SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \tau_ {\mathcal {R L}} = \left\{\left(h _ {1}, h _ {2}, a _ {l}, a _ {r}, b _ {l}, b _ {r}, c, d\right) \mid a _ {l} a _ {r} + b _ {l} c + b _ {r} c + d ^ {2} - a _ {l} a _ {r} d ^ {2} - b _ {l} b _ {r} c ^ {2} + a _ {l} b _ {r} c d \right. \\ \left. + a _ {r} b _ {l} c d - 1 = 0 \right\}, \tag {42} \\ \end{array}
$$
is the DTB curve for the 2-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$ . As in this case, for the parameter values belonging to  $\tau_{\mathcal{R}\mathcal{L}}$ , the points of the 2-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$  tend to  $\pm \infty$ , and the corresponding eigenvalue tends to one. Moreover, for  $(1 - d)h_{1} + c h_{2} \neq 0$ , the BCB curves of  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$  can be computed as
$$
\xi_ {\mathcal {R L}} ^ {1} = \left\{\left(h _ {1}, h _ {2}, a _ {l}, b _ {l}, c, d\right) \mid a _ {l} + d + a _ {l} d - b _ {l} c + 1 = 0 \right\},
$$
$$
\xi_ {\mathcal {R L}} ^ {2} = \left\{\left(h _ {1}, h _ {2}, a _ {r}, b _ {r}, c, d\right) \mid a _ {r} + d + a _ {r} d - b _ {r} c + 1 = 0 \right\}. \tag {43}
$$
Note that here the condition  $(1 - d)h_{1} + ch_{2}\neq 0$  guarantees a regular BCB in the sense that only one periodic point of  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$  collides with the switching boundary; for more details see [6]. Besides,
$$
\begin{array}{l} \mathcal {F} _ {\mathcal {R L}} = \left\{(h _ {1}, h _ {2}, a _ {l}, a _ {r}, b _ {l}, b _ {r}, c, d) \mid a _ {l} a _ {r} + b _ {l} c + b _ {r} c + d ^ {2} + a _ {l} a _ {r} d ^ {2} + b _ {l} b _ {r} c ^ {2} \right. \\ \left. - a _ {l} b _ {r} c d - a _ {r} b _ {l} c d + 1 = 0 \right\}, \tag {44} \\ \end{array}
$$
is the DFB curve of the 2-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$
Remark 5. One can see that for  $(1 - d)h_{1} + c h_{2}\neq 0$  the DFB curves of the fixed points  $\mathcal{O}_{\mathcal{L}}$  and  $\mathcal{O}_{\mathcal{R}}$  ( $\mathcal{F}_{\mathcal{L}}$  and  $\mathcal{F}_{\mathcal{R}}$ ) and the BCB boundaries of the 2-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$  ( $\xi_{\mathcal{R}\mathcal{L}}^{1}$  and  $\xi_{\mathcal{R}\mathcal{L}}^{2}$ ) are the same. In this case, the DFB of the fixed points can lead to the (attracting) 2-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}}$ .
A.1.4 3-cycles of the map (11) and their bifurcations
Here, we investigate the existence, stability and bifurcation structure of maximal or basic 3-cycles. Note that for the continuous map (11), basic  $n$ -cycles  $\mathcal{O}_{\mathcal{R}\mathcal{L}^{n - 1}}$  ( $n\geq 3$ ) exist in pairs with their complementary cycles  $(\mathcal{O}_{\mathcal{R}\mathcal{L}^{n - 2}\mathcal{R}})$ , and they appear via BCBs such that one of them may be attracting and the other repelling [6, 20, 38]. In this case, a BCB of basic cycles demonstrates a non-smooth fold bifurcation which includes a stable basic orbit and an unstable nonbasic orbit [6, 5, 19]. Furthermore, the complementary orbits can have nonempty stability regions such that, similar to the basic orbits, they are bounded by curves of BCBs, DTBs and DFBs [6, 5].
Basic 3-cycles  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$  and their complementary cycles  $\mathcal{O}_{\mathcal{R}^2\mathcal{L}}$ . The basic 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$  can be obtained from the equation  $T_{\mathcal{L}} \circ T_{\mathcal{L}} \circ T_{\mathcal{R}}(z_1, z_2) = (z_1, z_2)^{\mathrm{T}}$  where
$$
\begin{array}{l} T _ {\mathcal {L}} \circ T _ {\mathcal {L}} \circ T _ {\mathcal {R}} (z _ {1}, z _ {2}) = \left( \begin{array}{c c} a _ {r} (a _ {l} ^ {2} + b _ {l}   c) + b _ {r} (a _ {l}   c + c   d) & c (a _ {l} ^ {2} + b _ {l}   c) + d (a _ {l}   c + c   d) \\ b _ {r} (d ^ {2} + b _ {l}   c) + a _ {r} (a _ {l}   b _ {l} + b _ {l}   d) & d (d ^ {2} + b _ {l}   c) + c (a _ {l}   b _ {l} + b _ {l}   d) \end{array} \right) \left( \begin{array}{c} z _ {1} \\ z _ {2} \end{array} \right) \\ + \binom {h _ {1} (b _ {l} c + a _ {l} (a _ {l} + 1) + 1) + h _ {2} (a _ {l} c + c (d + 1))} {h _ {1} (b _ {l} d + b _ {l} (a _ {l} + 1)) + h _ {2} (b _ {l} c + d (d + 1) + 1)}. \tag {45} \\ \end{array}
$$
If  $\left(I - J_{\mathcal{L}}^{2}J_{\mathcal{R}}\right)$  is invertible, then the solution  $(z_{1},z_{2})^{\mathsf{T}} = \left(z_{1}^{(1)},z_{2}^{(1)}\right)^{\mathsf{T}}$  is
$$
\left(z _ {1} ^ {(1)}, z _ {2} ^ {(1)}\right) ^ {T} = \left(\frac {\left((1 - d) h _ {1} + c h _ {2}\right) G _ {1}}{G}, \frac {G _ {2}}{G}\right) ^ {\top}, \tag {46}
$$
$$
G _ {1} = a _ {l} ^ {2} d ^ {2} + a _ {l} ^ {2} d + a _ {l} ^ {2} - 2 a _ {l} b _ {l} c d - a _ {l} b _ {l} c + a _ {l} d ^ {2} + a _ {l} d + a _ {l} + b _ {l} ^ {2} c ^ {2} - b _ {l} c d + b _ {l} c + d ^ {2} + d + 1,
$$
$$
G = - a _ {l} ^ {2} a _ {r} - d ^ {3} - c \left(a _ {l} b _ {l} + a _ {l} b _ {r} + a _ {r} b _ {l} + d (2 b _ {l} + b _ {r})\right) + \left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) ^ {2} + 1,
$$
$$
\begin{array}{l} G _ {2} = h _ {2} + b _ {l} h _ {1} + d h _ {2} + d ^ {2} h _ {2} + a _ {l} b _ {l} h _ {1} + b _ {l} c h _ {2} + b _ {l} d h _ {1} - a _ {l} ^ {2} a _ {r} h _ {2} + b _ {r} d ^ {2} h _ {1} - a _ {r} b _ {l} ^ {2} c h _ {1} \\ - a _ {l} ^ {2} a _ {r} d h _ {2} + a _ {l} b _ {r} d ^ {2} h _ {1} + b _ {l} b _ {r} c ^ {2} h _ {2} - a _ {r} b _ {l} ^ {2} c ^ {2} h _ {2} - a _ {l} ^ {2} a _ {r} d ^ {2} h _ {2} + a _ {l} ^ {2} b _ {r} d ^ {2} h _ {1} + b _ {l} ^ {2} b _ {r} c ^ {2} h _ {1} \\ + a _ {l} a _ {r} b _ {l} h _ {1} - a _ {l} b _ {r} c h _ {2} - a _ {r} b _ {l} c h _ {2} + a _ {r} b _ {l} d h _ {1} + b _ {l} b _ {r} c h _ {1} - b _ {r} c d h _ {2} + a _ {l} a _ {r} b _ {l} c h _ {2} \\ + a _ {l} a _ {r} b _ {l} d h _ {1} - a _ {l} b _ {r} c d h _ {2} - b _ {l} b _ {r} c d h _ {1} + 2 a _ {l} a _ {r} b _ {l} c d h _ {2} - 2 a _ {l} b _ {l} b _ {r} c d h _ {1}. \tag {47} \\ \end{array}
$$
$$
T _ {\mathcal {R}} (z _ {1} ^ {(1)}, z _ {2} ^ {(1)}) = \left(z _ {1} ^ {(2)}, z _ {2} ^ {(2)}\right) ^ {\top} = \left(\frac {\left((1 - d) h _ {1} + c h _ {2}\right) K _ {1}}{G}, \frac {K _ {2}}{G}\right) ^ {T},
$$
$$
T _ {\mathcal {L}} \left(z _ {1} ^ {(2)}, z _ {2} ^ {(2)}\right) = \left(z _ {1} ^ {(3)}, z _ {2} ^ {(3)}\right) ^ {\top} = \left(\frac {\left((1 - d) h _ {1} + c h _ {2}\right) H _ {1}}{G}, \frac {H _ {2}}{G}\right) ^ {T}, \tag {48}
$$
$$
\begin{array}{l} K _ {1} = a _ {r} + d + a _ {l} a _ {r} + b _ {l} c + a _ {r} d + a _ {r} d ^ {2} + d ^ {2} + a _ {l} a _ {r} d - a _ {l} b _ {r} c - b _ {r} c d + a _ {l} a _ {r} d ^ {2} + b _ {l} b _ {r} c ^ {2} \\ - a _ {l} b _ {r} c d - a _ {r} b _ {l} c d + 1, \\ \end{array}
$$
$$
\begin{array}{l} K _ {2} = h _ {2} + b _ {r} h _ {1} + d h _ {2} + d ^ {2} h _ {2} + a _ {l} b _ {r} h _ {1} + b _ {r} c h _ {2} + b _ {l} d h _ {1} - a _ {l} ^ {2} a _ {r} h _ {2} + a _ {l} ^ {2} b _ {r} h _ {1} + b _ {l} d ^ {2} h _ {1} \\ + a _ {l} ^ {2} b _ {r} c h _ {2} + a _ {r} b _ {l} d ^ {2} h _ {1} + b _ {l} b _ {r} c ^ {2} h _ {2} - a _ {l} ^ {2} a _ {r} d ^ {2} h _ {2} + b _ {l} ^ {2} b _ {r} c ^ {2} h _ {1} - a _ {l} b _ {l} c h _ {2} - a _ {r} b _ {l} c h _ {2} + a _ {l} b _ {l} d h _ {1} \\ + b _ {l} b _ {r} c h _ {1} - b _ {l} c d h _ {2} + a _ {l} a _ {r} b _ {l} d h _ {1} - a _ {l} b _ {l} b _ {r} c h _ {1} - a _ {r} b _ {l} c d h _ {2} - b _ {l} b _ {r} c d h _ {1} + a _ {l} a _ {r} b _ {l} d ^ {2} h _ {1} \\ - a _ {l} b _ {l} b _ {r} c ^ {2} h _ {2} - a _ {r} b _ {l} ^ {2} c d h _ {1} + a _ {l} ^ {2} b _ {r} c d h _ {2} + a _ {l} a _ {r} b _ {l} c d h _ {2} - a _ {l} b _ {l} b _ {r} c d h _ {1} - a _ {l} ^ {2} a _ {r} d h _ {2}, \\ \end{array}
$$
$$
\begin{array}{l} H _ {1} = a _ {l} + d + a _ {l} a _ {r} + a _ {l} d + b _ {r} c + a _ {l} d ^ {2} + d ^ {2} + a _ {l} a _ {r} d - a _ {r} b _ {l} c - b _ {l} c d + a _ {l} a _ {r} d ^ {2} + b _ {l} b _ {r} c ^ {2} \\ - a _ {l} b _ {r} c d - a _ {r} b _ {l} c d + 1, \\ \end{array}
$$
$$
H _ {2} = h _ {2} + b _ {l} h _ {1} + d h _ {2} + d ^ {2} h _ {2} + b _ {l} ^ {2} c ^ {2} h _ {2} + a _ {r} b _ {l} h _ {1} + b _ {l} c h _ {2} + b _ {r} d h _ {1} - a _ {l} ^ {2} a _ {r} h _ {2} + b _ {l} ^ {2} c h _ {1}
$$
$$
\begin{array}{l} + b _ {l} d ^ {2} h _ {1} - a _ {l} ^ {2} a _ {r} d h _ {2} - b _ {l} c d h _ {2} + a _ {l} b _ {l} d ^ {2} h _ {1} + a _ {l} ^ {2} b _ {r} d h _ {1} - b _ {l} ^ {2} c d h _ {1} - a _ {l} ^ {2} a _ {r} d ^ {2} h _ {2} + b _ {l} ^ {2} b _ {r} c ^ {2} h _ {1} \\ + a _ {l} a _ {r} b _ {l} h _ {1} - a _ {l} b _ {l} c h _ {2} - a _ {l} b _ {r} c h _ {2} + a _ {l} b _ {r} d h _ {1} + a _ {l} a _ {r} b _ {l} c h _ {2} - a _ {l} b _ {l} b _ {r} c h _ {1} - a _ {l} b _ {l} c d h _ {2} \\ + a _ {l} a _ {r} b _ {l} d ^ {2} h _ {1} - a _ {l} b _ {l} b _ {r} c ^ {2} h _ {2} - a _ {r} b _ {l} ^ {2} c d h _ {1} + a _ {l} ^ {2} b _ {r} c d h _ {2} + a _ {l} a _ {r} b _ {l} c d h _ {2} - a _ {l} b _ {l} b _ {r} c d h _ {1}. \tag {49} \\ \end{array}
$$
Therefore, the existence region of the 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$  is given by
$$
\begin{array}{l} E _ {\mathcal {O} _ {\mathcal {R L} ^ {2}}} = \left\{(h _ {1}, h _ {2}, a _ {l}, b _ {l}, c, d) \mid \frac {\left((1 - d) h _ {1} + c h _ {2}\right) G _ {1}}{G} > 0, \quad \frac {\left((1 - d) h _ {1} + c h _ {2}\right) K _ {1}}{G} <   0, \right. \\ \left. \frac {\left((1 - d) h _ {1} + c h _ {2}\right) H _ {1}}{G} <   0 \right\}, \tag {50} \\ \end{array}
$$
where  $G, G_1, K_1$  and  $H_1$  are defined in (47) and (49). On the other hand, the characteristic polynomial of
$$
J _ {\mathcal {R L} ^ {2}} = J _ {\mathcal {L}} ^ {2} J _ {\mathcal {R}} = \left( \begin{array}{c c} a _ {r} (a _ {l} ^ {2} + b _ {l}   c) + b _ {r} (a _ {l}   c + c   d) & c (a _ {l} ^ {2} + b _ {l}   c) + d (a _ {l}   c + c   d) \\ b _ {r} (d ^ {2} + b _ {l}   c) + a _ {r} (a _ {l}   b _ {l} + b _ {l}   d) & d (d ^ {2} + b _ {l}   c) + c (a _ {l}   b _ {l} + b _ {l}   d) \end{array} \right),
$$
$$
\left. \mathcal {P} _ {\mathcal {O} _ {\mathcal {R L} ^ {2}}} (\lambda) = \lambda^ {2} - \left(a _ {l} ^ {2} a _ {r} + d ^ {3} + c \left(a _ {l} b _ {l} + a _ {l} b _ {r} + a _ {r} b _ {l} + d \left(2 b _ {l} + b _ {r}\right)\right)\right) \lambda + \left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) ^ {2}. \right. \tag {51}
$$
According to
$$
\mathcal {D} _ {\mathcal {R L} ^ {2}} = \left(a _ {l} d - b _ {l} c\right) ^ {2} \left(a _ {r} d - b _ {r} c\right),
$$
$$
\mathcal {P} _ {\mathcal {O} _ {\mathcal {R L} ^ {2}}} (1) = - a _ {l} ^ {2} a _ {r} - d ^ {3} - c \left(a _ {l} b _ {l} + a _ {l} b _ {r} + a _ {r} b _ {l} + d \left(2 b _ {l} + b _ {r}\right)\right)
$$
$$
\mathcal {P} _ {\mathcal {O} _ {\mathcal {R L} ^ {2}}} (- 1) = a _ {l} ^ {2} a _ {r} + d ^ {3} + c \left(a _ {l} b _ {l} + a _ {l} b _ {r} + a _ {r} b _ {l} + d \left(2 b _ {l} + b _ {r}\right)\right) + \left(a _ {r} d - b _ {r} c\right) \left(a _ {l} d - b _ {l} c\right) ^ {2} + 1, \tag {52}
$$
the stability region of the 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$  is given by
$$
\begin{array}{l} \mathcal {S} _ {\mathcal {R L} ^ {2}} = \left\{(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d) \in E _ {\mathcal {O} _ {\mathcal {R L} ^ {2}}} \mid - 1 <   (a _ {l} d - b _ {l} c) ^ {2} (a _ {r} d - b _ {r} c) <   1, \right. \\ - \left(a _ {l} d - b _ {l} c\right) ^ {2} \left(a _ {r} d - b _ {r} c\right) - 1 <   a _ {l} ^ {2} a _ {r} + d ^ {3} + c \left(a _ {l} b _ {l} + a _ {l} b _ {r} + a _ {r} b _ {l} + d \left(2 b _ {l} + b _ {r}\right)\right) \\ \left. <   \left(a _ {l} d - b _ {l} c\right) ^ {2} \left(a _ {r} d - b _ {r} c\right) + 1 \right\}. \tag {53} \\ \end{array}
$$
Furthermore, for  $(1 - d)h_{1} + c h_{2}\neq 0$  and  $G_{1},G_{2},K_{1},K_{2},H_{1},H_{2}\neq 0$ , the DTB curve for the 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$  is
$$
\begin{array}{l} \tau_ {\mathcal {R L} ^ {2}} = \left\{\left(h _ {1}, h _ {2}, a _ {l}, a _ {r}, b _ {l}, b _ {r}, c, d\right) \mid b _ {r} a _ {l} ^ {2} c d ^ {2} - a _ {r} a _ {l} ^ {2} d ^ {3} + a _ {r} a _ {l} ^ {2} - 2 b _ {r} a _ {l} b _ {l} c ^ {2} d + 2 a _ {r} a _ {l} b _ {l} c d ^ {2} \right. \\ \left. + a _ {l} b _ {l} c + b _ {r} a _ {l} c + b _ {r} b _ {l} ^ {2} c ^ {3} - a _ {r} b _ {l} ^ {2} c ^ {2} d + 2 b _ {l} c d + a _ {r} b _ {l} c + b _ {r} c d + d ^ {3} - 1 = 0 \right\}. \tag {54} \\ \end{array}
$$
$$
\text {F o r} (1 - d) h _ {1} + c h _ {2} \neq 0
$$
$$
\begin{array}{l} \xi_ {\mathcal {R L} ^ {2}} ^ {1} = \left\{\left(h _ {1}, h _ {2}, a _ {r}, b _ {r}, c, d\right) \mid K _ {1} = a _ {r} + d + a _ {l} a _ {r} + b _ {l} c + a _ {r} d + a _ {r} d ^ {2} + d ^ {2} + a _ {l} a _ {r} d \right. \\ \left. - a _ {l} b _ {r} c - b _ {r} c d + a _ {l} a _ {r} d ^ {2} + b _ {l} b _ {r} c ^ {2} - a _ {l} b _ {r} c d - a _ {r} b _ {l} c d + 1 = 0 \right\}, \\ \end{array}
$$
$$
\xi_ {\mathcal {R L} ^ {2}} ^ {2} = \left\{(h _ {1}, h _ {2}, a _ {r}, b _ {r}, c, d) \mid H _ {1} = a _ {l} + d + a _ {l} a _ {r} + a _ {l} d + b _ {r} c + a _ {l} d ^ {2} + d ^ {2} + a _ {l} a _ {r} d \right.
$$
$$
\left. - a _ {r} b _ {l} c - b _ {l} c d + a _ {l} a _ {r} d ^ {2} + b _ {l} b _ {r} c ^ {2} - a _ {l} b _ {r} c d - a _ {r} b _ {l} c d + 1 = 0 \right\}, \tag {55}
$$
are (regular) BCB curves of  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$ . Furthermore, the set
$$
\begin{array}{l} \mathcal {F} _ {\mathcal {R L ^ {2}}} = \left\{\left(h _ {1}, h _ {2}, a _ {l}, a _ {r}, b _ {l}, b _ {r}, c, d\right) \mid - b _ {r} a _ {l} ^ {2} c d ^ {2} + a _ {r} a _ {l} ^ {2} d ^ {3} + a _ {r} a _ {l} ^ {2} + 2 b _ {r} a _ {l} b _ {l} c ^ {2} d - 2 a _ {r} a _ {l} b _ {l} c d ^ {2} \right. \\ \left. + a _ {l} b _ {l} c + b _ {r} a _ {l} c - b _ {r} b _ {l} ^ {2} c ^ {3} + a _ {r} b _ {l} ^ {2} c ^ {2} d + 2 b _ {l} c d + a _ {r} b _ {l} c + b _ {r} c d + d ^ {3} + 1 = 0 \right\}, \tag {56} \\ \end{array}
$$
is the DFB curve of  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$ . As noted, the basic 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$  exists in a pair with its complementary cycle  $\mathcal{O}_{\mathcal{R}^2\mathcal{L}}$ . Moreover, the existence region of  $\mathcal{O}_{\mathcal{R}^2\mathcal{L}}$  can easily be found by interchanging the letters  $\mathcal{L}$  and  $\mathcal{R}$  in all notations of the equations (45)-(49) and considering
$$
z _ {1} ^ {(1)} <   0, \quad z _ {1} ^ {(2)} > 0, \quad z _ {1} ^ {(3)} > 0. \tag {57}
$$
Further, the stability region of the 3-cycle  $\mathcal{O}_{\mathcal{R}^2\mathcal{L}}$  for the parameter values satisfying (57) is given by
$$
\begin{array}{l} \mathcal {S} _ {\mathcal {R} ^ {2} \mathcal {L}} = \left\{\left(h _ {1}, h _ {2}, a _ {l / r}, b _ {l / r}, c, d\right) \mid - 1 <   \left(a _ {r} d - b _ {r} c\right) ^ {2} \left(a _ {l} d - b _ {l} c\right) <   1, \right. \\ - \left(a _ {r} d - b _ {r} c\right) ^ {2} \left(a _ {l} d - b _ {l} c\right) - 1 <   a _ {r} ^ {2} a _ {l} + d ^ {3} + c \left(a _ {r} b _ {r} + a _ {r} b _ {l} + a _ {l} b _ {r} + d \left(2 b _ {r} + b _ {l}\right)\right) \\ \left. <   \left(a _ {r} d - b _ {r} c\right) ^ {2} \left(a _ {l} d - b _ {l} c\right) + 1 \right\}. \tag {58} \\ \end{array}
$$
Notice that whenever the stable 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$  exists, its complementary orbit  $\mathcal{O}_{\mathcal{R}^2\mathcal{L}}$  also exists, but it is unstable. Furthermore, for  $(1 - d)h_{1} + c h_{2} \neq 0$  both the 3-cycles  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}$  and  $\mathcal{O}_{\mathcal{R}^2\mathcal{L}}$  appear at the same BCB curves (55). On the other hand, the DTB and DFB curves of the 3-cycle  $\mathcal{O}_{\mathcal{R}^2\mathcal{L}}$  are given by
$$
\begin{array}{l} \tau_ {\mathcal {R} ^ {2} \mathcal {L}} = \left\{(h _ {1}, h _ {2}, a _ {l}, a _ {r}, b _ {l}, b _ {r}, c, d) \mid b _ {l} a _ {r} ^ {2} c d ^ {2} - a _ {l} a _ {r} ^ {2} d ^ {3} + a _ {l} a _ {r} ^ {2} - 2 b _ {l} a _ {r} b _ {r} c ^ {2} d + 2 a _ {l} a _ {r} b _ {r} c d ^ {2} \right. \\ \left. + a _ {r} b _ {r} c + b _ {l} a _ {r} c + b _ {l} b _ {r} ^ {2} c ^ {3} - a _ {l} b _ {r} ^ {2} c ^ {2} d + 2 b _ {r} c d + a _ {l} b _ {r} c + b _ {l} c d + d ^ {3} - 1 = 0 \right\}, \tag {59} \\ \end{array}
$$
$$
\begin{array}{l} \mathcal {F} _ {\mathcal {R} ^ {2} \mathcal {L}} = \left\{\left(h _ {1}, h _ {2}, a _ {l}, a _ {r}, b _ {l}, b _ {r}, c, d\right) \mid - b _ {l} a _ {r} ^ {2} c d ^ {2} + a _ {l} a _ {r} ^ {2} d ^ {3} + a _ {l} a _ {r} ^ {2} + 2 b _ {l} a _ {r} b _ {r} c ^ {2} d - 2 a _ {l} a _ {r} b _ {r} c d ^ {2} \right. \\ \left. + a _ {r} b _ {r} c + b _ {l} a _ {r} c - b _ {l} b _ {r} ^ {2} c ^ {3} + a _ {l} b _ {r} ^ {2} c ^ {2} d + 2 b _ {r} c d + a _ {l} b _ {r} c + b _ {l} c d + d ^ {3} + 1 = 0 \right\}, \tag {60} \\ \end{array}
$$
A.1.5 Multiple attractor bifurcations (MABs) of the map (11)
To detect multiple attractor bifurcations for the map (11), a straightforward way is to determine the overlapping stability regions of different periodic orbits. For instance, as shown in Fig. 1A, for  $c = 0.8$ ,  $d = 0.2$ ,  $b_{l} = -0.4$ ,  $b_{r} = 0.5$ , two stability regions  $S_{\mathcal{R}\mathcal{L}}$  and  $S_{\mathcal{R}\mathcal{L}^2}$  overlap in the  $(a_l,a_r)$ -parameter plane (or in the  $(a_{11},a_{11} + w_{11})$ -parameter space for the 2d PLRNN (7)). Their overlapping region, displayed in yellow, reveals the structure of the  $(a_l,a_r)$ -parameter plane. This helps us to find various MABs. Assuming  $h_2 = 0$  and varying  $h_1$  from a negative value to a positive one, an MAB of the form
$$
\mathcal {O} _ {\mathcal {L}} ^ {s} \xleftrightarrow {h _ {1}} \mathcal {O} _ {\mathcal {R L}} ^ {s} + \mathcal {O} _ {\mathcal {R L} ^ {2}} ^ {s}, \tag {61}
$$
occurs in the overlapping region. An example of this kind of bifurcation is illustrated in Fig. S3A.
Moreover, Fig. S4 indicates, for  $c = 0.9$ ,  $d = 0.3$ ,  $b_{l} = -0.6$ ,  $b_{r} = -1.54$ , there are nonempty overlapping regions  $\mathcal{S}_{\mathcal{R}\mathcal{L}} \cap \mathcal{S}_{\mathcal{R}\mathcal{L}^2}$  and  $\mathcal{S}_{\mathcal{R}\mathcal{L}^2} \cap \mathcal{S}_{\mathcal{R}}$ . This leads to the occurrence of two different MABs given by (61) and
$$
\mathcal {O} _ {\mathcal {L}} ^ {s} \xleftrightarrow {h _ {1}} \mathcal {O} _ {\mathcal {R}} ^ {s} + \mathcal {O} _ {\mathcal {R L} ^ {2}}, \tag {62}
$$
for  $h_2 = 0$  and  $h_1$  changing from negative to positive values. Both of these bifurcations are shown in Fig. S3A and Fig. S3B, associated with the points  $P_1$  and  $P_2$  in Fig. S4. Note that in Fig. S4, all the points  $P_2$ ,  $P_3$  and  $P_4$  belong to the overlapping region  $S_{\mathcal{R}\mathcal{L}^2} \cap S_{\mathcal{R}}$  (in sky blue). These points are related to the parameter values  $c = 0.9$ ,  $d = 0.3$ ,  $b_l = -0.6$ ,  $b_r = -1.54$ ,  $a_r = -1.8$ ,  $h_2 = 0$ , and they only differ in the parameter  $a_l$ . In this case, one can see that fixing all parameters and changing only the parameter  $a_l$ , from  $P_2$  to  $P_4$ , the basins of attraction change. The corresponding basins of attraction for these three points are demonstrated in Fig. S3B (right) and Fig. S5 for  $h_1 = 0.5$  (after the bifurcation).
Figure S3: MAB at  $c = 0.9$ ,  $d = 0.3$ ,  $b_{l} = -0.6$ ,  $b_{r} = -1.54$ ,  $h_{2} = 0$ . A) Left: Bifurcation diagram for  $a_{l} = -0.44$  and  $a_{r} = -1.8$  corresponding to the point  $P_{1}$  in Fig. S4. Right: Multistability of the fixed point  $\mathcal{O}_{\mathcal{R}}^{s}$  and the 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}^s$  after the bifurcation and their basins of attraction at  $h_1 = 0.5$ . B) Left: Bifurcation diagram for  $a_{l} = -0.35$  and  $a_{r} = -2.2$  corresponding to the point  $P_{2}$  in Fig. S4. Right: Multistability of the 2-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}}^{s}$  and the 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}^s$  after the bifurcation and their basins of attraction at  $h_1 = 0.7$ .
Figure S4: Analytically calculated stability regions for a different parameter setting than used in Fig. 1. Left: Analytically calculated stability regions  $S_{\mathcal{R}}$ ,  $S_{\mathcal{RL}}$  and  $S_{\mathcal{RL}^2}$ , shown in green, red and blue, respectively, in the  $(a_{11}, a_{11} + w_{11})$ -parameter plane for  $a_{22} = 0.3$ ,  $w_{21} = -1.54$ . The overlapping regions  $S_{\mathcal{RL}} \cap S_{\mathcal{RL}^2}$  and  $S_{\mathcal{RL}^2} \cap S_{\mathcal{R}}$ , representing multi-stable regimes, are given in yellow and sky blue. Right: Bifurcation curves for the same parameter settings as determined by SCYFI. Note that SCYFI identifies additional structure (regions demarcated by gray curves) not included in our analytical derivations.
Figure S5: Multistability of the fixed point  $\mathcal{O}_{\mathcal{R}}^s$  and the 3-cycle  $\mathcal{O}_{\mathcal{R}\mathcal{L}^2}^s$  at  $c = 0.9$ ,  $d = 0.3$ ,  $b_{l} = -0.6$ ,  $b_{r} = -1.54$ ,  $a_{r} = -1.8$ ,  $h_2 = 0$  after the bifurcation and their basins of attraction at  $h_1 = 0.5$ . Left:  $a_{l} = 0.06$  (point  $P_{3}$  in Fig. S4); right:  $a_{l} = 0.26$  (point  $P_{4}$  in Fig. S4).
A.2 Proofs of theorems
A.2.1 Proof of theorem 1
Proof. Let  $\mathcal{L}(\theta)$  be some loss function employed for PLRNN training that decomposes in time as  $\mathcal{L} = \sum_{t=1}^{T} \mathcal{L}_t$ . Then
$$
\frac {\partial \mathcal {L}}{\partial \theta} = \sum_ {t = 1} ^ {T} \frac {\partial \mathcal {L} _ {t}}{\partial \theta},
$$
$$
\frac {\partial \mathcal {L} _ {t}}{\partial \theta} = \frac {\partial \mathcal {L} _ {t}}{\partial z _ {t}} \frac {\partial \boldsymbol {z} _ {t}}{\partial \theta}. \tag {63}
$$
Denoting the Jacobian of system (2) at time  $t$  by
$$
\boldsymbol {J} _ {t} := \frac {\partial F _ {\boldsymbol {\theta}} \left(\boldsymbol {z} _ {t - 1}\right)}{\partial \boldsymbol {z} _ {t - 1}} = \frac {\partial \boldsymbol {z} _ {t}}{\partial \boldsymbol {z} _ {t - 1}}, \tag {64}
$$
$$
\frac {\partial \boldsymbol {z} _ {t}}{\partial \theta} = \frac {\partial \boldsymbol {z} _ {t}}{\partial \boldsymbol {z} _ {t - 1}} \frac {\partial \boldsymbol {z} _ {t - 1}}{\partial \theta} + \frac {\partial^ {+} \boldsymbol {z} _ {t}}{\partial \theta} = \boldsymbol {J} _ {t} \frac {\partial \boldsymbol {z} _ {t - 1}}{\partial \theta} + \frac {\partial^ {+} \boldsymbol {z} _ {t}}{\partial \theta}, \tag {65}
$$
where  $\partial^{+}$  denotes the immediate partial derivative (see [48] for more details). Assume that  $\Gamma_{k}$  is a  $k$ -cycle ( $k \geq 1$ ) of (2). Thus,  $\Gamma_{k}$  is a set of temporally successive periodic points
$$
P _ {k} := \left\{\boldsymbol {z} _ {t ^ {* k}}, \boldsymbol {z} _ {t ^ {* k} - 1}, \dots , \boldsymbol {z} _ {t ^ {* k} - (k - 1)} \right\} = \left\{\boldsymbol {z} _ {t ^ {* k}}, F \left(\boldsymbol {z} _ {t ^ {* k}}\right), \dots , F _ {\boldsymbol {\theta}} ^ {k - 1} \left(\boldsymbol {z} _ {t ^ {* k}}\right) \right\}, \tag {66}
$$
such that all of them are fixed points of
$$
\boldsymbol {z} _ {t + k} = F _ {\boldsymbol {\theta}} ^ {k} \left(\boldsymbol {z} _ {t}\right) = F _ {\boldsymbol {\theta}} \left(F _ {\boldsymbol {\theta}} \left(F _ {\boldsymbol {\theta}} \left(\dots F _ {\boldsymbol {\theta}} \left(\boldsymbol {z} _ {t}\right) \dots\right)\right)\right), \tag {67}
$$
and  $k$  is the smallest such positive integer (for  $k = 1, \Gamma_1$  is a fixed point of  $F_{\theta}$ ). Similar to (65), the tangent vector  $\frac{\partial z_{t + k}}{\partial \theta}$  can be computed as
$$
\frac {\partial \boldsymbol {z} _ {t + k}}{\partial \theta} = \frac {\partial \boldsymbol {z} _ {t + k}}{\partial \boldsymbol {z} _ {t}} \frac {\partial \boldsymbol {z} _ {t}}{\partial \theta} + \frac {\partial^ {+} \boldsymbol {z} _ {t + k}}{\partial \theta} = \prod_ {r = 0} ^ {k - 1} \boldsymbol {J} _ {t + k - r} \frac {\partial \boldsymbol {z} _ {t}}{\partial \theta} + \frac {\partial^ {+} \boldsymbol {z} _ {t + k}}{\partial \theta}. \tag {68}
$$
Thus, for  $\pmb{z}_{t^{*k}} = F_{\pmb{\theta}}^{k}(\pmb{z}_{t^{*k}})$  we have
$$
\frac {\partial \boldsymbol {z} _ {t ^ {* k}}}{\partial \theta} = \prod_ {r = 0} ^ {k - 1} \boldsymbol {J} _ {t ^ {* k} - r} \frac {\partial \boldsymbol {z} _ {t ^ {* k}}}{\partial \theta} + \frac {\partial^ {+} \boldsymbol {z} _ {t ^ {* k}}}{\partial \theta}. \tag {69}
$$
Accordingly