Bifurcation and Chaos in a Nonlinear Population Model with Delayed Nonlinear Interactions

Abstract

This paper investigates the dynamic behavior of a second-order nonlinear rational difference equation modeling a population system with nonlinear interactions between current and previous population states. We derive analytical conditions for the stability of fixed points, explore codim-1 bifurcations, and compute the associated topological normal forms. The analysis also establishes the existence of period-2 solutions and reveals the potential for chaotic dynamics within specific parameter ranges. To validate the theoretical findings, we conduct numerical simulations and bifurcation analysis using the MATLAB package MatContM (version 5p4). Chaotic behavior is further confirmed through the computation of the largest Lyapunov exponent. The results offer new insights into the complex dynamics of delayed population models with nonlinear feedback, extending classical models and suggesting potential applications in stochastic systems and epidemiological modeling.

1. Introduction

Difference equations play a key role in describing discrete-time dynamical systems. They arise both as numerical solutions to differential equations and as discrete analogs of them. These equations have numerous applications across various disciplines, including biology, physics, ecology, economics, and more [1,2,3,4,5,6,7]. Despite their seemingly simple form, analyzing the dynamic behavior of many difference equations, such as finding general solutions, studying the stability of fixed points, identifying periodic solutions, and investigating bifurcations, can be quite challenging. Interest in the qualitative analysis of difference equations has grown in recent years. Many studies have examined different types of rational difference equations, including those in [8,9,10,11,12,13,14], and the references therein.

We consider the second-order nonlinear rational difference equation

$$y_{n+1}={\displaystyle \frac{y_n{y}_{n-1}+\alpha y_n}{y_n{y}_{n-1}+\alpha y_{n-1}+{\alpha }^2}},n=0,1,\dots ,$$

(1)

where $\alpha$ is a parameter, and the initial conditions $y_{-1}$ and $y_0$ are real numbers such that the solution is well-defined. It is important to note that Equation (1) is a special case of the general rational difference equation with quadratic terms

$$y_{n+1}={\displaystyle \frac{A{y}_n^2+B{y}_n{y}_{n-1}+C{y}_{n-1}^2+D{y}_n+E{y}_{n-1}+F}{a{y}_n^2+b{y}_n{y}_{n-1}+c{y}_{n-1}^2+d{y}_n+e{y}_{n-1}+f}},n=0,1,\dots ,$$

(2)

where $A,B,\dots ,f$ are parameters, and the initial conditions $y_{-1}$ and $y_0$ are real numbers ensuring a well-defined solution. Special cases of Equation (2) are of great interest due to their applications in fields such as ecology, biology, physics, and economics. The Beverton–Holt model, investigated in [15,16], is given by

$$y_{n+1}={\displaystyle \frac{\left(KR\right)y_n}{(R-1)y_n+K}},$$

(3)

and represents a special case of Equation (1). Analysis of the Beverton–Holt model reveals globally stable dynamics without chaotic behavior. A transcritical bifurcation occurs at $R=1$, where stability shifts between the extinction equilibrium $N=0$ and the carrying capacity equilibrium $N=K$. Hassell [17] analyzed the following generalized model

$$y_{n+1}={\displaystyle \frac{\lambda y_n}{{(a{y}_n+1)}^b}},$$

(4)

and studied its stability properties. He showed that the model may exhibit exponential damping, oscillatory damping, or stable limit cycle behavior, depending on the parameter b and the effective population growth rate $\lambda$. May [18] studied the logistic map

$$y_{n+1}=-a{y}_n^2+a{y}_n.$$

(5)

His pioneering work demonstrated the period-doubling route to chaos and established key principles of chaotic dynamics in discrete-time systems. The general second-order form

$$y_{n+1}={\displaystyle \frac{\beta y_n+\gamma y_{n-1}+\alpha }{B{y}_n+C{y}_{n-1}+A}}$$

(6)

was examined by Kulenovic and Ladas [19]. Their thorough analysis provided conditions for the local and global stability of fixed points and criteria for bounded solutions and identified parameter ranges that lead to periodic and chaotic behavior. Kostrov and Kudlak [20] studied the behavior of solutions for

$$x_{n+1}={\displaystyle \frac{\gamma y_{n-1}+\alpha }{D{y}_n{y}_{n-1}+B{y}_n+y_{n-1}}},n=0,1,\dots .$$

(7)

They showed that the equation exhibits a period-doubling bifurcation. In [21], Khyat studied various types of bifurcations for special cases of Equation (2). Saleh and Radad [22] studied the existence of period-doubling bifurcations in the rational difference equation:

$$y_{n+1}={\displaystyle \frac{\beta y_{n-1}+\alpha }{B{y}_n+C{y}_{n-1}+A}},$$

(8)

with positive parameters and initial conditions. Delayed ecological systems were examined by Cushing [23]:

$$y_{n+1}={\displaystyle \frac{r{y}_n}{a{y}_n+b{y}_{n-k}+1}},$$

(9)

where he derived stability-switching conditions for time-delayed population models. Al-Hdaibat et al. [24] discussed the general solution and dynamic behaviors of the following rational difference equation:

$$y_{n+1}={\displaystyle \frac{a{y}_{n-1}}{c{y}_n{y}_{n-1}+b}},n=0,1,\dots .$$

(10)

The rational difference equation

$$y_{n+1}={\displaystyle \frac{\alpha y_n}{\beta y_{n-1}+1}}$$

(11)

was studied by Elaydi in [25], where the fixed points and their stability were analyzed. In Equation (11), $y_n$ denotes the population size at time n, $\alpha$ represents the intrinsic growth rate under ideal conditions, and $\beta$ reflects density-dependent feedback due to environmental constraints such as resource competition. The presence of $y_{n-1}$ introduces a generational delay, capturing the influence of the previous generation on the current population. This model is particularly relevant for species with overlapping generations, such as the blowfly (Lucilia cuprina) [26,27].

Assuming that both intrinsic growth and environmental feedback are governed by the same biological factor, i.e., $\beta =\alpha$, Equation (11) simplifies to

$$y_{n+1}={\displaystyle \frac{\alpha y_n}{\alpha y_{n-1}+1}}.$$

This formulation implies that species with higher reproductive rates also face proportionally stronger density-dependent limitations, making the system more sensitive to variations in $\alpha$.

Further refinement can be made by scaling the feedback term with ${\alpha }^2$, yielding

$$y_{n+1}={\displaystyle \frac{\alpha y_n}{\alpha y_{n-1}+{\alpha }^2}},$$

which models scenarios where environmental constraints intensify quadratically with the intrinsic growth rate. This modification reflects situations where rapid reproduction leads to disproportionately greater resource depletion or environmental degradation.

Additionally, incorporating a nonlinear interaction term $y_n{y}_{n-1}$ captures direct intergenerational effects, such as crowding or competition between overlapping cohorts. This leads to the modified model (1), which exhibits richer dynamics, including periodic solutions, bifurcations, and chaotic behavior for certain parameter values. In Equation (1), $y_n$ denotes the state of the system (e.g., population size) at discrete time n, and $\alpha \in \mathbb{R}$ is a parameter representing intrinsic growth or feedback intensity. The next state $y_{n+1}$ depends on the current and previous states, $y_n$ and $y_{n-1}$, with initial conditions $y_{-1},y_0\in \mathbb{R}$ chosen to ensure the denominator remains nonzero. The term $y_n{y}_{n-1}$ models intergenerational interaction, while the additional terms involving $\alpha$ capture nonlinear growth and density-dependent effects. Similar nonlinear interaction mechanisms are well-documented in predator–prey models and have been shown to significantly influence system stability and the emergence of complex population cycles [28,29,30,31].

Compared to the classical models, which have been effectively applied to species like the blowfly [26,27], our proposed model (1) offers a more comprehensive representation of ecological dynamics by incorporating several novel features:

• We include a nonlinear interaction term $y_n{y}_{n-1}$, capturing intergenerational competition effects absent in traditional models.
• We include quadratic scaling of environmental feedback (${\alpha }^2$), reflecting more realistic nonlinear resource limitations linked to intrinsic growth rates.
• From a dynamical systems viewpoint, the model exhibits a richer dynamical behaviors, including limit-point, period-doubling, and Neimark–Sacker bifurcations, as well as chaotic dynamics, which are either absent or less pronounced in classical models.

Consequently, our model provides both a biologically enriched framework for studying delayed, nonlinear population dynamics and a mathematically robust structure for exploring complex behaviors in ecological systems with memory effects.

Although this study focuses on integer-order discrete dynamics, future extensions may benefit from incorporating fractional calculus, such as Caputo, $\psi$-Hilfer, and other generalized fractional derivatives, to better capture memory effects and anomalous diffusion. These fractional approaches have shown considerable promise in real-world modeling, particularly in fractional-order SIR models and ecological systems characterized by long-term dependencies. In particular, recent developments in $\psi$-Hilfer-type difference equations and fractional operators with non-singular kernels provide powerful tools for more accurate data fitting and a deeper understanding of intergenerational dynamics; see, for instance, [32,33,34,35].

In this paper, we explore the dynamic behaviors of Equation (1), showing that it has three fixed points, which correspond to biologically significant states such as extinction or stable coexistence. Using linear stability analysis, we identify when these points lose stability. We also examine bifurcations both numerically and with one-parameter bifurcation analysis, exploring transitions between stability, periodic solutions, and potentially chaotic dynamics as the bifurcation parameter varies. Furthermore, we explore the existence of period-2 solutions, where the populations oscillate in a cycle with a period of two time steps, reflecting alternating dynamics between predator and prey. This study offers new insights into the stability, chaos, and periodic behaviors in predator–prey systems with memory, extending existing ecological models by incorporating delayed interactions and contributing significantly to the understanding of nonlinear dynamics in ecological systems.

This paper is organized as follows. In Section 2, we calculate the stability of the fixed points for Equation (1). Section 3 investigates codim-1 bifurcations of the system. We compute the topological normal forms for each bifurcation. Additionally, we explore the existence of period-2 solutions in Equation (1) and show that these solutions are always unstable. To confirm our theoretical results, we perform numerical simulations and bifurcation analysis using the MATLAB package MatContM in Section 4. In Section 5, we analyze the chaotic behavior of Equation (1) using the largest Lyapunov exponent. This analysis confirms the findings from the earlier sections.

2. Existence and Stability of Fixed Points, Bifurcation Diagram

Equation (1) can be written as

$$y_{n+1}={\displaystyle \frac{y_n\left(y_{n-1}+\alpha \right)}{y_n{y}_{n-1}+\alpha \left(y_{n-1}+\alpha \right)}}={\displaystyle \frac{y_n}{{\displaystyle \frac{y_n{y}_{n-1}+\alpha \left(y_{n-1}+\alpha \right)}{\left(y_{n-1}+\alpha \right)}}}}={\displaystyle \frac{y_n}{\left({\displaystyle \frac{y_{n-1}}{y_{n-1}+\alpha }}\right)y_n+\alpha }}.$$

(12)

If we set $x_n={\displaystyle \frac{y_{n-1}}{y_{n-1}+\alpha }}$, then the second-order rational Equation (12) can be converted to the first-order system of rational difference equations

$$x_{n+1}={\displaystyle \frac{y_n}{y_n+\alpha }},y_{n+1}={\displaystyle \frac{y_n}{x_n{y}_n+\alpha }},n=0,1,2,\dots ,$$

(13)

with initial conditions $x_0,y_0\in \mathbb{R}$.

Assume that $\alpha >0$ and the initial values satisfy $x_0>0$ and $y_0>0$. Then, by induction, it follows that $x_n>0$ and $y_n>0$ for all $n\ge 0$. From the first equation in (13), and using the facts that $y_n>0$ and $\alpha >0$, we obtain

$$0<x_{n+1}={\displaystyle \frac{y_n}{y_n+\alpha }}<1.$$

Since $x_n>0$ for all $n\ge 0$, and the function ${\displaystyle \frac{y_n}{y_n+\alpha }}$ is continuous and bounded away from zero for $y_n$ bounded below, this implies the existence of a constant $m\in (0,1)$, independent of n, such that

$$0<m\le x_n<1\mathrm{for}\mathrm{all}n\ge 0.$$

Similarly, from the second equation in (13), we have

$$y_{n+1}={\displaystyle \frac{y_n}{x_n{y}_n+\alpha }}>0,$$

due to the positivity of $x_n$, $y_n$, and $\alpha$. Hence, the first quadrant (i.e., the set $\{(x,y)\in {\mathbb{R}}^2:x\ge 0,y\ge 0\}$) is positively invariant under the dynamics of System (13).

To estimate $y_{n+1}$, we write

$$y_{n+1}={\displaystyle \frac{y_n}{x_n{y}_n+\alpha }}={\displaystyle \frac{1}{x_n+{\displaystyle \frac{\alpha }{y_n}}}}<{\displaystyle \frac{1}{x_n}}\le {\displaystyle \frac{1}{m}},$$

where the last inequality follows from the uniform lower bound $x_n\ge m>0$. Thus, the sequence $\left\{y_n\right\}$ is bounded above by the constant $M:=1/m$. We summarize these findings in the following proposition.

Proposition 1.

Let $\alpha >0$, and suppose the initial conditions satisfy $x_0>0$ and $y_0>0$. Then, the solution of System (13) satisfies the following properties:

1. 

The solution remains strictly positive for all $n\ge 0$; that is,

$$x_n>0and{y}_n>0foralln\ge 0.$$

2. 

There exist constants $0<m<1$ and $M>0$, independent of n, such that

$$m\le x_n<1,0<y_n\le M,foralln\ge 0.$$

3. 

The first quadrant is positively invariant under the system dynamics.

Consequently, all solutions of System (13) with strictly positive initial conditions are globally defined, remain positive, and are uniformly bounded for all $n\ge 0$.

System (13) can be expressed in vector form as

$$x\mapsto f\left(x,\alpha \right),f:{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^2,$$

(14)

where $x={\left(x_1,x_2\right)}^{\mathrm{T}}$ is a state variable vector, $\alpha$ is a parameter, and $f\left(x,\alpha \right)={\left(f_1(x,\alpha ),f_2(x,\alpha )\right)}^{\mathrm{T}}$ with

$$f_1\left(x,\alpha \right)={\displaystyle \frac{x_2}{x_2+\alpha }},\mathrm{and}{f}_2(x,\alpha )={\displaystyle \frac{x_2}{x_1{x}_2+\alpha }}.$$

The mapping defined by this system is smooth (infinitely differentiable) on its natural domain, where the denominators $y_n+\alpha$ and $x_n{y}_n+\alpha$ are nonzero. Moreover, it is locally invertible with a smooth inverse in these regions and thus constitutes a local diffeomorphism. We analyze the dynamics of the system under these conditions.

The stability analysis of System (14) usually start with fixed points, i.e., the solutions of the equation

$$f(x^{*},\alpha )=x^{*}.$$

System (14) can have three fixed points

$$P_1(0,0),P_2\left({\displaystyle \frac{(\alpha -1)+\delta }{2\alpha }},{\displaystyle \frac{-(\alpha -1)+\delta }{2}}\right),\mathrm{and}{P}_3\left({\displaystyle \frac{(\alpha -1)-\delta }{2\alpha }},{\displaystyle \frac{-(\alpha -1)-\delta }{2}}\right),$$

(15)

where $\delta =\sqrt{1+2\alpha -3{\alpha }^2}$. Note that the fixed points $P_2$ and $P_3$ exist only if the system parameter $\alpha \in \left[{\displaystyle \frac{-1}{3}},1\right]\setminus \left\{0\right\}$.

The Jacobian matrix evaluated at the fixed point $x^{*}$ of System (14) is given by

$$A\left(\alpha \right)=\left(\begin{array}{cc}0& {\displaystyle \frac{\alpha }{{\left(x_2^{*}+\alpha \right)}^2}}\\ -{\displaystyle \frac{{\left(x_2^{*}\right)}^2}{{(x_1^{*}{x}_2^{*}+\alpha )}^2}}& {\displaystyle \frac{\alpha }{{\left(x_1^{*}{x}_2^{*}+\alpha \right)}^2}}\end{array}\right).$$

(16)

The characteristic equation of the Jacobian matrix A is given by

$$\mathbb{P}\left(\lambda \right)={\lambda }^2-\left(\mathrm{tr}A\right)\lambda +\det A,$$

(17)

where $\mathrm{tr}A={\displaystyle \frac{\alpha }{{\left(x_1^{*}{x}_2^{*}+\alpha \right)}^2}}$ and $\det A={\displaystyle \frac{\alpha {\left(x_2^{*}\right)}^2}{{\left[\left(x_2^{*}+\alpha \right)\left(x_1^{*}{x}_2^{*}+\alpha \right)\right]}^2}}$.

Suppose ${\lambda }_1$ and ${\lambda }_2$ are the roots of the characteristic Equation (17). The fixed point $x^{*}$ is called a sink if $\left|{\lambda }_1\right|<1$ and $\left|{\lambda }_2\right|<1$; in this case, it is locally asymptotically stable. Conversely, $x^{*}$ is called a source if $\left|{\lambda }_1\right|>1$ and $\left|{\lambda }_2\right|>1$, making it locally unstable. If one root satisfies $\left|\lambda \right|<1$ while the other satisfies $\left|\lambda \right|>1$, i.e., $\left|{\lambda }_1\right|<1<\left|{\lambda }_2\right|$ or $\left|{\lambda }_2\right|<1<\left|{\lambda }_1\right|$, then $x^{*}$ is a saddle point and is unstable. Finally, the fixed point is called nonhyperbolic if either $\left|{\lambda }_1\right|=1$ or $\left|{\lambda }_2\right|=1$.

By straightforward calculations, we obtain the following proposition regarding the existence and stability of fixed point $P_1$ of System (14).

Proposition 2.

Consider the fixed point $P_1$ of System (14). The stability properties of $P_1$ depend on the parameter $\alpha \in \mathbb{R}\setminus \left\{0\right\}$ as follows:

1. 

If $\alpha \in (-\infty ,-1)\cup (1,\infty )$, then $P_1$ is a sink and is locally asymptotically stable.

2. 

If $\alpha \in (-1,0)\cup (0,1)$, then $P_1$ is a saddle point and hence unstable.

3. 

If $\alpha =\pm 1$, then $P_1$ is a nonhyperbolic fixed point.

Proof. 

The Jacobian matrix of System (14) evaluated at the fixed point $P_1$ is

$$A\left(\alpha \right)=\left(\begin{array}{cc}0& {\displaystyle \frac{1}{\alpha }}\\ 0& {\displaystyle \frac{1}{\alpha }}\end{array}\right).$$

The characteristic equation of $A\left(\alpha \right)$ is

$$\mathbb{P}\left(\lambda \right)=\lambda \left(\lambda -{\displaystyle \frac{1}{\alpha }}\right).$$

Hence, the eigenvalues are

$${\lambda }_1=0,{\lambda }_2={\displaystyle \frac{1}{\alpha }}.$$

1.

For $\alpha \in (-\infty ,-1)\cup (1,\infty )$, we have

$$\left|{\lambda }_2\right|=\left|{\displaystyle \frac{1}{\alpha }}\right|<1,$$

and $|{\lambda }_1|=0<1$. Since $|{\lambda }_{1,2}|<1$, the fixed point $P_1$ is a sink and thus locally asymptotically stable.

2.

For $\alpha \in (-1,0)\cup (0,1)$, it follows that

$$\left|{\lambda }_2\right|=\left|{\displaystyle \frac{1}{\alpha }}\right|>1,$$

while $|{\lambda }_1|=0<1$. The presence of an eigenvalue with magnitude greater than one indicates $P_1$ is a saddle point and therefore unstable. Note that $\alpha =0$ is excluded since the Jacobian becomes singular.

3.

At the boundary values $\alpha =\pm 1$, the eigenvalues are

$${\lambda }_2={\displaystyle \frac{1}{\pm 1}}=\pm 1,{\lambda }_1=0,$$

with $|{\lambda }_2|=1$. Hence, the fixed point $P_1$ is nonhyperbolic since at least one eigenvalue is equal to one.

□

The Jury stability criterion for a two-dimensional discrete-time dynamical system (see, for example, [25]) is given by the following theorem.

Theorem 1.

The nature of a fixed point of a two-dimensional discrete-time dynamical system is characterized as follows:

1. 

The fixed point is locally asymptotically stable if and only if the following Jury conditions are satisfied:

$$\begin{array}{cc}\hfill \left(\mathrm{J}1\right)& |det(A\left)\right|<1,\hfill \\ \hfill \left(\mathrm{J}2\right)& 1-\mathrm{tr}\left(A\right)+det\left(A\right)>0,\hfill \\ \hfill \left(\mathrm{J}3\right)& 1+\mathrm{tr}\left(A\right)+det\left(A\right)>0.\hfill \end{array}$$

2. 

If condition (J1) fails, then the fixed point is an unstable source, that is, both eigenvalues satisfy $\left|\lambda \right|>1$.

3. 

If condition (J1) holds but either condition (J2) or (J3) fails, then the fixed point is a saddle point; specifically, one eigenvalue satisfies $\left|\lambda \right|<1$, and the other satisfies $\left|\lambda \right|>1$.

4. 

If any of the conditions (J1), (J2), or (J3) hold with equality, then the fixed point is nonhyperbolic.

This theorem can be used to determine the nature of the fixed points $P_2$ and $P_3$, leading to the following propositions.

Proposition 3.

Let $P_2$ be the fixed point of System (14). The local stability of $P_2$ depends on the real parameter $\alpha \in \mathbb{R}\setminus \left\{0\right\}$ as follows:

1. 

If $\alpha \in \left(-0.2955977425,0\right)\cup \left(0,1\right)$, then $P_2$ is a sink and is locally asymptotically stable.

2. 

If $\alpha \in \left(-{\displaystyle \frac{1}{3}},-0.324717958\right)$, then $P_2$ is a source and is unstable.

3. 

If $\alpha \in \left(-0.324717958,-0.2955977425\right)$, then $P_2$ is a saddle point and is unstable.

4. 

If $\alpha =-{\displaystyle \frac{1}{3}}$ or $\alpha =-0.2955977425$, then $P_2$ is nonhyperbolic.

Proof. 

To analyze the local stability of the fixed point $P_2$, we examine the Jacobian matrix of System (14) evaluated at $P_2$, which is given by

$$A\left(\alpha \right)=\left(\begin{array}{cc}0& {\displaystyle \frac{4\alpha }{{\left(\alpha +1+\delta \right)}^2}}\\ -{\displaystyle \frac{{\left(-\alpha +1+\delta \right)}^2}{4}}& \alpha \end{array}\right).$$

The trace and determinant of this matrix are

$$\mathrm{tr}\left(A\right)=\alpha ,\det \left(A\right)={\displaystyle \frac{\alpha {\left(-\alpha +1+\delta \right)}^2}{{\left(\alpha +1+\delta \right)}^2}}.$$

We apply the Jury stability test given in Theorem 1:

1.

When $\alpha \in \left(-0.2955977425,0\right)\cup \left(0,1\right)$, all three conditions (J1), (J2), and (J3) are satisfied. Thus, $P_2$ is locally asymptotically stable.

2.

When $\alpha \in \left(-{\displaystyle \frac{1}{3}},-0.324717958\right)$, condition (J1) fails ($|det(A\left)\right|>1$). Hence, $P_2$ is an unstable source.

3.

When $\alpha \in \left(-0.324717958,-0.2955977425\right)$, condition (J1) holds but condition (J3) fails ($1+\mathrm{tr}\left(A\right)+det\left(A\right)<0$), which implies that $P_2$ is a saddle point and unstable.

4.

When $\alpha =-{\displaystyle \frac{1}{3}}$ or $\alpha =-0.2955977425$, at least one of the Jury conditions holds with equality. Therefore, the fixed point $P_2$ is nonhyperbolic in these cases.

□

Proposition 4.

Let $P_3$ be the fixed point of System (14). The local stability of $P_3$ depends on the real parameter $\alpha \in \mathbb{R}\setminus \left\{0\right\}$ as follows:

1. 

If $\alpha \in \left(0.5698402912,1\right)$, then $P_3$ is a sink and is locally asymptotically stable.

2. 

If $\alpha \in \left(-{\displaystyle \frac{1}{3}},0\right)\cup \left(0,0.5698402912\right)$, then $P_3$ is a source and is unstable.

3. 

If $\alpha =-{\displaystyle \frac{1}{3}}$, $\alpha =0.5698402912$, or $\alpha =1$, then $P_3$ is nonhyperbolic.

Proof. 

To analyze the local stability of the fixed point $P_3$, we consider the Jacobian matrix of System (14) evaluated at $P_3$, given by

$$A\left(\alpha \right)=\left(\begin{array}{cc}0& {\displaystyle \frac{4\alpha }{{\left(-\alpha -1+\delta \right)}^2}}\\ -{\displaystyle \frac{{\left(\alpha -1+\delta \right)}^2}{4}}& \alpha \end{array}\right).$$

The trace and determinant of $A\left(\alpha \right)$ are

$$\mathrm{tr}\left(A\right)=-\alpha ,\det \left(A\right)={\displaystyle \frac{\alpha {\left(\alpha -1+\delta \right)}^2}{{\left(-\alpha -1+\delta \right)}^2}}.$$

We apply the Jury stability criteria given in Theorem 1 to analyze the behavior of the fixed point $P_3$ depending on the value of $\alpha$:

1.

For $\alpha \in \left(0.5698402912,1\right)$, all three Jury conditions are satisfied. Therefore, $P_3$ is locally asymptotically stable (a sink).

2.

For $\alpha \in \left(-{\displaystyle \frac{1}{3}},0\right)\cup \left(0,0.5698402912\right)$, condition (J1) fails since $|det(A\left)\right|>1$. Hence, $P_3$ is unstable and acts as a source. Note that $\alpha =0$ is excluded since the Jacobian becomes singular.

3.

At the values $\alpha =-{\displaystyle \frac{1}{3}}$, $\alpha =0.5698402912$, or $\alpha =1$, at least one of the Jury conditions holds with equality, and thus the fixed point $P_3$ is nonhyperbolic.

□

While following the curves of the fixed points $P_1$, $P_2$, and $P_3$, five codim-1 bifurcations related to changes in stability can occur:

1.

A limit point (fold, $LP$) bifurcation occurs when ${\alpha }_{LP}={\displaystyle \frac{-1}{3}}$.

2.

Two period-doubling (flip, $PD$) bifurcations can occur:

(i)

$P{D}_1$ occurs when ${\alpha }_{P{D}_1}=-1$,

(ii)

$P{D}_2$ occurs when ${\alpha }_{P{D}_2}=-0.2955977425$.

3.

A Neimark–Sacker ($NS$) bifurcation occurs when ${\alpha }_{NS}=0.5698402912$.

4.

A pitchfork ($PF$) bifurcation occurs when ${\alpha }_{PF}=1$.

Combining the results obtained, we can construct the bifurcation diagram of System (14), as shown in Figure 1.

3. Bifurcation Analysis

While varying the parameter $\alpha$ of System (14), we generically encounter three codimension-one bifurcations associated with stability changes of the fixed points $P_1$, $P_2$, and $P_3$: (i) a limit point ($LP$), (ii) a period-doubling ($PD$), and (iii) a Neimark–Sacker ($NS$) bifurcation. A nongeneric situation arises at a pitchfork ($PF$) bifurcation when ${\alpha }_{PF}=1$, where the matrix $A-I_2$ becomes rank-deficient (here, $I_2$ denotes the $2\times 2$ identity matrix).

To analyze the local dynamics and bifurcation structure of System (14), we employ normal form theory for discrete-time dynamical systems (see [6,36]). In this context, it is essential to assume that the system is sufficiently smooth to ensure the existence of topological normal forms; see, for example, [36], Chapter 4. For our system, this smoothness assumption is satisfied, as established in the following lemma.

Lemma 1.

Let $\alpha \in \mathbb{R}$ be fixed. Consider the function $f:D\times \mathbb{R}\to {\mathbb{R}}^2$ as defined by System (14), and the domain is

$$D=\left\{{(x_1,x_2)}^{\mathrm{T}}\in {\mathbb{R}}^2:x_2\ne -\alpha ,x_1{x}_2\ne -\alpha \right\}.$$

Then $f(x,\alpha )\in C^{\infty }\left(D\right)$, that is, the system is infinitely differentiable with respect to the state variable x on D.

Proof. 

We verify the differentiability of each component of $f(x,\alpha )$ on the domain D. Let $x={(x_1,x_2)}^{\mathrm{T}}\in D$. The first-order partial derivatives are

$${\displaystyle \frac{\partial f_1}{\partial x_1}}=0,{\displaystyle \frac{\partial f_1}{\partial x_2}}={\displaystyle \frac{\alpha }{{(x_2+\alpha )}^2}},{\displaystyle \frac{\partial f_2}{\partial x_1}}={\displaystyle \frac{-x_2^2}{{(x_1{x}_2+\alpha )}^2}},{\displaystyle \frac{\partial f_2}{\partial x_2}}={\displaystyle \frac{\alpha }{{(x_1{x}_2+\alpha )}^2}}.$$

These expressions are rational functions whose denominators are nonzero on D, and thus they are continuous. Hence, $f(x,\alpha )\in C^1\left(D\right)$. To prove higher-order differentiability, we proceed by induction. Suppose that $f(x,\alpha )\in C^k\left(D\right)$ for some $k\in \mathbb{N}$. Then all partial derivatives up to order k can be written in the form

$${\displaystyle \frac{P\left(x\right)}{{(x_2+\alpha )}^m}},\mathrm{or}{\displaystyle \frac{Q\left(x\right)}{{(x_1{x}_2+\alpha )}^n}},$$

for some polynomials $P\left(x\right)$, $Q\left(x\right)$, and integers $m,n\in \mathbb{N}$. Differentiating such expressions yields rational functions whose denominators remain nonzero on D and hence are continuous. Thus, $f(x,\alpha )\in C^{k+1}\left(D\right)$. By induction, $f(x,\alpha )\in C^k\left(D\right)$ for all $k\in \mathbb{N}$, which implies $f(x,\alpha )\in C^{\infty }\left(D\right)$. □

Assume that for some $\alpha ={\alpha }_c$, System (14) undergoes a codim-1 bifurcation at the fixed point $x^{*}$. The Taylor expansion of $f(x^{*}+x,{\alpha }_c)$ around $x^{*}$ can be expressed as

$$f\left(x^{*}+x,{\alpha }_c\right):=x^{*}+A\left({\alpha }_c\right)x+{\displaystyle \frac{1}{2}}B(x,x)+{\displaystyle \frac{1}{6}}C(x,x,x)+\dots ,$$

where the dots represent higher-order terms in x, $A\left({\alpha }_c\right)$ is the Jacobian matrix evaluated at $x^{*}$ as given in Equation (16), and $B(x,x)$ and $C(x,x,x)$ are vectors with two components, defined as follows:

$$B(u,v):=\left(\begin{array}{c}{\displaystyle \sum _{j,k=1}^2}{\displaystyle \frac{{\partial }^2{f}_1\left(x^{*},{\alpha }_c\right)}{\partial {\xi }_j\partial {\xi }_k}}{u}_j{v}_k\\ {\displaystyle \sum _{j,k=1}^2}{\displaystyle \frac{{\partial }^2{f}_2\left(x^{*},{\alpha }_c\right)}{\partial {\xi }_j\partial {\xi }_k}}{u}_j{v}_k\end{array}\right)=\left(\begin{array}{c}{\displaystyle \frac{-2{\alpha }_c{u}_2{v}_2}{{(x_2^{*}+{\alpha }_c)}^3}}\\ {\displaystyle \frac{2\left({\left(x_2^{*}\right)}^3{u}_1{v}_1-{\alpha }_c{x}_1^{*}{u}_2{v}_2-{\alpha }_c{x}_2^{*}(u_1{v}_2+u_2{v}_1)\right)}{{(x_1^{*}{x}_2^{*}+{\alpha }_c)}^3}}\end{array}\right),$$

$$C(u,v,w):=\left(\begin{array}{c}{\displaystyle \sum _{j,k,l=1}^2}{\displaystyle \frac{{\partial }^3{f}_1\left(x^{*},{\alpha }_c\right)}{\partial {\xi }_j\partial {\xi }_k\partial {\xi }_l}}{u}_j{v}_k{w}_l\\ {\displaystyle \sum _{j,k,l=1}^2}{\displaystyle \frac{{\partial }^3{f}_2\left(x^{*},{\alpha }_c\right)}{\partial {\xi }_j\partial {\xi }_k\partial {\xi }_l}}{u}_j{v}_k{w}_l\end{array}\right)=\left(\begin{array}{c}{\displaystyle \frac{6{\alpha }_c{u}_2{v}_2{w}_2}{{(x_2^{*}+{\alpha }_c)}^4}}\\ {\displaystyle \frac{-2g(u,v,w)}{{(x_1^{*}{x}_2^{*}+{\alpha }_c)}^4}}\end{array}\right),$$

where $u={(u_1,u_2)}^{\mathrm{T}}$, $v={(v_1,v_2)}^{\mathrm{T}}$, $w={(w_1,w_2)}^{\mathrm{T}}$, and the scalar function $g(u,v,w)$ is given by

$$\begin{array}{cc}\hfill g(u,v,w):=& 3{\left(x_2^{*}\right)}^4{u}_1{v}_1{w}_1-3{\alpha }_c{\left(x_1^{*}\right)}^2{u}_2{v}_2{w}_2-2{\alpha }_c{x}_1^{*}{x}_2^{*}(u_1{v}_2{w}_2+u_2{v}_1{w}_2+u_2{v}_2{w}_1)\hfill \\ \hfill & -3{\alpha }_c{\left(x_2^{*}\right)}^2(u_1{v}_1{w}_2+u_1{v}_2{w}_1+u_2{v}_1{w}_1)+{\alpha }_c^2(u_1{v}_2{w}_2+u_2{v}_1{w}_2+u_2{v}_2{w}_1).\hfill \end{array}$$

3.1. Limit Point Bifurcation

When the parameter $\alpha$ crosses the critical value ${\alpha }_{LP}={\displaystyle \frac{-1}{3}}$ corresponding to the LP bifurcation, the fixed points $P_2$ and $P_3$ of System (14) collide at the fixed point $x^{LP}=\left(2,{\displaystyle \frac{2}{3}}\right)$ and subsequently disappear, see Figure 1. At this point, the Jacobian matrix $A\left({\alpha }_{LP}\right)$ evaluated at $x^{LP}$ has a simple eigenvalue equal to 1. Therefore, the corresponding critical eigenspace is one-dimensional and spanned by an eigenvector $q\in {\mathbb{R}}^2$ satisfying

$$\left(A-I_2\right)q=0.$$

(18)

Let $p\in {\mathbb{R}}^2$ be the adjoint eigenvector, which satisfies

$$\left(A^{\mathrm{T}}-I_2\right)p=0,$$

(19)

where $A^{\mathrm{T}}$ is the transpose of matrix A. To compute the vectors p and q, we solve the following $3\times 3$ bordered systems:

$$\left(\begin{array}{cc}A-I_2& w_{\mathrm{bor}}\\ v_{\mathrm{bor}}^{\mathrm{T}}& 0\end{array}\right)\left(\begin{array}{c}q\\ s\end{array}\right)=\left(\begin{array}{c}\mathbf{0}\\ 1\end{array}\right),$$

(20)

and

$$\left(\begin{array}{cc}{A}^{\mathrm{T}}-I_2& v_{\mathrm{bor}}\\ w_{\mathrm{bor}}^{\mathrm{T}}& 0\end{array}\right)\left(\begin{array}{c}p\\ s\end{array}\right)=\left(\begin{array}{c}\mathbf{0}\\ 1\end{array}\right),$$

(21)

where $\mathbf{0}={\left(0,0\right)}^{\mathrm{T}}$, $s\in \mathbb{R}$ is the same in both systems, and the vectors $w_{\mathrm{bor}}$, $v_{\mathrm{bor}}\in {\mathbb{R}}^2$ are chosen such that the bordered matrices $M\left(x^{LP},{\alpha }_{LP}\right):=\left(\begin{array}{cc}A-I_2& w_{\mathrm{bor}}\\ v_{\mathrm{bor}}^{\mathrm{T}}& 0\end{array}\right)$ and $M^{\mathrm{T}}\left(x^{LP},{\alpha }_{LP}\right):=\left(\begin{array}{cc}{A}^{\mathrm{T}}-I_2& v_{\mathrm{bor}}\\ w_{\mathrm{bor}}^{\mathrm{T}}& 0\end{array}\right)$ are nonsingular. The vectors p and q are normalized such that

$$〈q,q〉=〈p,q〉=1.$$

(22)

Now, let ${\alpha }_{LP}={\displaystyle \frac{-1}{3}}$, $x^{LP}=\left(2,{\displaystyle \frac{2}{3}}\right)$, and choose $v_{\mathrm{bot}}={\left(-3,1\right)}^{\mathrm{T}}$, $w_{\mathrm{bot}}={\left(-{\displaystyle \frac{4}{9}},1\right)}^{\mathrm{T}}$. Then the bordered matrices become

$$M\left(x^{LP},{\alpha }_{LP}\right)=\left(\begin{array}{ccc}-1& -3& -{\displaystyle \frac{4}{9}}\\ -{\displaystyle \frac{4}{9}}& -{\displaystyle \frac{4}{3}}& 1\\ -3& 1& 0\end{array}\right),M^{\mathrm{T}}\left(x^{LP},{\alpha }_{LP}\right)=\left(\begin{array}{ccc}-1& -{\displaystyle \frac{4}{9}}& -3\\ -3& -{\displaystyle \frac{4}{3}}& 1\\ -{\displaystyle \frac{4}{9}}& 1& 0\end{array}\right),$$

which are nonsingular. Therefore, the vectors q and p satisfy (20) and (21) and are given (up to scalar multiples) by

$$\begin{array}{cc}\hfill q& \sim {\left(-{\displaystyle \frac{3}{10}},{\displaystyle \frac{1}{10}}\right)}^{\mathrm{T}},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill p& \sim {\left(-{\displaystyle \frac{36}{97}},{\displaystyle \frac{81}{97}}\right)}^{\mathrm{T}}.\hfill \end{array}$$

(24)

Using the normalization condition (22), the vectors can be written as

$$\begin{array}{cc}\hfill q& ={\left({\displaystyle \frac{3}{\sqrt{10}}},-{\displaystyle \frac{1}{\sqrt{10}}}\right)}^{\mathrm{T}},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill p& ={\left({\displaystyle \frac{4\sqrt{10}}{21}},-{\displaystyle \frac{3\sqrt{10}}{7}}\right)}^{\mathrm{T}}.\hfill \end{array}$$

(26)

Then we compute

$$B(q,q)={\left({\displaystyle \frac{9}{5}},{\displaystyle \frac{2}{5}}\right)}^{\mathrm{T}},$$

(27)

For parameter values $\alpha$ close to ${\alpha }_{LP}$, the restriction of System (14) to a parameter-dependent center manifold is locally smoothly equivalent to the topological normal form:

$${\xi }_1\to {\beta }_1\left(\alpha \right)+{\xi }_1+b_1\left(\alpha \right){\xi }_1^2+\mathcal{O}\left({\xi }_1^3\right),{\xi }_1\in \mathbb{R},$$

(28)

where ${\beta }_1\left({\alpha }_{LP}\right)=0$,

$$b_1\left({\alpha }_{LP}\right)={\displaystyle \frac{1}{2}}〈p,B(q,q)〉={\displaystyle \frac{3\sqrt{10}}{35}}\approx 0.2710523708\ne 0.$$

(29)

and the $\mathcal{O}$-terms may also depend on $\alpha$.

3.2. Period-Doubling Bifurcation

When the parameter $\alpha$ cross the critical values ${\alpha }_{P{D}_1}=-1$ and ${\alpha }_{P{D}_2}=-0.2955977425$ corresponding to the PD bifurcations, the Jacobian matrix evaluated at $P_1$ and $P_2$, respectively, have a simple eigenvalue ${\lambda }_1=-1$ and no other eigenvalues on the unit circle. Therefore, the corresponding critical eigenspace is one-dimensional and spanned by an eigenvector $q\in {\mathbb{R}}^2$ satisfying

$$\left(A+I_2\right)q=0.$$

(30)

Let $p\in {\mathbb{R}}^2$ be the adjoint eigenvector, which satisfies

$$\left(A^{\mathrm{T}}+I_2\right)p=0,$$

(31)

where $A^{\mathrm{T}}$ is the transpose of matrix A.

For the first $PD$ point ($P{D}_1$), we can compute the vectors p and q by solving the following $3\times 3$ bordered systems:

$$\left(\begin{array}{cc}A+I_2& w_{\mathrm{bor}}\\ v_{\mathrm{bor}}^{\mathrm{T}}& 0\end{array}\right)\left(\begin{array}{c}q\\ s\end{array}\right)=\left(\begin{array}{c}\mathbf{0}\\ 1\end{array}\right),$$

(32)

and

$$\left(\begin{array}{cc}{A}^{\mathrm{T}}+I_2& v_{\mathrm{bor}}\\ w_{\mathrm{bor}}^{\mathrm{T}}& 0\end{array}\right)\left(\begin{array}{c}p\\ s\end{array}\right)=\left(\begin{array}{c}\mathbf{0}\\ 1\end{array}\right),$$

(33)

where $\mathbf{0}={\left(0,0\right)}^{\mathrm{T}}$, $s\in \mathbb{R}$ in (32) and (33) are the same, and the bordering vectors $w_{\mathrm{bor}}$, $v_{\mathrm{bor}}\in {\mathbb{R}}^2$ are chosen such that the matrices

$$M\left(x^{P{D}_1},{\alpha }_{P{D}_1}\right):=\left(\begin{array}{cc}A+I_2& w_{\mathrm{bor}}\\ v_{\mathrm{bor}}^{\mathrm{T}}& 0\end{array}\right)\mathrm{and}{M}^{\mathrm{T}}\left(x^{P{D}_1},{\alpha }_{P{D}_1}\right):=\left(\begin{array}{cc}{A}^{\mathrm{T}}+I_2& v_{\mathrm{bor}}\\ w_{\mathrm{bor}}^{\mathrm{T}}& 0\end{array}\right)$$

are nonsingular. Choose $v_{\mathrm{bot}}={\left(1,1\right)}^{\mathrm{T}}$ and $w_{\mathrm{bot}}={\left(0,1\right)}^{\mathrm{T}}$. Then the matrices

$$M\left(x^{P{D}_1},{\alpha }_{P{D}_1}\right)=\left(\begin{array}{ccc}1& -1& 0\\ 0& 0& 1\\ 1& 1& 0\end{array}\right),M^{\mathrm{T}}\left(x^{P{D}_1},{\alpha }_{P{D}_1}\right)=\left(\begin{array}{ccc}1& 0& 1\\ -1& 0& 1\\ 0& 1& 0\end{array}\right)$$

are nonsingular. Therefore the vectors q and p that satisfy (32) and (33) are

$$\begin{array}{cc}\hfill q& \sim {\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)}^{\mathrm{T}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill p& \sim {\left(0,1\right)}^{\mathrm{T}}.\hfill \end{array}$$

(35)

Normalizing the vectors p and q such that

$$〈q,q〉=〈p,q〉=1,$$

(36)

the vectors p and q can then be written as

$$\begin{array}{cc}\hfill q& ={\left({\displaystyle \frac{\sqrt{2}}{2}},{\displaystyle \frac{\sqrt{2}}{2}}\right)}^{\mathrm{T}},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill p& ={\left(0,\sqrt{2}\right)}^{\mathrm{T}}.\hfill \end{array}$$

(38)

Then

$$\begin{array}{cc}\hfill B(q,q)& ={\left(-1,0\right)}^{\mathrm{T}},\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill C(q,q,q)& ={\left({\displaystyle \frac{-3\sqrt{2}}{2}},{\displaystyle \frac{-3\sqrt{2}}{2}}\right)}^{\mathrm{T}},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\left(I_2-A\right)}^{-1}& =\left(\begin{array}{cc}1& {\displaystyle \frac{-1}{2}}\\ 0& {\displaystyle \frac{1}{2}}\end{array}\right),\hfill \end{array}$$

(41)

and hence

$$B\left(q,{\left(I_2-A\right)}^{-1}B(q,q)\right)={\left(0,0\right)}^{\mathrm{T}}.$$

For parameter values $\alpha$ close to ${\alpha }_{P{D}_1}$, the restriction of System (14) to a parameter-dependent center manifold is locally smoothly equivalent to the topological normal form

$${\xi }_2\to -\left(1+{\beta }_2\left(\alpha \right)\right){\xi }_2+c\left(\alpha \right){\xi }_2^3+\mathcal{O}\left({\xi }_2^4\right),{\xi }_2\in \mathbb{R},$$

(42)

where ${\beta }_2\left({\alpha }_{P{D}_1}\right)=0$,

$$c\left({\alpha }_{P{D}_1}\right)={\displaystyle \frac{1}{6}}〈p,c(q,q,q)+3B\left(q,{\left(I_2-A\right)}^{-1}B(q,q)\right)〉=-{\displaystyle \frac{1}{2}}\ne 0$$

(43)

and the $\mathcal{O}$-terms may also depend on $\alpha$.

Similarly, we compute the normal form coefficient (43) for the second $PD$ point ($P{D}_2$). For the $P{D}_2$ point, we have the critical parameter value ${\alpha }_{P{D}_2}=-0.2955977425$, which corresponds to the point $x^{P{D}_2}=\left(1.543689012,0.8392867550\right)$. Let $v_{\mathrm{bot}}={\left(0.7176623958,0.7176623958\right)}^{\mathrm{T}}$ and $w_{\mathrm{bot}}={\left(0.5844718998,0.8297416624\right)}^{\mathrm{T}}$. Then the matrices

$$M\left(x^{P{D}_2},{\alpha }_{P{D}_2}\right)=\left(\begin{array}{ccc}1& -1& 0.5844718998\\ -0.7044022578& 0.704402257& 0.8297416624\\ 0.7176623958& 0.7176623958& 0\end{array}\right)$$

and

$$M^{\mathrm{T}}\left(x^{P{D}_2},{\alpha }_{P{D}_2}\right)=\left(\begin{array}{ccc}1& -0.7044022578& 0.7176623958\\ -1& 0.704402257& 0.7176623958\\ 0.5844718998& 0.8297416624& 0\end{array}\right)$$

are nonsingular. Therefore, the vectors q and p that satisfy (32) and (33) are

$$\begin{array}{cc}\hfill q& \sim {\left(0.6967064221,0.6967064219\right)}^{\mathrm{T}},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill p& \sim {\left(0.5674051319,0.8055129384\right)}^{\mathrm{T}}.\hfill \end{array}$$

(45)

Using the normalization condition (36), the vectors p and q can then be written as

$$\begin{array}{cc}\hfill q& ={\left(0.707106781021522,0.707106780818536\right)}^{\mathrm{T}},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill p& ={\left(0.584471900355116,0.829741663228833\right)}^{\mathrm{T}}.\hfill \end{array}$$

(47)

Then

$$\begin{array}{cc}\hfill B(q,q)& ={\left(1.83928675808550,1.54368901137504\right)}^{\mathrm{T}},\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill C(q,q,q)& ={\left(-7.17637531500112,-5.68211246826260\right)}^{\mathrm{T}},\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill {\left(I_2-A\right)}^{-1}& =\left(\begin{array}{cc}2.19148788679499& -1.69148788721414\\ -1.19148788560351& 1.69148788552265\end{array}\right),\hfill \end{array}$$

(50)

and hence

$$B\left(q,{\left(I_2-A\right)}^{-1}B(q,q)\right)={\left(1.09155295490263,2.10305666505445\right)}^{\mathrm{T}}.$$

Therefore, for parameter values $\alpha$ close to ${\alpha }_{P{D}_2}$, the restriction of System (14) to a parameter-dependent center manifold is locally smoothly equivalent to the topological normal form

$${\xi }_2\to -\left(1+{\beta }_2\left(\alpha \right)\right){\xi }_2+c\left(\alpha \right){\xi }_2^3+\mathcal{O}\left({\xi }_2^4\right),{\xi }_2\in \mathbb{R},$$

(51)

where ${\beta }_2\left({\alpha }_{P{D}_2}\right)=0$,

$$c\left({\alpha }_{P{D}_2}\right)={\displaystyle \frac{1}{6}}〈p,c(q,q,q)+3B\left(q,{\left(I_2-A\right)}^{-1}B(q,q)\right)〉=-0.2933579789\ne 0$$

(52)

and the $\mathcal{O}$-terms may also depend on $\alpha$.

The sign of the normal form coefficient $c\left({\alpha }_{PD}\right)$ allows us to predict the direction of the bifurcation of the period-2 cycle that bifurcates from the $PD$ point. If $c\left({\alpha }_{PD}\right)>0$, the bifurcation is supercritical, and the period-2 cycle is stable. If $c\left({\alpha }_{PD}\right)<0$, the bifurcation is subcritical, and the period-2 cycle is unstable. Therefore, since $c\left({\alpha }_{P{D}_1}\right)$ and $c\left({\alpha }_{P{D}_2}\right)$ are always negative, the $PD$ bifurcations are subcritical, and unstable period-2 cycles are born from the $P{D}_1$ and $P{D}_2$ points.

3.3. Neimark–Sacker Bifurcation

When the parameter $\alpha$ crosses the critical value ${\alpha }_{\mathrm{NS}}=0.569840$, corresponding to the Neimark–Sacker (NS) bifurcation, the Jacobian matrix evaluated at $P_3:=x^{\mathrm{NS}}=\left(-1.324719057,-0.3247179075\right)$ has a simple pair of complex eigenvalues:

$$\lambda ={\mathrm{e}}^{i{\theta }_{\mathrm{NS}}}=0.28492+i0.9585519007\mathrm{and}\overline{\lambda }={\mathrm{e}}^{-i{\theta }_{\mathrm{NS}}}=0.28492-i0.9585519007,$$

where $0<{\theta }_{\mathrm{NS}}<\pi$, and ${\mathrm{e}}^{ki{\theta }_{\mathrm{NS}}}-1\ne 0$ for $k=1,2,3,4$ (no strong resonances).

Assume that $q,p\in {\mathbb{C}}^2$ are two eigenvectors of A and its transposed matrix $A^T$ corresponding to $\lambda$ and $\overline{\lambda }$, respectively, i.e., $Aq=\lambda q$ and $A^{T}p=\overline{\lambda }p$. Then, for $\alpha ={\alpha }_{\mathrm{NS}}$, we have

$$\begin{array}{cc}\hfill q& \sim {\left(-0.2833490603+i0.9532668127,-0.1048603543\right)}^T,\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill p& \sim {\left(-0.5027718030+i0.1494439080,-i4.974429739\right)}^T.\hfill \end{array}$$

(54)

To achieve normalization, we can take the normalized vectors as

$$\begin{array}{cc}\hfill q& ={\left(-0.283349060279590+i0.953266812631335,-0.104860354292447\right)}^T,\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill p& ={\left(i0.524512123421591,-4.76824633366354-i1.41731370468605\right)}^T.\hfill \end{array}$$

(56)

Then,

$$\begin{array}{cc}\hfill c(q,q,\overline{q})& ={\left(-1.09196400447135,-0.0347763175644534-i0.0389991886419276\right)}^T,\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill B\left(q,{\left(I_2-A\right)}^{-1}B(q,\overline{q})\right)& ={\left(0.343730050221157,-0.0024343506407263+i0.0442608565237924\right)}^T,\hfill \end{array}$$

(58)

and hence

$$B\left(\overline{q},{\left({\mathrm{e}}^{2i{\theta }_{\mathrm{NS}}}{I}_2-A\right)}^{-1}B(q,q)\right)=\left(\begin{array}{c}-0.0729022102397733+i0.3108076700740190\\ -0.0166993490232101+i0.0532084611653178\end{array}\right).$$

(59)

For parameter values $\alpha$ close to ${\alpha }_{\mathrm{NS}}$, the restriction of System (14) to a parameter-dependent center manifold is locally smoothly equivalent to the topological normal form

$$w\to {\mathrm{e}}^{i\theta \left(\alpha \right)}\left(1+\beta \left(\alpha \right)\right)w+c_1\left(\beta \left(\alpha \right)\right)w{\left|w\right|}^2+\mathcal{O}\left({\left|w\right|}^4\right),$$

where w is a complex variable, $\beta \left({\alpha }_{\mathrm{NS}}\right)=0$, $\theta \left({\alpha }_{\mathrm{NS}}\right)={\theta }_{\mathrm{NS}}$, and

$$\begin{array}{cc}{c}_1\left(0\right)\hfill & ={\displaystyle \frac{1}{2}}〈p,c(q,q,\overline{q})+2B\left(q,{\left(I_2-A\right)}^{-1}B(q,\overline{q})\right)+B\left(\overline{q},{\left({\mathrm{e}}^{2i{\theta }_{\mathrm{NS}}}{I}_2-A\right)}^{-1}B(q,q)\right)〉\hfill \\ \hfill & =0.143042091402532-i0.159649497677570.\hfill \end{array}$$

(60)

Using $c_1\left(0\right)$, we can determine the direction of the appearance of the invariant curve in System (14). First, we need to compute the real number L (i.e., the first Lyapunov coefficient for the NS bifurcation):

$$L_1=\mathrm{Re}\left({\mathrm{e}}^{-i{\theta }_{\mathrm{NS}}}·c_1\left(0\right)\right)\ne 0.$$

(61)

If $L_1$ is negative, the bifurcation is supercritical and the invariant curve is stable. When $L_1$ is positive, it is subcritical and the invariant curve is unstable. From (60) and (61), the first Lyapunov coefficient for the NS bifurcation is equal to

$$L_1=-0.112276776762226<0.$$

(62)

Therefore, the NS bifurcation is supercritical, and the bifurcating periodic solution is asymptotically stable.

3.4. The Existence of Period-2 Cycles

We now investigate the existence of period-2 cycles in System (14). After straightforward computations, we obtain a formula for cycles of period-2 given by

$$x\mapsto f^2\left(x,\alpha \right),f^2:{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^2,$$

(63)

where

$$f^2\left(x,\alpha \right)=\left({\displaystyle \frac{x_2}{\alpha x_1{x}_2+x_2+{\alpha }^2}},{\displaystyle \frac{x_2^2+\alpha x_2}{\alpha x_1{x}_2^2+x_2^2+{\alpha }^2{x}_1{x}_2+{\alpha }^2{x}_2+{\alpha }^3}}\right).$$

The system (63) can have four fixed points, given by

$$\left\{\begin{array}{cc}\hfill & P_1^2(0,0),P_2^2\left({\displaystyle \frac{(\alpha -1)+\delta }{2\alpha }},{\displaystyle \frac{-(\alpha -1)+\delta }{2}}\right),P_3^2\left({\displaystyle \frac{(\alpha -1)-\delta }{2\alpha }},{\displaystyle \frac{-(\alpha -1)-\delta }{2}}\right),\hfill \\ \hfill \\ \hfill & P_4^2\left({\displaystyle \frac{\gamma \left({\alpha }^3-{\alpha }^2-3\alpha -1\right)+\left({\alpha }^5-{\alpha }^4+6{\alpha }^2+5\alpha +1\right)}{\gamma (2{\alpha }^2+\alpha )-\left(3{\alpha }^3+4{\alpha }^2+\alpha \right)}},{\displaystyle \frac{-\left({\alpha }^2+2\alpha +1\right)+\gamma }{2\alpha }}\right),\hfill \\ \hfill \\ \hfill & P_5^2\left({\displaystyle \frac{\gamma \left({\alpha }^3-{\alpha }^2-3\alpha -1\right)-\left({\alpha }^5-{\alpha }^4+6{\alpha }^2+5\alpha +1\right)}{\gamma (2{\alpha }^2+\alpha )+\left(3{\alpha }^3+4{\alpha }^2+\alpha \right)}},{\displaystyle \frac{-\left({\alpha }^2+2\alpha +1\right)-\gamma }{2\alpha }}\right),\hfill \end{array}\right.$$

(64)

where $\delta =\sqrt{1+2\alpha -3{\alpha }^2}$ and $\gamma =\sqrt{{\alpha }^4+2{\alpha }^2+4\alpha +1}$. The fixed points $P_1^2$, $P_2^2$, and $P_3^2$ are the fixed points of System (14) and can be ignored. The fixed points $P_4^2$ and $P_5^2$ exist only if the system parameter $\alpha$ lies in the intervals $(-\infty ,-1)\cup (-0.2955977425,0)\cup (0,\infty )$. Note that, $\alpha =-1$ and $\alpha =-0.2955977425$ correspond to the critical parameter values ${\alpha }_{P{D}_1}$ and ${\alpha }_{P{D}_2}$, respectively, which correspond to the $PD$ bifurcations of System (14). The Jacobian matrix of System (63) evaluated at the fixed point $x^{*}$ is given by

$$A_2\left(\alpha \right)=\left(\begin{array}{cc}-{\displaystyle \frac{\alpha {\left(x_2^{*}\right)}^2}{{\left(\alpha x_1^{*}{x}_2^{*}+x_2^{*}+{\alpha }^2\right)}^2}}& {\displaystyle \frac{{\alpha }^2}{{\left(\alpha x_1^{*}{x}_2^{*}+x_2^{*}+{\alpha }^2\right)}^2}}\\ \\ {\displaystyle \frac{\alpha {\left(x_2^{*}\right)}^2\left({\left(x_2^{*}\right)}^2+2\alpha x_2^{*}+{\alpha }^2\right)}{{\left(\alpha x_1^{*}{\left(x_2^{*}\right)}^2+{\left(x_2^{*}\right)}^2+{\alpha }^2\left(x_1^{*}{x}_2^{*}+x_2^{*}\right)+{\alpha }^3\right)}^2}}& {\displaystyle \frac{\alpha \left(\alpha {\left(x_2^{*}\right)}^2-{\left(x_2^{*}\right)}^2+2{\alpha }^2{x}_2^{*}+{\alpha }^3\right)}{{\left(\alpha x_1^{*}{\left(x_2^{*}\right)}^2+{\left(x_2^{*}\right)}^2+{\alpha }^2\left(x_1^{*}{x}_2^{*}+x_2^{*}\right)+{\alpha }^3\right)}^2}}\end{array}\right).$$

(65)

The eigenvalues of the Jacobian matrix are given by

$${\lambda }_{1,2}=-{\displaystyle \frac{1}{2\alpha }}\left({\alpha }^4-{\alpha }^3+2\alpha +1\right)\pm {\displaystyle \frac{1}{2\alpha }}\sqrt{{\alpha }^8-2{\alpha }^7+{\alpha }^6+4{\alpha }^5-6{\alpha }^4-10{\alpha }^3+4\alpha +1}.$$

(66)

One of the eigenvalues always greater than 1 for $\alpha \in (-\infty ,-1)\cup (-0.2955977425,0)\cup (0,\infty )$. Thus, the period-2 cycles in System (63) are unstable. Figure 2 shows that for different values of $\alpha$ and initial value for the state variables, computed using the formulas for $P_4^2$ and $P_5^2$, a solution of System (14) converges to unstable period-2 cycles.

4. Numerical Simulation

In this section, we present numerical simulations of System (14) to validate our theoretical findings.

4.1. Numerical Bifurcation Analysis

We perform a bifurcation analysis using the Matlab package MatContM, which is based on numerical continuation methods. First, we continue the fixed-point curve with the free parameter $\alpha$ starting at $\alpha =-0.333333$ and initial values for the state variables $x_1=2$ and $x_2=0.666667$. Four bifurcation points are detected, as shown in

Table 1. Continuation of fixed points of System (14) using the software package MatContM.

Label	${\mathit{x}}_1$	${\mathit{x}}_2$	$\mathit{\alpha }$	Normal Form Coefficient
LP	2	0.666667	$-0.333333$	0.2710524
PD1	1.543689	0.839287	$-0.295598$	$-0.2933580$
BP	0	0	1	—
NS	$-1.324718$	$-0.324718$	0.569840	$-0.1122768$
PD2	0	0	$-1$	$-0.5$

, including LP, PD₂, the branching point (BP), and the NS point. Next, we select the $BP$ point as initial data for the continuation and continue the fixed-point curve with respect to $\alpha$ as the free system parameter. This leads us to detect a second PD point, PD₁, as indicated in Table 1. The numerical data displayed in the Matlab command window are summarized in Table 1.

The solution branches computed using MatContM are plotted in Figure 3. Note that the branching point ($BP$) corresponds to a $PF$ bifurcation, where two additional branches of fixed points bifurcate from this point, as shown in Figure 1. Additionally, the numerical data presented in Table 1 are consistent with the results derived in Section 2 and Section 3.

4.2. Stability Analysis of the 2-Cycles

We perform a numerical stability analysis of the period-2 cycles using a continuation method implemented in MatContM. Specifically, we use the following initial data for the period-2 orbits:

$$\left(\alpha ,x_1,x_2\right)=(0.5,1.137458524,-0.3625413910)\mathrm{and}\left(\alpha ,x_1,x_2\right)=(-2,0.2807764062,1.280776406),$$

and continue each cycle with $\alpha$ as the free parameter. The continuation results are shown as solid curves in Figure 4. Along these curves, we identify bifurcation points, including branch points ($BP$) and period-doubling bifurcations ($PD$). It is evident that the period-2 cycles emerge from the $P{D}_1$ and $P{D}_2$ bifurcation points on the fixed-point curves of System (14), which are represented by dashed lines in Figure 4. These bifurcation origins are labeled $B{P}_1$ and $B{P}_2$ on the period-2 cycle branches. Moreover, the presence of $PD$ bifurcations along the period-2 cycle curves suggests the existence of higher-period cycles. Throughout the continuation process, we monitor the eigenvalues of the corresponding Jacobians and consistently observe at least one eigenvalue with magnitude greater than 1 or less than $-1$. This indicates that the period-2 cycles bifurcating from the $PD$ points of System (14) are unstable.

4.3. Analysis of the NS Bifurcation

In Section 2, we show that the fixed point $P_3$ is locally stable for values of the parameter $\alpha$ in the interval ${\alpha }_{NS}:=0.569840<\alpha <1$ and $\alpha \ne 0$. At $\alpha ={\alpha }_{NS}$, the fixed point $P_3$ loses its stability through an NS bifurcation. As demonstrated in Equation (62), the first Lyapunov coefficient associated with this bifurcation is negative, which implies that an attracting invariant closed curve emerges from the fixed point when $-1/3<\alpha <{\alpha }_{NS}$.

Figure 5a,b confirm that for $\alpha >{\alpha }_{NS}$, the fixed point $P_3$ remains locally stable and acts as an attractor, as further illustrated in Figure 6a,b. Conversely, when $\alpha <{\alpha }_{NS}$, Figure 5c–f show that the fixed point becomes unstable, and simultaneously, an attracting invariant closed curve bifurcates from it, as depicted in Figure 6c–f.

These computational results support the theoretical analysis of the NS bifurcation discussed in Section 2 and Section 3.

5. Analysis by Lyapunov Exponent

The largest Lyapunov exponent $\sigma$ quantifies the exponential rate at which nearby trajectories diverge, serving as a key indicator of chaos in dynamical systems. For System (13), we approximate $\sigma$ numerically by tracking the evolution of tangent vectors along trajectories. Starting from an initial condition $(x_0,y_0)$, we iterate both the system dynamics and the Jacobian matrix $A_n$ simultaneously. At each step n, the Jacobian $A_n$ is given by

$$A_n=\left(\begin{array}{cc}0& {\displaystyle \frac{\alpha }{{(y_n+\alpha )}^2}}\\ -{\displaystyle \frac{y_n^2}{{(x_n{y}_n+\alpha )}^2}}& {\displaystyle \frac{\alpha }{{(x_n{y}_n+\alpha )}^2}}\end{array}\right),n=0,1,2,\dots N.$$

The Lyapunov exponent is then estimated as the average exponential growth rate:

$$\sigma \left(\alpha \right)\approx {\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}ln\parallel A_n{v}_n\parallel ,$$

where $v_n$ is a unit vector indicating the direction of maximal expansion at step n. This renormalization-based scheme, as described in [37], converges to the largest Lyapunov exponent by progressively aligning the tangent vector with the dominant expanding direction. We vary the parameter $\alpha$ over the interval $[-1.5,3]$ with a step size of ${10}^{-4}$, using $N={10}^4$ iterations to capture transitions between chaotic behavior ($\sigma >0$) and periodic or stable dynamics ($\sigma \le 0$), as illustrated in Figure 7a. Although transient dynamics are not explicitly discarded, the long iteration horizon ensures convergence. Numerical stability is preserved through stepwise normalization of the tangent vector.

By comparing Figure 7a with the bifurcation diagram in Figure 1, we observe the following behavior:

• For $\alpha \in [-1.5,-1)$, the exponent $\sigma$ is negative and gradually approaches zero at the period-doubling bifurcation point $P{D}_1$, where an unstable period-2 cycle is born.
• In the interval $\alpha \in (-1,-0.3)$, $\sigma$ becomes positive, indicating chaotic dynamics. It then decreases and approaches zero near $\alpha =-0.3$, corresponding to the $P{D}_2$ bifurcation.
• For $\alpha >-0.3$, $\sigma$ becomes negative, reflecting the stability of the fixed point $P_2$ and the presence of attracting invariant closed curves near the Neimark–Sacker ($NS$) bifurcation point.
• The Lyapunov exponent also approaches zero near $\alpha \approx 0.5$ and $\alpha =1$, which correspond to the $NS$ and branch point ($BP$) bifurcations, respectively.

From this analysis, we conclude that the system exhibits chaotic behavior for $\alpha \in (-1,-0.3)$, where $\sigma >0$. This is further illustrated in Figure 7b, which shows a chaotic cycle around $\alpha =-0.5$.

6. Conclusions

This paper investigates the dynamic behavior of the second-order nonlinear rational difference Equation (1), which models a population system with nonlinear interactions between the current and previous population states. We derive analytical conditions for the stability of fixed points, explore codim-1 bifurcations, and compute the corresponding topological normal forms. Our analysis also establishes the existence of period-2 solutions, offering deeper insights into the system’s dynamic structure.

The nonlinear cross-term $y_n{y}_{n-1}$ is interpreted as representing intergenerational interaction, adding significant ecological relevance that is often absent in purely mathematical models. Compared to classical population models such as the Beverton–Holt equation [15] and the Pielou logistic delay equation [26,27], the proposed model exhibits a richer array of dynamical behaviors, including period-doubling bifurcations and chaotic dynamics, particularly in the range $-1<\alpha <-0.3$, as verified through Lyapunov exponent analysis.

To support the theoretical results, we performed numerical simulations and bifurcation analysis using the Matlab package MatContM, which employs numerical continuation techniques. The findings suggest that the model provides a biologically meaningful and mathematically versatile framework for studying delayed population dynamics. Furthermore, it opens avenues for future research, including extensions to stochastic systems and applications in epidemiological models with memory and nonlinear interaction effects. In particular, future work could explore fractional-order extensions, such as Caputo and $\psi$-Hilfer difference operators, to capture memory-dependent effects and enhance the model’s applicability to real-world data.