Exploring Chaos in Fractional Order Systems: A Study of Constant and Variable-Order Dynamics

Abstract

Fractional calculus generalizes well-known differentiation and integration to noninteger orders, allowing a more accurate framework for modeling complex dynamical behaviors. The application of fractional-order systems is quite wide in engineering, biology, and physics because they inherently capture the memory effects and long-range dependencies. Out of these, fractional jerk chaotic systems have gained attention regarding their applications in secure communication, signal processing, and control systems. This work develops a comparative analysis of a fractional jerk system that includes constant- and variable-order derivatives to contribute to chaos–stability analysis. Additionally, this study uncovers novel chaotic behaviors, further expanding our understanding of complex dynamical systems. The results yield new insights into using variable-order dynamics to enable chaotic systems to better adapt to real applications.

1. Introduction

Fractional calculus generalizes conventional integer-order differentiation and integration to noninteger orders. Recently, it has received much attention since it can model more complex dynamic behaviors than traditional models. As seen in [1,2,3], fractional-order systems provide a more flexible and realistic approach to modeling physical, biological, and engineering processes, especially those with memory effects and long-range dependence [4,5,6,7,8,9].

The most interesting property among the characteristics of fractional systems is chaotic behavior, which might be used in secure communication, encryption, signal processing, and modeling of biological systems [10,11,12]. Being a third-order dynamical system, the so-called jerk system represents one of the simplest paradigmatic examples in studying chaotic dynamics. In fractional order, it exhibits even richer properties, revealing new insights into nonlinear behavior, bifurcations, and synchronization phenomena.

Fractional-order systems extend classical models by incorporating memory effects and long-range dependencies, making them ideal for describing real-world phenomena in engineering, biology, and physics [13,14,15]. Unlike integer-order models, fractional systems provide greater flexibility and accuracy in capturing complex dynamics, particularly chaotic behavior [16].

The solution to these fractional systems is necessary for understanding their stability, bifurcations, and chaotic properties. Some of the fractional jerk chaotic systems, because of their enhanced dynamical properties, have found emerging applications in secure communication, signal processing, and control systems [17,18,19]. This article identifies their constant and variable-order behaviors, offering fresh perspectives on chaos and stability that enhance practical applications.

The novelty of the paper is the comparative analysis of fractional jerk chaotic systems with constant and variable-order derivatives, giving new insights into how dynamic changes in the fractional order affect chaos and stability and bifurcations. Unlike traditional approaches, this work explores the influence of variable-order dynamics, enhancing our understanding of how evolving memory effects shape chaotic behavior. The paper illustrates, in this direction, the potentials of variable-order systems, therefore extending the contribution of fractional systems in secure communications, encryption, signal processing, and control systems. It allows a new avenue toward modeling real-world complex problems with the presence of long memory effects numerically and via computational validation.

In this study, we simulate and investigate dynamical behavior in the fractional jerk system [20] in the following way:

$$\left\{\begin{array}{cc}\hfill {\mathcal{D}}_0^{\alpha }{x}_1\left(t\right)& =x_2,\hfill \\ \hfill {\mathcal{D}}_0^{\alpha }{x}_2\left(t\right)& =x_3,\hfill \\ \hfill {\mathcal{D}}_0^{\alpha }{x}_3\left(t\right)& =a{x}_1+b{x}_2^2-x_2-c{x}_3.\hfill \end{array}\right.$$

(1)

subject to the following constraints:

$$\begin{array}{cc}\hfill x_1\left(0\right)& =0.1,x_2\left(0\right)=0.1,x_3\left(0\right)=0.1.\hfill \end{array}$$

(2)

where $a=4.2$, $b=1$, and $c=2$.

The fractional jerk system is particularly powerful for the study of chaotic behavior, since fractional derivatives introduce additional degrees of freedom, which can impact system stability and bifurcation structures. It is applicable in control engineering, wherein synchronization of chaos is crucial in secure communication; mechanical and electrical systems, wherein nonlinear dynamics play a key role in oscillatory behavior; and biomedical signal processing, wherein memory effects are significant while modeling physiological processes. An analysis of the fractional jerk system provides deeper insights into the dynamics between memory and chaos; therefore, it is a valuable tool for applied and theoretical research.

Variable-order derivatives are better at modeling systems with disorders that change over time than constant-order derivatives because they are more flexible and can be tuned. Variable-order derivatives enable us to have an evolving memory effect as a system state or a function of an environmental condition, unlike constant-order derivatives, which assume a uniform memory effect throughout system evolution. With dynamic control of the prior-state effect, this kind of flexibility strengthens system stability, thereby inhibiting chaos. In applications like secure communication and signal processing, the capability enables systems to adjust performance adaptively in different situations and improve responsiveness and robustness. Apart from that, variable-order derivatives can describe real-world phenomena if the order of the nonlocality changes with time, hence being extremely useful in engineering and biological systems that demand large control of chaotic dynamics.

In this paper, we present comparative studies of the chaotic behavior that may arise both for the considered fractional-order variable and constant-order jerk system. Variable-order fractional derivatives represent one of the current developments of a fractal-type property: the order of differentiation might vary dynamically and be dependent either on time or on state in general. With a focus on understanding the changes due to variations that chaos goes through, one seeks in-depth comprehension of how fractional dynamics exert such influence on system stability and predictability.

2. Definitions

Definition 1

([21]). The Liouville–Caputo (LC) derivative with constant order α is given as follows:

$${}_{ }{}^{\mathit{LC}}D_t^{\alpha }Q\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}{\int }_0^t{\displaystyle \frac{1}{{(t-w)}^{\alpha }}}{\displaystyle \frac{d}{dw}}Q\left(w\right)dw,0<\alpha <1.$$

(3)

Definition 2

([22]). The Liouville–Caputo (LC) fractional derivative of variable-order $\alpha \left(t\right)$ is given as follows:

$${}_{\text{0\hspace{1em} }}{}^{\mathit{LCV}}D_0^{\alpha \left(t\right)}Q\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha (t\left)\right)}}{\int }_0^t{(t-w)}^{-\alpha \left(t\right)}{Q}^{\prime }\left(w\right)dw,0<\alpha \left(t\right)<1.$$

(4)

where $\mathsf{\Gamma }(·)$ denotes the Gamma function.

3. Algorithms for Fractional Systems with Constant vs. Variable Order

Numerical schemes are generally unavoidable to approximate solutions of nonlinear ordinary and partial differential equations for which no closed-form analytical solution is available. However, fractional differential equations (FDEs) are much more challenging in numerical analysis due to their noninteger order derivatives. The Euler approach, Adams–Bashforth method, Adomian decomposition method, homotopy analysis method, Laplace–Adomian decomposition method, and some polynomial interpolation methods have been used to solve FDEs [23,24,25,26,27]. These are considered in the literature.

In applications where either computational efficiency or stability is more important, simpler methods such as the Euler method and similar methods may be preferred, since their implementations are simpler and they are less computationally demanding. However, where precision is in question and, at the same time, computational resources are available, the more advanced techniques to provide better performance for the approximation of fractional derivatives will be the Lagrange two-step interpolation method [28,29]. These kinds of methods provide a satisfactory trade-off between precision and computational burden, and thus, they are valuable tools in analyzing fractional systems, especially in such domains where a high degree of precision is called for.

This section focuses on approximating solutions in [22,30,31] for models using Caputo fractional operators through efficient numerical techniques. The system is as follows:

$${}^{\ast }D_{0,t}^{\alpha }v\left(t\right)=Q\left(v\left(t\right)\right),t\in [0,a],v\left(0\right)=v_0,$$

(5)

where * denotes one of the fractional-order operators under consideration and $v_0$ is the initial state of the system using Definition 2. Consider an equispaced over $[0,a]$ with step size $\Delta t=0.05$. Let $v_r$ approximate $v\left(t\right)$ at $t=t_r$.

3.1. Constant Order

By using the numerical method in [31], Equation (5) can be reformulated as follows:

$$v\left(t\right)=v\left(0\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right)}}{\int }_0^t{(t-w)}^{Q-1}\vartheta (v,w)dw,$$

(6)

where $t=t_{n+1}$; Equation (6) is then:

$$v_{n+1}=v\left(t_{n+1}\right)=v\left(0\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right)}}{\int }_0^{t_{n+1}}{(t_{n+1}-w)}^{Q-1}\vartheta (v,w)dw,$$

(7)

where $t=t_n$; Equation (6) is then:

$$\begin{array}{cc}\hfill v_n& =v\left(t_n\right)=v\left(0\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right)}}{\int }_0^{t_n}{(t_n-w)}^{Q-1}\vartheta (v,w)dw,\hfill \end{array}$$

(8)

From Equations (7) and (8)

$$\begin{array}{cc}\hfill v\left(t_{n+1}\right)-v\left(t_n\right)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right)}}\left[{\int }_0^{t_{n+1}}{(t_{n+1}-w)}^{Q-1}\vartheta (v,w)dw-{\int }_0^{t_n}{(t_n-w)}^{Q-1}\vartheta (v,w)dw\right].\hfill \end{array}$$

(9)

The solution of Equation (5) is obtained using Lagrange polynomial interpolation as follows:

$$\begin{array}{cc}\hfill v_{n+1}& ={\displaystyle \frac{q{h}^Q}{\mathsf{\Gamma }(Q+2)}}\sum _{k=0}^r\vartheta (t_k,v_k)\left[{(n-k+1)}^Q(n-k+2+2Q)-{(n-k)}^Q(n-k+2+2Q)\right]\hfill \\ \hfill & -{\displaystyle \frac{hQ}{\mathsf{\Gamma }(Q+2)}}\sum _{k=0}^r\vartheta (t_{k-1},v_{k-1})\left[{(n-k+1)}^{Q+1}-{(n-k)}^Q(n-k+1+Q)\right].\hfill \end{array}$$

(10)

The 3D constant-order fractional system with Caputo derivative has solutions given by:

$$\begin{array}{cc}\hfill C_{{x_1}_{n+1}\left(t\right)}& ={\displaystyle \frac{q{h}^Q}{\mathsf{\Gamma }(Q+2)}}\sum _{k=0}^r\vartheta (t_k,{x_1}_k)\left[{(n-k+1)}^Q(n-k+2+2Q)-{(n-k)}^Q(n-k+2+2Q)\right]\hfill \\ \hfill & -{\displaystyle \frac{hQ}{\mathsf{\Gamma }(Q+2)}}\sum _{k=0}^r\vartheta (t_{k-1},{x_1}_{k-1})\left[{(n-k+1)}^{Q+1}-{(n-k)}^Q(n-k+1+Q)\right],\hfill \end{array}$$

(11)

and ${\vartheta }_1(t,x_1)=x_2$.

$$\begin{array}{cc}\hfill C_{{x_2}_{n+1}\left(t\right)}& ={\displaystyle \frac{q{h}^Q}{\mathsf{\Gamma }(Q+2)}}\sum _{k=0}^r\vartheta (t_k,{x_2}_k)\left[{(n-k+1)}^Q(n-k+2+2Q)-{(n-k)}^Q(n-k+2+2Q)\right]\hfill \\ \hfill & -{\displaystyle \frac{hQ}{\mathsf{\Gamma }(Q+2)}}\sum _{k=0}^r\vartheta (t_{k-1},{x_2}_{k-1})\left[{(n-k+1)}^{Q+1}-{(n-k)}^Q(n-k+1+Q)\right],\hfill \end{array}$$

(12)

and ${\vartheta }_2(t,x_2)=x_3$.

$$\begin{array}{cc}\hfill C_{{x_3}_{n+1}\left(t\right)}& ={\displaystyle \frac{q{h}^Q}{\mathsf{\Gamma }(Q+2)}}\sum _{k=0}^r\vartheta (t_k,{x_3}_k)\left[{(n-k+1)}^Q(n-k+2+2Q)-{(n-k)}^Q(n-k+2+2Q)\right]\hfill \\ \hfill & -{\displaystyle \frac{hQ}{\mathsf{\Gamma }(Q+2)}}\sum _{k=0}^r\vartheta (t_{k-1},{x_3}_{k-1})\left[{(n-k+1)}^{Q+1}-{(n-k)}^Q(n-k+1+Q)\right],\hfill \end{array}$$

(13)

and ${\vartheta }_3(t,x_3)=a{x}_1+b{x}_2^2-x_2-c{x}_3$.

3.2. Variable Order

By using the numerical method in [30], Equation (5) can be reformulated given by:

$$v\left(t\right)-v\left(0\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right(t\left)\right)}}{\int }_0^{t}f(w,v\left(w\right)){(t-w)}^{Q\left(t\right)-1}dw.$$

(14)

At $t=t_{n+1}$, Equation (14) is formulated as follows:

$$v\left(t_{n+1}\right)-v\left(0\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right(t\left)\right)}}\sum _{r=0}^n{\int }_{t_r}^{t_{r+1}}f(w,v\left(w\right)){(t_{n+1}-w)}^{Q\left(t\right)-1}dw.$$

(15)

Using the two-step Lagrange polynomial interpolation, we approximate the function $f(w,v(w\left)\right)$ over the interval $[t_k,t_{k+1}]$ as follows:

$$M_k\left(w\right)\simeq {\displaystyle \frac{f(t_{\xi },v_{\xi })}{h}}(w-t_{\xi -1})-{\displaystyle \frac{f(t_{\xi -1},v_{\xi -1})}{h}}(w-t_{\xi }).$$

(16)

Now, considering Equations (15) and (16), we have the following:

$$v_{n+1}\left(t\right)=v_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right(t\left)\right)}}\sum _{\xi =0}^n\left({\displaystyle \frac{f(t_{\xi },v_{\xi })}{h}}{\int }_{t_{\xi }}^{t_{\xi +1}}(t-t_{\xi -1}){(t_{n+1}-t)}^{Q\left(t\right)-1}dt\right.$$

$$\left.-{\displaystyle \frac{f(t_{\xi -1},v_{\xi -1})}{h}}{\int }_{t_{\xi }}^{t_{\xi +1}}(t-t_{\xi }){(t_{n+1}-t)}^{Q\left(t\right)-1}dt\right).$$

(17)

We now define the following expressions:

$$A_{Q\left(t\right),\xi ,1}=h^{Q\left(t\right)+1}{\displaystyle \frac{{(n+1-\xi )}^{Q\left(t\right)}(n-\xi +2+Q\left(t\right))-{(n-\xi )}^{Q\left(t\right)}(n-\xi +2+2Q\left(t\right))}{Q\left(t\right)\mathsf{\Gamma }\left(Q\right(t)+1)}},$$

(18)

$$A_{Q\left(t\right),\xi ,2}=h^{Q\left(t\right)+1}{\displaystyle \frac{{(n+1-\xi )}^{Q\left(t\right)+1}-{(n-\xi )}^{Q\left(t\right)}(n-\xi +1+Q\left(t\right))}{Q\left(t\right)\mathsf{\Gamma }\left(Q\right(t)+1)}}.$$

(19)

From Equation (17) and (18):

$$v_{n+1}\left(t\right)=v\left(0\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right(t\left)\right)}}\sum _{\xi =0}^n\left({\displaystyle \frac{h^{Q\left(t\right)}f(t_{\xi },v_{\xi })}{Q\left(t\right)\mathsf{\Gamma }\left(Q\right(t)+1)}}{(n+1-\xi )}^{Q\left(t\right)}\times (n-\xi +2+Q\left(t\right))\right.$$

$$-{(n-\xi )}^{Q\left(t\right)}(n-\xi +2+2Q\left(t\right))-{\displaystyle \frac{h^{Q\left(t\right)}f(t_{\xi -1},v_{\xi -1})}{Q\left(t\right)\mathsf{\Gamma }\left(Q\right(t)+1)}}{(n+1-\xi )}^{Q\left(t\right)+1}$$

$$\left.-{(n-\xi )}^{Q\left(t\right)}(n-\xi +1+Q\left(t\right))\right).$$

(20)

Thus, the solutions for the variable-order system with a Caputo derivative are given by:

$${vC}_{{x_1}_{n+1}\left(t\right)}={x_1}_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right(t\left)\right)}}\sum _{\xi =0}^n{\displaystyle \frac{h^{Q\left(t\right)}{\vartheta }_1(t_{\xi },{x_1}_{\xi },{x_2}_{\xi },{x_3}_{\xi })}{Q\left(t\right)\mathsf{\Gamma }\left(Q\right(t)+1)}}\times \left({(n+1-\xi )}^{Q\left(t\right)}\right.$$

$$\left.-{(n-\xi )}^{Q\left(t\right)}(n-\xi +2+2Q\left(t\right))\right),$$

(21)

and ${\vartheta }_1(t,x_1)=x_2$.

$${vC}_{{x_2}_{n+1}\left(t\right)}={x_2}_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right(t\left)\right)}}\sum _{\xi =0}^n{\displaystyle \frac{h^{Q\left(t\right)}{\vartheta }_1(t_{\xi },{x_1}_{\xi },{x_2}_{\xi },{x_3}_{\xi })}{Q\left(t\right)\mathsf{\Gamma }\left(Q\right(t)+1)}}\times \left({(n+1-\xi )}^{Q\left(t\right)}\right.$$

$$\left.-{(n-\xi )}^{Q\left(t\right)}(n-\xi +2+2Q\left(t\right))\right),$$

(22)

and ${\vartheta }_2(t,x_2)=x_3$.

$${vC}_{{x_3}_{n+1}\left(t\right)}={x_3}_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(Q\right(t\left)\right)}}\sum _{\xi =0}^n{\displaystyle \frac{h^{Q\left(t\right)}{\vartheta }_1(t_{\xi },{x_1}_{\xi },{x_2}_{\xi },{x_3}_{\xi })}{Q\left(t\right)\mathsf{\Gamma }\left(Q\right(t)+1)}}\times \left({(n+1-\xi )}^{Q\left(t\right)}\right.$$

$$\left.-{(n-\xi )}^{Q\left(t\right)}(n-\xi +2+2Q\left(t\right))\right),$$

(23)

and ${\vartheta }_3(t,x_3)=a{x}_1+b{x}_2^2-x_2-c{x}_3$.

4. Numerical Simulation

This section explains how system dynamics change with a difference of $\alpha$. The plots in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 are the dynamics of a dynamical system for various parameter settings and functional forms of $\alpha$. The phase space plots (first rows) provide the trajectories for different planes ($x_1$-$x_2$, $x_2$-$x_3$, $x_1$-$x_3$), and the time series plots (bottom rows) show the time evolution of the system variables ($x_1$, $x_2$, $x_3$). Time-varying $\alpha$ results in a great deal of chaos and complexity, as seen in Figure 1, Figure 5, and Figure 9 with $\alpha =0.95-0.05·cos(t/5)$, $\alpha =1/(1+e^{-t})$, and $\alpha =tan(t+1)$, respectively. These functional forms create chaotic oscillations and random trajectories, with chaotic behavior dominating for tangent modulation. On the other hand, sustained $\alpha$ (e.g., $\alpha =0.98$ in Figure 2, $\alpha =0.95$ in Figure 6, and $\alpha =0.99$ in Figure 10) results in periodic and stable dynamics, characterized by smooth closed orbits in phase space and recurrent oscillations in the time series. Logistic growth $\alpha$ Figure 5 displays convergence to stability from chaotic dynamics that $\alpha$ asymptotically approaches a constant.

Figure 13, Figure 14 and Figure 15 show the dynamics for two values of the parameter $\alpha$: a periodic time modulation ($\alpha =0.96-0.03·cos(t/5)$) and a constant value ($\alpha =0.96$). The periodic modulation of time $\alpha$ in Figure 13 and Figure 15 generates chaotic dynamics, as confirmed by aperiodic trajectories in phase space with intersecting tracks and oscillating time series with variable amplitude and frequency. The quasiperiodicity of the cosine term brings in complexity, resulting in aperiodic dynamics. The periodic and stable dynamics are shown in Figure 14 and Figure 16, with $\alpha =0.96$ fixed. The phase space orbits are smooth, closed loops that exhibit a stable limit cycle, and the time series plots exhibit regular oscillations with constant frequencies and amplitudes. This analogy captures the sensitivity of the system to the functional form $\alpha$, in which time-dependent modulation induces chaos and a constant value stabilizes the system, with implications for control in dynamical systems.

Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 exhibit novel chaotic and dynamic patterns in a time-varying parameter-dependent system, $sin(t/7)$, ${\displaystyle \frac{1}{1+exp(-2+t/8)}}$, and $tanh(t+1)$. Parameterizations add nonlinearity and time modulation and lead to complex phase portraits and oscillatory time-series patterns. The phase-space chaotic attractors have complex shapes and densities, which represent sensitivity to the initial conditions and parameter variations. The time series plots also determine transitions from chaotic to periodic behavior, with the oscillations increasing or stabilizing with time. This task is particularly novel, as it combines nonlinear dynamics with time-varying parameters and offers insight into complex systems like biological rhythms or climate models. The dynamics that arise between periodicity and chaos emphasize the system’s rich and unstable dynamics.

The results suggest that time-dependent fractional orders ($\alpha$) introduce chaos and complexity, while the constant $\alpha$ stabilizes with a periodic output. Chaotic behavior of modulated $\alpha$ creates unpredictable, aperiodic paths, whereas constant $\alpha$ yield smooth, closed orbits. Periodic-chaotic transitions are a demonstration of the system’s sensitivity to fractional parameters, and they indicate its application in simulating biological rhythms, climate phenomena, and secure communication. Chaotic behavior has to be understood to develop complex systems in the real world because chaos can be engineered to create engineering, cryptography, and nonlinear control systems.

The time evolution of Lyapunov exponents (LE1, LE2, LE3) of variable and constant fractional orders $\alpha$ is presented in Figure 23, Figure 24 and Figure 25. Variable fractional orders, in all three cases, exhibit more intense transient behaviors compared to their constant counterparts because they are time-varying. In Figure 23, $0.95-0.05·cos(t/5)$ gives periodic oscillations, with LE1 settling at close to $+1$, LE2 at 0, and LE3 at approximately $-4$. Similarly, in Figure 24, with $1/(1+exp(-2+t/8\left)\right)$, LE1 settles just above $+1$, with continuing chaos, while LE2 and LE3 settle at 0 and $-4$, respectively. Figure 25, with $\alpha =tanh(t+1)$, illustrates the same stabilization, as LE1 is chaotic, LE2 has quasiperiodic behavior, and LE3 shows contraction in some dimensions of the attractor. In all the figures, the fractional orders that are fixed stabilize faster than variable ones, but the asymptotic behavior is the same. LE1 is positive in both cases and indicates chaotic dynamics, LE2 approaches 0, and LE3 approaches a negative value, indicating state-space contraction. Variable orders (oscillatory cosine, exponential sigmoid, and hyperbolic tangent) affect the transient behavior but ultimately approach behavior similar to the constant fractional order systems. These results demonstrate the equivalence of fractional-order variability with system stability and introduce knowledge on how chaos or stability can be regulated in dynamical systems.

5. Numerical Solutions

This section presents a comparative study of a system using constant and variable fractional orders. This investigation analyzes the system response under varying definitions $\alpha \left(t\right)$ to consider the impact of memory effects on the temporal development of variables. This study explores the interaction between constant-order (C) and variable-order (V) elements using numerical simulations, underscoring the significance of time-dependent fractional dynamics in characterizing the behavior, stability, and complexity of systems.

Table 1. Comparison of C and V for $\alpha =1$ and $\alpha =0.95-0.05·cos(t/5)$.

t	${\mathit{C}}_{{\mathit{x}}_1}$	${\mathit{V}}_{{\mathit{x}}_1}$	${\mathit{C}}_{{\mathit{x}}_2}$	${\mathit{V}}_{{\mathit{x}}_2}$	${\mathit{C}}_{{\mathit{x}}_3}$	${\mathit{V}}_{{\mathit{x}}_3}$
0.1	1.0932 × ${10}^{-1}$	1.1245 × ${10}^{-1}$	1.0626 × ${10}^{-1}$	1.0671 × ${10}^{-1}$	3.9601 × ${10}^{-2}$	2.2078 × ${10}^{-2}$
0.2	1.2005 × ${10}^{-1}$	1.2457 × ${10}^{-1}$	1.0718 × ${10}^{-1}$	1.0426 × ${10}^{-1}$	−1.9971 × ${10}^{-2}$	−4.1661 × ${10}^{-2}$
0.3	1.3057 × ${10}^{-1}$	1.3525 × ${10}^{-1}$	1.0249 × ${10}^{-1}$	9.5204 × ${10}^{-2}$	−7.2662 × ${10}^{-2}$	−9.3524 × ${10}^{-2}$
0.4	1.4038 × ${10}^{-1}$	1.4410 × ${10}^{-1}$	9.2847 × ${10}^{-2}$	8.1158 × ${10}^{-2}$	−1.1916 × ${10}^{-1}$	−1.3645 × ${10}^{-1}$
0.5	1.4900 × ${10}^{-1}$	1.5081 × ${10}^{-1}$	7.8847 × ${10}^{-2}$	6.3263 × ${10}^{-2}$	−1.5986 × ${10}^{-1}$	−1.7175 × ${10}^{-1}$
0.6	1.5603 × ${10}^{-1}$	1.5512 × ${10}^{-1}$	6.1058 × ${10}^{-2}$	4.2439 × ${10}^{-2}$	−1.9493 × ${10}^{-1}$	−2.0006 × ${10}^{-1}$
0.7	1.6111 × ${10}^{-1}$	1.5688 × ${10}^{-1}$	4.0043 × ${10}^{-2}$	1.9499 × ${10}^{-2}$	−2.2439 × ${10}^{-1}$	−2.2170 × ${10}^{-1}$

and

Table 2. Comparison of C and V for $\alpha =0.95$, $\alpha =1/(1+exp(-t\left)\right)$.

t	${\mathit{C}}_{{\mathit{x}}_1}$	${\mathit{V}}_{{\mathit{x}}_1}$	${\mathit{C}}_{{\mathit{x}}_2}$	${\mathit{V}}_{{\mathit{x}}_2}$	${\mathit{C}}_{{\mathit{x}}_3}$	${\mathit{V}}_{{\mathit{x}}_3}$
1	1.5461 × ${10}^{-1}$	1.0528 × ${10}^{-1}$	−4.6564 × ${10}^{-2}$	−4.0319 × ${10}^{-2}$	−2.6415 × ${10}^{-1}$	−1.4560 × ${10}^{-1}$
2	−9.6185 × ${10}^{-3}$	3.4253 × ${10}^{-2}$	−2.2333 × ${10}^{-1}$	−8.5266 × ${10}^{-2}$	−3.7468 × ${10}^{-2}$	−5.7021 × ${10}^{-2}$
3	−1.6461 × ${10}^{-1}$	−2.5476 × ${10}^{-2}$	−5.3530 × ${10}^{-2}$	−5.0005 × ${10}^{-2}$	2.8885 × ${10}^{-1}$	4.7622 × ${10}^{-2}$
4	−5.1768 × ${10}^{-2}$	−2.3635 × ${10}^{-2}$	2.1918 × ${10}^{-1}$	3.1486 × ${10}^{-2}$	1.8046 × ${10}^{-1}$	6.3339 × ${10}^{-2}$
5	1.7941 × ${10}^{-1}$	3.5612 × ${10}^{-2}$	1.6917 × ${10}^{-1}$	5.8622 × ${10}^{-2}$	−2.4392 × ${10}^{-1}$	−3.6333 × ${10}^{-2}$
6	1.5734 × ${10}^{-1}$	6.3555 × ${10}^{-2}$	−1.8629 × ${10}^{-1}$	−2.2307 × ${10}^{-2}$	−3.3700 × ${10}^{-1}$	−1.1683 × ${10}^{-1}$
7	−2.6922 × ${10}^{-1}$	−4.1635 × ${10}^{-3}$	−2.6922 × ${10}^{-1}$	−1.0130 × ${10}^{-1}$	1.5979 × ${10}^{-1}$	−2.0649 × ${10}^{-2}$

show how a system with two different values of $\alpha$ changes over time. They show how the constant-order components (C) and variable-order components (V) interact with three variables ($x_1,x_2,\mathrm{and}{x}_3$). In the first example ($\alpha =1,\alpha =0.95-0.05·cos(t/5)$), the system creates smooth periodic modulations, and the components C and V evolve gradually, for instance, the monotonous increase of $C_{x_1}$ and $V_{x_1}$ and the reduction of $C_{x_3}$ and $V_{x_3}$. In contrast, the second example ($\alpha =0.95,\alpha ={\displaystyle \frac{1}{1+exp(-t)}}$) is characterized by more intricate nonlinear dynamics, with an aperiodic oscillation of V, especially for $x_1$ and $x_3$, characteristic of chaotic behavior. Constant-order components (C) show smoother dynamics, while variable-order components (V) show transient and chaotic dynamics.

The comparison between C and V for $\alpha =0.99$ and $\alpha =tanh(t+1)$ is presented in

Table 3. Comparison of C and V for $\alpha =0.99$, $\alpha =tanh(t+1)$.

t	${\mathit{C}}_{{\mathit{x}}_1}$	${\mathit{V}}_{{\mathit{x}}_1}$	${\mathit{C}}_{{\mathit{x}}_2}$	${\mathit{V}}_{{\mathit{x}}_2}$	${\mathit{C}}_{{\mathit{x}}_3}$	${\mathit{V}}_{{\mathit{x}}_3}$
1	1.6063 × ${10}^{-1}$	1.4367 × ${10}^{-1}$	−3.8731 × ${10}^{-2}$	−4.5189 × ${10}^{-2}$	−2.7526 × ${10}^{-1}$	−2.3366 × ${10}^{-1}$
2	−3.7173 × ${10}^{-3}$	−8.3611 × ${10}^{-4}$	−2.4818 × ${10}^{-1}$	−2.0085 × ${10}^{-1}$	-5.7797 × ${10}^{-2}$	−4.3764 × ${10}^{-2}$
3	−2.0052 × ${10}^{-1}$	−1.5927 × ${10}^{-1}$	−8.0047 × ${10}^{-2}$	−6.0809 × ${10}^{-2}$	3.4659 × ${10}^{-1}$	2.7873 × ${10}^{-1}$
4	−8.9017 × ${10}^{-2}$	−6.7934 × ${10}^{-2}$	2.8041 × ${10}^{-1}$	2.3353 × ${10}^{-1}$	2.7171 × ${10}^{-1}$	2.1559 × ${10}^{-1}$
5	2.4279 × ${10}^{-1}$	2.4279 × ${10}^{-1}$	2.8496 × ${10}^{-1}$	2.8496 × ${10}^{-1}$	−2.9542 × ${10}^{-1}$	−2.9542 × ${10}^{-1}$
6	2.8804 × ${10}^{-1}$	2.4213 × ${10}^{-1}$	−2.3374 × ${10}^{-1}$	−2.0177 × ${10}^{-1}$	−5.8337 × ${10}^{-1}$	−4.9237 × ${10}^{-1}$
7	−1.4437 × ${10}^{-1}$	−1.2749 × ${10}^{-1}$	−4.9013 × ${10}^{-1}$	−4.2551 × ${10}^{-1}$	1.8122 × ${10}^{-1}$	1.5147 × ${10}^{-1}$

. The values show a high degree of agreement in all state variables $x_1,x_2,$ and $x_3$. Although slight differences appear at certain time steps, the overall consistency supports the validity of the variable-order model.

The system is sensitive to the time-dependent fractional order $\alpha \left(t\right)$, with sinusoidal fluctuations causing periodic behavior and sigmoid functions causing chaotic dynamics. Variable-order derivatives are implemented using a discretized Caputo technique, updating $\alpha \left(t\right)$ at each step, and using quadrature techniques and adaptive time-stepping ensures precision, even when $\alpha \left(t\right)$ varies quickly.

6. Conclusions

This paper extends fractional calculus through the systematic comparison of constant- and variable-order derivatives in jerk systems, exhibiting their unique capacity to model real-world phenomena with memory effects and nonlocal interactions. Through numerical simulations, we unveil previously unknown chaotic regimes, which appear uniquely in variable-order systems with dynamic multistability transitions. Our results bring us to the conclusion that a changing order produces chaotic and irregular dynamics, with unpredictable trajectories and nonperiodic oscillations. The steady and periodic behavior is the result of constant order. How the system produces constant and smooth oscillations and closed orbits is characteristic of regular and predictable dynamics. The relevance of fractional dynamics in the description and management of intricate events is brought to the fore by this. While variable order leads to more severe transients, it finally settles down to behave like constant orders in the long term. LE1 has a positive value, showing chaos, but LE2 and LE3 show neutral and contracting dynamics, respectively. This shows how the variable order has an effect on the final system stability, but does not change it. In future work, we plan to explore new fractional models, as discussed in [32,33,34], and comparative analysis with other numerical methods in [35,36,37].