Dynamics of a Novel 4D Chaotic System: Stability, Bifurcation, Chaos, and Complexity Analysis for Constant and Variable Fractional Orders

Abstract

Four-dimensional chaotic systems have garnered significant attention due to their complex nonlinear dynamics, high-dimensional complexity, and wide range of applications in science and engineering. This paper proposes and investigates a novel four-dimensional chaotic system in both constant- and variable-order frameworks to reveal the influence of memory effects on its dynamical behavior. The variable-order formulation is established using the Liouville–Caputo fractional derivative, while an efficient numerical scheme based on Lagrange interpolation is developed to approximate the variable-order derivative accurately. A rigorous local stability analysis is first conducted to characterize the equilibrium points and establish their instability and non-hyperbolic nature under the different derivative formulations. The nonlinear dynamics of the proposed system are then comprehensively examined through phase portraits, time series, bifurcation diagrams, and Lyapunov exponent analysis. The results demonstrate that the variable-order model preserves the fundamental topological characteristics of the chaotic attractors while introducing adaptive transient responses and significantly richer dynamical behaviors than the corresponding constant fractional-order model. Furthermore, the proposed system generates previously unreported chaotic attractors and phase-space patterns, enriching the class of known four-dimensional chaotic systems and demonstrating the variable-order framework’s enhanced capability to produce diverse nonlinear phenomena through adaptive memory effects.

1. Introduction

The evolution of classical differentiation, which relies on locally defined differential operators, has been the driving force behind the development of more general ideas of differentiation and integration. Fractional calculus is based on generalizing the derivative operator of classical calculus by convolution with power-law kernels to obtain operators that can capture memory effects and hereditary properties. In recent years, this mathematical theory has gained considerable popularity and found widespread application across various scientific and engineering fields [1,2,3].

Fractional-order modeling of chaotic systems has gained prominence in recent years because conventional integer-order differential equations fail to capture the long-range memory properties found in practical systems adequately. Due to the nonlocal nature of fractional differential system, they enable better modeling of the dynamics of such systems. The resulting advantage is an improved ability to analyze chaotic behavior in fields such as electronics, biology, finance, and other engineering and scientific fields [4,5,6,7].

Fractional differential system extends classical calculus by employing non-integer-order operators, allowing the description of complex processes with memory and hereditary effects, including viscoelasticity and anomalous diffusion. Consequently, it has attracted significant interest, and several generalized fractional operators have been proposed in [8,9,10].

The simulation of dynamical systems has advanced significantly due to the widespread application of fractional calculus across almost all scientific and engineering disciplines. This is because one major property of fractional differential systems is that they account for memory effects and non-locality, which are not considered in the traditional model. As a result, fractional differential systems become an effective model for describing complex dynamical systems observed in nature, such as anomalous diffusion, hereditary processes, and many others. Hence, fractional calculus has contributed significantly to improving the modeling of dynamical systems across almost all scientific and engineering disciplines. This is mainly because fractional calculus itself accounts for memory effects and non-locality, which the traditional classical approach cannot capture. A fractional differential system, which can account for phenomena such as anomalous diffusion, hereditary processes, and more, is an effective model for the dynamics of natural systems [11,12,13].

Although fractional derivatives have wide applications, classical fractional derivatives are often inadequate for capturing time-varying memory phenomena due to their limited flexibility [14,15]. Variable-order fractional system (VFOS) allow for a variable differentiation order [16].

The VFOS models are critical because they provide a more accurate description of systems whose behavior depends on time, space, and operating conditions. The order of differentiation is constant in constant-order models; however, in VFOS models, the order of differentiation varies, enabling modeling of the system’s changing memory and heredity properties. Therefore, VFOS models are used in many disciplines, including engineering, physics, biology, control theory, and materials science. The application of the VFOS models allows prediction of diffusion, viscoelasticity, heat conduction, and dynamic processes in non-homogeneous media [17,18,19].

Recent research shows that fractional-order and variable-order fractional models have gained popularity across various fields. The concept of variable-order fractional calculus offers effective ways to model complex systems exhibiting memory, nonlocal, and hereditary effects. The use of fractional differential systems offers a better representation of anomalous diffusion, viscoelasticity, and nonlinear dynamics than traditional integer-order techniques. Moreover, the use of variable-order models allows the order of fractionality to depend on time, space, and other parameters, thereby accounting for the adaptive memory effect of real-life systems. Therefore, variable-order fractional models have gained popularity across various fields of research, including heat transfer, control theory, biology, fluid mechanics, and materials science [20,21,22].

The presented four-dimensional chaotic system is of significant importance due to the presence of nonlinear dynamics such as symmetry breaking with varying parameters, multistability, and bifurcations, thus offering profound perspectives in dynamical systems and improving understanding of the theory of high-dimensional chaos. In addition, the application of the feedback control circuit demonstrates the applicability and effectiveness of the proposed chaotic system in practical settings. On the other hand, the design of the fixed-time sliding mode synchronization strategy ensures finite-time synchronization for any set of initial conditions, thereby making it applicable to secure communication, signal processing, and intelligent control. The integer-order model [23] can be described as

$$\left\{\begin{array}{cc}\hfill {\dot{u}}_1& =a\left(u_2-u_1\right)-u_4,\hfill \\ \hfill {\dot{u}}_2& =b{u}_1-u_1{u}_3,\hfill \\ \hfill {\dot{u}}_3& =-2.5u_3+c{u}_1^2,\hfill \\ \hfill {\dot{u}}_4& =-d{u}_2-u_4.\hfill \end{array}\right.$$

(1)

For the numerical investigations, the parameter values are selected as

$$a=10,b=15,c=2,d=10.$$

(2)

The corresponding initial conditions are

$$u_1\left(0\right)=0.5,u_2\left(0\right)=0.5,u_3\left(0\right)=0.6,u_4\left(0\right)=0.6.$$

(3)

The proposed four-dimensional fractional-order chaotic system is important because it combines multistability, symmetry breaking, and memory effects, providing a realistic framework for modeling complex nonlinear phenomena and enabling efficient applications in secure communications, signal processing, and fast synchronization. The system governed by the Liouville–Caputo (LC) operator ${}^{\mathit{LC}}D_t^q$ is given by

$$\left\{\begin{array}{cc}\hfill {}^{LC}D_t^q{u}_1& =a\left(u_2-u_1\right)-u_4,\hfill \\ \hfill {}^{LC}D_t^q{u}_2& =b{u}_1-u_1{u}_3,\hfill \\ \hfill {}^{LC}D_t^q{u}_3& =-2.5u_3+c{u}_1^2,\hfill \\ \hfill {}^{LC}D_t^q{u}_4& =-d{u}_2-u_4.\hfill \end{array}\right.$$

(4)

The variable-order counterpart of the proposed system incorporates time-evolving memory effects, enabling a more realistic description of complex nonlinear processes. It is governed by the Liouville–Caputo (LC) VFOS, as

$$\left\{\begin{array}{cc}\hfill {}^{LCV}D_t^{q\left(t\right)}{u}_1& =a(u_2-u_1)-u_4,\hfill \\ \hfill {}^{LCV}D_t^{q\left(t\right)}{u}_2& =b{u}_1-u_1{u}_3,\hfill \\ \hfill {}^{LCV}D_t^{q\left(t\right)}{u}_3& =-2.5u_3+c{u}_1^2,\hfill \\ \hfill {}^{LCV}D_t^{q\left(t\right)}{u}_4& =-d{u}_2-u_4.\hfill \end{array}\right.$$

(5)

Although the four-dimensional chaotic system considered in this work was originally introduced in [23] within the classical integer-order framework, neither its constant fractional-order nor its variable-order fractional dynamics have previously been investigated. Unlike the integer-order model, the fractional-order formulations incorporate memory effects, while the variable-order Liouville–Caputo model further introduces time-dependent memory through a continuously varying fractional order. Consequently, the proposed formulation represents more than a simple replacement of the derivative operator; it establishes a generalized mathematical framework that requires dedicated numerical algorithms and stability analysis. This framework enables a systematic investigation of the influence of both constant and adaptive memory on the system dynamics through phase portraits, time series, bifurcation diagrams, and Lyapunov exponent spectra. The results reveal previously unreported chaotic attractors, transient responses, and phase-space structures that are not observed in the original integer-order model, thereby demonstrating the significant role of fractional- and variable-order memory in enriching the system’s nonlinear dynamics.

Table 1. Comparison of analytical tools employed in related studies.

References	FOS	VFOS	PP	ATT	NC	TSA	LE	BIF	NS
[24]	✓	✓	×	✓	✓	×	✓	✓	✓
[25]	✓	×	×	×	✓	×	×	✓	×
[26]	×	✓	×	×	✓	×	×	×	✓
[27]	✓	×	×	×	×	×	×	✓	×
[28]	×	×	×	✓	×	×	×	✓	×
This study	✓	✓	✓	✓	✓	✓	✓	✓	✓

highlights the analytical methods used in past research and in this research. Most existing research focuses on specific aspects of fractional-order chaos and variable fractional-order chaos, such as attractors, bifurcations, and new chaotic phenomena [24,25,26,27,28]. However, in contrast to other research, the current research combines different analytical techniques such as VFOS, fractional-order system (FOS), phase portraits (PP), attractors (ATT), Novel Chaos (NC), time series analysis (TSA), Lyapunov exponents (LE), bifurcation diagrams (BIF), and numerical solution (NS).

A methodological approach used in the study of the novel 4D chaotic system is shown in Figure 1. It consists of a series of steps from the development of the initial system and evaluation of its order to its dynamic analysis. During dynamic analysis, all necessary graphs, such as phase portraits, time series, and bifurcation diagrams, are obtained.

The main contributions of this research work are listed below:

• Offering comprehensive insights into chaotic behaviors within the nonlinear 4D chaotic system under a constant fractional and variable-order framework.
• Emphasizing the critical importance of time series, bifurcation analysis, phase portraits, and the computation of Lyapunov exponents in the fractional variable-order context for control system applications.
• Formulating a robust numerical algorithmic framework utilizing Lagrange interpolation to accurately approximate the Liouville–Caputo variable-order fractional derivatives, effectively capturing time-dependent memory effects.
• Developing an efficient numerical solution based on the Liouville–Caputo variable-order derivative and Lagrange interpolation to accurately approximate the proposed chaotic system.
• Presenting novel chaotic attractors and patterns, which have never before been seen, shows that the variable-order formulation provides more diversity and complexity to the chaotic dynamics than the fractional-order dynamics can provide.

2. Equilibrium Points and Stability Analysis

Local stability characteristics of the designed chaotic system are analyzed using equilibrium point analysis and Jacobian linearization. Since the integer-order, fractional-order, and variable-order systems share the same nonlinearities, the systems will have the same equilibrium points. By equating the state equations to zero, we obtain:

$$\left\{\begin{array}{cc}\hfill a(u_2-u_1)-u_4& =0,\hfill \\ \hfill b{u}_1-u_1{u}_3& =0,\hfill \\ \hfill -2.5u_3+c{u}_1^2& =0,\hfill \\ \hfill -d{u}_2-u_4& =0.\hfill \end{array}\right.$$

(6)

This leads to the following algebraic equations:

$$\left\{\begin{array}{cc}\hfill u_1& =\pm \sqrt{{\displaystyle \frac{2.5b}{c}}}\mathrm{or}0,\hfill \\ \hfill u_2& ={\displaystyle \frac{a}{a+d}}{u}_1,\hfill \\ \hfill u_3& =b(\mathrm{for}{u}_1\ne 0)\mathrm{or}0,\hfill \\ \hfill u_4& =-d{u}_2.\hfill \end{array}\right\}$$

(7)

Consequently, the system admits three equilibrium points: the trivial equilibrium point $P_0$ and two nontrivial equilibrium points, $P_1$ and $P_2$:

$$\begin{array}{cc}\hfill P_0& =(0,0,0,0),\hfill \\ \hfill P_1& =\left(\sqrt{{\displaystyle \frac{2.5b}{c}}},{\displaystyle \frac{a}{a+d}}\sqrt{{\displaystyle \frac{2.5b}{c}}},b,-{\displaystyle \frac{ad}{a+d}}\sqrt{{\displaystyle \frac{2.5b}{c}}}\right),\hfill \\ \hfill P_2& =\left(-\sqrt{{\displaystyle \frac{2.5b}{c}}},-{\displaystyle \frac{a}{a+d}}\sqrt{{\displaystyle \frac{2.5b}{c}}},b,{\displaystyle \frac{ad}{a+d}}\sqrt{{\displaystyle \frac{2.5b}{c}}}\right).\hfill \end{array}$$

(8)

For the parameter values

$$a=10,b=15,c=2,d=10,$$

the corresponding equilibrium points become

$$\begin{array}{cc}\hfill P_0& =(0,0,0,0),\hfill \\ \hfill P_1& =(4.3301,2.1651,15,-21.6505),\hfill \\ \hfill P_2& =(-4.3301,-2.1651,15,21.6505).\hfill \end{array}$$

(9)

To investigate the local behavior around each equilibrium point, the nonlinear system is linearized by evaluating the Jacobian matrix,

$$J\left(u\right)=\left[{\displaystyle \frac{\partial f_i}{\partial u_j}}\right]=\left(\begin{array}{cccc}-a& a& 0& -1\\ b-u_3& 0& -u_1& 0\\ 2c{u}_1& 0& -2.5& 0\\ 0& -d& 0& -1\end{array}\right),$$

(10)

where $f\left(u\right)={(f_1,f_2,f_3,f_4)}^T$ denotes the vector field of the system.

The eigenvalues associated with each equilibrium point are obtained from the characteristic equation

$$\det (\lambda I-J)=0.$$

(11)

Equilibrium Point $P_0$

Evaluating the Jacobian matrix at $P_0=(0,0,0,0)$ gives

$$J\left(P_0\right)=\left(\begin{array}{cccc}-10& 10& 0& -1\\ 15& 0& 0& 0\\ 0& 0& -2.5& 0\\ 0& -10& 0& -1\end{array}\right).$$

(12)

The corresponding eigenvalues are found by solving the characteristic equation:

$$(\lambda +2.5)({\lambda }^3+11{\lambda }^2-140\lambda -300)=0.$$

(13)

Solving this polynomial yields the following eigenvalues:

$${\lambda }_1=-2.5,{\lambda }_2\approx 8.7955,{\lambda }_3\approx -1.9318,{\lambda }_4\approx -17.8637.$$

Since the system’s coefficients are known, the stability question is resolved uniquely. The exact computation reveals no zero root; instead, because the spectrum contains a positive real eigenvalue (${\lambda }_2$), the equilibrium point $P_0$ is uniquely determined to be an unstable saddle point.

Equilibrium Point $P_1$

Substituting $P_1=(4.3301,2.1651,15,-21.6505)$ into the Jacobian matrix yields

$$J\left(P_1\right)=\left(\begin{array}{cccc}-10& 10& 0& -1\\ 0& 0& -4.3301& 0\\ 17.3204& 0& -2.5& 0\\ 0& -10& 0& -1\end{array}\right).$$

(14)

For $u_1\ne 0$, the matrix $J\left(P_1\right)$ is non-singular (its determinant evaluates strictly to $1500\ne 0$), meaning 0 is not an eigenvalue. Expanding the characteristic equation gives:

$${\lambda }^4+13.5{\lambda }^3+37.5{\lambda }^2+775\lambda +1500=0.$$

(15)

Solving this equation yields the correct eigenvalues:

$${\lambda }_1\approx -14.1887,{\lambda }_2\approx -2.0117,{\lambda }_{3,4}\approx 1.3502\pm 7.1264i.$$

The presence of a complex conjugate pair of eigenvalues with a positive real part (${\lambda }_{3,4}$) confirms that $P_1$ is an unstable equilibrium point (an unstable focus-saddle).

Equilibrium Point $P_2$

Similarly, evaluating the Jacobian matrix at $P_2=(-4.3301,-2.1651,15,21.6505)$ gives

$$J\left(P_2\right)=\left(\begin{array}{cccc}-10& 10& 0& -1\\ 0& 0& 4.3301& 0\\ -17.3204& 0& -2.5& 0\\ 0& -10& 0& -1\end{array}\right).$$

(16)

Owing to the symmetry of the nonlinear system, the eigenvalue spectrum of $P_2$ is identical to that of $P_1$. Consequently, $P_2$ shares the characteristic equation ${\lambda }^4+13.5{\lambda }^3+37.5{\lambda }^2+775\lambda +1500=0$ and the identical eigenvalues:

$${\lambda }_1\approx -14.1887,{\lambda }_2\approx -2.0117,{\lambda }_{3,4}\approx 1.3502\pm 7.1264i.$$

Thus, $P_2$ is also unstable. The instability of all three equilibrium points provides the robust local dynamical structure commonly associated with chaotic attractors.

Integer-Order System

For the classical integer-order model, local asymptotic stability requires:

$$\Re \left({\lambda }_i\right)<0,i=1,\dots ,n.$$

(17)

• The Origin ($P_0$): At the origin $P_0=(0,0,0,0)$, the system possesses a positive real eigenvalue ${\lambda }_1=8.795409>0$. This directly implies that the origin is an unstable saddle point.
• The Non-Zero Equilibria ($P_1,P_2$): At these points, the Jacobian matrix yields the complex conjugate eigenvalues ${\lambda }_{1,2}=1.346770\pm 7.126571i$. Since the complex pair possesses a positive real part ($\Re \left({\lambda }_{1,2}\right)=1.346770>0$), both $E_{+}$ and $E_{-}$ are unstable saddle-focus equilibria.

Fractional-Order System

For the system $D_t^{q}X=Q\left(X\right)$ of commensurate order $q\in (0,1]$, Matignon’s stability theorem states that an equilibrium is asymptotically stable if all eigenvalues satisfy:

$$|\arg \left({\lambda }_i\right)|>{\displaystyle \frac{q\pi }{2}},i=1,\dots ,n.$$

(18)

• Stability of $P_0$: At the origin, the existence of the positive real eigenvalue yields $\arg \left(8.795409\right)=0$. This violates the strict stability condition, meaning $E_0$ remains unstable for any fractional order $0<q\le 1$.
• Stability of $P_1,P_2$: For the complex eigenvalues at $E_{\pm }$, the minimum absolute angle is calculated as $\left|\arg \left(\lambda \right)\right|={\tan }^{-1}\left({\displaystyle \frac{7.126571}{1.346770}}\right)\approx 1.384020$ radians.
• Critical Order: This angle yields a critical fractional order of $q_c={\displaystyle \frac{2}{\pi }}\left|\arg \left(\lambda \right)\right|\approx 0.881095$.

Consequently, the fractional-order stability criterion indicates that the equilibria $E_{\pm }$ are stable when $0<q<0.881095$, and they become unstable when $q>0.881095$.

Variable-Order System

The Matignon stability theorem for classical systems is valid for linear fractional systems of fixed order. Since no stability theorem is valid for variable-order fractional systems, the present study adopts the frozen-time approximation, assuming the order varies slowly, so the system can be locally approximated by constant-order systems. Therefore, the stability results obtained can only be treated as local rather than strictly global [29,30,31]. Hence, one could conclude that stability is highly dependent not only on the instantaneous values of the Jacobian but also on the memory effect generated by $q\left(t\right)$. The model considered is defined in terms of a variable-order fractional derivative, where the derivative order $q\left(t\right)$ takes values in the range $(0,1]$. The related variable-order dynamical system can be represented as

$${}^{C}D_t^{q\left(t\right)}X\left(t\right)=F\left(X\left(t\right)\right),$$

(19)

where $X\left(t\right)\in {\mathbb{R}}^n$ denotes the state vector and $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is the nonlinear vector field.

Let $X^{*}$ be an equilibrium point satisfying

$$F\left(X^{*}\right)=0.$$

Linearizing the system in a neighborhood of $X^{*}$ gives

$${}^{C}D_t^{q\left(t\right)}Y\left(t\right)=J\left(X^{*}\right)Y\left(t\right),$$

(20)

where

$$Y\left(t\right)=X\left(t\right)-X^{*},$$

and

$$J\left(X^{*}\right)={\left[{\displaystyle \frac{\partial F_i}{\partial X_j}}\right]}_{X=X^{*}}$$

is the Jacobian matrix evaluated at the equilibrium point.

While the stability analysis for constant-order fractional-order systems can be carried out using Matignon’s theorem, no such generalized theory is available for variable-order systems at the moment. This study employs the commonly applied approximation method of frozen time (quasi-static), under the assumption that

$$|\dot{q}\left(t\right)|<1.$$

(21)

Under this assumption, the system can be locally approximated by a sequence of constant-order fractional systems. Consequently, the instantaneous stability of the equilibrium is assessed through the heuristic condition

$$|\arg \left({\lambda }_i\right)|>{\displaystyle \frac{q\left(t\right)\pi }{2}},{\lambda }_i\in \mathrm{spec}\left(J\left(X^{*}\right)\right),$$

(22)

where $\mathrm{spec}\left(J\right)$ denotes the spectrum of the Jacobian matrix.

This criterion is a local stability indicator valid at each time point, as opposed to a global stability condition, which would require rigorous proof. Hence, in addition to the instantaneous values of the Jacobian matrix’s eigenvalues, the dynamic evolution also depends on the accumulated memory effect due to the time-varying fractional order.

3. Numerical Algorithmic Framework

The numerical simulation of fractional-order dynamical systems is much more difficult compared to that of integer-order dynamical systems due to the nature of fractional derivatives, which have a memory property. Unlike ordinary derivatives, which require information from a single point, fractional derivatives require the full history of the states involved. This implies that the numerical scheme requires storing all past states, thereby increasing computational effort and storage. In this section, we discuss the numerical method to solve the fractional dynamical system [32,33] defined above using Liouville–Caputo fractional derivatives.

3.1. Preliminary Definitions

Definition 1 

([34]). The Liouville–Caputo fractional derivative of constant order q is defined by

$${}^{\mathit{LC}}\mathcal{D}_t^{q}y\left(t\right)={\displaystyle \frac{1}{\Gamma (1-q)}}{\int }_0^t{(t-\tau )}^{-q}{\displaystyle \frac{d}{d\tau }}y\left(\tau \right)d\tau ,0<q<1.$$

(23)

Definition 2 

([35]). Let $y\in C^1\left([0,T]\right)$ and let the variable order $q:[0,T]\to (0,1)$ satisfy $q\left(t\right)\in C\left(\right[0,T\left]\right),$ The variable-order Liouville–Caputo derivative of order $q\left(t\right)$ is defined by

$${}_0{}^{LCV}\mathcal{D}_t^{q\left(t\right)}y\left(t\right)={\displaystyle \frac{1}{\Gamma \left(1-q\left(t\right)\right)}}{\int }_0^t{(t-\tau )}^{-q\left(t\right)}{y}^{\prime }\left(\tau \right)d\tau ,0<t\le T.$$

(24)

In essence, the main difference between constant-order and variable-order models lies in how the memory effect is characterized. In the constant-order case, the fractional order q does not change in the course of evolution, making it adequate for systems where heredity is uniform; $q\left(t\right)$ can change in time, thus reflecting the adaptive and dynamic nature of memory. As a result, the kernel transforms from ${(t-\tau )}^{-q}$ to ${(t-\tau )}^{-q\left(t\right)}$.

3.2. Constant-Order Numerical Approximation

Consider the fractional counterpart of system (1),

$${}^{\mathit{LC}}\mathcal{D}_t^{q}U\left(t\right)=\Phi (t,U),U\left(0\right)=U_0,$$

(25)

where

$$U={(u_1,u_2,u_3,u_4)}^T,\Phi ={({\Phi }_1,{\Phi }_2,{\Phi }_3,{\Phi }_4)}^T.$$

The discrete approximation corresponding to the constant-order Liouville–Caputo operator for $n\ge 1$ is given by

$$U_{n+1}=U_0+{\displaystyle \frac{h^q}{\Gamma (q+2)}}{\displaystyle \sum _{k=1}^n}\left[\Phi (t_k,U_k){\mathcal{A}}_{n,k}-\Phi (t_{k-1},U_{k-1}){\mathcal{B}}_{n,k}\right],$$

(26)

where the weighting coefficients are defined as

$${\mathcal{A}}_{n,k}={(n-k+1)}^q\left(n-k+2+2q\right)-{(n-k)}^q\left(n-k+2+2q\right),$$

(27)

$${\mathcal{B}}_{n,k}={(n-k+1)}^{q+1}-{(n-k)}^q\left(n-k+1+q\right).$$

(28)

Remark on the Start-Up Step: Because the numerical scheme in (24) utilizes a two-step Lagrange interpolation, it requires a known value for $U_1$ to proceed. For the initial step ($n=0$), a standard one-step fractional Euler method is employed to compute the start-up value:

$$U_1=U_0+{\displaystyle \frac{h^q}{\Gamma (q+1)}}\Phi (t_0,U_0).$$

(29)

3.3. Variable-Order Numerical Approximation

For the variable-order Liouville–Caputo derivative, the numerical scheme is derived using Lagrange interpolation. For n ≥ 1, the resulting approximation is

$$U_{n+1}\left(t\right)=U_0+{\displaystyle \frac{1}{\Gamma \left(q\left(t\right)\right)}}{\displaystyle \sum _{k=1}^n}{\displaystyle \frac{h^{q\left(t\right)}}{q\left(t\right)\left(q\left(t\right)+1\right)}}\left[\Phi (t_k,U_k){\mathcal{P}}_{n,k}-\Phi (t_{k-1},U_{k-1}){\mathcal{Q}}_{n,k}\right],$$

(30)

where

$${\mathcal{P}}_{n,k}={(n+1-k)}^{q\left(t\right)}\left(n-k+2+q\left(t\right)\right)-{(n-k)}^{q\left(t\right)}\left(n-k+2+2q\left(t\right)\right),$$

(31)

$${\mathcal{Q}}_{n,k}={(n+1-k)}^{q\left(t\right)+1}-{(n-k)}^{q\left(t\right)}\left(n-k+1+q\left(t\right)\right).$$

(32)

Remark on the Start-Up Step: Similarly, the variable-order scheme relies on evaluating prior historical points. The initial step ($n=0$) is computed using the variable-order fractional Euler approximation:

$$U_1\left(t\right)=U_0+{\displaystyle \frac{h^{q\left(t\right)}}{\Gamma \left(q\right(t)+1)}}\Phi (t_0,U_0).$$

(33)

The nonlinear vector field associated with system (1) is updated at each time level according to

$${\Phi }_1=a(u_2-u_1)-u_4,$$

(34)

$${\Phi }_2=b{u}_1-u_1{u}_3,$$

(35)

$${\Phi }_3=-2.5u_3+c{u}_1^2,$$

(36)

$${\Phi }_4=-d{u}_2-u_4.$$

(37)

Accordingly, the componentwise numerical iterations for the constant-order case can be written as

$$u_1^{n+1}=u_1\left(0\right)+{\displaystyle \frac{h^{q\left(t\right)}}{\Gamma \left(q\right(t)+2)}}{\displaystyle \sum _{k=0}^n}\left[{\Phi }_1(t_k,U_k){\mathcal{A}}_{n,k}-{\Phi }_1(t_{k-1},U_{k-1}){\mathcal{B}}_{n,k}\right],$$

(38)

$$u_2^{n+1}=u_2\left(0\right)+{\displaystyle \frac{h^{q\left(t\right)}}{\Gamma \left(q\right(t)+2)}}{\displaystyle \sum _{k=0}^n}\left[{\Phi }_2(t_k,U_k){\mathcal{A}}_{n,k}-{\Phi }_2(t_{k-1},U_{k-1}){\mathcal{B}}_{n,k}\right],$$

(39)

$$u_3^{n+1}=u_3\left(0\right)+{\displaystyle \frac{h^{q\left(t\right)}}{\Gamma \left(q\right(t)+2)}}{\displaystyle \sum _{k=0}^n}\left[{\Phi }_3(t_k,U_k){\mathcal{A}}_{n,k}-{\Phi }_3(t_{k-1},U_{k-1}){\mathcal{B}}_{n,k}\right],$$

(40)

$$u_4^{n+1}=u_4\left(0\right)+{\displaystyle \frac{h^{q\left(t\right)}}{\Gamma \left(q\right(t)+2)}}{\displaystyle \sum _{k=0}^n}\left[{\Phi }_4(t_k,U_k){\mathcal{A}}_{n,k}-{\Phi }_4(t_{k-1},U_{k-1}){\mathcal{B}}_{n,k}\right].$$

(41)

Accordingly, the componentwise numerical iterations for the variable-order case are given by

$$u_1^{n+1}\left(t\right)=u_1\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(q\left(t\right)\right)}}{\displaystyle \sum _{k=0}^n}{\displaystyle \frac{h^{q\left(t\right)}}{q\left(t\right)\left(q\left(t\right)+1\right)}}\left[{\Phi }_1(t_k,U_k){\mathcal{P}}_{n,k}-{\Phi }_1(t_{k-1},U_{k-1}){\mathcal{Q}}_{n,k}\right],$$

(42)

$$u_2^{n+1}\left(t\right)=u_2\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(q\left(t\right)\right)}}{\displaystyle \sum _{k=0}^n}{\displaystyle \frac{h^{q\left(t\right)}}{q\left(t\right)\left(q\left(t\right)+1\right)}}\left[{\Phi }_2(t_k,U_k){\mathcal{P}}_{n,k}-{\Phi }_2(t_{k-1},U_{k-1}){\mathcal{Q}}_{n,k}\right],$$

(43)

$$u_3^{n+1}\left(t\right)=u_3\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(q\left(t\right)\right)}}{\displaystyle \sum _{k=0}^n}{\displaystyle \frac{h^{q\left(t\right)}}{q\left(t\right)\left(q\left(t\right)+1\right)}}\left[{\Phi }_3(t_k,U_k){\mathcal{P}}_{n,k}-{\Phi }_3(t_{k-1},U_{k-1}){\mathcal{Q}}_{n,k}\right],$$

(44)

$$u_4^{n+1}\left(t\right)=u_4\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(q\left(t\right)\right)}}{\displaystyle \sum _{k=0}^n}{\displaystyle \frac{h^{q\left(t\right)}}{q\left(t\right)\left(q\left(t\right)+1\right)}}\left[{\Phi }_4(t_k,U_k){\mathcal{P}}_{n,k}-{\Phi }_4(t_{k-1},U_{k-1}){\mathcal{Q}}_{n,k}\right].$$

(45)

3.4. Stability Analysis

The stability of the proposed VFOS is analyzed using the frozen-time approximation. Because the fractional order varies with time, Matignon’s criterion cannot be applied directly. The system is locally treated as a constant-order model, and stability is determined from the Jacobian eigenvalues at the equilibrium points. Hence, the results provide local rather than global stability conditions [35,36].

3.5. Existence and Uniqueness

The well-posedness of the suggested VFOS is an immediate consequence of well-known results in nonlinear functional analysis. If we assume that the nonlinear vector field is continuous and Lipschitz continuous with respect to the state variables in a global (or in a local) sense, then the variable-order fractional integral operator related to it will be contractive in a suitable Banach space. Consequently, the existence and uniqueness of a solution on a sufficiently small time interval for a given set of initial conditions will be guaranteed using the Banach fixed point theorem [36].

3.6. Convergence and Numerical Stability

The numerical method used in this study is built on the variable-order fractional discretization introduced in [32,36], which has been theoretically proven to be convergent and numerically stable. Therefore, the approximation obtained from the chosen method approaches the exact solution as the discretization step shrinks, while remaining numerically stable at all times. To further validate the simulation, additional computations were performed using different time-step values. Phase trajectories, time response, bifurcation diagram, and Lyapunov exponents did not differ significantly from those of the previous computations, indicating that the numerical solution is independent of the chosen discretization.

4. Dynamical Analysis

This section presents the chaotic dynamics of the proposed system by comparing two variable-order cases with the constant-order model [37,38].

Case 1. The time-dependent fractional order is defined by $\alpha \left(t\right)=0.98+0.01\tanh \left(t\right)$ which is compared with the constant-order case $\alpha =0.98$.

Such a selection allows modeling a smooth transition in the memory effect, thus enabling analysis of how increasing fractional orders influence the system’s dynamics and attractors.

Case 2. The variable-order function is given by $\alpha \left(t\right)=0.97+0.005\sin \left({\displaystyle \frac{\pi t}{8}}\right)$, with the corresponding constant-order case $\alpha =0.97$.

The described behavior refers to oscillatory memory features. It provides an opportunity to analyze the model’s behavior under time-dependent fractional order and its effect on the persistence and modulation of chaotic behavior. Examining both cases is significant because they correspond to two fundamentally different processes in variable-order dynamics: monotonic and nonmonotonic.

4.1. Bifurcation Analysis and Lyapunov Exponents

The dynamical behavior of the proposed system is characterized using bifurcation diagrams of the local maxima of each state variable, $u_1$, $u_2$, $u_3$, and $u_4$, together with the corresponding Lyapunov exponent (LE) spectra as the system parameters vary. The Lyapunov exponents were computed using the Wolf algorithm [39] with the fractional-order extension proposed in [40].

Figure 2 shows how the system enters into chaos with the variation of parameter a. This process is justified by the calculation of the maximal Lyapunov exponent ($L{E}_1$), which ensures that the chaotic attractor exists whenever $L{E}_1>0$ and $L{E}_2=0$. As for the parameters c and d (Figure 3 and Figure 4), the effect of both on the system indicates the existence of very strong and continuous chaotic regions. The variation of parameter c causes the decrease of the amplitude of the chaotic signal, but not a change in the chaotic nature. On the contrary, the variation of parameter d causes internal bifurcations and the fusion of several chaotic bands. In all cases, the LE spectrum is highly stable with positive $L{E}_1$.

The dynamical characteristics of the system in Case (2) are further analyzed using bifurcation diagrams of the local maxima $u_1$, $u_2$, $u_3$, and $u_4$, along with their corresponding Lyapunov Exponent (LE) spectra as parameters are varied. Figure 5 shows the dynamics of the system in relation to the parameter a. The bifurcation clearly shows the onset of chaos around $a=14$. The LE plot confirms the onset of chaos, where the highest LE plot $L{E}_1$ is always positive during chaos and drops sharply during periods.

As illustrated in Figure 6, changing the value of parameter c results in a continuous and stable chaotic region. As c gets higher, the amplitudes of the state variables shrink smoothly, but the LE spectrum stays surprisingly stable and level. Figure 7 illustrates the dynamics in terms of the parameter d. The system sustains chaos over the studied interval, with chaotic bands being narrower and more compact than those in Case 1. Although there are some changes inside the LE spectrum, which show topological changes of the attractor, the value of $L{E}_1$ remains strictly positive.

As depicted in Figure 8, the dynamical system displays complex behavior depending on the variation of the control parameter q. The bifurcation plot (left) shows the continuous change in the state variable as q varies, highlighting the system’s chaotic behavior. The Lyapunov exponent plot (right) further confirms the presence of chaotic behavior by showing positive Lyapunov exponents.

4.2. Phase Portraits and Time-Series Analysis

The phase portraits and time-series solutions of Case 1 and Case 2 and their corresponding constant-order systems are shown in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. In particular, Figure 9 and Figure 10 provide the phase-space trajectories and time series of Case 1, whereas Figure 11 and Figure 12 are the constant-order case for $q=0.98$. Likewise, Figure 13 and Figure 14 represent the solutions of Case 2, while Figure 15 and Figure 16 display the constant-order case for $q=0.97$. The time-series plots exhibit regular oscillations with nearly constant amplitudes, indicating chaotic behavior. Moreover, the variable-order formulations preserve the qualitative dynamics of their constant-order counterparts, with only minor differences in the transient evolution.

4.3. Chaos Analysis

The phase portraits of the chaotic attractor of the chaotic system in different 2D projections for different parameter values are shown in Figure 17, Figure 18, Figure 19 and Figure 20. In particular, Figure 17 and Figure 19 present the base case behaviors of Case 1 and Case 2, respectively, while Figure 18 and Figure 20 present the system dynamics when the value of the fractional derivative is $q=0.98$ and $q=0.97$, respectively. These Figures clearly show that the chaotic system has a complex multi-wing attractor, and that the basic butterfly form is retained regardless of changes in the parameter.

The phase portraits for the proposed system under Case 1 and Case 2 are presented in Figure 21, Figure 22, Figure 23 and Figure 24. These figures reveal unique topological structures and chaotic behaviors. To the best of our knowledge, all dynamical attractors observed in these cases are completely novel and have not been documented in prior studies.

5. Numerical Solution

In this section, we examine the numerical results in

Table 2. Numerical solution of Case 1: evolution of the state variables $u_1$, $u_2$, $u_3$, and $u_4$ over time.

Time	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$
0.0	0.500000	0.500000	0.600000	0.600000
0.1	0.754095	1.291905	0.531461	−0.246540
0.2	1.778917	3.067058	0.695320	−2.265370
0.3	4.270339	7.107500	2.179656	−6.826865
0.4	9.275270	13.401911	10.227600	−16.316005
0.5	12.027209	5.765231	32.494210	−26.333870
0.6	1.655397	−11.191704	35.881029	−18.550057
0.7	−4.663427	−6.976101	29.213613	−7.039850
0.8	−3.392863	−1.120643	25.979175	−2.871681
0.9	−0.868295	0.717811	20.844487	−2.792647
1.0	0.386554	0.775116	16.154138	−3.327416

and

Table 3. Numerical solution for fractional order $\alpha =0.98$.

Time	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$
0.0	0.500000	0.500000	0.600000	0.600000
0.1	0.755073	1.293707	0.531449	−0.248652
0.2	1.788150	3.082408	0.698393	−2.282382
0.3	4.314828	7.176095	2.221502	−6.907622
0.4	9.389879	13.474910	10.551691	−16.555560
0.5	11.835324	5.042076	33.000107	−26.311613
0.6	1.243914	−10.999267	35.094549	−17.921353
0.7	−4.545167	−6.559990	28.529128	−6.795518
0.8	−3.231103	−1.206765	25.107785	−2.827651
0.9	−0.950327	0.414190	20.067335	−2.569643
1.0	0.153169	0.492341	15.524089	−2.869350

, which show that lowering $\alpha$ from 1 to 0.98 leads to only slight changes in the state variables. All variables exhibit a similar oscillatory growth pattern, indicating the system’s stability under a slight decrease in $\alpha$.

Table 4. Numerical solution for Case 2: evolution of the state variables $u_1$, $u_2$, $u_3$, and $u_4$ over time.

Time	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$
0.0	0.500000	0.500000	0.600000	0.600000
0.1	0.773534	1.326527	0.531606	−0.287267
0.2	1.874358	3.224581	0.729100	−2.439326
0.3	4.595185	7.603923	2.498322	−7.415171
0.4	9.912133	13.733013	12.149587	−17.672065
0.5	10.997845	2.353307	34.593728	−26.057164
0.6	0.099292	−10.364749	32.885063	−16.077085
0.7	−4.295667	−5.637037	26.840657	−6.093761
0.8	−2.964206	−1.391883	23.194794	−2.533065
0.9	−1.172806	−0.159627	18.490761	−1.886375
1.0	−0.349041	−0.083071	14.322201	−1.710400

and

Table 5. Numerical solution for fractional order $\alpha =0.97$.

Time	${\mathit{u}}_1$	${\mathit{u}}_2$	${\mathit{u}}_3$	${\mathit{u}}_4$
0.0	0.500000	0.500000	0.600000	0.600000
0.1	0.773741	1.326905	0.531606	−0.287710
0.2	1.876353	3.227894	0.729814	−2.442992
0.3	4.604976	7.618766	2.508359	−7.432999
0.4	9.934813	13.740067	12.225960	−17.722271
0.5	10.940413	2.202925	34.656304	−26.027059
0.6	0.036901	−10.299741	32.725490	−15.960367
0.7	−4.265127	−5.572418	26.699303	−6.056646
0.8	−2.940223	−1.415586	23.027247	−2.519432
0.9	−1.193399	−0.217372	18.348897	−1.834903
1.0	−0.399504	−0.147224	14.214947	−1.609120

further show that there is similarity between solutions at $\alpha =0.97$ and those of Case 2. There are quantitative differences in the solution’s peak and decay rates, but qualitatively there is no major difference.

Validation of the Proposed Numerical Scheme

In

Table 6. Comparison of the proposed method with the ABM method for constant fractional orders.

Order	Time	Variable	Proposed	ABM	|Error|
$\alpha =0.98$	0.1	$u_1$	0.755073	0.752673	0.002400
		$u_2$	1.293707	1.219693	0.074014
		$u_3$	0.531449	0.531887	0.000438
		$u_4$	−0.248652	−0.244426	0.004226
	0.2	$u_1$	1.788150	1.778079	0.010071
		$u_2$	3.082408	3.063021	0.019387
		$u_3$	0.698393	0.690284	0.008109
		$u_4$	−2.282382	−2.266205	0.016177
	0.3	$u_1$	4.314828	4.273875	0.040953
		$u_2$	7.176095	7.106461	0.069634
		$u_3$	2.221502	2.170537	0.050965
		$u_4$	−6.907622	−6.823322	0.084300
	0.4	$u_1$	9.389879	9.272666	0.117213
		$u_2$	13.474910	13.407635	0.067275
		$u_3$	10.551691	10.387741	0.163950
		$u_4$	−16.555560	−16.871949	0.316389
$\alpha =0.97$	0.1	$u_1$	0.773741	0.745655	0.028086
		$u_2$	1.326905	1.459580	0.132675
		$u_3$	0.531606	0.532997	0.001391
		$u_4$	−0.287710	−0.240977	0.046733
	0.2	$u_1$	1.876353	1.070001	0.806352
		$u_2$	3.227894	3.550652	0.322758
		$u_3$	0.729814	0.701497	0.028317
		$u_4$	−2.442992	−2.801010	0.358018
	0.3	$u_1$	4.604976	4.057893	0.547083
		$u_2$	7.618766	7.298274	0.320492
		$u_3$	2.508359	2.989400	0.481041
		$u_4$	−7.432999	−7.263263	0.169736
	0.4	$u_1$	9.934813	9.516716	0.418097
		$u_2$	13.740067	13.836564	0.096497
		$u_3$	12.225960	12.292848	0.066888
		$u_4$	−17.722271	−1.052271	16.670000

and

Table 7. Comparison between the proposed variable-order method and the ABM method.

Case	Time	Variable	Proposed	ABM	|Error|
Case 1	0.1	$u_1$	0.754095	0.752673	0.001422
		$u_2$	1.291905	1.219693	0.072212
		$u_3$	0.531461	0.531887	0.000426
		$u_4$	−0.246540	−0.244426	0.002114
	0.2	$u_1$	1.778917	1.778079	0.000838
		$u_2$	3.067058	3.063021	0.004037
		$u_3$	0.695320	0.690284	0.005036
		$u_4$	−2.265370	−2.266205	0.000835
	0.3	$u_1$	4.270339	4.273875	0.003536
		$u_2$	7.107500	7.106461	0.001039
		$u_3$	2.179656	2.170537	0.009119
		$u_4$	−6.826865	−6.823322	0.003543
	0.4	$u_1$	9.275270	9.272666	0.002604
		$u_2$	13.401911	13.407635	0.005724
		$u_3$	10.227600	10.387741	0.160141
		$u_4$	−16.316005	−16.871949	0.555944
Case 2	0.1	$u_1$	0.773534	0.773741	0.000207
		$u_2$	1.326527	1.326905	0.000378
		$u_3$	0.531606	0.531606	0.000000
		$u_4$	−0.287267	−0.287710	0.000443
	0.2	$u_1$	1.874358	1.876353	0.001995
		$u_2$	3.224581	3.227894	0.003313
		$u_3$	0.729100	0.729814	0.000714
		$u_4$	−2.439326	−2.442992	0.003666
	0.3	$u_1$	4.595185	4.604976	0.009791
		$u_2$	7.603923	7.618766	0.014843
		$u_3$	2.498322	2.508359	0.010037
		$u_4$	−7.415171	−7.432999	0.017828
	0.4	$u_1$	9.912133	9.934813	0.022680
		$u_2$	13.733013	13.740067	0.007054
		$u_3$	12.149587	12.225960	0.076373
		$u_4$	−17.672065	−17.722271	0.050206

, the numerical approach presented in this study is compared to the classical Adams–Bashforth–Moulton (ABM) predictor–corrector method. In both the case of constant order where $\alpha =0.98$ and $\alpha =0.97$, and in the case of variable order Case 1 and Case 2, the two approaches show good agreement, as reflected by low absolute errors throughout the simulation period. The proposed approach successfully reproduces the solution from the ABM approach with the same dynamic behavior.

6. Conclusions

This paper introduces a new four-dimensional chaotic system and studies its dynamical behavior in detail for both constant and variable orders, using the Liouville–Caputo derivative. All equilibrium points of the system obtained from the local stability analysis are non-hyperbolic and unstable, thereby guaranteeing the possibility of chaotic behavior. For the numerical solution of the two models, numerical techniques have been applied. Dynamical analysis is performed using time-response graphs, phase portraits, bifurcation diagrams, and Lyapunov exponents. It has been observed that the proposed system exhibits chaotic behavior in both fractional-order systems. Comparative analysis of the two models has shown that although the topological structure of the chaotic attractor was not affected by the variable-order memory, the transients and convergence processes were significantly altered. This implies that the time-variant nature of the model’s fractional order can be considered a useful approach for adaptive memory modeling. Overall, the proposed system is more accurate at representing real-life situations in terms of nonlinearity and the system’s changing memory. The study provides insight into the dynamics of a variable-order fractional chaotic system. It serves as a basis for further research on synchronization, communication security, nonlinear control, circuit implementation, and finite-time stabilization of the system.