Perfect Projective Synchronization of a Class of Fractional-Order Chaotic Systems Through Stabilization near the Origin via Fractional-Order Backstepping Control

Abstract

This study introduces a novel control strategy aimed at achieving projective synchronization in incommensurate fractional-order chaotic systems (IFOCS). The approach integrates the mathematical framework of fractional calculus with the recursive structure of the backstepping control technique. A key feature of the proposed method is the systematic use of the Mittag–Leffler function to verify stability at every step of the control design. By carefully constructing the error dynamics and proving their asymptotic convergence, the method guarantees the overall stability of the coupled system. In particular, stabilization of the error signals around the origin ensures perfect projective synchronization between the master and slave systems, even when these systems exhibit fundamentally different fractional-order chaotic behaviors. To illustrate the applicability of the method, the proposed fractional order backstepping control (FOBC) is implemented for the synchronization of two representative systems: the fractional-order Van der Pol oscillator and the fractional-order Rayleigh oscillator. These examples were deliberately chosen due to their structural differences, highlighting the robustness and versatility of the proposed approach. Extensive simulations are carried out under diverse initial conditions, confirming that the synchronization errors converge rapidly and remain stable in the presence of parameter variations and external disturbances. The results clearly demonstrate that the proposed FOBC strategy not only ensures precise synchronization but also provides resilience against uncertainties that typically challenge nonlinear chaotic systems. Overall, the work validates the effectiveness of FOBC as a powerful tool for managing complex dynamical behaviors in chaotic systems, opening the way for broader applications in engineering and science.

1. Introduction

Chaotic systems have long fascinated researchers across disciplines due to their complex dynamics, extreme sensitivity to initial conditions, and nonlinear behavior. Despite their inherent unpredictability, chaotic systems are not merely mathematical curiosities—they arise naturally in various real-world applications, ranging from electrical circuits, biological processes, and secure communication systems to weather modeling and robotics. As a result, understanding and controlling chaos has become a vital topic in nonlinear system theory, particularly when synchronization is desired between chaotic systems, either for analysis, comparison, or real-time control.

The synchronization of chaotic systems has evolved significantly since the pioneering works of Pecora and Carroll in the early 1990s, who introduced the concept of synchronizing two identical chaotic systems using a unidirectional coupling scheme [1]. Since then, a wide array of synchronization types have emerged, including complete, lag, generalized, and projective synchronization. Among these, projective synchronization offers a compelling framework wherein the state variables of the response (slave) system synchronize with those of the drive (master) system up to a constant scaling factor [2]. This synchronization mode is especially relevant in secure communications [3,4,5] and control applications [6,7,8] where signal scaling is advantageous or even necessary.

While much of the early work in chaotic synchronization focused on systems of integer order, the rise of fractional calculus has opened new frontiers. Fractional-order systems—where derivatives are defined in non-integer orders—offer a richer and more flexible framework for modeling memory-dependent and hereditary phenomena, frequently encountered in viscoelastic materials [9], electrical impedance [10], and anomalous diffusion [11]. In particular, the IFOCS systems, where each system component may have a distinct fractional order, present a more realistic and generalized model for complex dynamical systems [12,13,14].

Despite their modeling potential, controlling and synchronizing such IFOCS remains a challenging task. These may stem from unknown parameters, environmental influences, or internal structural variations. To ensure robust and reliable synchronization in such contexts, it is essential to develop control strategies that can handle the nonlinear, memory-laden nature of fractional dynamics while also compensating for uncertainties and perturbations [15,16,17].

Several methodologies have been proposed in the literature to address synchronization of fractional-order chaotic systems, including adaptive control [18], sliding mode control [19,20], fuzzy logic-based controllers [21], and observer-based techniques [22]. However, most of these approaches either assume commensurate fractional orders or require detailed knowledge of system parameters—conditions that are often difficult to satisfy in real applications. Moreover, relatively few studies have specifically addressed projective synchronization in the context of incommensurate systems.

One promising method that has gained attention in nonlinear control is the backstepping control technique. Originally developed for strict-feedback systems, backstepping allows systematic design of stabilizing controllers through recursive procedures [23,24]. When extended to the fractional-order domain—commonly referred to as FOBC, this method provides a powerful tool for stabilizing complex nonlinear systems, especially when combined with appropriate fractional-order stability criteria [25,26,27].

In the present study, we propose a novel FOBC based approach to achieve projective synchronization between two fundamentally different IFOCS. The key innovation lies in the integration of the Mittag-Leffler function—a generalization of the exponential function commonly used in fractional dynamics—to analyze and ensure the stability of the synchronization error dynamics. Unlike traditional Lyapunov methods, this approach is better suited to the non-local and history-dependent nature of fractional-order systems.

To validate the proposed control scheme, we implement it on a synchronization problem involving two classical yet distinct systems: the fractional-order Van-der Pol oscillator and the Rayleigh oscillator. These systems are known for their rich nonlinear dynamics and are often used as benchmarks in chaos studies. Our simulation results demonstrate that the FOBC strategy achieves accurate and robust synchronization of the systems’ error dynamics, even under varying initial conditions and perturbations.

The contributions of this paper can be summarized as follows:

• A novel FOBC framework is developed for perfectly projective synchronization of IFOCS.
• Mittag-Leffler stability theory is effectively employed to rigorously prove global asymptotic stability of the synchronization errors.
• A practical synchronization example is provided using the Van-der Pol and Rayleigh systems, showcasing the method’s robustness and applicability.
• Simulation studies validate the effectiveness and reliability of the proposed control strategy across different initial conditions and non-identical fractional-order derivatives that are not close to 1.

Although numerous studies have investigated the control of fractional-order chaotic systems, only a limited number have addressed projective synchronization in the presence of incommensurate dynamics within a single, coherent framework. The present work aims to fill this gap by proposing a robust solution that combines theoretical soundness with practical applicability.

The paper is organized as follows. Section 2 provides a concise introduction to fractional-order operators and recalls essential stability tools, namely Lyapunov theory and Mittag-Leffler functions, which are instrumental in analyzing fractional-order chaotic systems. Section 3 introduces the master–slave configuration under consideration, explaining both the system modeling and the design of the FOBC. A rigorous stability analysis is also provided to prove the effectiveness of the proposed control strategy. Section 4 illustrates the approach through numerical simulations and detailed performance evaluations. Finally, Section 5 concludes with a summary of the main contributions, practical insights, and potential directions for future research.

2. Overview of Fractional-Order Operators

Although fractional-order calculus has been known in mathematics for centuries, its relevance to practical engineering and industrial applications has only recently gained significant attention. By extending differentiation and integration to non-integer orders, fractional calculus provides a powerful generalization of classical calculus, offering richer modeling and analysis tools.

2.1. Fractional-Order Derivative

In fractional calculus, the fundamental operator is denoted as ${}_{b}D_t^{\alpha }$, and is defined as

$${}_{0}D_t^{\alpha }z\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{d^{\alpha z\left(t\right)}}{d{t}^{\alpha }}}\hfill & \mathrm{if}\Re \left(\alpha \right)>0\hfill \\ {\int }_b^t{\left(z\left(t\right)d\tau \right)}^{\alpha }\hfill & \mathrm{if}\Re \left(\alpha \right)<0\hfill \end{array}\right.$$

(1)

Here, b and t specify the limits of the operation, while $\alpha$ represents the fractional order, which can take any real value. Several definitions of this operator have been developed, with the most common being those of Riemann–Liouville, Caputo and Grünwald–Letnikov [28].

In this study, we adopt the Caputo definition of the fractional derivative, which is expressed as

Caputo Definition:

$${}_{0}D_t^{\alpha }z\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_0^t{\displaystyle \frac{z\left(\tau \right)}{{(t-\tau )}^{\alpha +1-n}}}d\tau ,\alpha \in (n-1,n]$$

(2)

where t $\in {\mathbb{R}}^{+}$, $n\in \mathbb{N}$, $\alpha \in {\mathbb{R}}^{+}\setminus 0$, $z\left(t\right)$ is a causal function, and $\Gamma (·)$ denotes the Gamma function:

$$\Gamma \left(\tau \right)={\int }_0^{\infty }{t}^{\tau -1}{e}^{-t}dt$$

(3)

For simplicity, in the remainder of this work the fractional-order operator will be written as $D^{\alpha }$.

2.2. Stability Analysis of IFOCS

Stability is a central issue in the study of dynamical systems, especially for chaotic systems where small disturbances can cause large deviations in behavior. Among the different techniques available, the Lyapunov method remains a cornerstone in both integer- and fractional-order system analysis. Additionally, in the fractional-order context, the Mittag-Leffler stability criterion is particularly important, as it accounts for the memory and hereditary properties inherent to these systems.

2.2.1. Stability in Lyapunov Creterion

Consider a nonlinear fractional-order system of the form:

$$D^{\alpha }z\left(t\right)=\psi (z,t)$$

(4)

where $\psi \left(t\right)$ is a smooth nonlinear function, $z\left(t\right)$ is the state vector, and t denotes time.

The Lyapunov approach analyzes stability by constructing a scalar Lyapunov function $V\left(z\right)$, which decreases along system trajectories. If $z=0$ is an equilibrium of the system, and for $\alpha =1$ the system reduces to its integer-order form, stability is established as follows: if $V\left(z\right)$ is positive definite $V\left(0\right)=0$, $V\left(z\right)>0$ for $z\ne 0$ and its derivative satisfies $\dot{V}\left(z\right)\le 0$, then the equilibrium is stable. If instead $\dot{V}\left(z\right)<0$, the equilibrium is asymptotically stable. In this case, $V\left(z\right)$ is referred to as a Lyapunov candidate function [29].

2.2.2. Stability in Mittag-Leffler Sense

For fractional-order systems, the Mittag-Leffler criterion provides an alternative and often more suitable approach.

Lemma 1 

([30]). For a real, continuous, and differentiable function $z\left(t\right)$, with $q=2^n;n\in \mathbb{N}$ and $0<\alpha <1$, the following inequality holds:

$$D^{\alpha }{z}^q\left(t\right)\le p{z}^{q-1}\left(t\right)D^{\alpha }z\left(t\right)$$

(5)

In particular, for $q=2$:

$${\displaystyle \frac{1}{2}}{D}^{\alpha }{z}^2\left(t\right)\le z\left(t\right)D^{\alpha }z\left(t\right)$$

(6)

Theorem 1 

([30]). For $z\left(t\right)\in \mathbb{R}$ and $0<\alpha <1$, assuming $z(0,0)=0$, the system in Equation (6) is stable under the control input $e\left(t\right)=d\left(z\right)$ if:

$$z^{q-1}\left(t\right)D^{\alpha }z\left(t\right)=z^{q-1}\left(t\right)z(z,d\left(z\right))\le 0$$

(7)

For $q=2^n$, if $z^{q-1}\left(t\right)z(z,d\left(z\right))$$<0$, then the system is asymptotically stable in the sense of Mittag-Leffler.

3. System Summary and Problem Definition

The second type of stability involves the synchronization of multiple systems, where their behaviors must align when operating steadily. For this alignment to happen, the behavior of one system needs to be controlled to match that of another. In the next sections, this synchronization is achieved through the design of the FOBC. Here, synchronization is illustrated within a master–slave system setup. Let us look at the master system, which resembles system (10) but without a specified control mechanism as follows:

$$\left\{\begin{array}{c}{D}^{{\kappa }_1}{x}_1\left(t\right)=h_1{x}_2+f_1\left(X_1,t\right)+{\beta }^T{F}_1(X_1,t)\\ D^{{\kappa }_2}{x}_2\left(t\right)=h_2{x}_3+f_2\left(X_2,t\right)+{\beta }^T{F}_2(X_2,t)\\ ⋮\\ D^{{\kappa }_i}{x}_i\left(t\right)=h_i{x}_{i+1}+f_i\left(X_i,t\right)+{\beta }^T{F}_i(X_i,t)\\ ⋮\\ D^{{\kappa }_n}{x}_n\left(t\right)=h_n+f_n\left(X_n,t\right)+{\beta }^T{F}_n(X_n,t)\end{array}\right.$$

(8)

Meanwhile, the slave system (11), which is similar to system (10), incorporates control:

$$\left\{\begin{array}{c}{D}^{{\kappa }_1}{y}_i\left(t\right)=h_1{y}_2+{\tilde{f}}_1\left(Y_1,t\right)+{\tilde{\beta }}^T{\tilde{F}}_1(Y_1,t)\\ D^{{\kappa }_2}{y}_i\left(t\right)=h_2{y}_3+{\tilde{f}}_2\left(Y_2,t\right)+{\tilde{\beta }}^T{\tilde{F}}_2(Y_2,t)\\ ⋮\\ D^{{\kappa }_i}{y}_i\left(t\right)=h_i{y}_{i+1}+{\tilde{f}}_i\left(Y_i,t\right)+{\tilde{\beta }}^T{\tilde{F}}_i(Y_i,t)\\ ⋮\\ D^{{\kappa }_n}{y}_n\left(t\right)=h_n+{\tilde{f}}_n\left(Y_n,t\right)+{\tilde{\beta }}^T{\tilde{F}}_n(Y_n,t)+u\left(t\right)\end{array}\right.$$

(9)

In this context, $x_i$ and $y_i$ represent the states for the master and slave systems, respectively, while $Y_i=[y_1,y_2,\dots ,y_i]$. The control signal designed for synthesis is denoted as $u\left(t\right)$, and the vectors $\beta \in R^r$ and $\tilde{\beta }\in R^r$ contain known constant parameters related to the modeling of the two systems. The functions $f_i$, $F_i$, ${\tilde{f}}_i$, and ${\tilde{F}}_i$ (where $i=0,\dots ,n$) are recognized smooth nonlinear functions, while $h_i$ signifies the constants within the system structure, ensuring $h_i\ne 0$, and ${\kappa }_i$ indicates the fractional orders of the subsystems with values constrained as $0<{\kappa }_i<1$.

Theorem 2. 

The control strategy that ensures the master system (10) and the slave system (11) synchronize asymptotically, which means that $e(\infty )={lim}_{t\to \infty }(x\left(t\right)-y\left(t\right))=0$, can be expressed as

$$\begin{array}{c}u\left(t\right)=c_n{z}_n+f_n(X_n,t)-{\tilde{f}}_n(Y_n,t)-\\ D^{{\kappa }_n}{v}_{n-1}+h_{n-1}{z}_{n-1}+{\beta }^T{F}_n(X_n,t)-{\tilde{\beta }}^T{\tilde{F}}_n(Y_n,t)\end{array}$$

(10)

In this context, $v_j$ ($j=1,\dots ,n-1$) symbolizes the designed virtual controls. The constants $c_i$ ($i=1,\dots ,n$) are parameters that have been predetermined by the designer.

The error terms are represented as $e_i=x_i-y_i$, where a change of variables occurs such that $z_1=e_1$ and for $j=2$ to n, $z_j=e_j-v_{j-1}$.

Proof of Theorem 2. 

Consider the error states, denoted as $e_i$, which represent the difference between the states of the master system (10) and those of the slave system (11). They are defined as

$$\begin{array}{c}{e}_i=x_i-y_i;i=1,\dots ,n.\end{array}$$

(11)

Then, we have

$$\begin{array}{c}{D}^{{\kappa }_n}{e}_i=D^{{\kappa }_n}{x}_i-D^{{\kappa }_n}{y}_i,i=1,\dots ,n.\end{array}$$

(12)

For the range $i=1$ to $n-1$, Equation (14) can be transformed into the following form:

$$\begin{array}{c}\left\{\begin{array}{c}{D}^{{\kappa }_1}{e}_1\left(t\right)=h_1(x_2-y_2)+f_1(X_1,t)-{\tilde{f}}_1(Y_1,t)+{\beta }^T{F}_1(X_1,t)-{\tilde{\beta }}^T{\tilde{F}}_1(Y_1,t)\hfill \\ D^{{\kappa }_2}{e}_2\left(t\right)=h_i(x_3-y_3)+f_2(X_2,t)-{\tilde{f}}_2(Y_2,t)+{\beta }^T{F}_2(X_2,t)-{\tilde{\beta }}^T{\tilde{F}}_2(Y_2,t)\hfill \\ ⋮\hfill \\ D^{{\kappa }_i}{e}_i\left(t\right)=h_i(x_{i+1}-y_{i+1})+f_i(X_i,t)-{\tilde{f}}_i(Y_i,t)+{\beta }^T{F}_i(X_i,t)-{\tilde{\beta }}^T{\tilde{F}}_i(Y_i,t)\hfill \\ ⋮\hfill \\ D^{{\kappa }_n}{e}_n\left(t\right)=f_n(X_n,t)-{\tilde{f}}_n(Y_n,t)+{\beta }^T{F}_n(X_n,t)-{\tilde{\beta }}^T{\tilde{F}}_n(Y_n,t)-u\left(t\right)\hfill \end{array}\right.\end{array}$$

(13)

Thus, we can express it as

$$\begin{array}{c}\left\{\begin{array}{c}{D}^{{\kappa }_1}{e}_1\left(t\right)=h_1{e}_2+f_1(X_1,t)-{\tilde{f}}_1(Y_1,t)+{\beta }^T{F}_1(X_1,t)-{\tilde{\beta }}^T{\tilde{F}}_1(Y_1,t)\hfill \\ ⋮\hfill \\ D^{{\kappa }_2}{e}_2\left(t\right)=h_2{e}_3+f_2(X_2,t)-{\tilde{f}}_2(Y_2,t)+{\beta }^T{F}_2(X_2,t)-{\tilde{\beta }}^T{\tilde{F}}_2(Y_2,t)\hfill \\ ⋮\hfill \\ ⋮\hfill \\ D^{{\kappa }_i}{e}_i\left(t\right)=h_i{e}_{i+1}+f_i(X_i,t)-{\tilde{f}}_i(Y_i,t)+{\beta }^T{F}_i(X_i,t)-{\tilde{\beta }}^T{\tilde{F}}_i(Y_i,t)\hfill \\ ⋮\hfill \\ D^{{\kappa }_n}{e}_n\left(t\right)=f_n(X_n,t)-{\tilde{f}}_n(Y_n,t)+{\beta }^T{F}_n(X_n,t)-{\tilde{\beta }}^T{\tilde{F}}_n(Y_n,t)-u\left(t\right)\hfill \end{array}\right.\end{array}$$

(14)

The challenge of synchronizing two systems in a master–slave setup will be approached as a stability problem, which will analyze the dynamics of the error states between the systems. The control mechanism will be designed to ensure the errors $e_i$ diminish towards zero as $t\to \infty$, demonstrating that the two systems achieve asymptotic synchronization. The FOBC approach will be utilized for this synchronization task, in line with the methodology employed for the stabilization issue previously described. Stabilizing these error states around the origin ensures that the trajectories of the slave system perfectly follow those of the master system, thereby guaranteeing projective synchronization with a scaling factor equal to one. While the general definition of projective synchronization may involve arbitrary nonzero scaling factors, the unity case already provides a rigorous framework to validate the effectiveness of the proposed FOBC strategy. A discussion of the possible extension to general scaling factors is included to highlight the implications and potential directions for further research.

Let us define $e_1=z_1\Rightarrow D^{{\kappa }_1}{z}_1=D^{{\kappa }_1}{e}_1$, which leads to

$$\begin{array}{c}{D}^{{\kappa }_1}{z}_1\left(t\right)=h_1{e}_2+f_1\left(X_1\right)-{\tilde{f}}_1(Y_1,t)+{\beta }^T{F}_1(X_1,t)-{\tilde{\beta }}^T{\tilde{F}}_1(Y_1,t)\end{array}$$

(15)

Next, let $e_2$ represent the control input, and denote $v_1$ as the virtual control input for the first subsystem. Define $z_2$ as the difference between these inputs: $z_2=e_2-v_1$. Consequently, we can rewrite $e_2$ as $e_2=z_2+v_1$. Therefore, Equation (17) can be reformulated as follows:

$$\begin{array}{c}{D}^{{\kappa }_1}{z}_1\left(t\right)=h_1(z_2+v_1)+f_1(X_1,t)-{\tilde{f}}_1(Y_1,t)+{\beta }^T{F}_1(X_1,t)-{\tilde{\beta }}^T{\tilde{F}}_1(Y_1,t)\end{array}$$

(16)

Now, let us consider a candidate Lyapunov function for the initial subsystem as follows:

$$\begin{array}{c}{V}_1\left(t\right)={\displaystyle \frac{1}{2}}{z}_1^2\end{array}$$

(17)

As derived from Lemma 1, applying the fractional-order derivative to Equation (19) leads to this inequality:

$$\begin{array}{c}{D}^{{\kappa }_1}{V}_1\left(t\right)\le z_1{D}^{{\kappa }_1}{z}_1\end{array}$$

(18)

This then simplifies to

$$\begin{array}{c}{D}^{{\kappa }_1}{V}_1\left(t\right)\le z_1\left(h_1(z_2+v_1)+f_1(X_1,t)-{\tilde{f}}_1(Y_1,t)+{\beta }^T{F}_1(X_1,t)-{\tilde{\beta }}^T{\tilde{F}}_1(Y_1,t)\right)\end{array}$$

(19)

To ensure stability, in line with the previously mentioned Mittag-Leffler criterion, the value of $v_1$ should be chosen according to Equation (22):

$$\begin{array}{c}{v}_1={\displaystyle \frac{1}{h_1}}(-c_1{z}_1-{\beta }^T{F}_1(X_1,t)+{\tilde{\beta }}^T{\tilde{F}}_1(Y_1,t)-f_1(X_1,t)+\widetilde{f_1}(Y_1,t))\end{array}$$

(20)

From this, we arrive at

$$\begin{array}{c}{D}^{{\kappa }_1}{V}_1\left(t\right)\le h_1{z}_1{z}_2-c_1{z}_1^2\end{array}$$

(21)

In the next step, as stated in Theorem 1, the term $h_1{z}_1{z}_2$ will diminish when the error state $z_2$ approaches zero. This virtual control $v_1$ effectively drives the state $z_1$ towards zero over time, which improves the stability of subsystem (18). The approach applied to the first subsystem is repeated for the second subsystem, leading to

$$\begin{array}{c}{D}^{{\kappa }_2}{z}_2\left(t\right)=h_2{e}_3+{\beta }^T{F}_2(X_2,t)-{\tilde{\beta }}^T{\tilde{F}}_2(Y_2,t)+f_2\left(X_2\right)-{\tilde{f}}_2(Y_2,t)\end{array}$$

(22)

With $e_2=z_2+v_1$, we have $D^{{\kappa }_2}{z}_2=D^{{\kappa }_2}{e}_2+D^{{\kappa }_2}{v}_1$. Thus, this results in

$$\begin{array}{c}{D}^{{\kappa }_2}{z}_2\left(t\right)=h_2(z_3+v_2)+f_2(X_2,t)-{\tilde{f}}_2(Y_2,t)+{\beta }^T{F}_2(X_2,t)-{\tilde{\beta }}^T{\tilde{F}}_2(Y_2,t)\end{array}$$

(23)

In this particular subsystem, let us denote $z_3$ as the result of subtracting the virtual control input $v_2$ from the control input $e_3$.

The candidate Lyapunov function can be expressed as follows:

$$V_2\left(t\right)=D^{{\kappa }_1-{\kappa }_2}{V}_1\left(t\right)+{\displaystyle \frac{1}{2}}{z}_2^2$$

(24)

where: $D^{{\kappa }_2}\left(D^{{\kappa }_1-{\kappa }_2}{V}_1\right)=D^{{\kappa }_2}{V}_1\left(t\right)\le h_1{z}_1{z}_2-c_1{z}_1^2$

According to Lemma 1 and Theorem 1, when we take the fractional derivative of order ${\kappa }_2$ of $V_2$, we find

$$\begin{array}{c}{D}^{{\kappa }_2}{V}_2\left(t\right)\le h_1{z}_1{z}_2-c_1{z}_1^2+z_2{D}^{{\kappa }_2}{z}_2\\ \Rightarrow D^{{\kappa }_2}{V}_2\left(t\right)\le h_1{z}_1{z}_2-c_1{z}_1^2+z_2(h_2(z_3+v_2)+f_2\left(X_2,t\right)-{\tilde{f}}_2\left(Y_2,t\right)+\\ {\beta }^T{F}_2(X_2,t)-{\tilde{\beta }}^T{\tilde{F}}_2(Y_2,t)-D^{{\kappa }_2}{v}_1)\end{array}$$

(25)

As explained in the stabilization section earlier, the second virtual control $v_2$ can be defined as

$$\begin{array}{c}{v}_2={\displaystyle \frac{1}{h_2}}(-c_2{z}_2-f_2\left(X_2,t\right)+{\tilde{f}}_2\left(Y_2,t\right)-\\ {\beta }^T{F}_2(X_2,t)+{\tilde{\beta }}^T{\tilde{F}}_2(Y_2,t)-h_1{z}_1+D^{{\kappa }_2}{v}_1)\end{array}$$

(26)

By inserting the virtual control expression (28) into inequality (27), we arrive at

$$\begin{array}{c}{D}^{{\kappa }_2}{V}_2\left(t\right)\le h_2{z}_2{z}_3-c_1{z}_1^2-c_2{z}_2^2\end{array}$$

(27)

In every following step, the coupled term $h_{i-1}{z}_{i-1}{z}_i$ will be removed. The virtual controls $v_i$ direct the state $z_i$ towards zero in steady-state, demonstrating the stability of the $i^{th}$ subsystem.

Likewise, for the $(n-1)$ subsystem, the virtual control $v_{n-1}$ can be represented as

$$\begin{array}{c}{v}_{n-1}={\displaystyle \frac{1}{h_{n-1}}}(-c_{n-1}{z}_{n-1}-f_{n-1}\left(X_{n-1},t\right)+\\ {\tilde{f}}_{n-1}\left(Y_{n-1},t\right)-{\beta }^T{F}_{n-1}(X_{n-1},t)+{\tilde{\beta }}^T{\tilde{F}}_{n-1}(Y_{n-1},t)\\ -h_{n-2}{z}_{n-2}+D^{{\kappa }_{n-1}}{v}_{n-2})\end{array}$$

(28)

For the complete system, at the last stage, we define a Lyapunov function as follows:

$$V_n\left(t\right)=D^{\left({\kappa }_{n-1}\right)-{\kappa }_n}{V}_{n-1}\left(t\right)+{\displaystyle \frac{1}{2}}{z}_n^2$$

(29)

where: $D^{{\kappa }_n}\left(D^{\left({\kappa }_{n-1}\right)-{\kappa }_n}{V}_{n-1}\right)=D^{{\kappa }_{n-1}}{V}_{n-1}\left(t\right)\le h_{n-1}{z}_{n-1}{z}_n-c_1{z}_1^2-c_2{z}_2^2-\dots -c_{n-1}{z}_{n-1}^2$.

By applying Lemma 1 and Theorem 1, we find the fractional derivative of order ${\kappa }_n$ for $V_n$ as follows:

$$\begin{array}{c}{D}^{{\kappa }_n}{V}_n\left(t\right)\le D^{{\kappa }_{n-1}}{V}_{n-1}\left(t\right)+z_n{D}^{{\kappa }_n}{z}_n\\ \Rightarrow D^{{\kappa }_n}{V}_n\left(t\right)\le h_{n-1}{z}_{n-1}{z}_n-c_1{z}_1^2-c_2{z}_2^2-\dots -c_{n-1}{z}_{n-1}^2+z_n{D}^{{\kappa }_n}{z}_n\\ \Rightarrow D^{{\kappa }_n}{V}_n\left(t\right)\le h_{n-1}{z}_{n-1}{z}_n-c_1{z}_1^2-c_2{z}_2^2-\dots -c_{n-1}{z}_{n-1}^2+\\ z_n(f_n\left(X_n,t\right)-{\tilde{f}}_n\left(Y_n,t\right)+{\beta }^T{F}_n(X_n,t)-{\tilde{\beta }}^T{\tilde{F}}_n(Y_n,t)+\\ -u\left(t\right)-D^{{\kappa }_n}{v}_{n-1})\end{array}$$

(30)

To ensure the stability of the entire system, the control input $u\left(t\right)$ needs to be set as follows:

$$\begin{array}{c}u\left(t\right)=c_n{z}_n+f_n\left(X_n,t\right)-{\tilde{f}}_n\left(Y_n,t\right)+{\beta }^T{F}_n(X_n,t)-\\ {\tilde{\beta }}^T{\tilde{F}}_n(Y_n,t)-D^{{\kappa }_n}{v}_{n-1}+h_{n-1}{z}_{n-1}\end{array}$$

(31)

By implementing the FOBC law (33), we can simplify inequality (32) to:

$$\begin{array}{c}{D}^{{\kappa }_n}{V}_n\left(t\right)\le -\sum _{i=1}^n{c}_i{z}_i^2\end{array}$$

(32)

This inequality and the fractional Lyapunov–Mittag-Leffler argument used in Theorem 1 imply that the composite Lyapunov function $V_n\left(t\right)$ is nonincreasing and that the quadratic sum ${\sum }_{i=1}^n{c}_i{z}_i^2$ is integrable over time. Under standard regularity assumptions (continuity and local Lipschitz conditions of the nonlinear functions, boundedness of all system signals), the fractional-order stability theory ensures that ${\displaystyle \underset{t\to \infty }{lim}{z}_i\left(t\right)=0,\forall i=1,\dots ,n.}$

Recalling the change of variables $z_1=e_1$ and $z_j=e_j-v_{j-1}$ for $j\ge 2$, we deduce that ${\displaystyle e_j\left(t\right)=z_j\left(t\right)+v_{j-1}\left(t\right)\Rightarrow \underset{t\to \infty }{lim}{e}_j\left(t\right)=\underset{t\to \infty }{lim}{v}_{j-1}\left(t\right).}$ Consequently, the asymptotic evolution of the tracking errors $e_j$ depends on the steady-state values of the virtual controls $v_{j-1}$. When the reference trajectory is zero (the stabilization around the origin ‘0’, which is precisely the case studied in this work) or constant, each virtual control $v_{j-1}$ contains only stabilizing and compensating terms that vanish as $X\left(t\right)-Y\left(t\right)\to 0$; hence, ${\displaystyle \underset{t\to \infty }{lim}{v}_j\left(t\right)=0, \mathrm{and} \mathrm{therefore}\underset{t\to \infty }{lim}{e}_j\left(t\right)=0.}$ In this case, complete asymptotic synchronization between the master and slave systems is achieved.

In contrast, when the reference is nonzero or time-varying, the virtual controls $v_j\left(t\right)$ tend to the corresponding steady-state or reference-related values that reproduce the reference dynamics. More precisely, if the reference signal $r\left(t\right)$ (or the master trajectory) remains bounded with bounded derivatives, the virtual controls satisfy ${\displaystyle \underset{t\to \infty }{lim}{v}_j\left(t\right)=v_j^{*}\left(r\left(t\right)\right)}$, where $v_j^{*}$ represents the ideal control law that perfectly tracks the reference trajectory. Accordingly, the tracking errors $e_j\left(t\right)$ approach zero in the sense of bounded steady-state convergence, i.e., ${\displaystyle \underset{t\to \infty }{lim}{e}_j\left(t\right)=0}$, for a constant reference, and $e_j\left(t\right)\to 0$ uniformly when $r\left(t\right)$ is bounded and smooth. □

This analysis demonstrates that the proposed FOBC law guarantees asymptotic or uniform convergence of all error states, while the virtual controls naturally capture the steady-state behavior required to follow any admissible reference trajectory. Furthermore, since each $h_i$ is nonzero and finite, division by $h_i$ in the recursive expressions for $v_i$ is well defined, thereby avoiding the singularities often encountered in classical integer-order backstepping schemes.

To illustrate, Figure 1 demonstrates the synchronization achieved by the two master-slave systems through FOBC control.

4. Results

In this section, we address the problem of stabilization and synchronization between two FOCS using the FOBC approach. The first system under consideration is a fractional-order variant of the classical Van der Pol oscillator, while the second is constructed in a similar way from the Rayleigh oscillator. All numerical experiments are carried out in the Matlab/Simulink environment.

The simulations are performed over a time horizon of $T=2$ s with a sampling step of $T_s=0.0001$ s.

The Rayleigh oscillator serves as a well-known nonlinear model that exhibits self-sustained oscillations as a result of an internal energy exchange mechanism. Originally introduced by Lord Rayleigh to describe the vibrations of a clarinet reed, this oscillator has since been applied to a wide range of physical and engineering problems [31]. Its classical mathematical form is expressed by

$$\ddot{x}-ϵ(1-{\dot{x}}^2)\dot{x}+{\omega }^{2}x=0$$

(33)

where: x is the state variable (e.g., displacement); $\dot{x}$ and $\ddot{x}$ denote velocity and acceleration, respectively, $ϵ$ controls the level of nonlinearity and the strength of energy input or damping, $\omega$ represents the natural frequency.

The damping term, $(-ϵ(1-{\dot{x}}^2)\dot{x})$, changes sign depending on the velocity. At small velocities, it acts as negative damping (energy injection), whereas at higher velocities it produces positive damping (energy dissipation). This balance leads to the appearance of a stable limit cycle, which characterizes self-sustained oscillations.

To apply the FOBC strategy, we first derive a non-commensurate fractional-order extension of the Rayleigh oscillator, hereafter referred to as the Fractional-Order Rayleigh Oscillator (FORO). This transformation is introduced as follows: ${\rho }_1=x,{\rho }_2=D^{{\alpha }_1}x,{\rho }_3=\dot{x},{\rho }_4=D^{{\alpha }_3}\dot{x}$ with the constraints $({\alpha }_1+{\alpha }_2=1)$ and $({\alpha }_3+{\alpha }_4=1)$. The resulting system takes the following form:

$$\left\{\begin{array}{c}{D}^{{\alpha }_1}{\rho }_1={\rho }_2\\ D^{{\alpha }_2}{\rho }_2={\rho }_3\\ D^{{\alpha }_3}{\rho }_3={\rho }_4\\ D^{{\alpha }_4}{\rho }_4=ϵ(1-{\rho }_3^2){\rho }_3-{\omega }^2{\rho }_1\end{array}\right.$$

(34)

with $0<{\alpha }_i<1$; $i=1,\dots ,4.$ where ($0<{\alpha }_i<1$) for ($i=1,\dots ,4$).

For illustration, consider the parameters ($ϵ=0.5$), ($\omega =1$), and fractional orders (${\alpha }_1=0.9$), (${\alpha }_2=0.$1), (${\alpha }_3=0.6$), (${\alpha }_4=0.4$), with initial conditions (${\rho }_1$(0) = 0.03), (${\rho }_2\left(0\right)=0.06$), (${\rho }_3\left(0\right)=-0.04$), and (${\rho }_4\left(0\right)=0.5$). Under these conditions, the simulated trajectories illustrate the transition from transient dynamics to the establishment of a stable limit cycle. Such results provide valuable insight into the oscillator’s dynamical response and confirm the validity of the theoretical framework.

Finally, the structure of system (36) can be expressed in the same general form as the master system (10), where the parameters are specified as ($h_i=1$) for ($i=1,\dots ,4$); ($f_j=0$); (${\beta }^T{F}_j=0$) for ($j=1,2,3$); (${\beta }^T{F}_4=ϵ(1-{\rho }_3^2){\rho }_3$); and ($f_4=-{\omega }^2{\rho }_1$).

Figure 2 shows the 3D phase portrait of the uncontrolled FORO oscillator, highlighting the trajectories of the state variables ${\rho }_i$ for $i=1,\dots ,4$.

Moreover, the Van-der Pol (VDP) oscillator is thoroughly characterized in [32], with its most widely accepted model expressed as follows:

$$\left\{\begin{array}{c}{\dot{\xi }}_1={\xi }_2\\ {\dot{\xi }}_2=\mu (1-{\xi }_1^2){\xi }_2-{\xi }_1+u\left(t\right)\end{array}\right.$$

(35)

In this model, the states are denoted as ${\xi }_i$, while $u\left(t\right)$ represents the control input.

The system can be reformulated as a fractional order Van-der Pol (FOVDP) oscillator using non-integer order by introducing the following state definitions: $y_1={\xi }_1$; $y_2=D^{{\kappa }_1}{\xi }_1$; $y_3={\xi }_2$; and $y_4=D^{{\kappa }_3}{\xi }_2$ where ${\kappa }_1+{\kappa }_2=1$ and ${\kappa }_3+{\kappa }_4=1$.

Consequently, the system (37) is expressed as

$$\left\{\begin{array}{c}{D}^{{\kappa }_1}{y}_1=y_2\\ D^{{\kappa }_2}{y}_2=y_3\\ D^{{\kappa }_3}{y}_3=y_4\\ D^{{\kappa }_4}{y}_4=\mu (1-y_1^2)y_3-y_1+u\left(t\right)\end{array}\right.$$

(36)

where $0<{\kappa }_i<1$, ${\kappa }_1=0.9;{\kappa }_2=0.1;{\kappa }_3=0.6;{\kappa }_4=0.4$.

The initial conditions selected are $y_1\left(0\right)=0.01$, $y_2\left(0\right)=-0.01$, $y_3\left(0\right)=0.2$, and $y_4\left(0\right)=0.05$, with $\mu =1$.

The structure of system (38) is similar to that of slave-system (11), defined as follows: $h_i=1$ for $i=1,\dots ,4$; ${\tilde{f}}_i=0$; ${\beta }^T{\tilde{F}}_i=0$ for $j=1,2,3$; ${\beta }^T{\tilde{F}}_4=\mu (1-{\xi }_1^2){\xi }_2$; and ${\tilde{f}}_4=-{\xi }_1$.

The 3D phase portrait of the FOVDP oscillator and the trajectories of the state variables $y_1$, $y_2$, $y_3$ and $y_4$, are displayed in Figure 3.

Within the synchronization challenge of the master–slave setup, the system represented by Equation (10) acts as the master, while the slave is defined by Equation (11). According to Theorem 2, these systems achieve asymptotic synchronization through the FOBC signal (39):

$$\begin{array}{c}u\left(t\right)=c_4{z}_4+ϵ(1-{\rho }_3^2){\rho }_3-{\omega }^2{\rho }_1-\mu (1-y_1^2)y_3-y_1-D^{{\kappa }_4}{v}_3+h_3{z}_3\end{array}$$

(37)

The simulation results are represented in Figure 4, Figure 5 and Figure 6.

In Figure 4, the two-dimensional phase plane shows the synchronized state variables $x_i$ and ${\rho }_i$ in a masterslave configuration using FOBC control. Figure 5 highlights the success of the suggested approach for the steady state synchronizing of states between the master FORO system and the slave FOVDP system. Chaotic system synchronizing. Figure 6 shows the control signal $u\left(t\right)$ for synchronizing the FORO-FOVDP systems and illustrates the synchronization errors connected to the FOBC control, therefore providing more understanding of the whole synchronization process. Given that chaotic systems are very responsive to their beginning states, starting the slave FOVDP confirms the success of the FOBC control technique. A system with non-zero starting values lets it closely follow the actions of the master FORO system even when starting from rather different beginning circumstances.

5. Conclusions

This research introduced a control method based on fractional-order backstepping to attain projective synchronization in IFOCS that are not equivalent. The design approach included the Mittag-Leffler function, ensuring that the error dynamics remained stable near the origin, thus guaranteeing the overall stability of the system and synchronizing the master and slave systems effectively. The method was tested on fractional-order Van der Pol and Rayleigh oscillators, successfully achieving precise synchronization under different initial conditions. These results demonstrate the strength and dependability of the proposed method and highlight how fractional-order control techniques can effectively tackle the challenges of nonlinear chaotic systems. Future research could investigate applying this framework to a broader range of chaotic models, multi-agent networks, or practical engineering systems that need strong resistance to uncertainties.