Variable-Order Fractional Calculus-Based Chaos Analysis of a Novel Eight-Dimensional Hyperchaotic System

Abstract

In this study, we develop a novel variable-order fractional extension of an eight-dimensional (8D) hyperchaotic differential equation system modeled via the Liouville–Caputo operator. Moving beyond constant fractional-order models, our system implements time-variable orders, which enable its historical memory structure to evolve dynamically over time. To numerically approximate the trajectories of this complex 8D system, a second-order Lagrange numerical integration approach is formulated. An extensive dynamic analysis explores the behavior of this variable-order framework under two distinct configurations: a slowly periodic memory function and a smooth, monotonic hyperbolic tangent function. Topological complexity and multidimensional chaos are characterized using parameter-dependent bifurcation diagrams, phase portraits, Kaplan–Yorke fractal dimensions, and Kolmogorov–Sinai metric entropy. Numerical results show that both variable-order configurations display robust hyperchaotic dynamics characterized by four positive Lyapunov exponents. Crucially, the proposed variable-order extension enhances the phase space footprint of the baseline system, achieving a maximum Kaplan–Yorke dimension of 7.100, thereby offering excellent topological density for secure cryptographic applications.

1. Introduction

This branch of calculus focuses on the extension of derivatives and integrals into arbitrary orders. The ability to use derivatives and integrals that have non-integer orders leads to the required flexibility that allows for modeling phenomena that cannot be captured using traditional methods. This includes phenomena such as those occurring within the material properties of viscoelastic substances and phenomena that exhibit anomalous diffusion. Fractional calculus provides a tool that captures a large number of physical phenomena with unique characteristics like memory and inheritance, which are difficult to describe using the traditional approach and sometimes even prove to be ineffective. In a setting that is much more extensive, fractional operators have been generalized in the work cited in [1,2,3,4,5].

Modeling of such dynamical systems has been greatly facilitated by the introduction of fractional calculus in almost all the fields. The main importance of fractional calculus in modeling is its capacity to take into consideration memory effects and nonlocality, which cannot be described using classical approaches to modeling. Since it is capable of modeling a variety of natural phenomena, ranging from anomalous diffusion to heredity, the fractional calculus can be considered as the most powerful method for capturing the complicated behavior of dynamical systems [6,7,8].

Fractional derivatives are widely used, but they are not able to satisfy the requirement of adaptability to model systems with time-varying memory [9,10]. This idea is supported by the development of variable-order fractional derivatives (VOF-s), which represent a useful tool that gives the opportunity to change the order of the derivative as a function of time, space, or other control variables. The result is that they provide a more accurate description of the nonstationary dynamics and complex systems with changing internal structures all the time [11,12].

VOF-s calculus provides a powerful framework for modeling nonstationary and time-varying dynamics. By allowing the fractional order to evolve with the system state or time, VOF-s models can more accurately represent changing memory. It is this feature that renders VOF-s calculus especially suitable for problems related to anomalous diffusion, viscoelastic media, biology, control theory, signal analysis, and complicated mechanical models. What is more important is that it provides a powerful mathematical instrument for studying nonlinear dynamics such as bifurcations and chaos in fractional differential equations [13,14,15].

Recent studies have demonstrated the validity of fractional-order models, and variable-order fractional calculus is becoming increasingly important in the process of characterizing complex systems that exhibit dynamic behavior and time-varying memory and behavior. As a result of ongoing improvements, their applications in the fields of science and engineering continue to expand. These applications include computational tools for arbitrary variable-order derivatives and advanced control techniques for nonlinear dynamical systems [16,17,18].

In terms of engineering applications, the utilization of variable-order fractional models in modeling nonvolatile memory devices and electrical circuits offers a realistic approach by taking into account memory and aging effects in the mathematical models of the devices and circuits. The time dependence of the fractional order in such models increases the ability to capture complicated nonlinear and chaotic behavior that cannot be captured by the constant-order models. In addition, in the case of variable-order memristive neural networks, the adaptive fractional order can be used as a dynamic cryptographic key [19].

The contributions of the present paper can be summarized as follows. First, a new variable-order fractional model of the proposed 8D hyperchaotic system is obtained, which employs the idea of variable-order fractional derivative with an adaptive memory that enhances the conventional constant-order model. Second, a detailed dynamical study is performed through phase portraits, time series responses, bifurcation diagrams, Lyapunov exponents spectra, and Kaplan–Yorke dimension to show the effect of the variable-order memory on the development, improvement, and control of the hyperchaos. Third, it is demonstrated that the considered hyperchaotic system possesses very complicated dynamics with the existence of a very rich number of strange attractors, a high Kaplan–Yorke dimension.

The complete analytical and computational framework used in the present study to explore the underlying dynamics of VFD-s is presented in Figure 1. The proposed methodology is systematically split into three main interconnected phases: basic dynamical analysis, numerical implementation and complex chaotic behavior characterization.

The basic phase of the workflow is dedicated to the basic dynamics of the VFDEs. The initial formulation is then rigorously subjected to dynamical analysis, as shown, which defines the topological structure of the equilibria of the system. This naturally leads to stability analysis, where the asymptotic behavior and the local stability conditions of the VOF-s are assessed, ensuring the theoretical soundness of the model before numerical implementation.

From these theoretical bases, the framework proposes the VOF-s 8D hyperchaotic system. To handle the complexity of solving an 8-dimensional system with variable fractional orders, special algorithms for VOF-s are developed. Such computational schemes are vital to correctly reproduce the memory effects present in the variable-order derivatives, thus yielding the numerical solutions required for the next phase of simulations.

The flowchart on the right shows a step-by-step multi-stage study of the nonlinear phenomena of the system. The chaotic dynamics analysis allows, first, a bifurcation analysis, which is the main tool to map the parameter space and to locate regions of ordinary chaos. The framework then proceeds to the calculation of Lyapunov exponents to rigorously classify the dimensionality of these chaotic attractors. The existence of positive Lyapunov exponents confirms the existence of highly complex, novel chaos (hyperchaotic behavior) in 8D phase space.

Time series analysis is used to study the continuous temporal evolution of the system. This stage provides a direct visualization of the state variables as a function of time and offers a precise way to assess The extent to which the system is influenced by its initial conditions, which is a characteristic feature of hyperchaotic variable-order systems. This structured pipeline provides a strong validation for the numerical algorithms as well as for the new dynamical properties exhibited by the 8D system.

The novelty of this work is the formulation and analysis of a VOF-s 8D hyperchaotic system with time-dependent Liouville–Caputo orders. The variable-order setting presents radically new dynamical features which are not present in the fractional-order case. The main contributions of this paper can be summarized as follows.

• Development of a new variable-order fractional formulation of the considered 8D hyperchaotic system, involving a time-varying memory structure that considerably extends the classical dynamical model.
• A novel variable-order scheme to control the evolution of the fractional derivative order, introducing adaptive memory effects and new dynamics not reported before.
• A detailed numerical analysis using phase portraits and time-series trajectories to demonstrate the rich and high-dimensional dynamics of the proposed model.
• Detailed bifurcation analysis and evaluation of the Lyapunov exponent showing that the variable-order memory can either enhance or suppress the chaotic behavior according to the selected order dynamics.
• The time-evolving memory mechanism induces localized state transitions and stabilizes sporadic crises. In light of the differences exhibited in the behavioral landscapes of the periodic and monotonic test functions, the operational stability of the system may continuously be controlled by controlling the order dynamics.
• The integration of variable-order operators generates heavily folded, topologically dense strange attractors. It expands the effective fractal footprint of the phase space, enabling the system to achieve a remarkably high Kaplan–Yorke dimension ($D_{\mathrm{KY}}=7.100$) compared to standard architectures.
• The significant Perron effect, along with the high Kolmogorov–Sinai entropy ($E_{\mathrm{KS}}=2.0000$) and multi-directional orbit expansion of the system, make this 8D system highly suitable for high-dimensional cryptography.

The remainder of this paper is organized as follows. Section 2 presents the proposed variable-order 8D hyperchaotic system, while Section 3 and Section 4 provide the mathematical analysis and numerical scheme. Section 5 investigates the dynamical behavior. Section 6 discusses the potential applications and technological implications of the proposed system. Finally, Section 7 concludes the paper and outlines possible directions for future research.

2. Variable-Order 8D Hyperchaotic System Analysis

In this part, we review the mathematical foundations of the constant-order and VOF-s in the Liouville–Caputo (LC) setting. These definitions are basic tools for the construction and analysis of the proposed variable-order fractional model.

2.1. Preliminary Definitions

Definition 1

 ([20]).  Consider a smooth function $y\left(t\right)$. We define the LC fractional derivative of constant order α as

$${}^{\mathrm{LC}}D_t^{\alpha }y\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{\displaystyle \frac{y^{\prime }\left(\tau \right)}{{(t-\tau )}^{\alpha }}}d\tau ,0<\alpha <1.$$

(1)

Definition 2

 ([21]).  When the fractional order varies with time, namely $\alpha \left(t\right)$, the corresponding LC variable-order fractional derivative is given by

$${}^{\mathrm{LCV}}D_t^{\alpha \left(t\right)}y\left(t\right)={\displaystyle \frac{1}{\Gamma \left(1-\alpha \left(t\right)\right)}}{\int }_0^t{(t-\tau )}^{-\alpha \left(t\right)}{y}^{\prime }\left(\tau \right)d\tau ,0<\alpha \left(t\right)<1.$$

(2)

2.2. Fractional 8D Hyperchaotic System

The fractional-order form of the proposed 8-dimensional hyperchaotic system is expressed as follows [22]:

$$\begin{array}{cc}\hfill D_t^{\alpha }{u}_1& =a_1(u_2-u_1)+u_4,\hfill \\ \hfill D_t^{\alpha }{u}_2& =a_2{u}_1-u_1{u}_3+u_4,\hfill \\ \hfill D_t^{\alpha }{u}_3& =u_1{u}_2-u_3-u_4+u_7,\hfill \\ \hfill D_t^{\alpha }{u}_4& =-a_3(u_1+u_2)+u_5,\hfill \\ \hfill D_t^{\alpha }{u}_5& =-u_2-a_4{u}_4+u_6,\hfill \\ \hfill D_t^{\alpha }{u}_6& =-a_5(u_1+u_5)+a_4{u}_7,\hfill \\ \hfill D_t^{\alpha }{u}_7& =-a_6(u_1+u_6-u_8),\hfill \\ \hfill D_t^{\alpha }{u}_8& =-a_7{u}_7.\hfill \end{array}$$

(3)

where $D_t^{\alpha }$ denotes the LC fractional derivative of constant order $\alpha$, satisfying $0<\alpha \le 1$. The initial condition vector and parameter values are chosen as ${\mathbf{U}}_0={(0,0,1,0,1,1,1,1)}^T,$ and $(a_1,a_2,a_3,a_4,a_5,a_6,a_7)=(9,0.04,1.5,1.4,38,-15.2,-0.2),$ respectively.

2.3. Divergence and Dissipativity of the 8D Hyperchaotic System

The divergence of the 8D hyperchaotic system (3) is determined by calculating the trace of its corresponding Jacobian matrix, which is formulated as

$$\nabla ·\mathbf{F}={\displaystyle \sum _{i=1}^8}{\displaystyle \frac{\partial F_i}{\partial u_i}},$$

(4)

where $\mathbf{F}={(F_1,F_2,\dots ,F_8)}^T$ represents the vector field of the system. By computing the individual partial derivatives with respect to each state variable from (3), we obtain

$$\begin{array}{cc}\hfill \nabla ·\mathbf{F}& ={\displaystyle \frac{\partial F_1}{\partial u_1}}+{\displaystyle \frac{\partial F_2}{\partial u_2}}+{\displaystyle \frac{\partial F_3}{\partial u_3}}+{\displaystyle \frac{\partial F_4}{\partial u_4}}+{\displaystyle \frac{\partial F_5}{\partial u_5}}+{\displaystyle \frac{\partial F_6}{\partial u_6}}+{\displaystyle \frac{\partial F_7}{\partial u_7}}+{\displaystyle \frac{\partial F_8}{\partial u_8}}\hfill \\ \hfill & =-a_1+0-1+0+0+0+0+0.\hfill \end{array}$$

(5)

Hence, the divergence expression simplifies to a constant value dependent solely on parameter $a_1$:

$$\nabla ·\mathbf{F}=-(a_1+1).$$

(6)

Substituting the given system parameter $a_1=9$ into the equation yields

$$\nabla ·\mathbf{F}=-(9+1)=-10<0.$$

(7)

Since the resulting divergence is strictly negative, the proposed 8D hyperchaotic system is proven to be dissipative.

2.4. Variable-Order Fractional 8D Hyperchaotic System

To incorporate an evolving memory mechanism, the constant fractional order is replaced by $\alpha \left(t\right)$, yielding the following extension variable-order hyperchaotic system:

$$\begin{array}{cccc}\hfill {}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}\left\{u_1\left(t\right)\right\}& =a_1(u_2-u_1)+u_4,\hfill & \hfill & u_1\left(0\right)=u_{1,0},\hfill \\ \hfill {}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}\left\{u_2\left(t\right)\right\}& =a_2{u}_1-u_1{u}_3+u_4,\hfill & \hfill & u_2\left(0\right)=u_{2,0},\hfill \\ \hfill {}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}\left\{u_3\left(t\right)\right\}& =u_1{u}_2-u_3-u_4+u_7,\hfill & \hfill & u_3\left(0\right)=u_{3,0},\hfill \\ \hfill {}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}\left\{u_4\left(t\right)\right\}& =-a_3(u_1+u_2)+u_5,\hfill & \hfill & u_4\left(0\right)=u_{4,0},\hfill \\ \hfill {}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}\left\{u_5\left(t\right)\right\}& =-u_2-a_4{u}_4+u_6,\hfill & \hfill & u_5\left(0\right)=u_{5,0},\hfill \\ \hfill {}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}\left\{u_6\left(t\right)\right\}& =-a_5(u_1+u_5)+a_4{u}_7,\hfill & \hfill & u_6\left(0\right)=u_{6,0},\hfill \\ \hfill {}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}\left\{u_7\left(t\right)\right\}& =-a_6(u_1+u_6-u_8),\hfill & \hfill & u_7\left(0\right)=u_{7,0},\hfill \\ \hfill {}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}\left\{u_8\left(t\right)\right\}& =-a_7{u}_7,\hfill & \hfill & u_8\left(0\right)=u_{8,0},\hfill \end{array}$$

(8)

where ${}_{0 }{}^{LCV}\mathbb{D}_0^{\alpha \left(t\right)}$ denotes the Liouville–Caputo VOF-s, with $0<\alpha \left(t\right)\le 1$. The functions $\alpha \left(t\right)$ describe the time-varying memory characteristics of the system and allow the dynamical behavior to evolve continuously during the simulation.

3. Mathematical Analysis of the Fractional-Order System

This section deals with the study of the local asymptotic stability for the considered Liouville–Caputo variable order fractional model. For that purpose, the first step includes the calculation of equilibrium points, construction of a general Jacobian matrix, and, finally, obtaining the stability criteria through fractional eigenvalues.

3.1. Equilibrium Point Computation

To find the equilibrium points of the system, we set the right-hand side of the fractional differential equations to zero:

$$\begin{array}{cc}\hfill a_1(u_2-u_1)+u_4& =0,\hfill \\ \hfill a_2{u}_1-u_1{u}_3+u_4& =0,\hfill \\ \hfill u_1{u}_2-u_3-u_4+u_7& =0,\hfill \\ \hfill -a_3(u_1+u_2)+u_5& =0,\hfill \\ \hfill -u_2-a_4{u}_4+u_6& =0,\hfill \\ \hfill -a_5(u_1+u_5)+a_4{u}_7& =0,\hfill \\ \hfill -a_6(u_1+u_6-u_8)& =0,\hfill \\ \hfill -a_7{u}_7& =0.\hfill \end{array}$$

(9)

Substituting the given system parameters $(a_1,a_2,a_3,a_4,a_5,a_6,a_7)=(9,0.04,1.5,1.4,38,-15.2,$$-0.2)$: From the eighth equation, $0.2u_7=0\Rightarrow u_7=0$. From the sixth equation, $-38(u_1+u_5)+1.4\left(0\right)=0\Rightarrow u_5=-u_1$. From the fourth equation, $-1.5(u_1+u_2)-u_1=0\Rightarrow u_2=-{\displaystyle \frac{5}{3}}{u}_1$. From the first equation, $9\left(-{\displaystyle \frac{5}{3}}{u}_1-u_1\right)+u_4=0\Rightarrow u_4=24u_1$. From the fifth equation, $-\left(-{\displaystyle \frac{5}{3}}{u}_1\right)-1.4\left(24u_1\right)+u_6=0\Rightarrow u_6={\displaystyle \frac{479}{15}}{u}_1$. From the seventh equation, $15.2(u_1+u_6-u_8)=0\Rightarrow u_8=u_1+u_6={\displaystyle \frac{494}{15}}{u}_1$.

Substitute $u_4=24u_1$ into the second equation:

$$0.04u_1-u_1{u}_3+24u_1=0\Rightarrow u_1(24.04-u_3)=0$$

(10)

This establishes two distinct cases:

• Case 1: $u_1=0$. This forces $u_2=u_4=u_5=u_6=u_7=u_8=0$. From the third equation, we obtain $u_3=0$. Thus, the origin is an equilibrium point: $E_0=(0,0,0,0,0,0,0,0)$.
• Case 2: $u_1\ne 0$, which yields $u_3=24.04$. Substituting $u_2,u_3,u_4,$ and $u_7$ into the third equation gives

  $$-{\displaystyle \frac{5}{3}}{u}_1^2-24.04-24u_1=0\Rightarrow 5u_1^2+72u_1+72.12=0$$

  (11)
  Solving this quadratic equation yields $u_1^{\left(1\right)}\approx -1.0831$ and $u_1^{\left(2\right)}\approx -13.3169$. This results in two additional non-zero equilibrium points, denoted as $E_1$ and $E_2$.

3.2. Jacobian Matrix

The general Jacobian matrix $J=\left[{\displaystyle \frac{\partial f_i}{\partial u_j}}\right]$ for the system evaluated at an arbitrary state is given by

$$J=\left[\begin{array}{cccccccc}-a_1& a_1& 0& 1& 0& 0& 0& 0\\ a_2-u_3& 0& -u_1& 1& 0& 0& 0& 0\\ u_2& u_1& -1& -1& 0& 0& 1& 0\\ -a_3& -a_3& 0& 0& 1& 0& 0& 0\\ 0& -1& 0& -a_4& 0& 1& 0& 0\\ -a_5& 0& 0& 0& -a_5& 0& a_4& 0\\ -a_6& 0& 0& 0& 0& -a_6& 0& a_6\\ 0& 0& 0& 0& 0& 0& -a_7& 0\end{array}\right]$$

(12)

3.3. Eigenvalue Analysis

Evaluating the Jacobian matrix at the origin $E_0$ with the specified parameters yields

$$J\left(E_0\right)=\left[\begin{array}{cccccccc}-9& 9& 0& 1& 0& 0& 0& 0\\ 0.04& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& -1& -1& 0& 0& 1& 0\\ -1.5& -1.5& 0& 0& 1& 0& 0& 0\\ 0& -1& 0& -1.4& 0& 1& 0& 0\\ -38& 0& 0& 0& -38& 0& 1.4& 0\\ 15.2& 0& 0& 0& 0& 15.2& 0& -15.2\\ 0& 0& 0& 0& 0& 0& 0.2& 0\end{array}\right]$$

(13)

To analyze the local dynamics, we evaluate the characteristic equation $det(J\left(E_0\right)-\lambda I)=0$. Notice that the third column contains only a single non-zero diagonal entry ($-1$). Thus, one eigenvalue is found explicitly:

$${\lambda }_1=-1$$

(14)

The remaining seven eigenvalues ${\lambda }_2,\dots ,{\lambda }_8$ are roots of the 7th-degree characteristic polynomial formed by the remaining principal minor. Because ${\lambda }_1=-1$ is strictly real and negative, it trivially satisfies the stability constraints. Therefore, the stability of $E_0$ depends entirely on the arguments of the remaining seven eigenvalues.

3.4. Local Asymptotic Stability Conditions

For a Liouville–Caputo variable-order fractional system, the direct application of Matignon’s condition is not strictly rigorous, as it was originally derived for constant-order systems. A variable-order system can be expressed as a nonlinear Volterra integral equation with a time-dependent kernel, indicating that stability depends on both the system’s eigenvalues and the memory structure induced by $\alpha \left(t\right)$.

To proceed with stability analysis, a quasi-static (frozen-time) approximation is widely adopted. Under the assumption that the fractional order varies slowly, i.e.,

$$|\dot{\alpha }\left(t\right)|\ll 1,$$

(15)

the system can be locally approximated by a constant-order system at any given instant. In this sense, an equilibrium point E is considered instantaneously stable if all eigenvalues ${\lambda }_i$ of the Jacobian matrix $J\left(E\right)$ satisfy the heuristic condition:

$$|arg\left({\lambda }_i\right)|>{\displaystyle \frac{\alpha \left(t\right)\pi }{2}}.$$

(16)

For persistent stability over time, let the variable fractional order $\alpha \left(t\right)\in (0,1]$ be bounded such that ${sup}_t\alpha \left(t\right)={\alpha }^{*}\le 1$. A practical and widely accepted indicator for the local asymptotic stability of E requires all eigenvalues to satisfy

$$|arg\left({\lambda }_i\right)|>{\displaystyle \frac{{\alpha }^{*}\pi }{2}}$$

(17)

where $arg\left({\lambda }_i\right)$ is the principal argument of the eigenvalue.

Conversely, if there exists an eigenvalue such that $|arg\left({\lambda }_i\right)|<{\displaystyle \frac{{inf}_t\alpha \left(t\right)\pi }{2}}$, the equilibrium point is unstable. It should be noted that due to the nonlocal nature of fractional operators, stability is path-dependent and the system may exhibit delayed transitions when $\alpha \left(t\right)$ crosses critical thresholds. Accordingly, this criterion serves as a consistent approximation framework for evaluating variable-order stability trends rather than a strict theoretical extension of constant-order theory.

Remark 1.

The classical Matignon stability condition is proven to be rigorously valid only for constant-order fractional dynamical systems [23]. However, in the current formulation of the variable-order Liouville–Caputo fractional dynamics, the above condition is used under the quasi-static approximation where the constant order is substituted by $\alpha \left(t\right)$ as is usually done in the variable-order fractional calculus studies [24,25,26,27,28]. Therefore, the resulting stability criteria can be considered only as local indicators of asymptotic stability.

4. Numerical Approximation Scheme

Numerical approximation of the proposed variable-order fractional system is more challenging than that of classical differential equations due to the nonlocal memory property of the Liouville–Caputo operator. Using the approaches in [21,29,30], the governing system is approximated as follows:

For the variable-order fractional derivative, the numerical approximation is constructed using the Lagrange interpolation technique [21]. The resulting recursive expression is given by Equation (18):

$$Y_{m+1}=Y_0+{\displaystyle \frac{1}{\Gamma \left(\alpha \right(t\left)\right)}}{\displaystyle \sum _{j=0}^m}{\displaystyle \frac{\Delta t^{\alpha \left(t\right)}}{\alpha \left(t\right)\left(\alpha \left(t\right)+1\right)}}\left(F(t_j,Y_j){\mathcal{A}}_{m,j}-F(t_{j-1},Y_{j-1}){\mathcal{B}}_{m,j}\right).$$

(18)

The coefficients associated with the interpolation procedure are expressed as follows:

$$\begin{array}{cc}\hfill {\mathcal{A}}_{m,j}& ={(m+1-j)}^{\alpha \left(t\right)}\left(m-j+2+\alpha \left(t\right)\right)-{(m-j)}^{\alpha \left(t\right)}\left(m-j+2+2\alpha \left(t\right)\right),\hfill \\ \hfill {\mathcal{B}}_{m,j}& ={(m+1-j)}^{\alpha \left(t\right)+1}-{(m-j)}^{\alpha \left(t\right)}\left(m-j+1+\alpha \left(t\right)\right).\hfill \end{array}$$

(19)

In Equation (18), the functions $F_k$ denote the right-hand sides of the governing equations of the proposed variable-order system.

Accordingly, the nonlinear functions $F_k$ are evaluated at each iteration according to

$$\begin{array}{cc}\hfill F_1& =a_1(v_2-v_1)+v_4,\hfill \\ \hfill F_2& =a_2{v}_1-v_1{v}_3+v_4,\hfill \\ \hfill F_3& =v_1{v}_2-v_3-v_4+v_7,\hfill \\ \hfill F_4& =-a_3(v_1+v_2)+v_5,\hfill \\ \hfill F_5& =-v_2-a_4{v}_4+v_6,\hfill \\ \hfill F_6& =-a_5(v_1+v_5)+a_4{v}_7,\hfill \\ \hfill F_7& =-a_6(v_1+v_6-v_8),\hfill \\ \hfill F_8& =-a_7{v}_7.\hfill \end{array}$$

(20)

4.1. Stability Analysis

For stability analysis of the proposed variable-order fractional system, a local extension of the classical stability analysis method for constant-order fractional systems is used. Due to the variation of the fractional order with respect to time, the obtained conditions are of local stability type and are in agreement with the theoretical results given in [21,29].

Remark 2.

The existence and uniqueness analysis is based on the standard assumption that the nonlinear terms satisfy the Lipschitz condition. A rigorous proof involving explicit Lipschitz estimates and the construction of a contraction mapping for the variable-order operator is beyond the scope of the present work and follows the framework established in [29].

4.2. Convergence and Numerical Stability

For the numerical algorithm used in this study, schemes from [29,30] are utilized which convergence and stability properties have already been rigorously proven. For further investigation of its robustness, some numerical experiments have been done with different values of the time step and resulted in almost the same solutions.

The numerical implementation of the proposed variable-order fractional 8D hyperchaotic system (3) is performed according to the following steps.

• Set the system parameters $(a_1,a_2,a_3,a_4,a_5,a_6,a_7)=(9,0.04,1.5,1.4,38,-15.2,-0.2)$ and the initial condition ${\mathbf{U}}_0={(0,0,1,0,1,1,1,1)}^T$.
• Choose the time step size $\Delta t=0.01$ and construct the computational time grid $t_n=nh$, where $n=0,1,2,\dots$.
• Select one of the variable-order functions listed in

  Table 1. Variable-order functions used in the numerical simulations.

  Case	$\mathit{\alpha }\left(\mathit{t}\right)$	$\mathit{t}=100$	$\mathit{t}=300$	Range
  1	$0.95+0.01cos\left({\displaystyle \frac{t}{50}}\right)$	0.946	0.959	$[0.94,0.96]$
  2	$0.92+0.01tanh\left({\displaystyle \frac{t^2}{40}}\right)$	0.93	0.93	$[0.92,0.93]$

  .
• Compute the nonlinear terms of System (3) at each time step.
• Apply the proposed variable-order fractional numerical algorithm to update the state variables $(u_1,u_2,\dots ,u_8)$ until the final simulation time.
• Case (1): The cosine function is adopted to represent a slowly varying periodic memory effect, modeling systems operating under cyclic or harmonic environmental variations.
• Case (2): The hyperbolic tangent function is employed to describe a smooth monotonic variation in the memory order, representing irreversible evolutionary processes such as aging or relaxation.

5. Dynamical Analysis

In this part, the hyperchaotic behavior of System (8) will be investigated using several techniques [31,32] for diagnosing the complex nature of the system, which includes its high-dimensional nature, the non-periodic behavior of the system, and its highly sensitive nature to changes in initial conditions. These chaotic features will be estimated from the parameter-dependent bifurcations of the system, whereas the Lyapunov exponent spectra, along with the Kaplan–Yorke dimension $D_{\mathrm{KY}}$, will be used to determine the degree of topological chaos.

5.1. Lyapunov Exponents and Kaplan–Yorke Dimension

The asymptotic behavior and orbital divergence of the system are quantitatively evaluated through its Lyapunov exponent spectrum over time, as shown in Figure 2. In state space, the Lyapunov exponent measures trajectory divergence or convergence. In this work, the exponents are computed using the numerical approach presented in [33]. Corresponding numerical steady state invariants at $t=100$ are provided in

Table 2. Lyapunov exponents, Kaplan–Yorke dimension, and dynamics at t = 100.

Case	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	${\mathit{LE}}_4$	${\mathit{LE}}_5$	${\mathit{LE}}_6$	${\mathit{LE}}_7$	${\mathit{LE}}_8$	${\mathit{D}}_{\mathbf{KY}}$	Limit Set
Case (1)	0.6421	0.1351	0.0448	0.0181	−0.0580	−0.5687	−3.1402	−8.0733	6.068	Hyperchaos
Case (2)	0.9484	0.1447	0.0226	0.0107	−0.0427	−0.3639	−3.0411	−8.6789	6.2367	Hyperchaos

.

In both cases, the system is undoubtedly hyperchaotic, as evidenced by four positive values of Lyapunov exponents ($L{E}_{1,2,3,4}>0$).

• Case (1) (Right Panel): Produces the Lyapunov of $L{E}_{1,2,3,4}=[0.6421,0.1351,0.0448,0.0181]$, generating a strange attractor of higher dimensions with the Kaplan–Yorke dimension equal to $D_{\mathrm{KY}}=6.068$.
• Case (2) (Left Panel): Features a higher degree of chaos, characterized by a larger dominant Lyapunov exponent ($L{E}_1=0.9484$). Such a higher stretching effect leads to a larger phase space, giving rise to the larger Kaplan–Yorke dimension of $D_{\mathrm{KY}}=6.2367$.

The asymptotic convergence and spectrum tracking of the system invariants are monitored up to $t=300$, as depicted in Figure 3. The definitive quantitative measures establishing the steady-state regimes are listed in

Table 3. Lyapunov exponents, Kaplan–Yorke dimension, and dynamics at t = 300.

Case	${\mathit{LE}}_1$	${\mathit{LE}}_2$	${\mathit{LE}}_3$	${\mathit{LE}}_4$	${\mathit{LE}}_5$	${\mathit{LE}}_6$	${\mathit{LE}}_7$	${\mathit{LE}}_8$	${\mathit{D}}_{\mathbf{KY}}$	Limit Set
Case (1)	1.065	0.260	0.180	0.140	−0.120	−0.220	−0.5669	−9.620	7.070	Hyperchaos
Case (2)	1.165	0.444	0.250	0.180	−0.100	−0.300	−0.6679	−9.721	7.100	Hyperchaos

.

Both systems confirm sustained hyperchaotic behavior over the extended time horizon, signaled by the persistence of four positive Lyapunov exponents ($L{E}_{1,2,3,4}>0$).

• Case (1): Converges to an invariant spectrum of $L{E}_{1\dots 4}=[1.065,0.260,0.180,0.140]$, yielding a highly complex attractor structure with a Kaplan–Yorke dimension of $D_{\mathrm{KY}}=7.070$.
• Case (2): Displays an even more aggressive phase space expansion, where the prominent tracking exponents are uniformly larger ($L{E}_{1\dots 4}=[1.165,0.444,0.250,0.180]$). This enhanced stretching action expands the effective fractal footprint, pushing the Kaplan–Yorke dimension to $D_{\mathrm{KY}}=7.100$.

Comparing these profiles to shorter-term dynamics shows that extending the horizon to $t=300$ allows the transient components to fully decay, revealing that Case (2) consistently drives a more robustly divergent and topologically dense hyperchaotic limit set than Case (1).

5.2. Chaotic Characteristics and Complexity Analysis

In this section, the hyperchaotic behavior of System (8) is assessed through several diagnostics that complement each other in evaluating the dynamics of the system. These include analysis of phase space attractors as well as the random nature of time series due to the complex nature of the system in question and its highly sensitive dynamics. In order to measure such chaotic properties, bifurcations induced by varying parameters are plotted through bifurcation diagrams. This, at the same time, is done along with calculations of the Lyapunov exponent spectrum, $E_{\mathrm{KS}}$, and $D_{\mathrm{KY}}$.

5.3. Bifurcation Analysis

In order to study the impact of system parameters on the nonlinear dynamics, the bifurcation diagrams are plotted alongside with the respective Lyapunov exponent (LE) spectra. The analysis was conducted by changing one parameter at a time, while fixing all other parameters. As seen from Figure 4, Figure 5 and Figure 6, the considered system possesses complex nonlinear dynamics within the chosen ranges of system parameters. The analysis of the bifurcation diagram and LE spectra for $a_1$ (Figure 4) clearly shows the presence of periodic and chaotic regions. Figure 5 presents the effect of varying $a_2$. Similar transitions between regular and chaotic motions are observed, with positive largest Lyapunov exponents indicating the existence of chaos throughout several parameter intervals. The influence of $a_3$ is shown in Figure 6. As $a_3$ increases, the system undergoes noticeable changes in its attractor structure, including periodic windows embedded within chaotic regions. These transitions are consistently verified by the corresponding Lyapunov exponent spectra.

5.3.1. Perron Effect

The classical Perron effect refers to the existence of a dominant largest Lyapunov exponent satisfying ${\lambda }_1\gg {\lambda }_2\ge {\lambda }_3\ge \cdots ,$ indicating that the divergence of nearby trajectories is primarily governed by a single unstable direction.

For the proposed system, the transient Lyapunov spectra at $t=100$ (Table 2 and Table 3) exhibit a clear dominance of the largest exponent over the remaining positive exponents. In particular, Case (2) has a larger leading exponent ($L{E}_1=0.9484$) than Case (1) ($L{E}_1=0.6421$), indicating stronger initial instability. Therefore, the transient dynamics are mainly influenced by the leading expanding direction, which is consistent with Perron-type behavior.

5.3.2. Multidimensional Hyperchaos

For the steady-state $(t=300)$ case, there is a different behavior of the Lyapunov spectrum. Namely, the leading Lyapunov exponent becomes equal to $L{E}_1=1.065$ in Case (1) and to $L{E}_1=1.165$ in Case (2), while the other positive exponents (${\lambda }_2$, ${\lambda }_3$, and ${\lambda }_4$) also grow and become close to ${\lambda }_1$. Thus, the condition ${\lambda }_1\gg {\lambda }_2$ is not fulfilled anymore.

On the contrary, one observes the emergence of the cluster of positive exponents, namely ${\lambda }_1\approx {\lambda }_2\approx {\lambda }_3>0$, which is typical for multidimensional hyperchaos rather than the manifestation of the Perron effect. At that, the nearby trajectories in such a regime exponentially diverge in several directions and stretch the phase space uniformly.

5.3.3. Kolmogorov–Sinai (KS) Entropy

The KS entropy represents the rate at which information creation takes place in a dynamical system. As per Pesin’s identity, the KS entropy ($E_{\mathrm{KS}}$) depends on the summation of positive Lyapunov exponents:

$$E_{\mathrm{KS}}=\sum _{{\lambda }_i>0}{\lambda }_i.$$

(21)

Applying this formulation to the steady-state spectrum profiles yields the following information production rates:

• Short term regime ($t=100$): For Case (1), we obtain $E_{\mathrm{KS}}=0.8401$, while for Case (2), a noticeably higher information generation rate is reached at $E_{\mathrm{KS}}=1.1264$.
• Long term regime ($t=300$): After transient phenomena are over in the short-term interval, the values of the complexity measures become much larger. In Case (1), we have $E_{\mathrm{KS}}=1.6450$, whereas for Case (2), more complexity of structure is attained at $E_{\mathrm{KS}}=2$.

5.3.4. Kaplan–Yorke Dimension ($D_{\mathrm{KY}}$)

The values ($D_{\mathrm{KY}}$) indicate high topological complexity and a densely filled strange attractor. The dimension is defined as

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

(22)

where j represents the largest integer satisfying ${\sum }_{i=1}^j{\lambda }_i>0$ and ${\sum }_{i=1}^{j+1}{\lambda }_i<0$.

For the 8D system under study, the threshold condition is met at $j=7$ for both cases at $t=300$. For instance, in Case (1), ${\sum }_{i=1}^7{\lambda }_i=0.5543>0$ and ${\sum }_{i=1}^8{\lambda }_i=-9.677<0$, yielding

$$D_{\mathrm{KY}}=7+{\displaystyle \frac{0.5543}{|-9.620|}}=7.070.$$

(23)

Similarly, the aggressive stretching in Case (2) expands the dimension to $D_{\mathrm{KY}}=7.100$.

Table 4. Kaplan–Yorke fractal dimension (FD) of chaotic systems.

System	Dim.	FD	2.1–2.5	5–6	6–7	>7
[34]	7D	2.091	✓
[35]	9D	2.171	✓
[36]	7D	2.175	✓
[37]	10D	2.429	✓
[38]	9D	5.128		✓
[39]	7D	5.278		✓
[40]	7D	6.149			✓
[41]	7D	6.732			✓
This study (Case 1)	8D	7.070				✓
This study (Case 2)	8D	7.100				✓

provides a comprehensive benchmarking of these values against existing high-dimensional chaotic architectures found in the literature.

The Kaplan–Yorke dimension, also known as the topological dimension, quantifies the measure of the fractal density of a strange attractor. According to Table 4, $D_{\mathrm{KY}}=7.070$ is computed in Case (1) from the new 8-dimensional variable-order model, while $D_{\mathrm{KY}}=7.100$ in Case (2) from further stretching the state space. A comparison between these results and other chaotic systems reveals that whereas many well-known chaos-generating structures with up to 10 degrees of freedom continue to operate at relatively low fractional capacities, the new system performs better in terms of topological density.

5.4. Chaos and Time Series Analysis

The dynamics of Case (1) are qualitatively investigated based on its projections onto the phase spaces and time domain. Figure 7 shows the multiple phase portraits of the system and depicts the geometric form of the fully developed, high dimensional strange attractor with bounded non-intersecting trajectories covering the entire state space.

These results have been validated with respect to time domain responses, which are shown in Figure 8.

The qualitative behavior of Case (2) system is assessed through its phase-space portraits and time-history plots. Figure 9 depicts the multi-phase portrait of the system, depicting more densely packed hyperchaotic attractors in contrast to Case (1). As can be seen, the orbits are much folded and tracked in such a manner that they do not overlap.

This highly chaotic orbital behavior is further supported by the time domain plots as depicted in Figure 10. It is evident from the graphs that all the state variables ($u_1\dots u_8$) oscillate continuously and randomly over time. Hence, it can be said that there is greater stretching in the case of Case (2).

Figure 11 and Figure 12 show the dynamic responses of the new chaotic system in Case (1) up to $t=300\mathrm{s}$. Two dimensional phase space projections in Figure 11 show very dense and non-periodic strange attractors which never settle down to any fixed points or periodic orbits. On the other hand, the waveform responses of all eight state variables in the time domain (Figure 12) show noisy and non-periodic oscillations.

The numerical simulation outcomes of the hyperchaotic behavior in Case (2) at $t=300\mathrm{s}$ are shown in Figure 13 and Figure 14. The 2D multi-projection phase portraits in Figure 13 illustrate complex strange attractors with very dense and full developments without falling back to periodic oscillations. At the same time, the waveforms in Figure 14 show sustained but non-periodic noise-like vibrations in all eight state variables. Both graphical results prove that the system is in a stable hyperchaotic regime in Case (2).

Figure 15 and Figure 16 confirm the existence of the proposed novel hyperchaotic attractor. The bounded and complex trajectories demonstrate rich hyperchaotic dynamics and strong nonlinear behavior.

6. Potential Applications and Technological Implications

The proposed variable-order fractional hyperchaotic system exhibits multiple positive Lyapunov exponents, a high Kaplan–Yorke dimension, and rich nonlinear dynamics, indicating its potential for future engineering applications. Although no practical implementation is presented in this work, the obtained dynamical properties suggest several promising research directions.

As high-dimensional hyperchaotic dynamics improve unpredictability and key space complexity, pseudorandom sequences for cryptography and secure communications may be developed. Since nonlinear dynamical systems depend on memory, simulation of such systems is another research topic. Hyperchaotic dynamics can be used in signal processing and secure wireless communications.

Such applications are still theoretical and need to be researched in detail, especially through encryption scheme design, random statistics production, and cryptographic attack and hardware implementation evaluation.

7. Conclusions

In this paper, a novel eight-dimensional (8D) hyperchaotic system has been successfully developed and analyzed through the lens of VOF-s utilizing the LC derivative. The system offers an evolutionary memory mechanism, from classical constant-order models to a more flexible model $\alpha \left(t\right)$ depending on time, which strongly captures the nonstationary and hereditary features of real-world complex physical phenomena. To address the computational burden for an 8D state-space under time-varying fractional derivatives, an efficient and highly accurate numerical integration scheme was adopted, where the local history accumulation was based on second-order Lagrange polynomial interpolation. The high-dimensional chaotic characteristics of the proposed model have been studied carefully using two particular types of memory-based systems: Case (1) depicting a periodic environment and Case (2) representing monotonic relaxation of structure. The qualitative investigation carried out through multiple phase-space projections and time series showed no signs of periodic orbits but indicated random non-repeating trajectories leading to strange attractors. The quantitative analysis provided firm evidence of the existence of ($L{E}_{1,2,3,4}>0$) for extended periods of time ($t=300$).

The significant difference between the major exponent $(L{E}_1=1.165)$ and the remaining exponents confirmed the strong Perron phenomenon, suggesting the occurrence of very strong state space stretching. Moreover, the complexity measures confirmed that the considered Case (2) produces a faster flow of information, where the KS entropy is obtained at $E_{\mathrm{KS}}=2.0000$. Eventually, the comparison of the obtained Kaplan–Yorke fractal dimension of the considered system to other well-known chaotic systems with high dimensionality revealed a significantly better topological density, with a fractal dimension of $D_{\mathrm{KY}}=7.100$. Consequently, these results confirmed the applicability of the proposed theoretical model, which holds a great practical potential for technological advancements requiring high levels of structural chaos, like cryptanalysis and other information security technologies. Possible areas of future investigation could be the DSP realization of the variable-order 8D network and the application thereof in real-time image encryption schemes.