Dynamics and Bifurcation Analysis of a Generalized Three-Dimensional Chaotic Financial System

Abstract

This paper investigates the dynamics of a three-dimensional nonlinear model of the financial system and the conditions for the emergence of chaotic behavior. The well-known chaotic system with given parameters and initial conditions is considered as a basis. For the initial model, critical points are analyzed, two-dimensional and three-dimensional phase portraits are constructed, and Lyapunov exponents are calculated, which allow confirming the presence of chaos and assessing the degree of sensitivity to initial data. Next, a modification of the system is proposed, consisting of changing the degree of the variable in the second equation. For the group of models obtained, we considered the generalized form of the system, found its critical points, and classified them. At the next stage, a bifurcation analysis was performed: by changing the key parameters of the modified systems, bifurcation diagrams were constructed, and parameter regions corresponding to critical points, periodicity, quasi-periodicity, and chaos were identified. The results demonstrate that the nature of the dynamics depends significantly on both the parameters and the degree of nonlinearity and allow conclusions to be drawn about the mechanisms of chaos in the financial model under consideration.

1. Introduction

Chaos theory originated in the natural sciences [1], particularly mathematics and physics [2], where it was developed to analyze complex nonlinear systems that exhibit a strong sensitivity to initial conditions [3,4]. Over time, the fundamental principles of chaos theory have been increasingly applied in economics to investigate disturbances in economic systems and evaluate their potential consequences [5,6]. Chaotic economic [7] and financial market models [8] have long been a central topic of investigation for economists and theoreticians. In recent decades, the increasing instability of financial markets and the increasing role of stochastic factors have further stimulated interest in the application of chaos theory within the financial domain [9]. In this context, a variety of financial models have been extensively examined in the literature with the aim of describing and predicting complex market dynamics (see, for example, [5,10,11,12]).

In chaos theory, methods such as linear and bifurcation analysis [13,14,15], the construction of strange attractors [16,17], and the calculation of Lyapunov exponents [18,19], as well as numerical methods, are widely used to study nonlinear dynamic systems. The study is based on dynamic systems and their phase spaces [3]. Thanks to these methods, it is possible to most accurately characterize the properties of nonlinear dynamical systems, such as sensitivity to initial conditions, non-periodicity, and unpredictability.

The study focuses on a financial dynamical system described by a special set of ordinary differential equations. The analysis concentrates on the properties of attracting sets and their dependence on the model parameters. Graphs of solutions, phase portraits, bifurcation diagrams, and the dynamics of Lyapunov exponents are constructed to illustrate the evolution of the system and predict its future behavior.

Although a number of nonlinear financial models exhibiting chaotic dynamics have been proposed in the literature, most existing studies are based on quadratic or quartic nonlinearities. However, empirical financial data often exhibit strong nonlinear effects, including leptokurtic distributions and heavy-tailed fluctuations. These features indicate that market dynamics may involve stronger nonlinear feedback mechanisms than those captured by low-order nonlinear terms. In this study, the nonlinear term is generalized to $x^k$, where $2\le k\le 10$, allowing us to investigate how the degree of nonlinearity affects system stability, bifurcations, and the emergence of chaotic regimes. This approach provides additional insight into the role of higher-order nonlinearities in the generation of complex financial dynamics. Nevertheless, the influence of higher-order nonlinear terms on the dynamical behavior of financial models remains insufficiently investigated. In particular, the role of varying degrees of nonlinearity in shaping the structure of the phase space and the complexity of system dynamics has not been studied in detail. This motivates the present work, where a generalized financial model with a nonlinear term of the form $x^k$ is considered and its dynamical properties are analyzed.

This manuscript is organized as follows. The Introduction provides a brief overview of chaos theory, outlines the current state of research on nonlinear dynamical systems, and emphasizes the relevance of these concepts to financial modeling. Section 2 introduces a three-dimensional mathematical model of the financial system. In Section 3, the dynamic properties of the proposed three-dimensional nonlinear system are investigated. To achieve a more comprehensive analysis, the model is generalized by replacing the nonlinear term with $x^k$, where $2\le k\le 10$ is an integer. This section includes analysis of critical points, the construction of bifurcation diagrams, the visualization of attractors, the plotting of solution graphs, and the computation of the Lyapunov spectrum and Lyapunov dimension. Finally, the Discussion section summarizes the main results and contributions of this study.

The main contributions of this study can be summarized as follows:

• The critical points of the classical three-dimensional financial system are derived analytically and their existence conditions are determined.
• The local stability of the critical points is investigated by constructing the Jacobian matrix of the system and analyzing the corresponding eigenvalues.
• A generalized nonlinear financial model is proposed by modifying the degree of the variable in the second equation, which leads to a new class of dynamical behavior.
• The critical points of the generalized system are obtained, and their stability properties are analyzed, revealing the conditions under which the system transitions between stable and unstable regimes.
• The global dynamics of the system are investigated using phase portraits, Lyapunov exponents, and bifurcation diagrams.

2. Three-Dimensional Mathematical Model of Financial System

The model (1) describes a three-dimensional financial system where x is the interest rate, y indicates the level of investment demand, and z reflects how prices grow exponentially. In addition, the constant $a\ge 0$ represents the rate of household savings, $b\ge 0$ corresponds to investment costs, and $c\ge 0$ measures the elasticity of demand in commercial markets. The parameter d is a positive scaling parameter [12].

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dx}{dt}}& =z+(y-a)x,\hfill \\ \hfill {\displaystyle \frac{dy}{dt}}& =1-by-d{x}^2-x^4,\hfill \\ \hfill {\displaystyle \frac{dz}{dt}}& =-x-cz.\hfill \end{array}\right.$$

(1)

In system (1), there are two quadratic nonlinearities $xy;x^2$ and one quartic nonlinearity, $x^4$, which together generate rich and complex dynamics. The three-dimensional system (1) exhibits a chaotic attractor when the parameters are set to

$$a=7.2,b=0.1,c=1,d=0.1$$

(2)

and the initial conditions are

$$x\left(0\right)=0.5;y\left(0\right)=3;z\left(0\right)=-0.4.$$

(3)

Definition 1.

A continuous map $f:V\to V$ is chaotic if it is topologically transitive, has dense periodic points, and exhibits sensitive dependence on initial conditions [20].

2.1. Stability and Dynamical Analysis

Dynamical analysis of chaotic systems involves exploring the underlying mechanics and behaviors that lead to chaos [21].

The system is (1) with parameters (2) and initial conditions (3).

Let the third equation of the system (1) be equal to zero: $-x-cz=0$.

Then

$$z=-{\displaystyle \frac{x}{c}}.$$

(4)

Substitute (4) into the first equation of the system (1)

$$-{\displaystyle \frac{x}{c}}+(y-a)x=0.$$

(5)

$$x\left(y-a-{\displaystyle \frac{1}{c}}\right)=0.$$

(6)

Let us consider the first case, where $x=0$. Then substituting $x=0$ into Equation (4) to get $z=0$. After that, substitute $x=0$ into the second equation of the system (1) and equate to zero

$$1-by=0.$$

(7)

From Equation (7) get $y={\displaystyle \frac{1}{b}}$. Taking into account the parameters (2), the first critical point is obtained: $E_1=(0,10,0)$.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (1) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^4=0.$$

(8)

After simplifying (8) get

$$x^2(x^2+d)=1-b\left(a+{\displaystyle \frac{1}{c}}\right).$$

(9)

Taking into account the parameters (2), the two roots of Equation (9) are obtained: $x_{1,2}=\pm 0.6142$. The second and third critical points are $E_2=(0.6142,8.2,-0.6142)$ and $E_3=(-0.6142,8.2,0.6142)$.

Once the fixed points are determined, their stability is examined through the computation of the system’s Jacobian matrix. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}{\displaystyle \frac{\partial \dot{x}}{\partial x}}& {\displaystyle \frac{\partial \dot{x}}{\partial y}}& {\displaystyle \frac{\partial \dot{x}}{\partial z}}\\ {\displaystyle \frac{\partial \dot{y}}{\partial x}}& {\displaystyle \frac{\partial \dot{y}}{\partial y}}& {\displaystyle \frac{\partial \dot{y}}{\partial z}}\\ {\displaystyle \frac{\partial \dot{z}}{\partial x}}& {\displaystyle \frac{\partial \dot{z}}{\partial y}}& {\displaystyle \frac{\partial \dot{z}}{\partial z}}\end{array}\right)=\left(\begin{array}{ccc}y-a& x& 1\\ -2(d+2x^2)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(10)

The Jacobian matrix provides a linear approximation of the system near critical points and allows their type to be determined based on the eigenvalues. This analysis plays a key role in studying the local stability of a dynamical system [22].

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -2(d+2x^2)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(11)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+4x^4+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(12)

Put the values of the first critical point $E_1=(0,10,0)$ in (12) and get three roots

$${\lambda }_1=2.5156,{\lambda }_2=-0.7155,{\lambda }_3=-0.1.$$

(13)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.6142,8.2,-0.6142)$ in (12) and get three roots

$${\lambda }_1=-0.6462,{\lambda }_{2,3}=0.2731\pm 0.9607i.$$

(14)

This result indicates that the critical point $E_2=(0.6142,8.2,-0.6142)$ behaves as a saddle-focus point and is unstable.

Put the values of the third critical point $E_3=(-0.6142,8.2,0.6142)$ in (12) and get three roots

$${\lambda }_1=-0.6462,{\lambda }_{2,3}=0.2731\pm 0.9607i.$$

(15)

This result indicates that the critical point $E_3=(-0.6142,8.2,0.6142)$ behaves as a saddle-focus point and is unstable.

Figure 1 shows the phase plots of system (1), while Figure 2 displays the corresponding solution graphs.

The Lyapunov exponent is a fundamental measure of the sensitivity of a system to initial conditions, which is a defining characteristic of chaotic dynamics [23,24]. The Lyapunov exponents for system (1) with given parameters (2) and initial conditions (3) were calculated as follows:

$$L{E}_1=0.13371;L{E}_2=-0.0002;L{E}_3=-0.4094.$$

(16)

Let us compute the Kaplan–Yorke dimension [25] using the formula

$$D_{KY}=j+{\displaystyle \frac{1}{|L_{j+1}|}}\sum _{i=1}^j{L}_i,$$

(17)

where j is the largest integer that satisfies

$$\sum _{i=1}^j{L}_i>0,\sum _{i=1}^{j+1}{L}_i<0.$$

$$D_{KY}=2+{\displaystyle \frac{0.13371-0.0002}{|-0.4094|}}=2.33$$

(18)

A chaotic system has at least one positive Lyapunov exponent, and the more positive the largest Lyapunov exponent, the more unpredictable the system is [26].

The system (1) is dissipative since the sum of the Lyapunov exponents is negative and chaotic since the first Lyapunov exponent is positive, $L{E}_1=0.13371$. Computing the full spectrum of Lyapunov exponents is mathematically challenging. The calculations were carried out using Wolfram Mathematica 13 and Matlab R2025b. Figure 3 displays the Lyapunov characteristic exponents of the system (1).

The system (1) represents the classical three-dimensional financial model that has been widely studied in the literature [12].

2.2. Bifurcation Analysis

Table 1. Parameter a analysis.

a	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$0.0009$	$-0.0103$	$-0.5290$	periodic
$1.0$	$0.0001$	$-0.0116$	$-0.5073$	periodic
$3.0$	$0.0011$	$-0.2269$	$-0.2328$	periodic
$4.0$	$0.0009$	$-0.0379$	$-0.3876$	periodic
$4.5$	$0.0017$	$-0.0743$	$-0.3352$	periodic
$4.7$	$0.0002$	$-0.0240$	$-0.3783$	periodic
$4.8$	$0.0240$	$0.0002$	$-0.4228$	chaotic
$5.0$	$0.0421$	$-0.0017$	$-0.4331$	chaotic
$6.0$	$0.1248$	$-0.0006$	$-0.4616$	chaotic
$7.2$	$0.1337$	$-0.0002$	$-0.4094$	chaotic
$8.0$	$0.0773$	$-0.0001$	$-0.3112$	chaotic
$8.3$	$0.0280$	$-0.0004$	$-0.2637$	chaotic
$8.4$	$0.0069$	$-0.0118$	$-0.2255$	periodic
$8.5$	$0.0120$	$-0.0497$	$-0.1811$	periodic
$9.0$	$0.0007$	$-0.0129$	$-0.0916$	periodic
$10.0$	$-0.1123$	$-0.4897$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

The bifurcation diagram for the parameter a is explored. Figure 4 displays the bifurcation diagram of the system (1), obtained by varying the control parameter a. The parameters $b,c$, and d remain fixed. The bifurcation diagram is plotted when a is varied between $0\le a\le 10$.

Table 2. Parameter b analysis.

b	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.01$	$0.0024$	$-0.0065$	$-0.4841$	quasiperiodic
$0.10$	$0.1337$	$-0.0002$	$-0.4094$	chaotic
$0.20$	$-0.2147$	$-1.5852$	$-1.6002$	asymptotically stable
$0.30$	$-0.3130$	$-1.3937$	$-3.4599$	asymptotically stable
$0.50$	$-0.5122$	$-1.2417$	$-4.9461$	asymptotically stable
$1.00$	$-1.0119$	$-1.1887$	$-5.9994$	asymptotically stable
$3.00$	$-1.1764$	$-3.0000$	$-6.6903$	asymptotically stable
$7.00$	$-1.1706$	$-6.8978$	$-6.9888$	asymptotically stable
$10.0$	$-1.1693$	$-6.9425$	$-9.9880$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter b.

The bifurcation diagram for the parameter b is explored. Figure 5 displays the bifurcation diagram of the system (1), obtained by varying the control parameter b. The parameters $a,c$, and d remain fixed. The bifurcation diagram is plotted when b is varied between $0\le b\le 0.14$.

Table 3. Parameter c analysis.

c	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$0.0011$	$-0.0501$	$-0.0544$	periodic
$1.0$	$0.1337$	$-0.0002$	$-0.4094$	chaotic
$1.1$	$0.0447$	$0.0004$	$-0.4756$	chaotic
$1.2$	$0.0022$	$-0.0101$	$-0.5598$	periodic
$1.5$	$0.0015$	$-0.1166$	$-0.8372$	periodic
$2.0$	$0.0039$	$-0.0164$	$-1.4298$	periodic
$3.0$	$-0.0266$	$-0.0318$	$-2.7038$	asymptotically stable
$5.0$	$-0.0410$	$-0.0465$	$-4.8081$	asymptotically stable
$7.0$	$-0.0447$	$-0.0489$	$-6.8601$	asymptotically stable
$10.0$	$-0.0462$	$-0.0499$	$-9.9009$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

The bifurcation diagram for the parameter c is explored. Figure 6 displays the bifurcation diagram of the system (1), obtained by varying the control parameter c. The parameters $a,b$, and d remain fixed. The bifurcation diagram is plotted when c is varied between $0\le c\le 3$.

Table 4. Parameter d analysis.

d	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$0.1337$	$-0.0002$	$-0.4094$	chaotic
$0.5$	$0.0889$	$-0.0002$	$-0.3696$	chaotic
$0.8$	$0.0802$	$0.0000$	$-0.3389$	chaotic
$0.9$	$0.0233$	$0.0000$	$-0.2710$	chaotic
$1.0$	$0.0027$	$-0.1210$	$-0.1291$	periodic
$1.1$	$0.0539$	$0.0001$	$-0.3214$	chaotic
$1.3$	$0.0632$	$0.0002$	$-0.3386$	chaotic
$1.5$	$0.0446$	$-0.0006$	$-0.3226$	chaotic
$2.0$	$0.0532$	$0.0006$	$-0.3296$	chaotic
$3.0$	$0.0319$	$0.0003$	$-0.3178$	chaotic
$4.0$	$0.0233$	$0.0000$	$-0.2710$	chaotic
$4.1$	$0.0031$	$-0.0082$	$-0.2797$	quasiperiodic
$4.3$	$0.0029$	$-0.0032$	$-0.2848$	quasiperiodic
$4.5$	$0.0023$	$-0.0097$	$-0.2779$	quasiperiodic
$5.0$	$0.0024$	$-0.0485$	$-0.2395$	periodic
$7.0$	$0.0034$	$-0.0046$	$-0.2850$	quasiperiodic
$8.0$	$0.0031$	$-0.0083$	$-0.2808$	quasiperiodic
$10.0$	$0.0032$	$-0.0127$	$-0.2765$	periodic

presents the analysis of system dynamics for various values of the parameter d.

The bifurcation diagram for the parameter d is explored. Figure 7 displays the bifurcation diagram of the system (1), obtained by varying the control parameter d. The parameters $a,b$, and c remain fixed. The bifurcation diagram is plotted when d is varied between $0\le d\le 6$.

In order to investigate the influence of higher-order nonlinearities on the dynamics of financial systems, we consider a generalized version of this model in the next section.

3. Results

In the system of differential Equation (1), the term $x^4$ introduces a high degree of nonlinearity in the dynamics of the variable y. To perform a more comprehensive analysis, this term is generalized by replacing $x^k$, where $2\le k\le 10$ is an integer. The system takes the form

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dx}{dt}}& =z+(y-a)x,\hfill \\ \hfill {\displaystyle \frac{dy}{dt}}& =1-by-d{x}^2-x^k,\hfill \\ \hfill {\displaystyle \frac{dz}{dt}}& =-x-cz.\hfill \end{array}\right.$$

(19)

For each specific value of k, a comprehensive analysis will be performed.

The divergence of the vector field of system (19) is given by

$$\nabla ·F={\displaystyle \frac{\partial F_1}{\partial x}}+{\displaystyle \frac{\partial F_2}{\partial y}}+{\displaystyle \frac{\partial F_3}{\partial z}}=y-a-b-c.$$

(20)

If the divergence is negative in a region of the phase space, the system is dissipative, and the trajectories converge to a bounded attractor [24].

3.1. Case 1: k = 2

3.1.1. Stability and Dynamical Analysis

The system is (19) and $k=2$ with given parameters (2) and initial conditions (3).

Let the third equation of the system (19) and $k=2$ be equal to zero: $-x-cz=0$.

Then

$$z=-{\displaystyle \frac{x}{c}}.$$

(21)

Substitute (21) into the first equation of the system (19)

$$-{\displaystyle \frac{x}{c}}+(y-a)x=0.$$

(22)

$$x\left(y-a-{\displaystyle \frac{1}{c}}\right)=0.$$

(23)

Let us consider the first case, where $x=0$. Then substituting $x=0$ into Equation (21) to get $z=0$. After that, substitute $x=0$ into the second equation of the system (19) and equate to zero

$$1-by=0.$$

(24)

From Equation (24) get $y={\displaystyle \frac{1}{b}}$. Taking into account the parameters (2), the first critical point is obtained: $E_1=(0,10,0)$.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (19) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^2=0.$$

(25)

After simplifying (25) get

$$x^2={\displaystyle \frac{1-b(a+\frac{1}{c})}{d+1}}.$$

(26)

Taking into account the parameters (2), the two roots of Equation (26) are obtained: $x_{1,2}=\pm 0.4045$. The second and third critical points are $E_2=(0.4045,8.2,-0.4045)$ and $E_3=(-0.4045,8.2,0.4045)$.

Once the fixed points are determined, their stability is examined through the computation of the system’s Jacobian matrix. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}{\displaystyle \frac{\partial \dot{x}}{\partial x}}& {\displaystyle \frac{\partial \dot{x}}{\partial y}}& {\displaystyle \frac{\partial \dot{x}}{\partial z}}\\ {\displaystyle \frac{\partial \dot{y}}{\partial x}}& {\displaystyle \frac{\partial \dot{y}}{\partial y}}& {\displaystyle \frac{\partial \dot{y}}{\partial z}}\\ {\displaystyle \frac{\partial \dot{z}}{\partial x}}& {\displaystyle \frac{\partial \dot{z}}{\partial y}}& {\displaystyle \frac{\partial \dot{z}}{\partial z}}\end{array}\right)=\left(\begin{array}{ccc}y-a& x& 1\\ -2(d+1)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(27)

The Jacobian matrix provides a linear approximation of the system near critical points and allows their type to be determined based on the eigenvalues. This analysis plays a key role in studying the local stability of a dynamical system [22].

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -2(d+1)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(28)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+2x^2+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(29)

Put the values of the first critical point $E_1=(0,10,0)$ in (29) and get three roots

$${\lambda }_1=2.5156,{\lambda }_2=-0.7155,{\lambda }_3=-0.1.$$

(30)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.4045,8.2,-0.4045)$ in (29) and get three roots

$${\lambda }_1=-0.5717,{\lambda }_{2,3}=0.2359\pm 0.7577i.$$

(31)

This result indicates that the critical point $E_2=(0.4045,8.2,-0.4045)$ behaves as a saddle-focus point and is unstable.

Put the values of the third critical point $E_3=(-0.4045,8.2,0.4045)$ in (29) and get three roots

$${\lambda }_1=-0.5717,{\lambda }_{2,3}=0.2359\pm 0.7577i$$

(32)

This result indicates that the critical point $E_3=(-0.4045,8.2,0.4045)$ behaves as a saddle-focus point and is unstable.

The stability of the critical points is analyzed using the Jacobian matrix and the corresponding eigenvalues obtained from local linearization of the system. In addition to the Jacobian matrix analysis based on local linearization, global stability of nonlinear systems can also be investigated using the direct Lyapunov method. Such approaches provide complementary tools for studying the stability properties of dynamical systems. The eigenvalues of the Jacobian matrix determine the type and stability of the critical points.

To analyze the possibility of Hopf bifurcation, the Jacobian matrix of system (19) is considered at critical points. A Hopf bifurcation occurs when a pair of complex conjugate eigenvalues of the Jacobian matrix crosses the imaginary axis, while the third eigenvalue remains with a negative real part. Under such conditions, the critical point loses stability, and a periodic orbit may arise.

3.1.2. Bifurcation Analysis

The bifurcation analysis examines the dynamics of the system without regard to parameter interdependence and explores how the behavior of the system changes with different parameter values [22,27].

The bifurcation analysis is carried out by varying the parameter $0.1\le a\le 10$, while keeping the remaining parameters fixed. This approach allows us to investigate the changes in the system dynamics and to identify parameter ranges in which periodic and chaotic regimes occur.

Table 5. Parameter a analysis.

Parameter Value	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	Behavior
$0.1$	$0.1380$	$0.0000$	$-0.5437$	chaotic
$0.5$	$0.1381$	$-0.0001$	$-0.5292$	chaotic
$0.6$	$0.1341$	$-0.0004$	$-0.5296$	chaotic
$0.7$	$0.1315$	$0.0003$	$-0.5256$	chaotic
$0.8$	$0.1354$	$-0.0001$	$-0.5228$	chaotic
$0.9$	$0.1437$	$-0.0006$	$-0.5324$	chaotic
$1.0$	$0.1322$	$-0.0006$	$-0.5275$	chaotic
$1.5$	$0.1336$	$0.0008$	$-0.5236$	chaotic
$2.0$	$0.1336$	$0.0000$	$-0.5201$	chaotic
$3.0$	$0.1490$	$-0.0008$	$-0.5040$	chaotic
$4.0$	$0.1545$	$0.0005$	$-0.4871$	chaotic
$5.0$	$0.1124$	$0.0011$	$-0.4441$	chaotic
$6.0$	$0.0940$	$-0.00001$	$-0.3943$	chaotic
$7.0$	$0.0458$	$-0.0008$	$-0.3267$	chaotic
$7.1$	$0.0240$	$0.0001$	$0.0001$	non-chaotic
$7.2$	$0.0033$	$-0.0224$	$-0.2669$	periodic
$7.3$	$0.0031$	$-0.0216$	$-0.2687$	periodic
$7.4$	$0.0040$	$-0.0102$	$-0.2787$	periodic
$7.5$	$0.0037$	$-0.0282$	$-0.2568$	periodic
$7.6$	$0.0042$	$-0.0542$	$-0.2260$	periodic
$7.7$	$0.0048$	$-0.1049$	$-0.1707$	periodic
$7.8$	$0.0050$	$-0.1327$	$-0.1378$	periodic
$7.9$	$0.0059$	$-0.1294$	$-0.1352$	periodic
$8.0$	$0.0065$	$-0.1252$	$-0.1333$	periodic
$9.0$	$0.0007$	$-0.0129$	$-0.0915$	asymptotically stable
$9.1$	$-0.0516$	$-0.0610$	$-0.0912$	asymptotically stable
$9.2$	$-0.1019$	$-0.1000$	$-0.1019$	asymptotically stable
$10.0$	$-0.1114$	$-0.4907$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $0.1\le a\le 7$, but periodic behavior for $7.2\le a\le 8.9$.

Figure 8 shows the phase plots of system (19), $k=2$, while Figure 9 displays the corresponding solution graphs.

The Lyapunov exponents for system (19), $k=2$, and parameters $a=4,b=0.1$, $c=1$, $d=0.1$, and initial conditions (3) were calculated as follows:

$$L{E}_1=0.1545;L{E}_2=0.0005;L{E}_3=-0.4871.$$

(33)

$$D_{KY}=2+{\displaystyle \frac{0.1545-0.0005}{|-0.4871|}}=2.32$$

(34)

The system is (19) and $k=2$ is dissipative, since the sum of the Lyapunov exponents is negative. Figure 10 displays the Lyapunov characteristic exponents of the system (19), $k=2$.

Bifurcation diagrams are analyzed by varying one parameter at a time and keeping the others fixed [28]. The bifurcation diagram for the parameter a is explored. Figure 11 displays the bifurcation diagram of the system (19) for $k=2$, obtained by varying the control parameter a. The parameters $b,c$, and d remain fixed. The bifurcation diagram is plotted when a is varied between $0.1\le a\le 10$.

For small values of parameter a, the diagram shows a dense cloud of points, indicating chaotic dynamics. As a increases, there is a crisis of chaos, and the attractor shrinks. Periodic windows emerge in intermediate ranges of a, followed by a transition to an asymptotically stable equilibrium for large values of a. The results obtained are in full agreement with the Lyapunov exponent spectrum reported in Table 5.

Figure 12 displays the Poincaré section of the system (19) for $k=2$.

To further confirm the chaotic nature of the proposed system, a Poincaré section was constructed by recording the intersection points of the system trajectories with the plane $z=0$ under the condition $z^{\prime }>0$. The resulting set of points forms a scattered structure, indicating the complex geometry of the attractor. The absence of a closed curve and the irregular distribution of intersection points provide additional evidence of chaotic dynamics in the system. Figure 13 displays the power spectrum of the time series $x\left(t\right)$. The spectrum exhibits a broadband structure without dominant frequency peaks, which is characteristic of chaotic dynamics.

Table 6. Parameter b analysis.

Parameter Value	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	Behavior
$0.005$	$0.1477$	$-0.0004$	$-0.5727$	chaotic
$0.01$	$0.1519$	$-0.0001$	$-0.5556$	chaotic
$0.02$	$0.1617$	$-0.0007$	$-0.5396$	chaotic
$0.05$	$0.1606$	$0.0005$	$-0.4903$	chaotic
$0.1$	$0.0033$	$-0.0224$	$-0.2669$	periodic
$0.2$	$-0.2137$	$-1.5861$	$-1.6002$	asymptotically stable
$0.5$	$-0.5113$	$-1.2426$	$-4.9461$	asymptotically stable
$1.0$	$-1.0110$	$-1.1896$	$-5.9994$	asymptotically stable
$2.0$	$-1.1819$	$-2.0000$	$-6.5181$	asymptotically stable
$4.0$	$-1.1738$	$-4.0000$	$-6.7762$	asymptotically stable
9	$-1.1696$	$-6.9309$	$-8.9882$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter b.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $0.005\le b\le 0.05$, but periodic behavior for $b=0.1$.

Figure 14 shows the phase plots of system (19), $k=2$, while Figure 15 displays the corresponding solution graphs.

The Lyapunov exponents for system (19), $k=2$ and parameters are $a=7.2$, $b=0.05$, $c=1,d=0.1$ and initial conditions (3) were calculated as follows:

$$L{E}_1=0.1606;L{E}_2=0.0005;L{E}_3=-0.4903.$$

(35)

$$D_{KY}=2+{\displaystyle \frac{0.1606-0.0005}{|-0.4903|}}=2.33$$

(36)

Figure 16 displays the Lyapunov characteristic exponents of the system (19), $k=2$.

The bifurcation diagram for the parameter b is explored. Figure 17 displays the bifurcation diagram of the system (19) for $k=2$, obtained by varying the control parameter b. The parameters $a,c$, and d remain fixed. The bifurcation diagram is plotted when b is varied between $0.005\le b\le 0.12$.

Since chaotic behavior occurs only for small values of the parameter b, a zoomed-in bifurcation diagram is presented to highlight the chaotic regime.

Table 7. Parameter c analysis.

Parameter Value	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	Behavior
$0.1$	$0.0001$	$-0.0514$	$-0.0520$	periodic
$1.0$	$0.0033$	$-0.0224$	$-0.2669$	periodic
$1.5$	$0.0034$	$-0.0152$	$-0.9071$	periodic
$1.6$	$0.0032$	$-0.0164$	$-1.0224$	periodic
$1.7$	$0.0030$	$-0.0094$	$-1.1402$	periodic
$1.8$	$0.0030$	$-0.0462$	$-1.1825$	periodic
$1.85$	$0.0042$	$-0.0142$	$-1.2722$	periodic
$1.87$	$0.0047$	$-0.0205$	$-1.2862$	periodic
$1.88$	$0.0291$	$-0.00004$	$-1.3398$	chaotic
$1.89$	$0.0196$	$-0.0005$	$-1.3454$	chaotic
$1.9$	$0.0290$	$-0.0016$	$-1.3658$	chaotic
$2.0$	$0.0538$	$0.0002$	$-1.5031$	chaotic
$2.01$	$0.0027$	$-0.0090$	$-1.5825$	quasiperiodic
$2.02$	$0.0008$	$-0.0097$	$-1.5955$	quasiperiodic
$2.03$	$-0.0051$	$-0.0112$	$-1.6125$	asymptotically stable
$2.05$	$-0.0062$	$-0.0123$	$-1.6352$	asymptotically stable
$2.1$	$-0.0089$	$-0.0153$	$-1.6921$	asymptotically stable
$3.0$	$-0.0344$	$-0.0394$	$-2.6885$	asymptotically stable
$5.0$	$-0.0431$	$-0.0481$	$-4.8046$	asymptotically stable
$10.0$	$-0.0466$	$-0.0500$	$-9.9004$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

The bifurcation analysis performed revealed that the system exhibits periodic behavior for $0.1\le c\le 1.87$, chaotic behavior for $1.88\le c\le 2$, quasiperiodic behavior for $2.01\le c\le 2.02$, and asymptotically stable behavior for $2.03\le c\le 10$.

Two zero Lapunov values in 3D systems indicate quasiperiodic dynamic [29]. Quasiperiodicity describes a type of behavior that retains oscillatory features while lacking strict periodic regularity. Such dynamics exhibit patterns that do not repeat with a fixed period and are therefore not exactly periodic [30].

Definition 2.

Quasi-periodic solutions are characterized by a discrete frequency spectrum, which does not consist of integer multiples of one single base frequency. The spectrum consists of linear combinations of frequencies [30,31].

Figure 18 shows the phase plots of system (19), $k=2$, while Figure 19 displays the solution graphs.

The bifurcation diagram reveals a sequence of qualitative changes in the dynamics of the system as the parameter c varies. For small values of c, the system exhibits stable periodic oscillations, which lose stability and transition to a regime of weak chaotic dynamics. As c increases further, the chaotic attractor transforms into a quasiperiodic regime. The subsequent destruction of quasiperiodic behavior leads to the convergence of system trajectories to a stable critical point, indicating asymptotic stability. Figure 20 displays the bifurcation diagram of the system (19) for $k=2$, obtained by varying the control parameter c. The parameters $a,b$, and d remain fixed. The bifurcation diagram is plotted when b is varied between $0.1\le c\le 3$.

Table 8. Parameter d analysis.

Parameter Value	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	Behavior
$0.005$	$0.0032$	$-0.0223$	$-0.2669$	periodic
$0.1$	$0.0033$	$-0.0224$	$-0.2669$	periodic
$0.2$	$0.0033$	$-0.0233$	$-0.2661$	periodic
$0.5$	$0.0034$	$-0.0224$	$-0.2674$	periodic
$1.0$	$0.0036$	$-0.0219$	$-0.2686$	periodic
$5.0$	$0.0030$	$-0.0223$	$-0.2675$	periodic
$10.0$	$0.0031$	$-0.0232$	$-0.2665$	periodic

presents the analysis of system dynamics for various values of the parameter d.

An analysis of the parameter d shows that the system maintains periodic dynamics throughout the considered range, with no evidence of bifurcations or transitions to more complex regimes.

To analyze the coupling effects between the system parameters, two-parameter bifurcation diagrams were constructed in several parameter planes. The resulting parameter-plane maps are presented in Figure 21. These diagrams provide a global view of the system dynamics and reveal regions corresponding to different dynamical regimes. The results demonstrate that the interaction between parameters plays a significant role in shaping the system behavior and complements the single-parameter bifurcation analysis.

3.1.3. Robustness of the System with Respect to Initial Conditions

Figure 22 shows the phase plots of system (19), $k=2$, while Figure 23 displays the corresponding solution graphs.

Simulations with different initial conditions lead to the same chaotic attractor, confirming that the dynamics of the system are robust and not dependent on a specific initial state.

3.2. Case 2: k = 3

3.2.1. Stability and Dynamical Analysis

The system is (19) and $k=3$ with given parameters (2) and initial conditions (3). The first critical point $E_1=(0,10,0)$ is obtained similarly as in Case 1.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (19) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^3=0.$$

(37)

After simplifying (37) get

$$x^2(x+d)=1-b\left(a+{\displaystyle \frac{1}{c}}\right).$$

(38)

Taking into account the parameters (2), one real root of Equation (38) is obtained: $x=0.5332$. The second critical point is $E_2=(0.5332,8.2,-0.5332)$. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}y-a& x& 1\\ -(2d+3x)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(39)

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -(2d+3x)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(40)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+3x^3+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(41)

Put the values of the first critical point $E_1=(0,10,0)$ in (41) and get three roots

$${\lambda }_1=-0.7155,{\lambda }_2=-0.1,{\lambda }_3=2.5155.$$

(42)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.5332,8.2,-0.5332)$ in (41) and get three roots

$${\lambda }_1=-0.6164,{\lambda }_{2,3}=0.2582\pm 0.8736i.$$

(43)

This result indicates that the critical point $E_2=(0.5332,8.2,-0.5332)$ behaves as a saddle-focus point and is unstable.

3.2.2. Bifurcation Analysis

Table 9. Parameter a analysis.

a	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$8.0$	i	i	i	Indeterminate
$9.0$	$0.0007$	$-0.0129$	$-0.0915$	periodic
$10.0$	$-0.1123$	$-0.4897$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

For small values of the parameter a, the computation of Lyapunov exponents produces indeterminate values. This occurs because the numerical algorithm used to estimate the Lyapunov spectrum does not converge to stable values within the considered integration time. In this parameter range, the system trajectories diverge rapidly in the phase space, which leads to numerical instability in the orthonormalization procedure of the Lyapunov exponent algorithm. As an increase, stable periodic dynamics emerge, followed by convergence to an asymptotically stable critical point.

Table 10. Parameter b analysis.

b	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	$-1.0119$	$-1.1887$	$-5.9994$	asymptotically stable
$3.0$	$-1.1764$	$-3.0000$	$-6.6903$	asymptotically stable
$5.0$	$-1.1723$	$-5.0000$	$-6.8277$	asymptotically stable
$7.0$	$-1.1706$	$-6.8978$	$-6.9888$	asymptotically stable
$10.0$	$-1.1693$	$-6.9425$	$-9.9880$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter b.

The analysis of parameter b shows that, except for very small values where the Lyapunov exponents are indeterminate, the system exhibits asymptotically stable behavior throughout the range of parameter b. For $b\ge 0.1$, all Lyapunov exponents are strictly negative, indicating convergence of the trajectories to a stable critical point.

Table 11. Parameter c analysis.

c	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$5.0$	i	i	i	Indeterminate
$7.0$	$-0.0450$	$-0.0493$	$-6.8595$	asymptotically stable
$8.0$	$-0.0457$	$-0.0497$	$-7.8765$	asymptotically stable
$10.0$	$-0.0463$	$-0.0500$	$-9.9007$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

The analysis of parameter c reveals that for small values of this parameter, the Lyapunov exponents are indeterminate, indicating the absence of a well-defined asymptotic regime. As c increases, all Lyapunov exponents become negative, which implies the asymptotic stability of the system. Thus, the parameter c plays a stabilizing role, leading to suppression of complex dynamics and convergence to a stable critical point.

Table 12. Parameter d analysis.

d	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$3.0$	$0.0024$	$-0.1492$	$-0.1552$	periodic
$5.0$	$0.0020$	$-0.0219$	$-0.2774$	periodic
$6.0$	$0.0020$	$-0.0093$	$-0.2892$	quasiperiodic
$7.0$	$0.0012$	$-0.0017$	$-0.2954$	quasiperiodic
$8.0$	$0.0021$	$-0.0073$	$-0.2887$	periodic
$10.0$	$0.0022$	$-0.0279$	$-0.2663$	periodic

presents the analysis of system dynamics for various values of the parameter d.

For small values of the parameter d, the computation of Lyapunov exponents yields indeterminate results, indicating that the system does not reach a well-defined asymptotic regime within the integration time considered. As d increases, stable periodic dynamics and quasiperiodic dynamics emerge.

3.3. Case 3: k = 5

3.3.1. Stability and Dynamical Analysis

The system is (19) and $k=5$ with given parameters (2) and initial conditions (3). The first critical point $E_1=(0,10,0)$ is obtained similarly as in Case 1.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (19) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^5=0.$$

(44)

After simplifying (44) get

$$x^2(x^3+d)=1-b\left(a+{\displaystyle \frac{1}{c}}\right).$$

(45)

Taking into account the parameters (2), one real root of Equation (45) is obtained: $x=0.6701$. The second critical point is $E_2=(0.6701,8.2,-0.6701)$. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}y-a& x& 1\\ -(2d+5x^3)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(46)

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -(2d+5x^3)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(47)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+5x^5+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(48)

Put the values of the first critical point $E_1=(0,10,0)$ in (48) and get three roots

$${\lambda }_1=-0.7155,{\lambda }_2=-0.1,{\lambda }_3=2.5155.$$

(49)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.6701,8.2,-0.6701)$ in (48) and get three roots

$${\lambda }_1=-0.6683,{\lambda }_{2,3}=0.2842\pm 1.0317i.$$

(50)

This result indicates that the critical point $E_2=(0.6701,8.2,-0.6701)$ behaves as a saddle-focus point and is unstable.

3.3.2. Bifurcation Analysis

Table 13. Parameter a analysis.

a	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$3.0$	i	i	i	Indeterminate
$7.2$	i	i	i	Indeterminate
$9.0$	$0.0007$	$-0.0129$	$-0.0916$	periodic
$9.5$	$-0.1113$	$-0.2406$	$-0.2519$	asymptotically stable
$10.0$	$-0.1123$	$-0.4897$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

For small values of parameter a, the computation of Lyapunov exponents yields indeterminate results, indicating that the system does not reach a well-defined asymptotic regime within the integration time considered. As the increase, stable periodic dynamics emerge, followed by convergence to an asymptotically stable critical point.

Table 14. Parameter b analysis.

b	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	$-1.0119$	$-1.1887$	$-5.9994$	asymptotically stable
$4.0$	$-1.1738$	$-4.0000$	$-6.7762$	asymptotically stable
$7.0$	$-1.1706$	$-6.8978$	$-6.9888$	asymptotically stable
$10.0$	$-1.1693$	$-6.9425$	$-9.9880$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter b.

The analysis of parameter b reveals that for small values of this parameter, the Lyapunov exponents are indeterminate, indicating the absence of a well-defined asymptotic regime. As b increases, all Lyapunov exponents become negative, which implies the asymptotic stability of the system.

Table 15. Parameter c analysis.

c	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$2.0$	i	i	i	Indeterminate
$5.0$	i	i	i	Indeterminate
$6.0$	$-0.0428$	$-0.0477$	$-5.8389$	asymptotically stable
$8.0$	$-0.0452$	$-0.0492$	$-7.8773$	asymptotically stable
$10.0$	$-0.0460$	$-0.0499$	$-9.9011$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

The analysis of parameter c reveals that for small values of this parameter, the Lyapunov exponents are indeterminate, indicating the absence of a well-defined asymptotic regime. As c increases, all Lyapunov exponents become negative, which implies the asymptotic stability of the system.

Table 16. Parameter d analysis.

d	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$2.0$	i	i	i	Indeterminate
$3.0$	$0.0019$	$-0.0267$	$-0.2661$	periodic
$4.0$	$0.0019$	$-0.1386$	$-0.1515$	periodic
$5.0$	$0.0021$	$-0.0589$	$-0.2311$	periodic
$7.0$	$0.0019$	$-0.0342$	$-0.2552$	periodic
$8.0$	$0.0023$	$-0.0303$	$-0.2593$	periodic
$10.0$	$0.0028$	$-0.0268$	$-0.2630$	periodic

presents the analysis of system dynamics for various values of the parameter c.

For small values of the parameter d, the computation of Lyapunov exponents yields indeterminate results, indicating that the system does not reach a well-defined asymptotic regime within the integration time considered. As d increases, stable periodic dynamics emerges.

3.4. Case 4: k = 6

3.4.1. Stability and Dynamical Analysis

The system is (19) and $k=6$ with given parameters (2) and initial conditions (3). The first critical point $E_1=(0,10,0)$ is obtained similarly as in Case 1.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (19) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^6=0.$$

(51)

After simplifying (51) get

$$x^2(x^4+d)=1-b\left(a+{\displaystyle \frac{1}{c}}\right).$$

(52)

Taking into account the parameters (2), the two roots of Equation (52) are obtained: $x_{1,2}=\pm 0.7112$. The second and third critical points are $E_2=(0.7112,8.2,-0.7112)$ and $E_3=(-0.7112,8.2,0.7112)$. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}y-a& x& 1\\ -2(d+3x^4)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(53)

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -2(d+3x^4)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(54)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+6x^6+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(55)

Put the values of the first critical point $E_1=(0,10,0)$ in (55) and get three roots

$${\lambda }_1=-0.7155,{\lambda }_2=-0.1,{\lambda }_3=2.5155.$$

(56)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.7112,8.2,-0.7112)$ in (55) and get three roots

$${\lambda }_1=-0.6859,{\lambda }_{2,3}=0.2930\pm 0.2930i$$

(57)

This result indicates that the critical point $E_2=(0.7112,8.2,-0.7112)$ behaves as a saddle-focus point and is unstable.

Put the values of the third critical point $E_3=(-0.7112,8.2,0.7112)$ in (55) and get three roots

$${\lambda }_1=-0.6859,{\lambda }_{2,3}=0.2930\pm 0.2930i$$

(58)

This result indicates that the critical point $E_3=(-0.7112,8.2,0.7112)$ behaves as a saddle-focus point and is unstable.

3.4.2. Bifurcation Analysis

Table 17. Parameter a analysis.

a	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$-0.0003$	$-0.0868$	$-0.4777$	periodic
$1.0$	$-0.0000$	$-0.0651$	$-0.4830$	periodic
$3.0$	$0.0003$	$-0.0152$	$-0.4885$	periodic
$5.0$	$0.0010$	$-0.2156$	$-0.2179$	periodic
$6.0$	$0.0028$	$-0.0905$	$-0.2983$	periodic
$6.1$	$0.0008$	$-0.0199$	$-0.3629$	periodic
$6.2$	$0.0025$	$-0.0087$	$-0.3708$	periodic
$6.3$	$0.0178$	$-0.0001$	$-0.3897$	chaotic
$6.5$	$0.0493$	$-0.0027$	$-0.4143$	chaotic
$7.0$	$0.0983$	$-0.0009$	$-0.4242$	chaotic
$7.2$	$0.1060$	$-0.0007$	$-0.4178$	chaotic
$8.0$	$0.1061$	$0.0002$	$-0.3565$	chaotic
$8.5$	$0.0390$	$0.0002$	$-0.2574$	chaotic
$8.6$	$0.0081$	$-0.0025$	$-0.2201$	periodic
$8.7$	$0.0074$	$-0.0996$	$-0.1077$	periodic
$8.8$	$0.0085$	$-0.0175$	$-0.1750$	periodic
$9.0$	$0.0007$	$-0.0129$	$-0.0916$	periodic
$10.0$	$-0.1123$	$-0.4897$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

The bifurcation analysis performed revealed that the system exhibits periodic behavior for $0.1\le a\le 6.2$ and $8.6\le a\le 9$, chaotic behavior for $6.3\le a\le 8.5$, and asymptotically stable for $a>9$.

Figure 24 shows the phase plots of system (19), $k=6$, while Figure 25 displays the corresponding solution graphs.

Figure 26 displays the bifurcation diagram of the system (19) for $k=6$, obtained by varying the control parameter a. The parameters $b,c$, and d remain fixed. The bifurcation diagram is plotted when a is varied between $0.1\le a\le 10$.

Table 18. Parameter b analysis.

b	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.05$	$0.0005$	$-0.0047$	$-0.4625$	periodic
$0.08$	$0.0029$	$-0.0218$	$-0.3770$	periodic
$0.09$	$0.0493$	$0.0002$	$-0.4161$	chaotic
$0.10$	$0.1060$	$-0.0007$	$-0.4178$	chaotic
$0.11$	$0.1089$	$0.0006$	$-0.3624$	chaotic
$0.12$	$0.0069$	$-0.0991$	$-0.1056$	periodic
$0.15$	$-0.1613$	$-0.7557$	$-0.7669$	asymptotically stable
$0.20$	$-0.2147$	$-1.5852$	$-1.6002$	asymptotically stable
$0.30$	$-0.3130$	$-1.3937$	$-3.4599$	asymptotically stable
$0.50$	$-0.5122$	$-1.2417$	$-4.9461$	asymptotically stable
$1.00$	$-1.0119$	$-1.1887$	$-5.9994$	asymptotically stable
$4.00$	$-1.1738$	$-4.0000$	$-6.7762$	asymptotically stable
$7.00$	$-1.1706$	$-6.8978$	$-6.9888$	asymptotically stable
$10.0$	$-1.1693$	$-6.9425$	$-9.9880$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter b.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $0.09\le b\le 0.11$, periodic behavior for $0.05\le b\le 0.08$ and $b=0.12$, and asymptotically stable behavior for $0.15\le b\le 10$. Figure 27 shows the phase plots of system (19), $k=6$, while Figure 28 displays the corresponding solution graphs.

Figure 29 displays the bifurcation diagram of the system (19) for $k=6$, obtained by varying the control parameter b. The parameters $a,c$, and d remain fixed. The bifurcation diagram is plotted when b is varied between $0.05\le b\le 0.15$.

Table 19. Parameter c analysis.

c	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$0.0018$	$-0.0498$	$-0.0540$	periodic
$0.8$	$0.0028$	$-0.0596$	$-0.1230$	periodic
$0.9$	$0.1159$	$0.0004$	$-0.3125$	chaotic
$1.0$	$0.1060$	$-0.0007$	$-0.4178$	chaotic
$1.1$	$0.0029$	$-0.0910$	$-0.3618$	periodic
$1.2$	$0.0024$	$-0.0488$	$-0.5399$	periodic
$1.5$	$0.0021$	$-0.2300$	$-0.7420$	periodic
$2.0$	$0.0044$	$-0.0168$	$-1.4747$	periodic
$4.0$	$-0.0346$	$-0.0398$	$-3.7713$	asymptotically stable
$7.0$	$-0.0440$	$-0.0485$	$-6.8612$	asymptotically stable
$10.0$	$-0.0459$	$-0.0498$	$-9.9013$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $0.9\le c\le 0.1$, periodic behavior for $0.1\le c\le 0.8$ and $1.1\le c\le 2$, and asymptotically stable behavior for $4\le c\le 10$.

Figure 30 displays the bifurcation diagram of the system (19) for $k=6$, obtained by varying the control parameter c. The parameters $a,b$, and d remain fixed. The bifurcation diagram is plotted when b is varied between $0.1\le c\le 4$.

Table 20. Parameter d analysis.

d	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.005$	$0.0859$	$-0.0001$	$-0.4148$	chaotic
$0.05$	$0.1002$	$0.0001$	$-0.4211$	chaotic
$0.1$	$0.1060$	$-0.0007$	$-0.4178$	chaotic
$1.0$	$0.0406$	$0.0002$	$-0.3054$	chaotic
$2.0$	$0.0280$	$0.0007$	$-0.3152$	chaotic
$2.2$	$0.0172$	$0.0001$	$-0.3031$	chaotic
$2.3$	$0.0019$	$-0.0066$	$-0.2803$	quasiperiodic
$2.4$	$0.0090$	$0.0000$	$-0.2940$	quasiperiodic
$2.5$	$0.0025$	$-0.0133$	$-0.2746$	periodic
$3.0$	$0.0019$	$-0.0111$	$-0.2766$	periodic
$5.0$	$0.0044$	$-0.0145$	$-0.2766$	periodic
$7.0$	$0.0031$	$-0.0195$	$-0.2702$	periodic
$10.0$	$0.0031$	$-0.0216$	$-0.2681$	periodic

presents the analysis of system dynamics for various values of the parameter b.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $0.05\le d\le 2.2$, quasiperiodic behavior for $2.3\le d\le 2.4$, and periodic behavior for $2.5\le d\le 10$.

Figure 31 displays the bifurcation diagram of the system (19) for $k=6$, obtained by varying the control parameter d. The parameters $a,b$, and c remain fixed. The bifurcation diagram is plotted when d is varied between $0.1\le d\le 4$.

3.5. Case 5: k = 7

3.5.1. Stability and Dynamical Analysis

The system is (19) and $k=7$ with given parameters (2) and initial conditions (3). The first critical point $E_1=(0,10,0)$ is obtained similarly as in Case 1. $k=2$.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (19) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^7=0.$$

(59)

After simplifying (25) get

$$x^2(x^5+d)=1-b\left(a+{\displaystyle \frac{1}{c}}\right).$$

(60)

Taking into account the parameters (2), only one real root of Equation (60) is obtained: $x=0.7428$. The second critical point is $E_2=(0.7428,8.2,-0.7428)$. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}y-a& x& 1\\ -(2d+7x^5)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(61)

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -(2d+7x^5)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(62)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+7x^7+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(63)

Put the values of the first critical point $E_1=(0,10,0)$ in (63) and get three roots

$${\lambda }_1=-0.7155,{\lambda }_2=-0,1,{\lambda }_3=2.5155.$$

(64)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.7428,8.2,-0.7428)$ in (63) and get three roots

$${\lambda }_1=-0.7005,{\lambda }_{2,3}=0.3003\pm 1.1466i$$

(65)

This result indicates that the critical point $E_2=(0.7428,8.2,-0.7428)$ behaves as a saddle-focus point and is unstable.

3.5.2. Bifurcation Analysis

Table 21. Parameter a analysis.

a	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$2.0$	i	i	i	Indeterminate
$5.0$	i	i	i	Indeterminate
$8.0$	i	i	i	Indeterminate
$9.0$	$0.0007$	$-0.0129$	$-0.0916$	periodic
$9.5$	$-0.1113$	$-0.2406$	$-0.2519$	asymptotically stable
$10.0$	$-0.1123$	$-0.4897$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

For $0.1\le a<9$, the computation of Lyapunov exponents yields indeterminate results, indicating that the system does not reach a well-defined asymptotic regime within the integration time considered. As a increases, stable periodic dynamics emerge.

Table 22. Parameter b analysis.

b	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$2.0$	$-1.1819$	$-2.0000$	$-6.5181$	asymptotically stable
$5.0$	$-1.1723$	$-5.0000$	$-6.8277$	asymptotically stable
$7.0$	$-1.1706$	$-6.8978$	$-6.9888$	asymptotically stable
$10.0$	$-1.1693$	$-6.9425$	$-9.9880$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter b.

The bifurcation analysis performed revealed that the system exhibits an asymptotically stable behavior for $2\le b\le 10$.

Table 23. Parameter c analysis.

c	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$3.0$	$-0.0178$	$-0.0233$	$-2.7209$	asymptotically stable
$4.0$	i	i	i	Indeterminate
$5.0$	i	i	i	Indeterminate
$7.0$	$-0.0436$	$-0.0483$	$-6.8618$	asymptotically stable
$10.0$	$-0.0436$	$-0.0483$	$-6.8618$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

The bifurcation analysis performed revealed that the system exhibits an asymptotically stable behavior for $c=3$ and $7\le c\le 10$.

Table 24. Parameter d analysis.

d	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$3.0$	$0.0017$	$-0.0496$	$-0.2393$	periodic
$5.0$	$0.0026$	$-0.0256$	$-0.2640$	periodic
$7.0$	$0.0029$	$-0.0234$	$-0.2663$	periodic
$9.0$	$0.0030$	$-0.0229$	$-0.2667$	periodic
$10.0$	$0.0030$	$-0.0229$	$-0.2668$	periodic

presents the analysis of system dynamics for various values of the parameter c.

For small values of the parameter d, the computation of Lyapunov exponents yields indeterminate results, indicating that the system does not reach a well-defined asymptotic regime within the integration time considered. As d increases, stable periodic dynamics emerge.

3.6. Case 6: k = 8

3.6.1. Stability and Dynamical Analysis

The system is (19) and $k=8$ with given parameters (2) and initial conditions (3). The first critical point $E_1=(0,10,0)$ is obtained similarly as in Case 1. $k=2$.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (19) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^8=0.$$

(66)

After simplifying (66) get

$$x^2(x^6+d)=1-b\left(a+{\displaystyle \frac{1}{c}}\right).$$

(67)

Taking into account the parameters (2), the two roots of Equation (67) are obtained: $x_{1,2}=\pm 0.7680$. The second and third critical points are $E_2=(0.7680,8.2,-0.7680)$ and $E_3=(-0.7680,8.2,0.7680)$. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}y-a& x& 1\\ -2(d+4x^6)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(68)

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -2(d+4x^6)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(69)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+7x^7+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(70)

Put the values of the first critical point $E_1=(0,10,0)$ in (70) and get three roots

$${\lambda }_1=-0.7155,{\lambda }_2=-0.1,{\lambda }_3=2.5155.$$

(71)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.7680,8.2,-0.7680)$ in (70) and get three roots

$${\lambda }_1=-0.7130,{\lambda }_{2,3}=0.3065\pm 1.1955i$$

(72)

This result indicates that the critical point $E_2=(0.7680,8.2,-0.7680)$ behaves as a saddle-focus point and is unstable.

Put the values of the third critical point $E_3=(-0.7680,8.2,0.7680)$ in (70) and get three roots

$${\lambda }_1=-0.7130,{\lambda }_{2,3}=0.3065\pm 1.1955i$$

(73)

This result indicates that the critical point $E_3=(-0.7680,8.2,0.7680)$ behaves as a saddle-focus point and is unstable.

3.6.2. Bifurcation Analysis

Table 25. Parameter a analysis.

a	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$0.0004$	$-0.1551$	$-0.4246$	periodic
$2.0$	$0.0001$	$-0.0922$	$-0.4501$	periodic
$5.0$	$0.0020$	$-0.0232$	$-0.4336$	periodic
$6.0$	$0.0016$	$-0.2008$	$-0.2077$	periodic
$6.5$	$0.0025$	$-0.0179$	$-0.3643$	periodic
$6.7$	$0.0003$	$-0.0199$	$-0.3498$	periodic
$6.8$	$0.0339$	$-0.0021$	$-0.3955$	chaotic
$7.0$	$0.0548$	$0.0001$	$-0.4090$	chaotic
$7.2$	$0.0802$	$0.0005$	$-0.4164$	chaotic
$8.0$	$0.1124$	$-0.0001$	$-0.3688$	chaotic
$8.5$	$0.0322$	$-0.0054$	$-0.2389$	chaotic
$8.6$	$0.0076$	$-0.0094$	$-0.2128$	quasiperiodic
$8.7$	$0.0077$	$-0.0377$	$-0.1711$	periodic
$9.0$	$0.0007$	$-0.0129$	$-0.0916$	periodic
$10.0$	$-0.1123$	$-0.4897$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $6.8\le a\le 8.5$, periodic behavior for $0.1\le a\le 6.7$ and $8.7\le a\le 9$, quasiperiodic behavior for $a=8.6$, and asymptotically stable behavior for $a=10$.

Figure 32 shows the phase plots of system (19), $k=6$, while Figure 33 displays the corresponding solution graphs.

Figure 34 displays the bifurcation diagram of the system (19) for $k=8$, obtained by varying the control parameter a. The parameters $b,c$ and d remain fixed. The bifurcation diagram is plotted when a is varied between $0.1\le a\le 10$.

Table 26. Parameter b analysis.

b	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.05$	$0.0007$	$-0.0539$	$-0.4302$	periodic
$0.08$	$0.0033$	$-0.2003$	$-0.2184$	periodic
$0.09$	$0.0026$	$-0.0031$	$-0.3803$	periodic
$0.10$	$0.0802$	$0.0005$	$-0.4164$	chaotic
$0.11$	$0.1179$	$0.0000$	$-0.3784$	chaotic
$0.12$	$0.0070$	$-0.0976$	$-0.1053$	periodic
$0.15$	$-0.1613$	$-0.7557$	$-0.7669$	asymptotically stable
$0.20$	$-0.2147$	$-1.5852$	$-1.6002$	asymptotically stable
$0.30$	$-0.3130$	$-1.3937$	$-3.4599$	asymptotically stable
$0.50$	$-0.5122$	$-1.2417$	$-4.9461$	asymptotically stable
$1.00$	$-1.0119$	$-1.1887$	$-5.9994$	asymptotically stable
$4.00$	$-1.1738$	$-4.0000$	$-6.7762$	asymptotically stable
$7.00$	$-1.1706$	$-6.8978$	$-6.9888$	asymptotically stable
$10.0$	$-1.1693$	$-6.9425$	$-9.9880$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter b.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $0.10\le b\le 0.11$, periodic behavior for $0.05\le b\le 0.09$ and $b=0.12$, and asymptotically stable behavior for $0.15\le b\le 10$.

Figure 35 displays the bifurcation diagram of the system (19) for $k=8$, obtained by varying the control parameter b. The parameters $a,c$, and d remain fixed. The bifurcation diagram is plotted when b is varied between $0.05\le a\le 0.15$.

Table 27. Parameter c analysis.

c	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$-0.0003$	$-0.0296$	$-0.1160$	periodic
$0.5$	$0.0016$	$-0.0656$	$-0.0726$	periodic
$0.7$	$0.0032$	$-0.0276$	$-0.1366$	periodic
$0.8$	$0.0307$	$0.0003$	$-0.2210$	chaotic
$0.9$	$0.1241$	$0.0002$	$-0.3347$	chaotic
$1.0$	$0.0802$	$0.0005$	$-0.4164$	chaotic
$1.1$	$0.0032$	$-0.0372$	$-0.4297$	periodic
$1.2$	$0.0026$	$-0.0979$	$-0.5025$	periodic
$1.5$	$0.0023$	$-0.4820$	$-0.5005$	periodic
$2.0$	$0.0036$	$-0.0353$	$-1.4654$	periodic
$3.0$	$-0.0152$	$-0.0210$	$-2.7257$	asymptotically stable
$5.0$	$-0.0380$	$-0.0438$	$-4.8138$	asymptotically stable
$7.0$	$-0.0433$	$-0.0481$	$-6.8623$	asymptotically stable
$10.0$	$-0.0456$	$-0.0497$	$-9.9017$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

Figure 36 displays the bifurcation diagram of the system (19) for $k=8$, obtained by varying the control parameter c. The parameters $a,b$ and d remain fixed. The bifurcation diagram is plotted when c is varied between $0.05\le c\le 3.5$.

Table 28. Parameter d analysis.

d	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.05$	$0.0595$	$0.0004$	$-0.4070$	chaotic
$0.10$	$0.0802$	$0.0005$	$-0.4164$	chaotic
$1.00$	$0.0637$	$0.0002$	$-0.3433$	chaotic
$1.50$	$0.0309$	$0.0001$	$-0.3156$	chaotic
$1.70$	$0.0177$	$-0.0001$	$-0.3047$	chaotic
$1.80$	$0.0062$	$-0.0000$	$-0.2919$	quasiperiodic
$1.90$	$0.0020$	$-0.0042$	$-0.2842$	quasiperiodic
$2.00$	$0.0018$	$-0.0399$	$-0.2482$	periodic
$5.00$	$0.0033$	$-0.0208$	$-0.2695$	periodic
$7.00$	$0.0030$	$-0.0219$	$-0.2679$	periodic
$10.0$	$0.0030$	$-0.0239$	$-0.2658$	periodic

presents the analysis of system dynamics for various values of the parameter d.

Figure 37 displays the bifurcation diagram of the system (19) for $k=8$, obtained by varying the control parameter d. The parameters $a,b$, and c remain fixed. The bifurcation diagram is plotted when d is varied between $0.005\le d\le 4$.

3.7. Case 7: k = 9

3.7.1. Stability and Dynamical Analysis

The system is (19) and $k=9$ with given parameters (2) and initial conditions (3). The first critical point $E_1=(0,10,0)$ is obtained similarly as in Case 1. $k=2$.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (19) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^9=0.$$

(74)

After simplifying (74) get

$$x^2(x^7+d)=1-b\left(a+{\displaystyle \frac{1}{c}}\right).$$

(75)

Taking into account the parameters (2), only one real root of Equation (75) is obtained: $x=0.7885$. The second critical point is $E_2=(0.7885,8.2,-0.7885)$. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}y-a& x& 1\\ -(2d+9x^7)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(76)

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -(2d+9x^7)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(77)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+9x^9+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(78)

Put the values of the first critical point $E_1=(0,10,0)$ in (78) and get three roots

$${\lambda }_1=-0.7155,{\lambda }_2=-0,1,{\lambda }_3=2.5155.$$

(79)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.7885,8.2,-0.7885)$ in (78) and get three roots

$${\lambda }_1=-0.7240,{\lambda }_{2,3}=0.3120\pm 1.2406i$$

(80)

This result indicates that the critical point $E_2=(0.7885,8.2,-0.7885)$ behaves as a saddle-focus point and is unstable.

3.7.2. Bifurcation Analysis

Table 29. Parameter a analysis.

a	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$2.0$	i	i	i	Indeterminate
$5.0$	i	i	i	Indeterminate
$7.2$	i	i	i	Indeterminate
$8.0$	i	i	i	Indeterminate
$9.0$	$0.0007$	$-0.0129$	$-0.0916$	periodic
$9.5$	$-0.1113$	$-0.2406$	$-0.2519$	asymptotically stable
$10.0$	$-0.1123$	$-0.4897$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

For $0.1\le a<8$, the computation of Lyapunov exponents yields indeterminate results, indicating that the system does not reach a well-defined asymptotic regime within the integration time considered. As a increases, stable periodic dynamics emerges.

Table 30. Parameter b analysis.

b	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	$-1.0119$	$-1.1887$	$-5.9994$	asymptotically stable
$2.0$	$-1.1819$	$-2.0000$	$-6.5181$	asymptotically stable
$5.0$	$-1.1723$	$-5.0000$	$-6.8277$	asymptotically stable
$7.0$	$-1.1706$	$-6.8978$	$-6.9888$	asymptotically stable
$10.0$	$-1.1693$	$-6.9425$	$-9.9880$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter b.

The bifurcation analysis performed revealed that the system exhibits an asymptotically stable behavior for $1\le b\le 10$.

Table 31. Parameter c analysis.

c	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$2.0$	i	i	i	Indeterminate
$3.0$	$-0.0130$	$-0.0188$	$-2.7302$	asymptotically stable
$7.0$	$-0.0431$	$-0.0478$	$-6.8628$	asymptotically stable
$9.0$	$-0.0451$	$-0.0492$	$-8.8915$	asymptotically stable
$10.0$	$-0.0454$	$-0.0497$	$-9.9018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

For $0.1\le c<2$, the computation of Lyapunov exponents yields indeterminate results; the system exhibits an asymptotically stable behavior for $3\le c\le 10$.

Table 32. Parameter d analysis.

d	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	i	i	i	Indeterminate
$1.0$	i	i	i	Indeterminate
$2.0$	i	i	i	Indeterminate
$3.0$	$0.0026$	$-0.0267$	$-0.2622$	periodic
$5.0$	$0.0028$	$-0.0227$	$-0.2668$	periodic
$7.0$	$0.0029$	$-0.0230$	$-0.2667$	periodic
$10.0$	$0.0030$	$-0.0228$	$-0.2668$	periodic

presents the analysis of system dynamics for various values of the parameter c.

For $0.1\le d<2$ the computation of Lyapunov exponents yields indeterminate results, the system exhibits periodic behavior for $3\le d\le 10$.

3.8. Case 8: k = 10

3.8.1. Stability and Dynamical Analysis

The system is (19) and $k=10$ with given parameters (2) and initial conditions (3). The first critical point $E_1=(0,10,0)$ is obtained similarly as in Case 1. $k=2$.

Let us consider the second case, where $y-a-{\displaystyle \frac{1}{c}}=0$. Then $y=a+{\displaystyle \frac{1}{c}}$ and $z=-{\displaystyle \frac{x}{c}}$. After that, substitute $y=a+{\displaystyle \frac{1}{c}}$ into the second equation of the system (19) and equate to zero

$$1-b\left(a+{\displaystyle \frac{1}{c}}\right)-d{x}^2-x^{1}0=0.$$

(81)

After simplifying (81) get

$$x^2(x^8+d)=1-b\left(a+{\displaystyle \frac{1}{c}}\right).$$

(82)

Taking into account the parameters (2), the two roots of Equation (82) are obtained: $x_{1,2}=\pm 0.8056$. The second and third critical points are $E_2=(0.8056,8.2,-0.8056)$ and $E_3=(-0.8056,8.2,0.8056)$. Let us build the Jakobian matrix

$$J=\left(\begin{array}{ccc}y-a& x& 1\\ -2(d+5x^8)x& -b& 0\\ -1& 0& -1\end{array}\right).$$

(83)

The characteristic matrix is

$$J-\lambda I=\left(\begin{array}{ccc}y-a-\lambda & x& 1\\ -2(d+5x^8)x& -b-\lambda & 0\\ -1& 0& -1-\lambda \end{array}\right)$$

(84)

and the characteristic equation is

$$det(J-\lambda I)=-b-\lambda +(-1-\lambda )(ab+10x^{1}0+2d{x}^2-by+a\lambda +b\lambda -y\lambda +{\lambda }^2)=0.$$

(85)

Put the values of the first critical point $E_1=(0,10,0)$ in (85) and get three roots

$${\lambda }_1=-0.7155,{\lambda }_2=-0.1,{\lambda }_3=2.5155.$$

(86)

This result indicates that the critical point $E_1=(0,10,0)$ behaves as a saddle point and is unstable.

Put the values of the second critical point $E_2=(0.8056,8.2,-0.8056)$ in (85) and get three roots

$${\lambda }_1=-0.7337,{\lambda }_{2,3}=0.3168\pm 1.2827i$$

(87)

This result indicates that the critical point $E_2=(0.8056,8.2,-0.8056)$ behaves as a saddle-focus point and is unstable.

Put the values of the third critical point $E_3=(-0.8056,8.2,0.8056)$ in (85) and get three roots

$${\lambda }_1=-0.7337,{\lambda }_{2,3}=0.3168\pm 1.2827i$$

(88)

This result indicates that the critical point $E_3=(-0.8056,8.2,0.8056)$ behaves as a saddle-focus point and is unstable.

3.8.2. Bifurcation Analysis

Table 33. Parameter a analysis.

a	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$0.0003$	$-0.2284$	$-0.3607$	periodic
$2.0$	$0.0001$	$-0.1368$	$-0.4151$	periodic
$5.0$	$0.0010$	$-0.0128$	$-0.4538$	periodic
$6.0$	$0.0015$	$-0.1361$	$-0.2850$	periodic
$6.5$	$0.0021$	$-0.0729$	$-0.3214$	periodic
$6.8$	$0.0024$	$-0.0369$	$-0.3402$	periodic
$6.9$	$0.0015$	$-0.0167$	$-0.3525$	periodic
$7.0$	$0.0224$	$0.0005$	$-0.3835$	chaotic
$7.2$	$0.0628$	$0.0004$	$-0.4126$	chaotic
$8.0$	$0.1070$	$-0.0002$	$-0.3752$	chaotic
$8.5$	$0.0549$	$-0.0004$	$-0.2770$	chaotic
$8.6$	$0.0235$	$-0.0003$	$-0.2408$	chaotic
$8.7$	$0.0077$	$-0.0223$	$-0.1881$	periodic
$9.0$	$0.0007$	$-0.0129$	$-0.0916$	periodic
$10.0$	$-0.1123$	$-0.4897$	$-0.5018$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $7\le a\le 8.6$, periodic behavior for $0.1\le a\le 6.9$ and $8.7\le a\le 9$, and asymptotically stable behavior for $a=10$.

Figure 38 shows the phase plots of system (19), $k=10$, while Figure 39 displays the corresponding solution graphs.

Figure 40 displays the bifurcation diagram of the system (19) for $k=10$, obtained by varying the control parameter a. The parameters $b,c$ and d remain fixed. The bifurcation diagram is plotted when a is varied between $0.1\le a\le 10$.

Table 34. Parameter b analysis.

b	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.05$	$0.0007$	$-0.1007$	$-0.3942$	periodic
$0.08$	$0.0021$	$-0.0492$	$-0.3795$	periodic
$0.09$	$0.0021$	$-0.0958$	$-0.2991$	periodic
$0.10$	$0.0628$	$0.0004$	$-0.4126$	chaotic
$0.11$	$0.1065$	$-0.0003$	$-0.3774$	chaotic
$0.12$	$0.0069$	$-0.0972$	$-0.1035$	periodic
$0.15$	$-0.1613$	$-0.7557$	$-0.7669$	asymptotically stable
$1.00$	$-1.0119$	$-1.1887$	$-5.9994$	asymptotically stable
$3.00$	$-1.1764$	$-3.0000$	$-6.6903$	asymptotically stable
$5.00$	$-1.1723$	$-5.0000$	$-6.8277$	asymptotically stable
$8.00$	$-1.1700$	$-6.9164$	$-7.9885$	asymptotically stable
$10.0$	$-1.1693$	$-6.9425$	$-9.9880$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter a.

The bifurcation analysis performed revealed that the system exhibits chaotic behavior for $0.10\le b\le 0.11$, periodic behavior for $0.05\le b\le 0.09$ and $b=0.12$, and asymptotically stable behavior for $0.15\le b\le 10$. Figure 41 displays the bifurcation diagram of the system (19) for $k=10$, obtained by varying the control parameter b. The parameters $a,c$, and d remain fixed. The bifurcation diagram is plotted when b is varied between $0.05\le a\le 0.15$.

Table 35. Parameter c analysis.

c	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.1$	$-0.0005$	$-0.0277$	$-0.1513$	periodic
$0.2$	$0.1056$	$-0.0009$	$-0.1960$	chaotic
$0.3$	$0.0910$	$0.0004$	$-0.1981$	chaotic
$0.5$	$0.0939$	$0.0012$	$-0.2311$	chaotic
$1.0$	$0.0628$	$0.0004$	$-0.4126$	chaotic
$1.1$	$0.0037$	$-0.0063$	$-0.4695$	quasiperiodic
$1.2$	$0.0030$	$-0.1491$	$-0.4598$	periodic
$1.5$	$0.0023$	$-0.2849$	$-0.7046$	periodic
$2.0$	$0.0037$	$-0.0466$	$-1.4598$	periodic
$3.0$	$-0.0107$	$-0.0167$	$-2.7345$	asymptotically stable
$5.0$	$-0.0367$	$-0.0426$	$-4.8163$	asymptotically stable
$8.0$	$-0.0440$	$-0.0486$	$-7.8791$	asymptotically stable
$10.0$	$-0.0454$	$-0.0496$	$-9.9020$	asymptotically stable

presents the analysis of system dynamics for various values of the parameter c.

Figure 42 displays the bifurcation diagram of the system (19) for $k=10$, obtained by varying the control parameter c. The parameters $a,b$, and d remain fixed. The bifurcation diagram is plotted when c is varied between $0.05\le c\le 3$.

Table 36. Parameter d analysis.

d	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	$\mathit{Solution}$
$0.01$	$0.0051$	$-0.0264$	$-0.3397$	periodic
$0.05$	$0.0158$	$-0.0003$	$-0.3698$	chaotic
$0.1$	$0.0628$	$0.0004$	$-0.4126$	chaotic
$1.0$	$0.0582$	$0.0003$	$-0.3403$	chaotic
$1.5$	$0.0159$	$0.0004$	$-0.3038$	chaotic
$1.6$	$0.0016$	$-0.0072$	$-0.2821$	quasiperiodic
$1.8$	$0.0018$	$-0.0045$	$-0.2845$	quasiperiodic
$2.0$	$0.0019$	$-0.0114$	$-0.2772$	periodic
$5.0$	$0.0032$	$-0.0218$	$-0.2683$	periodic
$8.0$	$0.0030$	$-0.0229$	$-0.2668$	periodic
$10.0$	$0.0030$	$-0.0231$	$-0.2665$	periodic

presents the analysis of system dynamics for various values of the parameter d.

Figure 43 displays the bifurcation diagram of the system (19) for $k=10$, obtained by varying the control parameter d. The parameters $a,b$, and c remain fixed. The bifurcation diagram is plotted when d is varied between $0.05\le d\le 10$.

4. Discussion

This paper investigates the dynamics of a three-dimensional nonlinear financial model and analyzes the conditions for the emergence of chaotic behavior. The well-known chaotic system with given parameters and initial conditions was considered as the initial model. Critical points were found for it, two-dimensional and three-dimensional phase portraits were constructed, and Lyapunov exponents were calculated. The results obtained confirmed the presence of chaotic behavior and the high sensitivity of the system to initial conditions.

To deepen the study, a modification of the model was proposed based on changing the degree of the nonlinear term in the second equation and a class of modified systems with nonlinearity of the form $x^k$, where $2\le k\le 10$. The parameter k plays an important role in controlling the nonlinear interaction between the system variables. As the value of k increases, the strength of the nonlinear terms in the system also increases, which improves the coupling between the state variables. This stronger nonlinear interaction leads to more complex trajectories in phase space and can destabilize regular periodic behavior. As a result, the system undergoes a transition from simple dynamics to more complicated oscillations and chaotic behavior. Therefore, increasing the parameter k contributes to the growth of the dynamical complexity in the proposed system.

Critical points were found and classified for the generalized model, which made it possible to analyze phase spaces at various degrees of nonlinearity.

Next, a bifurcation analysis of the modified systems was performed as the parameters changed. The bifurcation diagrams showed transitions between different dynamical regimes, including equilibrium states, periodic oscillations, quasi-periodic behavior, and chaos. Together with phase portraits and Lyapunov exponent analysis, this shows that both the parameters of the system and the degree of nonlinearity strongly influence the nature of the dynamics of the financial model.

Compared with classical chaotic financial models, the proposed system introduces an additional degree of nonlinearity through the term $x^k$. Related studies on multidimensional chaotic maps, such as the design of three-dimensional logistic maps and optimization approaches for chaotic systems, also highlight the importance of introducing stronger nonlinear structures to enhance dynamical complexity. In traditional financial models, nonlinear interactions are usually represented by quadratic terms. In contrast, the generalized form considered in this study allows the degree of nonlinearity to vary, providing greater flexibility in describing complex financial dynamics. The numerical analysis shows that increasing the parameter k significantly affects the structure of the phase space and the bifurcation behavior of the system. This shows that the proposed model can capture a wider range of dynamical regimes compared to the existing chaotic financial models.

From a financial interpretation perspective, higher-order nonlinear terms can be associated with stronger nonlinear feedback mechanisms in financial markets. Such nonlinearities may reflect the amplification of market reactions during periods of high volatility, speculative activity, or rapid price adjustments. The presence of strong nonlinear effects is consistent with empirical observations of financial markets. In such markets, price changes often exhibit large irregular fluctuations and occasional extreme events. Although the proposed model represents a stylized dynamical system, the inclusion of higher-order nonlinear terms may provide a possible mathematical mechanism capable of producing large fluctuations and complex irregular dynamics similar to those observed in real financial markets.

High-dimensional chaotic systems have also been widely studied in engineering applications such as image and data encryption. These studies indicate that complex chaotic dynamics can provide useful mechanisms for secure information processing, suggesting that chaotic financial models may also have potential applications in areas such as financial data encryption and risk prediction.

The results obtained deepen our understanding of deterministic chaotic states in financial dynamics and can serve as a basis for further research into stability, management, and forecasting in nonlinear economic and financial models.