Exploring three periodic point dynamics in $2$D spatiotemporal discrete systems

1.   Introduction

Understanding the dynamics of complex systems is fundamental for grasping a wide array of natural phenomena, from chemical reactions to population dynamics ^[1,2,3,4,5]. Over the past few decades, significant attention has been devoted to the study of spatiotemporal systems, where both spatial and temporal dimensions interact to shape system behavior ^[6,7,8,9]. In particular, two-dimensional ($2$D) spatiotemporal discrete systems have attracted considerable interest for their applications in various fields, such as digital filtering, image processing, encryption, spatial dynamical systems, and numerical solutions for partial differential equations ^{[10,11,12,13,14,15,16,17,18,19]}.

Analyzing the dynamics of periodic points in spatiotemporal discrete systems is critical for identifying complex behaviors, such as stability transitions, bifurcations, and chaos. These phenomena are not only of theoretical interest, but also have significant implications across various fields. For example, in ecology, periodic cycles are essential in modeling predator-prey interactions, where stability transitions can predict population oscillations ^[4,20]. In engineering, such dynamics inform the design of synchronized networks critical for secure communications and reliable systems ^[5]. Similarly, in physics, wave propagation in discrete media, such as electrical networks and mechanical systems, relies on understanding spatiotemporal patterns and transitions ^[1]. Our results also intersect with neuroscience, where models like the Nagumo-FitzHugh equations describe neural excitations and transitions to complex states ^[21].

This work aims to contribute to these fields by establishing a rigorous mathematical framework for analyzing $3$-periodic point dynamics and their bifurcations, providing valuable tools for predicting and understanding intricate behaviors in spatiotemporal systems. Building on this extensive body of research, this paper focuses on the stability and bifurcations in $2$D spatiotemporal discrete systems, with a particular emphasis on periodic solutions of period three. We employ analytical and numerical techniques to examine the stability of these $3$-cycles, and investigate a range of bifurcation scenarios. This work extends the analysis presented in our earlier publication ^[22], continuing the exploration of bifurcations in $2$D spatiotemporal maps ^[23,24], and providing deeper insights into the complexity of such systems.

We are particularly interested in studying the $2$D spatiotemporal discrete system

where $m\in\mathbb{Z}$ and $n\in\mathbb{N}$ represent the spatial coordinate and the time, respectively. The function $f:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ is a nonlinear function with bounded variation.

From (1.1), we can define a $1$-D recurrence on the space

as follows: For an initial condition $[x]_{0} = \left(x_{m, 0}\right)_{m = -\infty}^{\infty}\in X$, called the "boundary condition", we recursively construct a sequence of solutions

by

Alternatively, if $F:X\longrightarrow X$ is the map defined by

for all $[x] = \left(x_{i}\right)_{i = -\infty}^{i = \infty}\in X$, then system (1.1) is equivalent to the following infinite-dimensional discrete dynamical system:

The map $F$ defined by (1.3)-(1.4) is said to be induced by system (1.1). Clearly, a sequence $\left\{ [x]_{n}, \, n = 0, 1, 2, \ldots\right\}$ is a solution of system (1.1) if and only if it is a solution of system (1.4) (see ^[23,24]).

In our previous work ^[23,24], we defined various forms of cycles for $k = 1$ and $k = 2$, and provided the necessary and sufficient conditions for their stability. In the present study, we define four types of $3$rd-order cycles and present the necessary and sufficient conditions for determining their stability.

The rest of the paper is structured as follows:

In Section 2, we analyze the properties of nonlinear dynamics, singularities, and basic bifurcations in two-dimensional spatiotemporal discrete systems. Definitions 1 and 2 introduce the concept of a cycle of order $k$. In Section 2.1, we defined the four types of $3$-periodic points considered in this study. Theorem 2.1 presents the main result on the stability of these cycles.

In Section 3, we investigate a spatiotemporal quadratic map. In Section 3.1, we review the results on cycles for $k = 1$ and $k = 2$, as well as their corresponding bifurcations. Theorems 3.1, 3.2, and 3.3 present these findings, while Figure 7 provides visual illustrations of the complexity of the nonlinear problem under investigation. Section 3.2 focuses on the existence of 3-cycles in the spatiotemporal quadratic map. The definitions and propositions in this section introduce the analytical expressions for the four types of third-order cycles considered in this study. In Section 3.4, we explore bifurcations in a spatiotemporal quadratic map based on the parameters $(a, b)$. This study examines bifurcation curves, such as fold and flip bifurcations, for various cycles. These curves define regions of stability, semi-stability, and instability within the parameter plane. The analysis highlights singular points where bifurcation curves intersect or become tangent, illustrating the emergence and modification of stability regions.

Finally, in Section 4 we draw relevant conclusions and discuss future perspectives for our research in this context.

2.   Singularities and basic bifurcations in $2$D spatiotemporal discrete systems

Consider the function $f^{[k]}:\, \mathbb{R}^{k+1}\longrightarrow\mathbb{R}$, defined recursively as follows:

Definition 1. For $k\in\mathbb{N}^{*}$, the map $F^{k}:X \rightarrow X$ is defined as

1. $F^{0}([x]): = I([x]) = [x]$,

2. $F^{k}([x]): = F\left(F^{k-1}([x])\right)$ for all $[x]\in X$.

Definition 2. A sequence $P_{k} = \left(x_{i}^{*}\right)_{i = -\infty}^{\infty}\in X$ is called a periodic point of period $k$ (or a $k$-cycle, $k$-periodic point) for the dynamical system (1.1) if

1. $F^{k}(P_{k}) = P_{k}$, i.e., $f^{[k]}(x_{j-k}^{*}, \ldots, x_{j}^{*}) = x_{j}^{*}$ for all $j\in\mathbb{Z}$,

2. $F^{r}(P_{k})\neq P_{k}$ for all $r < k$. That is, for every $r < k$, there exists at least one integer $j$ such that

The set $\left\{ P_{k}, F(P_{k}), \ldots, F^{k-1}(P_{k})\right\}$ is called a $k$-periodic orbit.

Remark 1. Due to the complexity of the calculations, we often focus on specific types of singularities, detailed as follows:

1. The $k$-cycle of the form $(x_{i\mod j})_{i = -\infty}^{\infty}$, where $j\in\mathbb{N}$, is denoted by $k_{j}$.

2. A 1-periodic point (also known as a fixed point or 1-cycle) $P_{1} = \left(x_{i}^{*}\right)_{i = -\infty}^{\infty}$ is of type $1_{1}$ if $x_{i}^{*} = x^{*}\in\mathbb{R}$ for all $i\in\mathbb{Z}$, and of type $1_{2}$ if there exist two real numbers $x^{*}$ and $y^{*}$ such that $x_{2i}^{*} = x^{*}$ and $x_{2i+1}^{*} = y^{*}$ for all $i\in\mathbb{Z}$.

Similarly, we can define a 1-periodic point of type $1_{n}$ for $n\in\mathbb{N}^{*}$.

3. A 2-periodic point (also called a 2-cycle) $P_{2} = \left(x_{i}^{*}\right)_{i = -\infty}^{\infty}$ is of horizontal type (H) or type $2_{1}$ if $x_{i}^{*} = x^{*}\in\mathbb{R}$ for all $i\in\mathbb{Z}$, and of diagonal type (D) or $2_{2}$ if there exist two real numbers $x^{*}$ and $y^{*}$ such that $x_{2i}^{*} = x^{*}$ and $x_{2i+1}^{*} = y^{*}$, $f(x^{*}, y^{*}) = x^{*}$, and $f(y^{*}, x^{*}) = y^{*}$ for all $i\in\mathbb{Z}$.

In our study, a 2-periodic point $P_{2} = \left(x_{i}^{*}\right)_{i = -\infty}^{\infty}$ is considered general if there exist two real numbers $x^{*}$ and $y^{*}$ such that $x_{2i}^{*} = x^{*}$ and $x_{2i+1}^{*} = y^{*}$, and $[x]^{*}$ is of neither type $2_{1}$ nor $2_{2}$.

2.1.   Types of 3-periodic points

A 3-periodic point (also called a 3-cycle) $P_{3} = \left(x_{i}^{*}\right)_{i = -\infty}^{\infty}$ can be classified into four types:

1. Horizontal type (H) or $\mathbf{3_{1}}$ type: This occurs when $x_{i}^{*} = x^{*}\in\mathbb{R}$ for all $i\in\mathbb{Z}$. In this case, there exist three real numbers $x^{*}$, $y^{*}$, and $z^{*}$ such that $x^{*}\neq f(x^{*}, x^{*}) = y^{*}$, $y^{*}\neq f(y^{*}, y^{*}) = z^{*}$, and $z^{*}\neq f(z^{*}, z^{*}) = x^{*}$ for all $i\in\mathbb{Z}$ (see Figure 1).

2. Diagonal type (D) or $\mathbf{3_{3+}}$ type: This occurs when there exist three real numbers $x^{*}$, $y^{*}$, and $z^{*}$ such that $z^{*} = x_{3i+2}^{*}\neq x_{3i}^{*} = x^{*}\neq y^{*} = x_{3i+1}^{*}\neq z^{*}$ and $f(x^{*}, y^{*}) = x^{*}, \, f(y^{*}, z^{*}) = y^{*}, \, f(z^{*}, x^{*}) = z^{*}$ for all $i\in\mathbb{Z}$ (see Figure 2).

3. Super diagonal type (SD) or $\mathbf{3_{2}}$ type: This occurs when there exist six real numbers $x^{*}, \, y^{*}, \, z^{*}, \, t^{*}, \, u^{*}, \, v^{*}$ such that $z^{*} = f(y^{*}, x^{*})\neq x_{3i}^{*} = x^{*}\neq y^{*} = x_{3i+1}^{*}\neq t^{*} = f(x^{*}, y^{*})\neq z^{*}$, $u^{*} = f(t^{*}, z^{*})\neq f(z^{*}, t^{*}) = v^{*}$, $f(v^{*}, u^{*}) = x^{*}$, and $f(u^{*}, v^{*}) = y^{*}$ for all $i\in\mathbb{Z}$ (see Figure 3).

4. Anti diagonal type (AD) or $\mathbf{3_{3-}}$ type: If there exist three real numbers $x^{*}$, $y^{*}$, and $z^{*}$ such that $z^{*} = x_{3i+2}^{*}\neq x_{3i}^{*} = x^{*}\neq y^{*} = x_{3i+1}^{*}\neq z^{*}$ and $f(x^{*}, y^{*}) = z^{*}, \, f(y^{*}, z^{*}) = x^{*}, \, f(z^{*}, x^{*}) = y^{*}$ for all $i\in\mathbb{Z}$ (see Figure 4).

Remark 2. A $3$-periodic point $P_{3} = \left(x_{i}^{*}\right)_{i = -\infty}^{\infty}$ is said to be general if there exist three real numbers $x^{*}$, $y^{*}$, and $z^{*}$ where $z^{*} = x_{3i+2}^{*}\neq x_{3i}^{*} = x^{*}\neq y^{*} = x_{3i+1}^{*}\neq z^{*}$ and $P_{3}$ is not type $3_{1}$, not type $3_{2}$, and not type $3_{3}$ (see Figure 5).

2.2.   Stability analysis

Next, we present the conditions for the local stability of the recurrence relation (1.1) at a $k$-cycle point based on the results established in ^[23,24].

Definition 3. A $k$-cycle $P_{k}$ of (1.1) is defined as:

(1) Stable if for every $\varepsilon > 0$ and $M\geq0$ there exists $\delta > 0$ such that $\left\Vert [x]-P_{k}\right\Vert < \delta$ implies $\left\Vert F^{m}([x])-P_{k}\right\Vert < \varepsilon$ for all $m\geq M$.

(2) Attracting (sink) if there exists $\delta > 0$ such that $\left\Vert [x]-P_{k}\right\Vert < \delta$ implies $\lim_{m\to+\infty}\left\Vert F^{m}([x])-P_{k}\right\Vert = 0$.

(3) Repulsive if there exists $\delta > 0$ such that $\left\Vert [x]-P_{k}\right\Vert < \delta$ implies $\lim_{m\to-\infty}\left\Vert F^{m}([x])-P_{k}\right\Vert = 0$.

(4) Asymptotically stable if it is both stable and attracting.

(5) Unstable if it is not stable.

Remark 3. A $k$-cycle $P_{k}$ of map (1.1) is said to be semi-stable if it is unstable, and for every $\varepsilon > 0$ and $M\geq0$ there exists $[x]\in X$ such that $\left\Vert F^{m}([x])-P_{k}\right\Vert < \varepsilon$ for all $m\geq M$.

The stability of a $k$-cycle $P_{k}$ is determined by analyzing the spectrum $\sigma(J_{P_{k}})$ of the Jacobian matrix $J_{P_{k}}$ at the $k$-cycle $P_{k}$. The matrix representing the Jacobian operator of the map $F^{k}$ at a $k$-periodic point $P_{k} = (x_{i}^{*})_{i = -\infty}^{\infty}$ is given by

where

We then have the following result:

Theorem 1. Let $P_{k}\in X$ be a $k$-periodic point of (1.1), and let $J_{P_{k}}$ be the Jacobian matrix of the map $F^{k}$ at $P_{k}$. Then:

(ⅰ) If

then $P_{k}$ is asymptotically stable.

(ⅱ) If

then $P_{k}$ is unstable. Moreover, if

then $P_{k}$ is semi-stable.

(ⅲ) If

then $P_{k}$ is repulsive.

Proof. The proof can be easily derived from [23, Proposition 1, p. 4] by using the fact that every $k$-periodic point of the map $F$ is a fixed point for $F^{k}$. □

Determining the stability of $k$-cycles through the spectrum of the Jacobian matrix is a significant mathematical challenge (see ^[23,24,25]). Next, we present a stability result for the $3$-cycle of type $3_{1}$, where the spectrum of the Jacobian matrix has been well characterized. Consider the function $h:\mathbb{R}^{4}\longrightarrow\mathbb{R}$ defined by $h: = f^{[3]}$, where

The Jacobian matrix $J_{P_{3}}$ of $F^{3}$ at the $3$-cycle $P_{3} = (x_{i}^{*})_{i = -\infty}^{\infty}\in X$ is given by

where

The brackets around $\Phi_{0}$ represent the $0$-$0$ component of the matrix $J_{P_{3}}$.

Theorem 2. Let $P_{3_{1}} = (x_{i}^{*} = x_{3_{1}}^{*})_{i = -\infty}^{\infty}$ be a 3-periodic point of type $3_{1}$ for system (3.1). Then, the spectrum of the Jacobian matrix $J_{P_{3_{1}}}$ is given by

where

Additionally, the $3$-cycle $P_{3_{1}}$ satisfies the following stability conditions:

1. Asymptotically stable if $|\Phi|+|\Psi|+|\Omega|+|\varTheta| < 1$,

2. Unstable if $|\Phi|+|\Psi|+|\Omega|+|\varTheta| > 1$,

3. Repulsive if $\big||\Phi|-|\Psi|+|\Omega|-|\varTheta|\big| > 1$.

Proof. From the definition of the Jacobian matrix $J_{P_{3_{1}}}$ associated with the $3$-periodic point, $J_{P_{3_{1}}}$ can be expressed in the following block-diagonal form:

Using this structure, $J_{P_{3_{1}}}$ can be rewritten as

where $\Phi, \Psi, \Omega$, and $\varTheta$ are defined as in the theorem, $I$ denotes the identity operator, and $S$ denotes the shift operator on the Hilbert space $X$. According to the polynomial spectral mapping theorem (see [26, Theorem 1, p. 53]), the spectrum of $J_{P_{3_{1}}}$ is given by

This corresponds to the formula for $\sigma(J_{P_{3_{1}}})$ provided in the theorem, i.e.

Furthermore, the spectral radius of $J_{P_{3_{1}}}$ is

and the minimum spectral radius is

Based on the spectral radius and the minimum spectral radius, the stability criteria (asymptotic stability, instability, and repulsiveness) directly follow from Theorem 1. □

2.3.   Bifurcation analysis

Consider the dynamics described by Eqs (1.1)–(1.4), which depend on two real parameters $a$ and $b$. Let $J_{P_{k}}$ denote the Jacobian matrix of the map $F^{k}$ at a $k$-cycle $P_{k}$. The $k$-th order bifurcation curves $\Lambda^{P_{k}}$ in the $(a, b)$ parameter plane are defined by the following system:

For specific configurations of the multiplier set $S\subset\mathbb{C}$, the following classical cases of bifurcation curves can be distinguished (as described in ^[27,28]):

● Fold bifurcation curve $\Lambda_{(k)_{0}}^{P_{k}}$: This corresponds to parameter points $(a, b)\in\mathbb{R}\times\mathbb{R}$ where $S = \{+1\}$ for a $k$-cycle.

● Flip bifurcation curve $\Lambda_{k}^{P_{k}}$: This corresponds to parameter points $(a, b)\in\mathbb{R}\times\mathbb{R}$ where $S = \{-1\}$ for a $k$-cycle $P_{k}\in X$.

3.   Study of a spatiotemporal quadratic map

In ^[29], the author asks some questions concerning noninvertible "spatio-discrete temporal" maps, and considers the following spatiotemporal quadratic map:

This map is of significant importance as it can serve as a fundamental building block for understanding nonlinear cases, similar to the role played by the logistic map.

In the following, we recall some results stated in the paper ^[23] concerning the existence of fixed points of type $1_{1}$ and $1_{2}$ in addition to the 2-periodic points of type $2_{1}$ for the $2$D spatiotemporal discrete system (3.1), depending on the variation of the parameters in the $(a, b)$-plane.

Furthermore, we examine various bifurcation scenarios that can arise in relation to this quadratic map. These bifurcations correspond to changes in the dynamic behavior of system (3.1) as the parameters $a$ and $b$ are varied.

The 2D bifurcation diagram Figure 6 represents the stability zones of the fixed point of type $1_{1}$ (noted $P_{1_{1}}$) in red, the fixed point of type $1_{2}$ (noted $P_{1_{2}}$) in green, and the 2-periodic point of type $2_{1}$ (noted $P_{2_{1}}$) in blue. The diagram is plotted in the $(a, b)$-parameter plane, where the values of the parameters $a$ and $b$ are varied.

By examining this bifurcation diagram, one can observe how the stability regions of these points change as the parameters $a$ and $b$ are adjusted. It provides insights into the parameter ranges where each point exhibits stability or undergoes bifurcations, allowing for a visual representation of the system's dynamics in the $(a, b)$-parameter space.

3.1.   Reminder of some results on cycles 1 and 2

As the parameters $a$ and $b$ vary across the set of real numbers, the map (3.1) can exhibit various equilibrium solutions. These equilibrium solutions correspond to the fixed points and 2-cycles listed in Remark 1.

Depending on the specific values of $a$ and $b$, the map (3.1) may possess stable fixed points or periodic orbits. These equilibrium solutions can be identified by analyzing the stability properties of the map at different parameter values. It is important to note that the specific behavior of the map (3.1) and the occurrence of equilibrium solutions may depend on the functional form and properties of the map itself, as well as the specific values of the parameters $a$ and $b$. Further analysis and numerical investigation may be required to determine the precise equilibrium solutions and their stability characteristics for different parameter values within the real number range.

3.1.1.   Fixed points of type $1_{1}$

For $b^{2}-2b+1-4a\geq0$, the map (3.1) has two fixed points of type $1_{1}$, given by

where

3.1.2.   Fixed points of type $1_{2}$

For $-3b^{2}+6b-3-4a\geq0$, the map (3.1) has two fixed points of type $1_{2}$, expressed as

where

3.1.3.   Periodic point of period 2 of type $2_{1}$

For $-3-2b+b^{2}-4a\geq0$, the map (3.1) has a 2-periodic orbit $\{P_{2_{1}}, P'_{2_{1}}\}$ of type $2_{1}$, represented by

where

In Figure 6, each colored region corresponds to the existence of at least one stable singularity of the map $f$ for specific values of the parameters $a$ and $b$. The red region indicates a stable fixed point of type $1_{1}$, the green region signifies a stable fixed point of type $1_{2}$, and the blue region represents the presence of an attractive $2$-periodic orbit $\{P_{2_{1}}, P'_{2_{1}}\}$. Conversely, the white region signifies the absence of stable cycles among these singularities. The analytically derived $(a, b)$-parametric plane shown in Figure 6 follows the methodology described in references ^[23,24].

3.1.4.   Bifurcation curves

The analytical studies presented in ^[23,24,25] allow for the determination of bifurcation curves in the $(a, b)$-plane that correspond to the fixed points of types $1_{1}$ and $1_{2}$, as well as the $2$-cycle of type $2_{1}$ in the quadratic map defined by equation (3.1). These references, particularly ^[23,24], offer detailed results and insights into the nature of these bifurcations.

The analytical expressions provided in these works precisely describe the locations and shapes of the bifurcation curves in the $(a, b)$-plane. These curves demarcate regions where the stability of fixed points and periodic orbits undergo qualitative changes as the parameters $a$ and $b$ are varied.

Theorem 3. Let $P_{1_{1}} = (x_{i} = x_{1_{1}}^{*})_{i = -\infty}^{i = \infty}$ be a fixed point of type $1_{1}$ for the map (3.1). Then:

1. The fold bifurcation curve associated with the fixed point $P_{1_{1}}$, denoted by $\Lambda_{(1)_{0}}^{P_{1_{1}}}$, is given in the $(a, b)$-plane by

2. The flip bifurcation curve associated with the fixed point $P_{1_{1}}$, denoted by $\Lambda_{1}^{P_{1_{1}}}$, is given in the $(a, b)$-plane by

Proof. Let $A: = f'_{y}(x_{1_{1}}^{*}, x_{1_{1}}^{*}) = b$ and $B: = f'_{x}(x_{1_{1}}^{*}, x_{1_{1}}^{*}) = -b+1-\sqrt{b^{2}-4a-2b+1}$. The Jacobian matrix $J_{P_{1_{1}}}$ at $P_{1_{1}}$ is

where $S$ is the shift operator ^[30,31]. According to the polynomial spectral mapping theorem (see Theorem 1, p. 53, in Halmos ^[26]), the spectrum of $J_{P_{1_{1}}}$ is

Here, $\sigma(I) = \{A\}$ and $\sigma(S) = \{z\in\mathbb{C}:|z| = 1\}$ (see ^[30,31]), so

Using the bifurcation conditions, $S = \{+1\}$ for fold bifurcations, and $S = \{-1\}$ for flip bifurcations, as outlined in Section 3.4, the equations for the bifurcation curves are derived as

and

This completes the proof. □

Theorem 4. Let $P_{1_{2}} = (x_{2i} = x_{1_{2}}^{*}, x_{2i+1} = y_{1_{2}}^{*})_{i = -\infty}^{i = \infty}$ be a fixed point of type $1_{2}$ for the map (3.1). Then:

1. The fold bifurcation curve associated with $P_{1_{2}}$, denoted by $\Lambda_{(1)_{0}}^{P_{1_{2}}}$, is given in the $(a, b)$-plane by

2. The flip bifurcation curve associated with $P_{1_{2}}$, denoted by $\Lambda_{1}^{P_{1_{2}}}$, is given in the $(a, b)$-plane by

Proof. Let $A: = f'_{y}(x_{1_{2}}^{*}, y_{1_{2}}^{*}) = b$, $B: = f'_{x}(x_{1_{2}}^{*}, y_{1_{2}}^{*}) = -b+1+\sqrt{b^{2}-4a-2b+1}$, and $B': = f'_{x}(y_{1_{2}}^{*}, x_{1_{2}}^{*}) = -b+1-\sqrt{b^{2}-4a-2b+1}$. The Jacobian matrix $J_{P_{1_{2}}}$ is given by

where $S_{BB'}$ is a weighted shift ^[26,31] with weight $(...\, , \, B'\, , \, (B)\, , \, B', \, ...)$. According to Theorem 6, p.6 in ^[23,25], we have

where

and

and the bifurcation equations follow from the conditions for the fold and flip bifurcations applied to the quadratic system (3.1). □

Theorem 5. Let $P_{2_{1}}$ be a $2$-cycle of type $2_{1}$. Then:

1. The fold bifurcation curve associated with the $2$-cycle $P_{2_{1}} = (x_{i} = x_{2_{1}}^{*})_{i = -\infty}^{i = \infty}$, denoted by $\Lambda_{(2)_{0}}^{P_{2_{1}}}$, is given in the $(a, b)$-plane by the equation

2. The flip bifurcation curve associated with the $2$-cycle $P_{2_{1}}$, denoted by $\Lambda_{2}^{P_{2_{1}}}$, is given in the $(a, b)$-plane by the equation

Proof. Let $P_{2_{1}}$ be a 2-cycle of type $2_{1}$, given by $P_{2_{1}} = (x_{i} = x_{2_{1}}^{*})_{i = -\infty}^{\infty}$. By definition, the 2-cycle $P_{2_{1}}$ is a fixed point of the second iterate map $F^{2}$, and the Jacobian matrix $J_{P_{2_{1}}}$ of $F^{2}$ at $P_{2_{1}}$ can be written as

where $A: = \left.\frac{\partial f^{[2]}(x, y, z)}{\partial z}\right|_{(x_{2_{1}}^{*}, x_{2_{1}}^{*}, x_{2_{1}}^{*})}$, $B: = \left.\frac{\partial f^{[2]}(x, y, z)}{\partial y}\right|_{(x_{2_{1}}^{*}, x_{2_{1}}^{*}, x_{2_{1}}^{*})}$, and $C: = \left.\frac{\partial f^{[2]}(x, y, z)}{\partial x}\right|_{(x_{2_{1}}^{*}, x_{2_{1}}^{*}, x_{2_{1}}^{*})}$.

The Jacobian matrix $J_{P_{2_{1}}}$ can then be expressed as

where $S$ is the shift operator. According to the polynomial spectral mapping theorem, the spectrum of $J_{P_{2_{1}}}$ is given by

where $\sigma(I) = \{A\}$ and $\sigma(S) = \{z\in\mathbb{C}:|z| = 1\}$. Hence, the spectrum becomes

For the fold bifurcation, the condition $S = \left\{ 1\right\}$ leads to the equation

which corresponds to the fold bifurcation curve

For the flip bifurcation, the condition $S = \left\{ -1\right\}$ leads to the equation

which corresponds to the flip bifurcation curve

□

In Figure 7a, the stability region of the fixed point $P_{2_{1}}$ (highlighted in red) is enclosed by the curves $\Lambda_{1}^{P_{1_{1}}}$ and $\Lambda_{(1)_{0}}^{P_{1_{1}}}$. Similarly, the stability region of the fixed point $P_{1_{2}}$ (depicted in green) is bounded by the curves $\Lambda_{(1)_{0}}^{II}$ and $\Lambda_{1}^{II}$.

Notably, in the region $\Delta$, bounded on the left by the curve $\Lambda_{(1)_{0}}^{P_{1_{1}}}$, no cycles exist. Upon crossing the curve $\Lambda_{(1)_{0}}^{P_{1_{1}}}$ from region $\Delta$, the spectrum $\sigma(J_{P_{1_{1}}})$ of the Jacobian matrix at the fixed point $P_{1_{1}}$ passes through the value $1$. This transition induces a tangent bifurcation, resulting in the birth of a $2$-cycle of type $2_{1}$.

Similarly, in the region between $\Lambda_{(1)_{0}}^{P_{1_{2}}}$ and $\Lambda_{1}^{P_{1_{2}}}$, bounded on the right by $\Lambda_{(1)_{0}}^{P_{1_{2}}}$, the spectrum of the Jacobian matrix $J_{P_{1_{2}}}$ passes through the value $-1$. This crossing results in a flip bifurcation, leading to the creation of a $2$-cycle of type $2_{1}$.

In Figure 7b, the stability region of the 2-periodic point $P_{2_{1}}$ is delimited by the curves $^{1}\Lambda_{2}^{P_{2_{1}}}$, $^{2}\Lambda_{2}^{P_{2_{1}}}$, $^{1}\Lambda_{(2)_{0}}^{P_{2_{1}}}$, $^{2}\Lambda_{(2)_{0}}^{P_{2_{1}}}$, $^{3}\Lambda_{(2)_{0}}^{P_{2_{1}}}$, and $^{4}\Lambda_{(2)_{0}}^{P_{2_{1}}}$. When the spectrum $\sigma(J_{P_{2_{1}}})$ crosses the value $1$, it leads to bifurcations characterized by the union of curves $^{1}\Lambda_{(2)_{0}}^{P_{2_{1}}}\cup{}^{2}\Lambda_{(2)_{0}}^{P_{2_{1}}}\cup{}^{3}\Lambda_{(2)_{0}}^{P_{2_{1}}}\cup{}^{4}\Lambda_{(2)_{0}}^{P_{2_{1}}}$, as described by Eq (3.2). Similarly, the bifurcations corresponding to the spectrum reaching $-1$ are characterised by the curves $^{1}\Lambda_{(2)_{0}}^{P_{2_{1}}}\cup{}^{2}\Lambda_{(2)_{0}}^{P_{2_{1}}}\cup{}^{3}\Lambda_{(2)_{0}}^{P_{2_{1}}}\cup{}^{4}\Lambda_{(2)_{0}}^{P_{2_{1}}}$, derived from equation (3.3). Similarly, the bifurcation curves corresponding to the spectrum reaching $-1$ are defined by $^{1}\Lambda_{2}^{P_{2_{1}}}\cup{}^{2}\Lambda_{2}^{P_{2_{1}}}$, derived from Eq (3.3).

Remark 4. In Figure 7a, the singular points $A$ and $B$ correspond to cases where a fold bifurcation curve $\Lambda_{(1)_{0}}^{P_{1_{1}}}$ is tangent to a Flip bifurcation curves $^{1, 2}\Lambda_{1}^{P_{1_{1}}}$. At these points, the multipliers satisfy $S_{1} = -S_{2} = \left\{ 1\right\}$, indicating a co-dimension-$2$ bifurcation. Points $C$ and $D$ have distinct origins: Point $C$ results from the intersection of two flip bifurcation curves $^{2}\Lambda_{1}^{P_{1_{1}}}$ and $^{3}\Lambda_{1}^{P_{1_{1}}}$ where $S_{1} = S_{2} = \left\{ -1\right\}$. Point $D$, however, emerges from the intersection of a fold bifurcation curve $\Lambda_{(2)_{0}}^{P_{2_{1}}}$ with a flip bifurcation curve $^{1}\Lambda_{1}^{P_{1_{1}}}$ where $S_{1} = -S_{2} = \left\{ 1\right\}$. All other intersections are solely caused by the projection of the bifurcation curves onto the $(a, b)$-plane.

As indicated in Figure 7a the singular point $C$ shown in Figure 7b involves, in addition to the bifurcation curves $^{2}\Lambda_{1}^{P_{1_{1}}}$ of the $1$-cycle, the fold bifurcation curves of the 2-cycle, $^{1, 2, 3}\Lambda_{(2)_{0}}^{P_{2_{1}}}$. At the singular point $K$, we observe the intersection of the fold bifurcation curves of the 2-cycle, $^{1}\Lambda_{(2)_{0}}^{P_{2_{1}}}$ and $^{2}\Lambda_{(2)_{0}}^{P_{2_{1}}}$. The singular point $E$ is of codimension greater than 3; it arises from the intersection of three bifurcation curves: The flip bifurcation of a 1-cycle, $^{1}\Lambda_{1}^{P_{1_{1}}}$, and the two Fold bifurcation curves of the 2-cycle, $^{2}\Lambda_{(2)_{0}}^{P_{2_{1}}}$ and $^{3}\Lambda_{(2)_{0}}^{P_{2_{1}}}$. The multipliers at this point satisfy $S_{1} = -S_{2} = \{1\}$. All other intersections are solely due to the projection of bifurcation curves onto the $(a, b)$-plane.

3.2.   Existence of $3$-cycles of the spatiotemporal quadratic map

The presence of a period-$3$ cycle in the logistic function has been well-established (see ^{[32,33,34,35]}). This finding suggests the existence of even longer periodic orbits ^[29,33]. The period-doubling route to chaos, a well-known phenomenon, is demonstrably dependent on the control parameter's strength.

Setting $b = 0$ in system (3.1) reduces it to the quadratic recurrence $x_{n+1} = x_{n}^{2}+a$. In the bifurcation diagram (Figure 8a), the $3$-period window of the quadratic map near $a = -1.7660$ is displayed. Similarly, if $b$ is close to $0$, for example, $b = 0.02$, could system (3.1) exhibit the same behavior?

Effectively, for $b = 0.02$ and $a \in [-1.8\, , \, -1.7]$, the dynamics of system (3.1) reveal a period-3 window, followed by a sequence of chaotic regimes (see Figure 8). This transition is marked by the emergence of a chaotic attractor, which arises after the destabilization of the $3$-cycle of type $3_1$, as illustrated in Figure 9. In the remainder of this paper, we provide a detailed exposition of the existence of $3$-cycles of different types, along with a stability and bifurcation analysis, including a geometric study of the foliation structure within the $(a, b)$-parameter plane.

3.2.1.   Case of $3$-cycles of type $3_{1}$ (H type)

A $3$-cycle of type $3_{1}$ is a solution to the equation $h(x, x, x, x)-x = 0$ where $f(x^{*}, x^{*})\neq x^{*}$. The polynomial $h(x, x, x, x)-x$ has degree $8$ and is divisible by $f(x, x)-x$. Defining $Q(X)$ as the quotient of these two expressions gives a polynomial of degree $6$, given by

If $x^{*}$ is a root of $h(x, x, x, x)-x$, then $y = f(x, x)$ is also a root. The polynomial $Q$ has 6 roots, which form two $3$-cycles of type $3_{1}$.

When $b = 0.002$, $a = -1.766$, system (3.1) exhibits a stable $3$-cycle of type $3_{1}$ denoted $P_{3_{1}}$ and given by $P_{3_{1}} = \left(x_{i}^{*} = -1.764267556\right)_{i = -\infty}^{\infty}$. The Jacobian matrix of $J_{P_{3_{1}}}$ can be obtained as (2.1)–(2.2), and the spectrum of the Jacobian matrix is given by

when the values of $\Phi, \Psi, \Omega$, and $\varTheta$ are approximately $0.0, -0.3044171416\times10^{-5}, -0.01909377224$, and $-0.7703056278$, respectively, so the spectrum $\sigma(J_{P_{3_{1}}})$ lies inside the unit disk. Therefore, the $3$-cycle $P_{3_{1}}$ is stable for the value $a = -1.766$ (as depicted in Figure 10a–10b), followed by a sequence of stable/unstable regimes. It is worth noting that the $3$-cycle $P_{3_{1}}$ becomes destabilized and gives rise to a strange attractor when $a = -1.740$ (see Figure 9). On the other hand, an unstable $3$-cycle orbit of type $3_{1}$ for system (3.1) is given by

The Jacobian matrix of $J_{P'_{3_{1}}}$ can be derived using Eqs (2.1)–(2.2), and its spectrum its given by

when the values of $\Phi, \Psi, \Omega$, and $\varTheta$ are approximately $0.0, -0.0004864819978, -0.1732908362$, and $2.246615678$, respectively, so the spectrum $\sigma(J_{P'_{3_{1}}})$ lies outside the unit disk (see Figure 11).

3.3.   Restriction of the study on space $X_{3}$

Analytically characterizing the stability of $3$-cycles other than type $3_{1}$ is challenging. Hence, we have limited the study of stability to the space $X_{3}$, defined as

Since we consider only $3$-cycles, we can construct a new Jacobian matrix of dimension $3\times3$. Let us consider the function $G:\mathbb{R}^{3}\rightarrow \mathbb{R}$ given by $G(x, y, z): = h(x, y, z, x)$. Then, we define the function $\widetilde{G}:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ by:

A general $3$-cycle is therefore a fixed point of type $1_{1}$, (i.e., $\widetilde{G}(x, y, z) = (x, y, z)$) which is neither a $1$-cycle nor a $2$-cycle. The Jacobian matrix of $\widetilde{G}$ at $P_{3} = (x_{3i} = x_{1}^{*}, x_{3i+1} = x_{2}^{*}, x_{3i+2} = x_{3}^{*})_{i = -\infty}^{i = \infty}$ is

3.3.1.   Study of $3$-cycle of type $3_{3-}$ (AD Type)

The cycle of type $3_{3-}$, denoted $P_{3_{3-}} = (x_{3i} = x^{*}, x_{3i+1} = y^{*}, x_{3i+2} = z^{*})_{i = -\infty}^{\infty}$, is defined by the following system of equations:

where $(x^{*}, y^{*}, z^{*})\neq(x^{*}, x^{*}, x^{*})$. In this case, the partial derivatives of $G(x^{*}, y^{*}, z^{*})$ are given by

Let $B: = \partial_{1}f(x^{*}, y^{*})\partial_{2}f(y^{*}, z^{*})+\partial_{1}f(y^{*}, z^{*})\partial_{2}f(z^{*}, x^{*})+\partial_{1}f(z^{*}, x^{*})\partial_{2}f(x^{*}, y^{*})$. Then,

The Jacobian matrix for this cycle is

The characteristic polynomial of the matrix $J_{P_{3_{3-}}}^{(3)}$ is

When $b = 0.002$ and $a = -1.766$, two anti-diagonal ($3_{3-}$ type) $3$-cycles are obtained, one of which is stable and the other unstable.

● The stable $3$-cycle of type $3_{3-}$ is given by

The Jacobian matrix is

and the moduli of the eigenvalues are $0.525488$, $0.525488$, and $0.52358$.

● The second $3$-cycle of type $3_{3-}$ is unstable, and is given by

The Jacobian matrix is

and the moduli of the eigenvalues are $2.4427$, $2.4427$, and $2.43782$.

The bifurcation occurs with $b = 0.002$ and $a\in(-1.755006, -1.755007)$. For the value $a = -1.755007$, we obtain

The Jacobian matrix is

The absolute values of the eigenvalues are $1.002995493509572$, $1.002995493509572$, and $1.0000000000002065$.

3.3.2.   Study of $3$-cycle of type $3_{3+}$ (D type)

The $3$-cycle of type $3_{3+}$, denoted by $P_{3_{3+}} = (x_{3i} = x^{*}, x_{3i+1} = y^{*}, x_{3i+2} = z^{*})_{i = -\infty}^{\infty}$, is defined by the system of equations

with $(x^{*}, y^{*}, z^{*})\neq(x^{*}, x^{*}, x^{*})$. In this case, the partial derivatives of $G(x^{*}, y^{*}, z^{*})$ are given by

Let $A = \partial_{1}f(x^{*}, y^{*})+\partial_{1}f(y^{*}, z^{*})+\partial_{1}f(z^{*}, x^{*})$ and $B = \partial_{2}f(x^{*}, y^{*})\partial_{2}f(y^{*}, z^{*})\partial_{2}f(z^{*}, x^{*})$. Then, the Jacobian matrix for this $3$-cycle is

For the function $f(x, y) = x^{2}+by+a$, $P_{3_{3+}} = (x_{3i} = x^{*}, x_{3i+1} = y^{*}, x_{3i+2} = z^{*})_{i = -\infty}^{\infty}$ is a $3$-cycle of type $3_{3+}$ if and only if $x^{*}$ (or $y^{*}$ or $z^{*}$) is a root of the following degree $6$ polynomial:

For numerical calculations, with $b = 0.002$ and $a = -1.766$, we find two unstable $3_{3+}$ type cycles:

1. The first $3$-cycle is given by $P_{3_{3+}} = (-0.920506, -0.918506, 1.92051)$. The Jacobian matrix is

with eigenvalue moduli $56.668, 6.23978$, and $6.19921$.

2. The second $3$-cycle is $P_{3_{3+}} = (-0.918507, 1.91851, 1.92051)$. The Jacobian matrix is

with eigenvalue moduli $56.6679, 56.491$, and $6.19923$.

3.3.3.   Analysis of the $3$-cycle of type $3_{2}$ (SD Type)

The standard definitions of cycles do not directly apply here due to the horizontal periodicity of order $2$. To analyze the 3-cycle of type $3_{2}$ (SD type), we use the following function:

The system of equations defining the 3-cycle of type $3_{2}$, denoted $P_{3_{2}} = (x_{2i} = x^{*}, x_{2i+1} = y^{*})_{i = -\infty}^{\infty}$, is

The Jacobian matrix at the 3-cycle $P_{3_{2}}$ is given by

Explicitly, it can be written as

where $K = by+x^{2}+a$ and $M = bx+y^{2}+a$.

This type of 3-cycle appears in the bifurcation diagram (see Figure 8) near the parameter values $b = 0.02$ and $a = -1.794$. For $b = 0.02$ and $a = -1.766$, two stable $3_{2}$-cycles and one unstable 3-cycle of type $3_{2}$ are observed

1. First stable $3$-cycle of type $3_{2}$:

The Jacobian matrix is

with eigenvalue modulus $0.7361$.

2. Second stable $3$-cycle of type $3_{2}$:

The Jacobian matrix is

with eigenvalue modulus $0.7361$.

3. Unstable $3$-cycle of type $3_{2}$:

The Jacobian matrix is

with eigenvalues having a modulus of $0.7361$.

3.4.   Bifurcation analysis

Figure 12 illustrates the bifurcation diagram of the spatiotemporal quadratic map $f$ in the $(a, \, b)$ parameter plane for the $3_{1}$-cycle in the case under consideration. The fold and flip bifurcations related to the fixed points of types $1_{1}$ and $1_{2}$ of $f$ have already been analyzed in Section 3.1.1 (see also ^[23,24]). The definitions of fold and flip bifurcations are provided in Section 2.3. Specifically, in the $(a, \, b)$ parameter plane, the flip bifurcation curves are denoted by $^{j}\Lambda_{3}^{3_{1}}$, where $j = 1, \, 2$, and the fold bifurcation curves are denoted by $^{j}\Lambda_{(3)_{0}}^{3_{1}}$, where $j = 1, \, 2$. The index $j$ differentiates between curves associated with the same cycle. The blue region represents the stability region of the $3_{1}$-cycle (see Figure 12a).

At the singular points $C_{i}$, for $i = 1, \, 2, \, 4$, the flip curves $^{j}\Lambda_{3}^{3_{1}}$ (for $j = 1, \, 2$) are tangent to the fold curves $^{j}\Lambda_{(3)_{0}}^{3_{1}}$ (for $j = 1, \, 2$). At these points, two of the multipliers are $S_{1} = -S_{2} = \{1\}$ (see, for example, ^[36]). At the singular point $C_{3}$, the fold curves $^{1}\Lambda_{(3)_{0}}^{3_{1}}$ and $^{2}\Lambda_{(3)_{0}}^{3_{1}}$ are tangential to each other (see Figure 12b).

The foliation of the parameter plane associated with the map $f$ at the $3_{1}$-cycle is qualitatively shown in Figure 13. A fold curve $^{1}\Lambda_{(3)_{0}}^{3_{1}}$ (respectively $^{2}\Lambda_{(3)_{0}}^{3_{1}}$) represents the junction of two sheets: one associated with a semi-stable $3_{1}$-cycle, and the other with an unstable $3_{1}$-cycle. The flip curve $^{1}\Lambda_{3}^{3_{1}}$ (respectively $^{2}\Lambda_{3}^{3_{1}}$) is located on the sheets related to the $3_{1}$-cycles. It consists of two segments that meet at the points $C_{i}$ (for $i = 1, \, 2, \, 4$), each segment being the beginning of a sheet: One is associated with a semi-stable $3_{1}$-cycle, and the other with an unstable $3_{1}$-cycle.

Figure 14 displays the bifurcation diagram of the spatiotemporal quadratic map $f$ in the $(a, \, b)$ parameter plane for the $3_{3-}$-cycle. The corresponding foliation is qualitatively depicted in Figure 15. The fold curves $^{j}\Lambda_{(3)_{0}}^{3_{3-}}$ (for $j = 1, \, 2$), which represent the junction of two sheets, correspond to the emergence of two $3_{3-}$ cycles: one stable or semi-stable and the other unstable. In Figure 15, the orange and blue regions indicate the stability or semi-stability of the $3_{3-}$ cycles, while the white regions correspond to the instability of the $3_{3-}$ cycles. The curves $^{j}\Lambda_{3}^{3_{3-}}$ (for $j = 1, \, 2$) represent two distinct branches of the flip bifurcation curves associated with the $3_{3-}$ cycles and their corresponding fold curves.

At the singular point $A$, the flip curves $^{j}\Lambda_{3}^{3_{3-}}$ (for $j = 1, \, 2$) are tangential to the corresponding fold curves $^{j}\Lambda_{(3)_{0}}^{3_{3-}}$. At this point, two of the three multipliers associated with $A$ are $S_{1} = -S_{2} = \{1\}$.

Figure 16 shows the bifurcation curves of the spatiotemporal quadratic map $f$ in the $(a, \, b)$ parameter plane for the $3_{+}$ cycle. The corresponding foliation is qualitatively illustrated in Figure 17. In this figure, the flip bifurcation curves are denoted by $^{j}\Lambda_{3}^{3_{3+}}$, while the fold bifurcation curves are represented by $^{j}\Lambda_{(3)_{0}}^{3_{3+}}$, where $j = 1, \, 2, \, 3$. The fold curves $\Lambda_{(3)_{0}}^{3_{3+}} = {}^{1}\Lambda_{(3)_{0}}^{3_{3+}}\cup{}^{2}\Lambda_{(3)_{0}}^{3_{3+}}\cup{}^{3}\Lambda_{(3)_{0}}^{3_{3+}}$ correspond to the junction of two sheets, signifying the emergence of two $3_{3+}$ cycles one stable or semi-stable, and the other unstable. The flip curves $\Lambda_{3}^{3_{3+}} = {}^{1}\Lambda_{3}^{3_{3+}}\cup{}^{2}\Lambda_{3}^{3_{3+}}\cup{}^{3}\Lambda_{3}^{3_{3+}}$ are associated with their corresponding fold curves of the same index $j$.

At the singular points $A_{i}$ (for $i = 1, \, 2$), the flip curves $^{j}\Lambda_{3}^{3_{3+}}$ (for $j = 1, \, 2$) are tangential to the fold curves $^{j}\Lambda_{(3)_{0}}^{3_{3+}}$, with two of the three multipliers at $A_{i}$ (for $i = 1, \, 2$) being $S_{1} = -S_{2} = \{1\}$. Conversely, at the singular point $A_{3}$, the flip curves $^{1}\Lambda_{3}^{3_{3+}}$ and $^{2}\Lambda_{3}^{3_{3+}}$ intersect, resulting in $S_{1} = S_{2} = \{-1\}$.

The red and blue regions in Figure 16 represent the stability or semi-stability regions of the $3_{3+}$ cycles, while the white regions indicate the instability regions of these cycles.

To complete the analysis of bifurcations for the spatiotemporal quadratic map $f$ in the $(a, \, b)$ parameter plane, we now present the bifurcation structure of the SD cycle, denoted as $3_{2}$. This study focuses on the case of the first box, as discussed in ^[28]. Figure 18 illustrates the bifurcation curves of the $3_{2}$ cycle, while the foliation is qualitatively depicted in Figure 19. In Figure 18, the flip bifurcation curves are denoted by $\Lambda_{3}^{3_{2}}$ and $\overline{\Lambda}_{3}^{3_{2}}$, while the fold bifurcation curves are labeled as $^{j, k}\Lambda_{(3)_{0}}^{3_{2}}$, where $j = 1, \, 2, \, 3$ differentiates the curves of the same cycle and $k = a$ or $k = b$ distinguishes the different branches of these fold bifurcation curves. The fold curves $\Lambda_{(3)_{0}}^{3_{2}} = {}^{1, \, a}\Lambda_{(3)_{0}}^{3_{2}}\cup{}^{1, \, b}\Lambda_{(3)_{0}}^{3_{2}}\cup{}^{2}\Lambda_{(3)_{0}}^{3_{2}}$ (and $^{3}\Lambda_{(3)_{0}}^{3_{2}}$) correspond to the junction of two sheets, indicating the emergence of four $3_{2}$ cycles, two stable or semi-stable and two unstable. The branches $^{1, \, a}\Lambda_{(3)_{0}}^{3_{2}}$ and $^{1, \, b}\Lambda_{(3)_{0}}^{3_{2}}$ merge, each giving rise to two $3_{2}$ cycles, one stable or semi-stable and the other unstable.

To understand the appearance and disappearance of the $3_{2}$ cycle and their stabilities, consider the cross-section shown in Figure 19-(ⅱ), where the parameter $a$ is fixed at $-1$ and $b$ increases from $-3$ to $0.5$. In Figure 19-(ⅰ), the points $K_{i}$ ($i = 1, 2, 3, 4$) indicate cycles related to fold bifurcations, where the appearance or disappearance of cycles occurs. As $b$ increases, $3_{2}$ cycles appear through fold bifurcations, leading to a total of four cycles (semi-stable or unstable). Before the point $K_{1}$ ($K_{1}\in{}^{2}\Lambda_{(3)_{0}}^{3_{2}}$), there is no SD cycle (considering the first box, see ^[28]). After crossing $K_{1}$, two SD cycles emerge, one semi-stable and one unstable. As $b$ continues to increase, we reach the point $K_{2}$ ($K_{2}\in{}^{3}\Lambda_{(3)_{0}}^{3_{2}}$), where two additional SD cycles appear, one unstable and one semi-stable. At points $K_{3}$ ($K_{3}\in{}^{1, \, a}\Lambda_{(3)_{0}}^{3_{2}}$) and $K_{4}$ ($K_{4}\in{}^{1, \, b}\Lambda_{(3)_{0}}^{3_{2}}$), the four SD cycles disappear, indicating the end of the $3_{2}$ cycles beyond the fold bifurcation curve.

The flip bifurcation curves associated with the fold curves are $\Lambda_{3}^{3_{2}} = {}^{1, \, a}\Lambda_{3}^{3_{2}}\cup{}^{2}\Lambda_{3}^{3_{2}}$ and $\overline{\Lambda}_{3}^{3_{2}} = {}^{1, \, b}\Lambda_{3}^{3_{2}}\cup{}^{3}\Lambda_{3}^{3_{2}}$. The flip curve $^{1, \, a}\Lambda_{3}^{3_{2}}$ is associated with the fold curve $^{1, \, a}\Lambda_{(3)_{0}}^{3_{2}}$, with the stability region (red region in Figure 18) bounded by these two curves. Similarly, the curve $^{2}\Lambda_{3}^{3_{2}}$ is associated with the fold curve $^{2}\Lambda_{(3)_{0}}^{3_{2}}$, with the stability region (blue region in Figure 18) bounded by these two curves. At the singular point $C$, the flip curve $\Lambda_{3}^{3_{2}}$ is tangential to the fold curve $\Lambda_{(3)_{0}}^{3_{2}}$, with two multipliers corresponding to $C$ given by $S_{1} = -S_{2} = \{1\}$.

Similarly, the flip bifurcation curve $\overline{\Lambda}_{3}^{3_{2}} = {}^{1, \, b}\Lambda_{3}^{3_{2}}\cup{}^{3}\Lambda_{3}^{3_{2}}$ consists of the branches $^{1, \, b}\Lambda_{3}^{3_{2}}$ and $^{3}\Lambda_{3}^{3_{2}}$. The flip curve $^{1, \, b}\Lambda_{3}^{3_{2}}$ is associated with the fold curve $^{3}\Lambda_{(3)_{0}}^{3_{2}}$, and the flip curve $^{3}\Lambda_{3}^{3_{2}}$ is associated with the fold curve $^{1, \, b}\Lambda_{(3)_{0}}^{3_{2}}$. The stability regions (orange region) are bounded by these four curves.

4.   Conclusions

This study highlights the intricate dynamics of $2$D spatiotemporal discrete systems, particularly focusing on the stability and bifurcations of periodic solutions such as $3$-cycles. The analysis of various types of 3-periodic points, categorized into four types (horizontal (H), super diagonal (SD), diagonal (D), and anti-diagonal (AD)), provides a comprehensive understanding of their stability conditions, which are determined by the spectral properties of the Jacobian matrix.

The study emphasizes the importance of bifurcation curves in illustrating how changes in parameters $a$ and $b$ can lead to qualitative shifts in system behavior, including the emergence of chaos or the stabilization of cycles. The findings reveal the intricate interplay between stability and bifurcations, particularly through the examination of a spatiotemporal quadratic map, which serves as a significant model for understanding nonlinear dynamics.

Key insights include the identification of stable and unstable $3$-cycles, the transition from stability to chaos as parameters vary, and the detailed mathematical framework that supports the analysis of these cycles. The paper also highlights the role of numerical results, such as bifurcation diagrams, in visualizing the stability regions of fixed points and periodic orbits.

In conclusion, this study contributes to the understanding of the dynamics of $2$D spatiotemporal discrete systems through bifurcation and periodicity analysis, and also suggests future research directions. These include potential applications to real-world phenomena such as pattern formation and epidemic propagation, thus providing valuable insights to the field of dynamical systems.

Author contributions

All authors contributed equally to the development of the research, analysis, and writing of the manuscript. All authors have read and agreed to the published version of the manuscript.

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Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.