq 1, - 1 \leq d \leq 1 \right\}. \tag {24}
$$
The fixed points are regular saddles for all parameters that belong to
$$
\left\{(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d) \in E _ {\mathcal {O} _ {\mathcal {L} / \mathcal {R}}} \mid a _ {l / r} + d > 1, a _ {l / r} d - a _ {l / r} - d + 1 <   b _ {l / r} c <   a _ {l / r} d \right\}, \tag {25}
$$
and in this case  $\lambda_1(\mathcal{O}_{\mathcal{L} / \mathcal{R}}) > 1$ ,  $0 < \lambda_{2}(\mathcal{O}_{\mathcal{L} / \mathcal{R}}) < 1$ . Furthermore, they are flip saddles (i.e., with one negative eigenvalue) if parameters are in
$$
\left\{(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d) \in E _ {\mathcal {O} _ {\mathcal {L} / \mathcal {R}}} \mid a _ {l / r} + d > 1, a _ {l / r} d <   b _ {l / r} c <   a _ {l / r} d + a _ {l / r} + d + 1 \right\} \bigcup
$$
$$
\left\{(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d) \in E _ {\mathcal {O} _ {\mathcal {L} / \mathcal {R}}} \mid d - a _ {l / r} - d + 1 <   b _ {l / r} c <   a _ {l / r} d + a _ {l / r} + d + 1, \right.
$$
$$
\left. 0 <   a _ {l / r} + d \leq 1, a _ {l / r} \right\}, \tag {26}
$$
for which  $\lambda_1(\mathcal{O}_{\mathcal{L} / \mathcal{R}}) > 1, -1 < \lambda_2(\mathcal{O}_{\mathcal{L} / \mathcal{R}}) < 0$  as well as in
$$
\left\{(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d) \in E _ {\mathcal {O} _ {\mathcal {L} / \mathcal {R}}} \mid a _ {l / r} + d \leq - 1, a _ {l / r} d <   b _ {l / r} c <   a _ {l / r} d - a _ {l / r} - d + 1 \right\} \bigcup
$$
$$
\begin{array}{l} \left\{(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d) \in E _ {\mathcal {O} _ {\mathcal {L} / \mathcal {R}}} \mid a _ {l / r} d + a _ {l / r} + d + 1 <   b _ {l / r} c <   a _ {l / r} d - a _ {l / r} + d - 1, \right. \\ \left. - 1 <   a _ {l / r} + d <   0 \right\} \tag {27} \\ \end{array}
$$
such that  $0 < \lambda_{1}(\mathcal{O}_{\mathcal{L} / \mathcal{R}}) < 1$ ,  $\lambda_{2}(\mathcal{O}_{\mathcal{L} / \mathcal{R}}) < -1$ .
When  $b_{l / r}c < 0$  and  $|a_{l / r} - d| < 2\sqrt{-b_{l / r}c}$ , the eigenvalues are complex conjugates and both  $\mathcal{O}_{\mathcal{L}}$  and  $\mathcal{O}_{\mathcal{R}}$  are spirally attracting (attracting focus) if  $a_{l / r}d - b_{l / r}c < 1$ . In this case, if  $a_{l / r} + d > 0$  then they are clockwise spiral, while for  $a_{l / r} + d < 0$  the spiralling motion will be counterclockwise. Moreover, for  $a_{l / r}d - b_{l / r}c > 1$  they are repelling foci. Finally, for  $a_{l / r}d - b_{l / r}c = 1$ , the fixed points are locally centers and they undergo a CB at the following boundaries:
$$
\mathcal {C} _ {\mathcal {L}} = \left\{(a _ {l}, b _ {l}, c, d) | b _ {l} c <   0, | a _ {l} - d | <   2 \sqrt {- b _ {l} c}, a _ {l} d - b _ {l} c = 1 \right\},
$$
$$
\mathcal {C} _ {\mathcal {R}} = \left\{\left(a _ {r}, b _ {r}, c, d\right) \mid b _ {r} c <   0, \left| a _ {r} - d \right| <   2 \sqrt {- b _ {r} c}, a _ {r} d - b _ {r} c = 1 \right\}. \tag {28}
$$
At these boundaries, the fixed points lose their stability with a pair of complex conjugate eigenvalues crossing the unit circle. For the parameters belonging to  $\mathcal{C}_{\mathcal{L} / \mathcal{R}}$ , the Jacobian  $J_{\mathcal{L} / \mathcal{R}}$  is a rotation matrix whose determinant is equal to 1. In this case,  $J_{\mathcal{L} / \mathcal{R}}$  can be determined by a rotation number which is either rational  $\binom{p}{q}$  or irrational  $(\rho)$ . Therefore, in some neighborhood of  $\mathcal{O}_{\mathcal{L} / \mathcal{R}}$ , there is a region filled with invariant ellipses such that they are periodic with period  $p$  (if the rotation number is a rational number  $\frac{p}{q}$ ) or quasiperiodic (if the rotation number is an irrational number  $\rho$ ); for more information see [60, 59]. For  $(1 - d)h_1 + ch_2 \neq 0$ , at the boundary
$$
\tau_ {\mathcal {L}} = \left\{\left. \left(h _ {1}, h _ {2}, a _ {l}, b _ {l}, c, d\right) \right| 1 - a _ {l} - d + a _ {l} d - b _ {l} c = 0 \right\}, \tag {29}
$$
the fixed point  $\mathcal{O}_{\mathcal{L}}$  undergoes a DTB, since, if the parameters tend to  $\tau_{\mathcal{L}}$ , then  $\mathcal{O}_{\mathcal{L}} \to \pm \infty$  and  $\lambda (\mathcal{O}_{\mathcal{L}}) \to 1$ . Similarly, for  $(1 - d)h_{1} + ch_{2} \neq 0$ , a DTB occurs for the fixed point  $\mathcal{O}_{\mathcal{R}}$  at the boundary
$$
\tau_ {\mathcal {R}} = \left\{\left. \left(h _ {1}, h _ {2}, a _ {r}, b _ {r}, c, d\right) \right| 1 - a _ {r} - d + a _ {r} d - b _ {r} c = 0 \right\}. \tag {30}
$$
A DTB of a fixed point results in its disappearance, as in this case the fixed point becomes virtual which may lead to changes in the global dynamics [6]. Furthermore, the BCB curves are given by
$$
\xi_ {\mathcal {L}} = \left\{\left(h _ {1}, h _ {2}, a _ {l}, b _ {l}, c, d\right) \mid (1 - d) (1 - a _ {l}) - b _ {l} c \neq 0, (1 - d) h _ {1} + c h _ {2} = 0 \right\}, \tag {31}
$$
$$
\xi_ {\mathcal {R}} = \left\{\left(h _ {1}, h _ {2}, a _ {r}, b _ {r}, c, d\right) \mid (1 - d) \left(1 - a _ {r}\right) - b _ {r} c \neq 0, (1 - d) h _ {1} + c h _ {2} = 0 \right\}. \tag {32}
$$
In addition, the DFB curves for the fixed points  $\mathcal{O}_{\mathcal{L}}$  and  $\mathcal{O}_{\mathcal{R}}$  are
$$
\mathcal {F} _ {\mathcal {L}} = \left\{\left. \left(h _ {1}, h _ {2}, a _ {l}, b _ {l}, c, d\right) \right| 1 + a _ {l} + d + a _ {l} d - b _ {l} c = 0 \right\},
$$
$$
\mathcal {F} _ {\mathcal {R}} = \left\{\left. \left(h _ {1}, h _ {2}, a _ {r}, b _ {r}, c, d\right) \right| 1 + a _ {r} + d + a _ {r} d - b _ {r} c = 0 \right\}. \tag {33}
$$
Remark 3. The existence regions  $E_{\mathcal{O}_{\mathcal{L}}}$  and  $E_{\mathcal{O}_{\mathcal{R}}}$  are bounded by the BCB curves  $\xi_{\mathcal{L}}$  and  $\xi_{\mathcal{R}}$ .
Remark 4. The stability regions  $S_{\mathcal{L} / \mathcal{R}}$  of fixed points (eq. (20)) are bounded by the DTB curves  $\tau_{\mathcal{L} / \mathcal{R}}$  (eqn. (29) and (30)), the DFB curves  $\mathcal{F}_{\mathcal{L} / \mathcal{R}}$  (eq. (33)), and the CB curves  $a_{l / r}d - b_{l / r}c = 1$ . For instance, for  $d = 1$ ,  $S_{\mathcal{L} / \mathcal{R}}$  are illustrated in Fig. S1(a). In this case, the stability regions only exist for  $b_{l / r}c < 0$  and  $a_{l / r} - d < -2\sqrt{-b_{l / r}c}$ . Moreover, as shown in Fig. S1(b), for  $d = 0.01$ , these stability regions can exist for both cases  $b_{l / r}c < 0$ ,  $a_{l / r} - d < -2\sqrt{-b_{l / r}c}$  (in blue), and  $b_{l / r}c < 0$ ,  $a_{l / r} - d > 2\sqrt{-b_{l / r}c}$  (in green), but not for  $b_{l / r}c \geq 0$ . Furthermore, if  $c = 1$ , there are stability regions  $S_{\mathcal{L} / \mathcal{R}}$  for the two cases  $b_{l / r}c < 0$ ,  $a_{l / r} - d < -2\sqrt{-b_{l / r}c}$  (in blue), and  $b_{l / r}c > 0$  (in purple); see Fig. S2(a). Finally, when  $c = 0$ , as explained in Remark 2, the stability regions have the form (24), i.e.
$$
\mathcal {S} _ {\mathcal {L} / \mathcal {R}} = \left\{\left(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d\right) \in E _ {\mathcal {O} _ {\mathcal {L} / \mathcal {R}}} \mid c = 0, b _ {l / r} \in \mathbb {R}, - 1 \leq a _ {l / r}, d \leq 1 \right\}. \tag {34}
$$
which are displayed in Fig. S2(b).
Figure S1: Stability regions  $S_{\mathcal{L} / \mathcal{R}}$ . Left: for  $d = 1$ ; right: for  $d = 0.01$ . The case  $b_{l / r}c < 0$ ,  $a_{l / r} - d < -2\sqrt{-b_{l / r}c}$  is plotted in blue, and the case  $b_{l / r}c < 0$ ,  $a_{l / r} - d > 2\sqrt{-b_{l / r}c}$  is drawn in green.
Figure S2: Stability regions  $S_{\mathcal{L} / \mathcal{R}}$ . Left: for  $c = 1$ ; the case  $b_{l / r}c < 0$ ,  $a_{l / r} - d < -2\sqrt{-b_{l / r}c}$  is plotted in yellow, and the case  $b_{l / r}c > 0$  is drawn in purple. Right: for  $c = 0$ .
Since the system (11) is a linear map in each sub-region  $\mathcal{L}$  and  $\mathcal{R}$ , there cannot be any  $n$ -cycle,  $n \geq 2$ , with all periodic points on only one linear side. So, all period- $n$  orbits have both letters  $\mathcal{L}$  and  $\mathcal{R}$  in their symbolic sequence.
A.1.3 2-cycles of the map (11) and their bifurcations
The 2-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$  of the map (11) is determined by solving the equation  $T_{\mathcal{L}}\circ T_{\mathcal{R}}(z_1,z_2) = (z_1,z_2)^{\mathsf{T}}$  where
$$
T _ {\mathcal {L}} \circ T _ {\mathcal {R}} \left(z _ {1}, z _ {2}\right) = \left( \begin{array}{l l} a _ {l} & a _ {r} + b _ {r} c \\ a _ {r} & b _ {l} + b _ {r} d \end{array} \quad & a _ {l} c + c d \right) \binom {z _ {1}} {z _ {2}} + \binom {c h _ {2} + h _ {1} (a _ {l} + 1)} {b _ {l} h _ {1} + h _ {2} (d + 1)}. \tag {35}
$$
In this case if  $\left(I - J_{\mathcal{L}}J_{\mathcal{R}}\right)$  is invertible, then the solution  $(z_{1},z_{2})^{\mathsf{T}} = \left(z_{1}^{(1)},z_{2}^{(1)}\right)^{\mathsf{T}}$  is given by
$$
\begin{array}{l} \left(z _ {1} ^ {(1)}, z _ {2} ^ {(1)}\right) ^ {\mathsf {T}} = \left(\frac {\left((1 - d) h _ {1} + c h _ {2}\right) \left(a _ {l} + d + a _ {l} d - b _ {l} c + 1\right)}{\left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) - c \left(b _ {l} + b _ {r}\right) - d ^ {2} - a _ {l} a _ {r} + 1}, \right. \\ \left. \frac {h _ {2} \left(1 + d - a _ {l} a _ {r} - b _ {r} c - a _ {l} a _ {r} d + a _ {r} b _ {l} c\right) + h _ {1} \left(b _ {l} + a _ {r} b _ {l} + b _ {r} d + a _ {l} b _ {r} d - b _ {l} b _ {r} c\right)}{\left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) - c \left(b _ {l} + b _ {r}\right) - d ^ {2} - a _ {l} a _ {r} + 1}\right). \tag {36} \\ \end{array}
$$
Also  $T_{\mathcal{R}}\big(z_1^{(1)},z_2^{(1)}\big) = \big(z_1^{(2)},z_2^{(2)}\big)^{\top}$  yields
$$
\begin{array}{l} \left(z _ {1} ^ {(2)}, z _ {2} ^ {(2)}\right) ^ {\top} = \left(\frac {\left((1 - d) h _ {1} + c h _ {2}\right) \left(a _ {r} + d + a _ {r} d - b _ {r} c + 1\right)}{\left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) - c \left(b _ {l} + b _ {r}\right) - d ^ {2} - a _ {l} a _ {r} + 1}, \right. \\ \left. \frac {h _ {2} \left(1 + d - a _ {l} a _ {r} - b _ {l} c - a _ {l} a _ {r} d + a _ {l} b _ {r} c\right) + h _ {1} \left(b _ {r} + a _ {l} b _ {r} + b _ {l} d + a _ {r} b _ {l} d - b _ {l} b _ {r} c\right)}{\left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) - c \left(b _ {l} + b _ {r}\right) - d ^ {2} - a _ {l} a _ {r} + 1}\right). \tag {37} \\ \end{array}
$$
Hence, the existence region of the 2-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$  is
$$
E _ {\mathcal {O} _ {\mathcal {R L}}} = \left\{(h _ {1}, h _ {2}, a _ {l}, b _ {l}, c, d) \mid \frac {\big ((1 - d) h _ {1} + c h _ {2} \big) \big (a _ {l} + d + a _ {l} d - b _ {l} c + 1 \big)}{\left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) - c \left(b _ {l} + b _ {r}\right) - d ^ {2} - a _ {l} a _ {r} + 1} > 0, \right.
$$
$$
\left. \frac {\left((1 - d) h _ {1} + c h _ {2}\right) \left(a _ {r} + d + a _ {r} d - b _ {r} c + 1\right)}{\left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) - c \left(b _ {l} + b _ {r}\right) - d ^ {2} - a _ {l} a _ {r} + 1} <   0 \right\}. \tag {38}
$$
The characteristic polynomial of  $J_{\mathcal{R}\mathcal{L}} = J_{\mathcal{L}}J_{\mathcal{R}} = \left( \begin{array}{cc}a_{l}a_{r} + b_{r}c & a_{l}c + cd\\ a_{r}b_{l} + b_{r}d & d^{2} + b_{l}c \end{array} \right)$  is given by
$$
\mathcal {P} _ {\mathcal {O} _ {\mathcal {R L}}} (\lambda) = \lambda^ {2} - \left(d ^ {2} + a _ {l} a _ {r} + b _ {l} c + b _ {r} c\right) \lambda + \left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right), \tag {39}
$$
$$
\mathcal {D} _ {\mathcal {R L}} = (a _ {r} d - b _ {r} c) (a _ {l} d - b _ {l} c),
$$
$$
\mathcal {P} _ {\mathcal {O} _ {\pi \mathcal {L}}} (1) = \left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) - c \left(b _ {l} + b _ {r}\right) - d ^ {2} - a _ {l} a _ {r} + 1,
$$
$$
\mathcal {P} _ {\mathcal {O} _ {\pi \mathcal {L}}} (- 1) = (a _ {r} d - b _ {r} c) (a _ {l} d - b _ {l} c) + c (b _ {l} + b _ {r}) + d ^ {2} + a _ {l} a _ {r} + 1,
$$
$$
\begin{array}{l} \lambda_ {1, 2} \left(\mathcal {O} _ {\mathcal {R L}}\right) = \frac {a _ {l} a _ {r} + c \left(b _ {l} + b _ {r}\right) + d ^ {2}}{2} \\ \pm \frac {\sqrt {\left(a _ {l} a _ {r} + b _ {l} c\right) ^ {2} + \left(b _ {r} c + d ^ {2}\right) ^ {2} + 2 \left(a _ {l} a _ {r} - b _ {l} c\right) \left(b _ {r} c - d ^ {2}\right) + 4 c d \left(a _ {l} b _ {r} + a _ {r} b _ {l}\right)}}{2}. \tag {40} \\ \end{array}
$$
Thus, the stability region of  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$  is
$$
\begin{array}{l} \mathcal {S} _ {\mathcal {R L}} = \left\{(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d) \in E _ {\mathcal {O} _ {\mathcal {R L}}} \mid - 1 <   (a _ {r} d - b _ {r} c) (a _ {l} d - b _ {l} c) <   1, \right. \\ \left. - \left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) - 1 <   c \left(b _ {l} + b _ {r}\right) + d ^ {2} + a _ {l} a _ {r} <   \left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) + 1 \right\}. \tag {41} \\ \end{array}
$$
In addition, for  $\left((1 - d)h_1 + ch_2\right)\left(a_l + d + a_ld - b_ld + c + 1\right)\neq 0$  , the set