Chaotic Dynamics and Numerical Solutions of the Stochastic Fractional-Order Lorenz System

INTRODUCTION

Non-linear systems often exhibit complexities that can be categorized as chaotic or periodic. All these characteristics pose significant difficulties for correct and rational prognosis, thus underscoring the nature of the task in such systems. To overcome these issues, discussing the models connected with fractional systems is necessary. Thereby, it is possible to get a deeper understanding of a system and, possibly, to develop more efficient methods to predict a system’s functioning by studying the details of its dynamical properties, especially the occurrence and behavior of chaos. This work focuses on identifying the factors relating to fractional systems and revealing the essential characteristics that define their actions.^[1-4]

Mathematical modeling has gained significant importance in scientific and engineering fields, particularly because it incorporates dynamical aspects that help describe complex real-world phenomena. In this context, fractional calculus has attracted growing interest.^[5-7] Fractional-order models are considered superior to classical integer-order models due to their ability to capture memory and hereditary characteristics in materials and processes.^[8,9] These memory effects and oscillatory behaviors play a crucial role when studying fractional versions of the Lorenz system, enriching both the formulation and the analysis of the model.

Chaotic systems such as the Lorenz system have been extensively studied in recent decades. Pehlivan and Uyaroglu^[10] introduced a new chaotic attractor within the broader Lorenz family, and several works have outlined the practical applications of chaos theory. In the context of communication systems, Cuomo et al.^[11] further demonstrated the role of chaos through the synchronization of Lorenz-based chaotic circuits. Lorenz’s foundational work also contributed significantly to non-linear dynamics, including applications in economics and mechanical chaos, thereby establishing the significance of chaotic behavior across disciplines.^[12]

Additional contributions include the work of Mahmoud et al.^[13] who examined the fundamental characteristics of complex Lorenz oscillators, and another study that explored the synchronization of complex Lorenz systems and their application in chaotic signal amplitude modulation.^[14] Lazzús et al.^[15] Conducted parameter-identification analysis for Lorenz-type models. Further research on chaos synchronization by Moon et al.^[16,17] expanded the understanding and potential applications of these systems.

Building on these developments, fractional Lorenz systems have emerged as powerful models for analyzing behaviors influenced by memory, fluctuations, and complex dynamical interactions. Their ability to incorporate historical effects makes them particularly relevant for studying systems where classical integer-order models are insufficient.

This research proposal extends previous studies by investigating the complex interactions of fractional Lorenz systems and exploring their applications in scientific and engineering fields. Several authors have examined feedback and sliding-mode control for chaotic systems. With the introduction of fractional calculus, researchers have explored fractional-order models, yet the field remains relatively open and continues to evolve to enhance its applicability.^[18-21]

To address existing gaps, new fractional operators have been developed. Notably, the Caputo–Fabrizio and Atangana–Baleanu–Caputo (ABC) operators have expanded the scope of fractional calculus and generated significant interest in their application to biological models and other complex systems.^[22-24] These developments have opened new directions in the analysis of fractional differential equations.

MATHEMATICAL PRELIMINARIES

Definition 1^[46] The ABC fractional derivative operator is given by:

where $B\left(\alpha \right)$ the normalization function, $\text{and}B\left(0\right)=B\left(1\right)=1$. The Mittag-Leffler function is given by $E_{\alpha }(.).$

Definition 2^[46] The AB fractional integral is defined as

Definition 3

Constraints on the function g(t)

For the ABC fractional derivative ${}_{}{}^{ABC}D{}_{0,t}^{\alpha }g\left(t\right)$, the function $g\left(t\right)$ must satisfy:

• $g\left(t\right)\in C^1$[0, T], i.e., continuously differentiable on the interval [0, T];

• Its classical derivative g′(t) is bounded or belongs to L¹(0, T);

• g(t) must satisfy initial conditions consistent with the model ($\text{e}.\text{g}.,g\left(0\right)=g_0).$

Definition 4

Why should the section include more than two definitions

To make Section 2 meaningful and justified as a standalone section, you may add:

• Assumptions on the non-linear function $\theta \left(t,g\left(t\right)\right)(e.g.,$ Lipschitz continuity).

• Growth conditions and boundedness conditions for the diffusion term $\Psi \left(t,g\left(t\right)\right).$

• Notation for Brownian motion $\text{B}\left(\text{t}\right)$ and its increment $\Delta B_n$.

• Basic properties of the ABC operator (linearity, limit cases, memory kernel behavior).

• The kernel form and its connection to the Mittag–Leffler memory law.

This gives the section depth and addresses the reviewer’s concern.

Definition 5

Constraints on the Fractional Order α

For both operators, the fractional order satisfies:

$0<\alpha <\text{1};$

When α → 1, the ABC derivative reduces to the standard Caputo derivative;

When α → 0, the operator approaches the identity operator.

The normalization function satisfies:

$B\left(\alpha \right)>0,B\left(0\right)=B\left(1\right)=1.$

FRACTIONAL-ORDER LORENZ’S SYSTEM (FLS)

The Atangana–Baleanu fractional operators in the sense of Caputo (ABC) and Riemann–Liouville type (AB integral) provide an essential mathematical framework for modeling real-world processes with nonlocality and non-singular memory kernels. In contrast to classical fractional operators whose kernels exhibit power-law singularity, the ABC derivative uses a Mittag–Leffler (M–L) kernel, which ensures smoother memory fading and avoids divergence at the origin. This characteristic is particularly suitable for dynamical systems and epidemiological models where the influence of past states decays gradually rather than abruptly. The normalization function in the ABC operator guarantees that the derivative recovers the classical first-order behavior when the fractional order approaches 1.

In the context of the present work, these operators allow us to incorporate long-range memory into the system dynamics while maintaining mathematical stability and better numerical behavior. The AB fractional integral complements the derivative by providing an integral operator that is fully consistent with the M–L kernel framework. This is crucial for deriving numerical schemes and establishing the well-posedness of the system under study. Additionally, the use of these operators enhances the ability to analyze bifurcations, chaotic behavior, and stability transitions in fractional-order non-linear systems, which are central themes of this paper. The smooth memory kernel also improves the robustness of numerical simulations, making the ABC/AB framework attractive for fractional-order modeling, including the systems analyzed.

The Lorenz oscillator is a three-dimensional flow characterized by chaos, named after Edward N. Lorenz. Lorenz created this system in 1963 using streamlined equations for atmospheric convection rolls. He has postulated the chaos theory, such as the butterfly effect, which refers to the sensitive dependence on initial conditions. In,^[47] Lorenz depicted that minor changes in the inputs of a given system could cause a massive change in the output after some time. The principle of the fluttering of a butterfly’s wings indicates that slight changes can cause significant changes in the system, including the formation or suppression of tornadoes. This idea focuses on how minor starts can culminate through a chain of events to a significant end.

Lorenz’s chaotic system is given by:

Where σ = 10, ρ = 28, $\beta =\frac{8}{3}$, and initial conditions $[x_1$(0),$x_2$(0), $x_3$ (0)] = (0.1, 0.1, 0.1).

The fractional-order Lorenz system involves memory effects and heredity characteristics, which are significant for depicting most real processes. Since they can model the difficulties and long memory structures displayed by several physical, biological, and engineering systems, they can capture the long memory and complex behavior of dynamics, ocean circulation, etc. Furthermore, fractional-order models can be used to improve the scheme and functionality of the control methods in systems containing damping and viscoelastic features since they better indicate the system dynamics. Many scholars studied the system and its fractional order to understand its dynamics, continual force, and synchronization optimization.^[48-50]

Where σ = 10, ρ = 28, $\beta =\frac{8}{3}$, and initial conditions $[x$(0), $x_2$(0), $x_3$ (0)] = (0.1, 0.1, 0.1).

STOCHASTIC FRACTIONAL-ORDER LORENZ’S SYSTEM (SFLS)

The Stochastic Fractional-Order Lorenz’s System (SFLS) combines the benefits of fractional calculus and stochastic processes to model systems that exhibit inherent randomness and memory effects effectively. Systems with inherent randomness and memory effects. Fractional-order derivatives provide a more accurate description of systems with long-range dependencies and hereditary properties. Incorporating stochastic elements allows the model to account for random fluctuations and uncertainties in real-world phenomena. This combination is beneficial in various applications, including atmospheric and climate modeling, ocean circulation, and engineering systems. It helps design robust control strategies by accurately representing system dynamics and random disturbances. Additionally, it is valuable in biomedical engineering for analyzing signals with fractal characteristics and noise, financial modeling for capturing market trends and fluctuations, secure communications for enhancing encryption, and material science for describing materials with viscoelastic properties under random loading conditions. Many scholars have worked on the system’s traditional integer form and the stochastic fractional-order form to explore its dynamics, continuous forcing, and optimum synchronization.^[51,52] In ecological modeling,^[25] authors have applied fractional derivative concepts^[26,27] to study bifurcation phenomena in fractional-order models,^[28,29] model physical problems,^[30-32] and obtain numerical solutions for fractional models using new methods.^[33,34] Furthermore, simulation studies have addressed infectious diseases incorporating social activities.^[35]

Due to inevitable stochastic disturbances, stochastic differential equations have attracted significant attention for their successes in various areas. Essential theories and references are available in the literature, which contains a vast number of papers investigating stochastic dynamical systems. In recent years, researchers have focused on random attractors and stochastic bifurcation in the Lorenz system.^[36-38] as well as stochastic resonance in chaotic systems influenced by white noise and harmonic forces.^[39] Analytical and numerical methods for these systems have also been developed, yet work on stochastic differential equations containing fractional derivatives remains limited. ^[40,41] Very few studies have addressed the existence and uniqueness of solutions for such equations, and only recently has the approximate controllability of fractional stochastic dynamic systems been established.^[42-45]

The dynamics described by stochastic fractional-order differential equations remain a relatively unexplored area. This research introduces a new perspective by presenting numerical solutions of the stochastic fractional-order Lorenz system with white noise in the Caputo sense, which is particularly relevant for fractal systems and stochasticity, where these solutions are essential for modeling and predicting systems with fractional dynamics. The ^DE convolutional maps of the system’s response to white noise provide fresh insights into chaos, synchronization, and stability, highlighting the real-world applicability of this field across multiple disciplines.

Where σ = 10, ρ = 28, $\beta =\frac{8}{3}$, and initial conditions ($x_1$(0), $x_2$(0), $x_3$ (0)) = (0.1, 0.1, 0.1).

here B(t) illustrates the standard Brownian motion and (${\sigma }_1$, ${\sigma }_2$,${\sigma }_3$) denote the white noise.

5. NUMERICAL SCHEME FOR STOCHASTIC FRACTIONAL SYSTEM

This section explores a novel scheme for the ABC fractional derivative (ABC-FD) as introduced in.^[53,54] We initiate the algorithm by presenting the general form of the differential equation for stochastic fractional differential equations :

with g($t_0$) = $t_0$ and Ψ(t, g(t)) illustrates the noise term. Let $z\left(.\right)$ be the positive increasing differentiable function. Then we write it as follows,

The above may be presented in the following way:

Put $tn+1=\left(n+1\right)\Delta t,$ we have,

Legend of Symbols Used in Equations (8) and (9)

g(t): Numerical approximation of the solution of the stochastic fractional system at time t.

$g\left(t_n\right)$ =$g^n$: Discrete approximation at the n-th time step.

$t_{\text{n}+1}$ = (n + 1)Δt: Time discretization with a uniform time step Δt.

$\text{z}\left(\text{t}\right)$: Auxiliary function derived from the Atangana–Baleanu fractional formulation or model transformation.

θ(t, g(t)): Drift (deterministic) component of the stochastic fractional system.

$\Psi \left(t,g\left(t\right)\right):$ Diffusion (noise) component of the stochastic fractional system.

B(t): Brownian motion (Wiener process) evaluated at time t.

$\rho \in$ (0, 1): Fractional order of the Atangana–Baleanu derivative in the Caputo sense (ABC derivative).

$AB\Gamma \left(\rho \right):$ Normalization function associated with the ABC fractional operator.

$\Gamma \left(·\right)$: Euler Gamma function.

$t\text{and}T$: Continuous time variables inside the convolution kernels.

$z\left(T\right)\theta \left(T,g\left(T\right)\right):$ Convolution term involving the drift function.

$z\left(T\right)\Psi \left(T,g\left(T\right)\right):$ Convolution term involving the diffusion function.

${\left(t-T\right)}^{\rho -1}$: Kernel appearing in the Atangana–Baleanu integral representation.

$B\left(t_{n+1}\right)-B\left(t_n\right)$ The Brownian increment is numerically approximated by:

$\Delta B_n\sim N\left(0,\Delta t\right)$

By using the Newton polynomial method, we may substitute the functions Θ and Ψ into the above equation, resulting in:

RESULTS AND DISCUSSION

The Lorenz butterfly has been illustrated with the deterministic system (3) when there is no noise and when α = 1, as shown in Figure 1. With the moderate noise level of 5, as revealed in Figure 2, the attractors are more scattered, which shows that the stochastic terms added large variability, although the overall chaotic structure remains. When increasing the noise level, Figure 3, where 10 reflects this fact: dispersion and trajectories’ peculiar irregularities are significantly higher at higher noise intensity. These observations again emphasize the influence of the fractional-order parameter α and the stochastic perturbations on the system, which maintains its chaotic nature. Figure 4 presents the plots for the state variables of the fractional-order Lorenz system with the noises added to them as inputs, namely, $x_1$, $x_2$, and $x_3$. In the deterministic case (red lines), the input and output show a preliminary high peak, after which they reach a steady state. When the noise level is raised to 0.05 (blue) and 0.10 (magenta), the oscillations become more apparent, illustrating the system’s sensitivity to noise. Figure 5 shows attractors for system (4) with α = 0.99 and no noise, maintaining clear chaotic structures. Figure 6 introduces moderate noise (5), resulting in increased variability but preserving the overall shape. In Figure 7, with higher (10), the attractors become more dispersed, reflecting the system’s sensitivity to noise. Figure 8 displays the time series plots for different noise levels, showing that higher noise intensities lead to greater fluctuations and blurred distinctions in system behavior. These observations highlight the significant impact of both the fractional-order parameter and noise on the system’s dynamics, emphasizing the necessity of considering these factors when modeling real-world systems that exhibit memory effects and random perturbations. In Figure 9, we plotted the attractors for system (4) when α = 0.95. It has been seen that for a signal of noise, it gives perfect signals of chaotic patterns. Figure 10 adds moderate noise (5), as it adds variability to the dataset, maintaining the attractor pattern. As can be seen in Figure 11, when the noise is 10, the attractors are dispersed, and the signal fluctuates intensely, indicating how sensitive the system is to noise. Figure 12 displays the time series plots. As shown in these results, the fractional-order parameter affects the dynamics of the system, as does the noise. Thus, both factors must be accounted for when studying real-world systems that exhibit memory effects and stochastic perturbations.

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Figure 1:

Plots of chaotic attractors from system (4) with α=1.

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Figure 2:

Plots of chaotic attractors from system (5) with α=1 and (${\sigma }_1$,${\sigma }_2$,${\sigma }_3$)=5.

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Figure 3:

Plots of chaotic attractors from system (5) with α=1 and (${\sigma }_1$,${\sigma }_2$,${\sigma }_3$)=10.

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Figure 4:

The displays a 2D graphic for system (5) with (${\sigma }_1$,${\sigma }_2$,${\sigma }_3$) various values.

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Figure 5:

Plots of chaotic attractors from system (4) with α=0.99.

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Figure 6:

Plots of chaotic attractors from system (5) with α=0.99 and (${\sigma }_1$,${\sigma }_2$,${\sigma }_3$)=5.

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Figure 7:

Plots of chaotic attractors from system (5) with α=0.99 and (${\sigma }_1$,${\sigma }_2$,${\sigma }_3$)=10.

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Figure 8:

The displays a 2D graphic for (${\sigma }_1$,${\sigma }_2$,${\sigma }_3$) various values.

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Figure 9:

Plots of chaotic attractors from system (4) with α=0.95.

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Figure 10:

Plots of chaotic attractors from system (5) with α=0.99 and (${\sigma }_1$,${\sigma }_2$,${\sigma }_3$)=5.

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Figure 11:

Plots of chaotic attractors from system (5) with α=0.99 and (${\sigma }_1$,${\sigma }_2$,${\sigma }_3$)=10.

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Figure 12:

It displays a 2D graphic for (σ1,σ2,σ3) various values.

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Tables 1-3 present the numerical solutions of the (SFLS) using a novel scheme for the ABC-FD under different noise conditions (σ1, σ2, σ3) = 0, 5, 10. The results show the state variables x, y, and z at different time points. In Table 1, without noise, the variables stabilize quickly. In Table 2, with moderate noise, there is noticeable variability, but the overall trends are maintained. Table 3, with higher noise, shows significant fluctuations, reflecting the system’s sensitivity to noise. These results demonstrate the novel scheme’s effectiveness in capturing the system’s dynamics under varying conditions, emphasizing the importance of considering noise in real-world applications.

CONCLUSION

The numerical solutions for the classical, fractional, and stochastic fractional-order Lorenz systems are critical for handling complex, non-linear dynamics and incorporating memory effects and random perturbations. These solutions help to open the understanding of the properties of chaotic behavior, synchronization, and long-term stability. They are mandatory for practical use, such as climate prediction, the control of technological processes, and business prediction. In this work, a general numerical method has been suggested and applied to the Lorenz system, its fractional-order version, and the SFLS. We apply the ABC-FD operator to solve these models and prove the method’s efficiency through the 3D and 2D graphical displays of the produced numerical solutions, where the multiplicative white noise effect is clear. The number of solutions illustrates this system’s effective transition between different noise values, which shows how the novel technique in this study will help capture the system’s noise response for applications in various fields. In future research, the Lorenz system with the generalized fractional derivatives or the stochastic counterpart to the system based on the added noise will be explored in more detail.