Finite-Time Projective Synchronization in Fractional-Order Inertial Memristive Neural Networks: A Novel Approach to Image Encryption

Abstract

This paper presents a novel finite-time projective synchronization (FTPS) control strategy for fractional-order inertial memristive neural networks (FOIMNNs), exploring its application in image encryption. A sufficient condition for ensuring FTPS in FOIMNNs is established and validated through numerical simulations. These simulations indicate that the proposed strategy provides reliable synchronization performance. Furthermore, an efficient method for image encryption was developed, potentially improving data security. Comparative analyses with existing methods suggest that this approach could offer incremental benefits in secure communication and data protection.

1. Introduction

Neural networks (NNs) play a crucial role in various fields, including associative memory [1] and secure communication [2,3,4]. However, traditional NNs may struggle to maintain robust memory and adaptability in tasks that require long-term dependencies and dynamic adaptability, especially in complex computational environments. The integration of memristor elements into NNs has led to the development of memristive neural networks (MNNs) [5], which enhance memory retention and adaptability. Memristors, as nonlinear resistors, imbue NNs with dynamic memory attributes, making them ideal for tasks requiring high adaptability and efficient memory storage.

In recent years, there has been growing interest in incorporating inertia into neural network models to better replicate the dynamics of biological systems. Inertia introduces a form of resistance to sudden changes in neuron states, which allows the network to exhibit more stable and realistic temporal behaviors. This characteristic is especially valuable in tasks that involve dynamic processing or require robustness against rapid fluctuations. Building on this idea, inertial memristive neural networks (IMNNs) represent a major advancement integrating both memristive and inertial properties into the architecture [6,7]. IMNNs not only benefit from the enhanced memory retention and adaptability provided by memristors but also leverage the inertia mechanism to simulate more realistic neural dynamics.

Fractional-order calculus generalizes traditional calculus by allowing non-integer-order differentiation and integration, making it an ideal tool for describing systems with memory effects and long-term dependencies, especially in neural networks. Unlike classical integer-order models, fractional-order systems provide a more accurate representation of neural processes, where the current state depends not only on the immediate past but also on the historical states. This characteristic is particularly valuable in modeling complex neural dynamics, such as those found in inertial memristive neural networks (IMNNs), where the ability to capture both current and historical neuron states is critical. In the context of neural networks, fractional-order calculus has proven to be instrumental in describing chaotic behaviors and enhancing synchronization strategies. For instance, previous studies have demonstrated the chaotic behaviors in systems like the fractional-order Chua circuit [8], the Chen system [9], and the Lü system [10]. These foundational works laid the groundwork for recent advancements in fractional-order modeling in neural networks, particularly in the synchronization of complex, dynamic systems such as FOIMNNs.

The academic literature has conducted extensive research on the diverse types of synchronization within neural networks, including complete synchronization [11], anti-synchronization [12], and lag synchronization [13]. However, most of the existing research on FOIMNNs has focused on asymptotic synchronization, where synchronization is achieved as time approaches infinity [14]. This approach may not be suitable for real-time applications that require quicker convergence. To address this limitation, finite-time projective synchronization (FTPS) has emerged as a promising alternative. FTPS ensures synchronization within a finite and predefined timeframe, offering practical benefits for applications that demand rapid response times, such as encryption and communication systems. While FTPS has seen limited exploration in FOIMNNs, it holds significant potential for real-world applications. Zheng et al. [15] established sufficient conditions for synchronization through linear feedback control, further advancing the understanding of FTPS in fractional-order systems.

Moreover, the use of FOIMNNs in image encryption has attracted significant interest due to their inherent randomness and complexity, making them particularly effective for cryptographic purposes where robust security is crucial. Through employing FTPS, a novel image encryption method can be developed, offering enhanced security and efficiency. The synchronous chaotic encryption system ensures that the driving sequence for image encryption is perfectly aligned with the response sequence for decryption, achieving lossless recovery of encrypted images. This newly proposed algorithm demonstrates superior performance in key metrics such as pixel correlation and entropy, where lower pixel correlation reflects reduced pixel dependency, and higher entropy suggests increased randomness, both contributing to enhanced encryption security. Recent studies by Yang et al. [2] and Hui et al. [4] further explored synchronization criteria in neural networks, applying them to secure image encryption systems with promising results.

In synchronization technology, timing uncertainties pose significant challenges, particularly in secure communication systems. Unpredictable timing can undermine encryption and decryption processes, affecting communication reliability. To address these challenges, this paper proposes a control strategy that incorporates fractional-order calculus, inertial effects, and memristive dynamics for FTPS in FOIMNNs. With the use of an appropriately designed controller, this approach ensures synchronization with well-defined timing parameters. This method aims to improve both the synchronization efficiency and security of encrypted communications. It contributes to the advancement of synchronization techniques, with a particular focus on protecting digital images from unauthorized access.

This research provides a perspective on the dynamics of FOIMNNs and their potential applications in secure communication and information processing. By applying fractional-order chaotic systems, our approach seeks to introduce greater complexity and unpredictability into the system, both of which are relevant factors for communication security. The key contributions of this study include the development of a fractional-order FTPS framework and its successful application in image encryption, demonstrating its effectiveness in ensuring the secure transmission and recovery of encrypted images. The main contributions of this study include:

• Extension of Fractional-Order Inertial Memristive Neural Networks: A new FOIMNN model is introduced. Both the inertial term and the state term are modeled with fractional-order dynamics. This model extends the application of fractional calculus in neural networks. It offers fewer restrictions on the fractional-order inertial term compared to in existing studies, such as [16,17]. The proposed model provides a more flexible framework for studying dynamic behaviors in neural networks.
• Finite-Time Projective Synchronization Control Strategy: A new FTPS control strategy is proposed for FOIMNNs. Synchronization is achieved within a finite timeframe. Numerical simulations are conducted to validate this approach, demonstrating its performance in a three-dimensional FOIMNN model.
• Application in Image Encryption: The proposed synchronization control strategy is applied to image encryption as an initial exploration into practical applications. FTPS is used for both encryption and decryptionto ensure synchronization between these processes. Metrics such as pixel correlation and entropy show favorable results, suggesting that FOIMNNs have potential utility in secure communication and data protection.

The structure of this paper is organized as follows: Section 2 introduces the FOIMNN model, outlining pertinent lemmas and assumptions. Section 3 discusses the theoretical derivation of FTPS for FOIMNNs, supplemented by two corollaries. Section 4 presents numerical simulations to substantiate the theorem’s applicability. Section 5 explores the practical application of FOIMNN synchronization in image protection. Finally, Section 6 concludes this paper, summarizing its findings and implications.

2. Preliminaries

This section introduces essential mathematical concepts related to the study. Initially, it defines fractional calculus, and then delves into the theoretical foundations of fractional differential equations. Subsequently, it explores solutions for discontinuous fractional differential equations, following Filippov’s interpretations, and concludes with the model assumptions required for this research.

2.1. Basic Knowledge

In this section, we outline the fundamental concepts relevant to our study. For a detailed explanation of the Riemann–Liouville fractional integral and the Caputo fractional derivative, please refer to Appendix A.

Definition 1

([18]). Let γ be a nonzero constant in $\mathbb{R}$. Two dynamical systems are considered: the leader system and the follower system, with solutions $x\left(t\right)$ and $y\left(t\right)$, respectively, originating from different initial conditions $x_0$ and $y_0$. If the distance between these solutions measured using the Euclidean norm satisfies

$$\underset{t\to \infty }{lim}\parallel x\left(t\right)-\gamma y\left(t\right)\parallel =0,$$

then there is global asymptotic projective synchronization between the leader and follower systems, where γ is referred to as the projective coefficient.

Lemma 1

([19]). Let $v\left(t\right):[t_0,+\infty )\to {\mathbb{R}}^{*}$ be a continuous, positive definite function. For $\sigma >0$ and $\varphi >0$, if

$${}_{\mathit{0}}{}^{C}D_t^{\alpha }v\left(t\right)\le -\sigma v\left(t\right)-\varphi \mathit{for}v\left(t\right)>0,$$

with $0<\alpha <1$, then the following results are obtained: For $\sigma >0$, for all $t\le {\widehat{t}}_1^{*}$, the function $v\left(t\right)$ approaches 0 in finite time ${\widehat{t}}_1^{*}$, meaning that there exists a time ${\widehat{t}}_1^{*}$ such that $v\left(t\right)=0$ for all $t\ge {\widehat{t}}_1^{*}$. Here, ${\widehat{t}}_1^{*}$ can be estimated as follows:

$${\widehat{t}}_1^{*}\le t_0+{\left({\displaystyle \frac{\Gamma (\alpha +1)v\left(t_0\right)}{\varphi }}\right)}^{{\displaystyle \frac{1}{\alpha }}}$$

(1)

Lemma 2

([20]). If there exists a positive definite Lyapunov function $V\left(x\right)$ such that $D^{\alpha }V\left(x\right)<0$ for all $t\ge t_0$, then $V\left(x\right)$ asymptotically approaches 0 as t approaches infinity.

Lemma 3

([21]). Consider a function $f\left(t\right)$, continuously differentiable over $t\in [0,\infty )$. For any α such that $0<\alpha <1$, the inequality presented below is valid:

$$D^{\alpha }\left|f\left(t\right)\right|\le \mathrm{sign}\left(f\left(t\right)\right)D^{\alpha }f\left(t\right).$$

Lemma 4

([22]). For functions $h_1\left(t\right),h_2\left(t\right)\in C^n([0,\infty ),{\mathbb{R}}^n)$ with constants ${\lambda }_1,{\lambda }_2\in \mathbb{R}$ and α satisfying $n-1<\alpha <n$,

$$D_t^{\alpha }({\lambda }_1{h}_1\left(t\right)+{\lambda }_2{h}_2\left(t\right))={\lambda }_1{D}_t^{\alpha }{h}_1\left(t\right)+{\lambda }_2{D}_t^{\alpha }{h}_2\left(t\right).$$

Lemma 5

([23]). Let $h\left(x\right)$ be a continuous and differentiable function for $h\left(t\right)\in C^n([0,\infty ),{\mathbb{R}}^n)$, and consider the Caputo fractional derivatives. If $0<q\le 1$ and $1<p<1+q\le 2$, the following holds:

$${\displaystyle \frac{d^{p-q}}{d{x}^{p-q}}}{\displaystyle \frac{d^q}{d{x}^q}}h\left(x\right)={\displaystyle \frac{d^p}{d{x}^p}}h\left(x\right).$$

Assumption 1.

For any $i=1,2,\dots ,n$, there exist constants $l_i,M_i>0$ such that for all $x,w\in \mathbb{R}$,

$$|f_i\left(x\right)-f_i\left(w\right)|\le l_i|x-w|\mathit{and}|f_i(·)|\le M_i.$$

2.2. System Descriptions

A class of FOIMNNs is defined as follows:

$$D_t^{\alpha }{x}_i\left(t\right)=-a_i{D}_t^{\beta }{x}_i\left(t\right)-b_i{x}_i\left(t\right)+\sum _{j=1}^n{h}_{ij}{x}_j\left(t\right)f_j\left(x_j\left(t\right)\right)+I_i$$

(2)

where $0\le \beta <1$, $1\le \alpha <2\beta$, $a_i>b_i$, and $t>t_0$ are established. The state of each neuron i (for $i=1,2,\dots ,n$) at time t is represented by $x_i\left(t\right)$. The following initial value complements system (2): $x_i\left(t_0\right)=x_{i0},x_{i0}\in \mathbb{R},i=1,2,\dots ,n$, with $a_i$ and $b_i$ as constants. $h_{ij}$ denotes the weights assigned to the links in memristive networks and is defined as follows:

$$h_{ij}\left(x_i\left(t\right)\right)=\left\{\begin{array}{cc}{\widehat{h}}_{ij},\hfill & |x_i\left(t\right)|\le T_i\hfill \\ {\check{h}}_{ij},\hfill & |x_i\left(t\right)|>T_i\hfill \end{array}\right.$$

and the switching jump time $T_i$, ${\widehat{h}}_{ij}$, and ${\check{h}}_{ij}$ denote the known values of memristive resistance, whereas $f_j(·)$ is the activation function for neuron j at time t.

Remark 1.

The fractional derivatives $D_t^{\alpha }$ and $D_t^{\beta }$ significantly influence the dynamic behavior of the state variables $x_i\left(t\right)$, capturing memory and inertia in FOIMNNs. Introducing α and β allows the model to better handle complex data patterns, especially in tasks like image encryption. Fractional derivatives offer a notable advantage over traditional integer-order models [24,25] in capturing complex dynamics.

Variable substitution is defined as

$$r_i\left(t\right)=D_t^{\beta }{x}_i\left(t\right)+x_i\left(t\right),i=1,2,\dots ,n.$$

(3)

An interpretation of an FOIMNNs is as follows:

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{r}_i\left(t\right)=-a_i{r}_i-(b_i-a_i)x_i\left(t\right)+\sum _{j=1}^n{h}_{ij}\left(x_j\left(t\right)\right)f_j\left(x_j\left(t\right)\right)+D_t^{\alpha -\beta }{x}_i\left(t\right)+I_i\hfill \\ D_t^{\beta }{x}_i\left(t\right)=r_i\left(t\right)-x_i\left(t\right)\hfill \end{array}\right.$$

(4)

where $i=1,2,\dots ,n$.

Let us define the set-valued maps

$$K\left[h_{ij}\left(x_i\left(t\right)\right)\right]=\left\{\begin{array}{cc}{\widehat{h}}_{ij},\hfill & |x_i\left(t\right)|<T_i\hfill \\ \mathrm{co}\{{\widehat{h}}_{ij},{\check{h}}_{ij}\},\hfill & |x_i\left(t\right)|=T_i\hfill \\ {\check{h}}_{ij},\hfill & |x_i\left(t\right)|>T_i\hfill \end{array}\right.$$

where $coE$ is the closure of the convex hull of the set $E\subseteq {\mathbb{R}}^n$.

Hereinafter, systems are discussed using the Filippov framework [26], as the presence of memristors introduces discontinuities in the system’s derivatives. To address this, Filippov theory is applied throughout the analysis [6,27,28,29].

Definition 2.

A function $x_i\left(t\right):[t_0,T)\to \mathbb{R}$, where $T>t_0$, qualifies as a solution to system (4), provided that it satisfies the initial condition $x_i\left(t_0\right)=x_{i0}$ and remains absolutely continuous on each compact subinterval of $[t_0,T)$.

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{r}_i\left(t\right)\in -a_i{r}_i-(b_i-a_i)x_i\left(t\right)+\sum _{j=1}^{n}K\left[h_{ij}\left(x_i\left(t\right)\right)\right]f_j\left(x_j\left(t\right)\right)+D_t^{\alpha -\beta }{x}_i\left(t\right)+I_i\hfill \\ D_t^{\beta }{x}_i\left(t\right)=r_i\left(t\right)-x_i\left(t\right)\hfill \end{array}\right.$$

(5)

Then, there exist measurable functions $H_{ij}\left(t\right)\in K\left[h_{ij}\left(x_i\left(t\right)\right)\right]$ such that

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{r}_i\left(t\right)=-a_i{r}_i-(b_i-a_i)x_i\left(t\right)+\sum _{j=1}^n{H}_{ij}\left(x_j\left(t\right)\right)f_j\left(x_j\left(t\right)\right)+D_t^{\alpha -\beta }{x}_i\left(t\right)+I_i\hfill \\ D_t^{\beta }{x}_i\left(t\right)=r_i\left(t\right)-x_i\left(t\right)\hfill \end{array}\right.$$

(6)

In this analysis, a response system is developed, associated with the driven system.

$$D_t^{\alpha }{y}_i\left(t\right)=-a_i{D}_t^{\beta }{y}_i\left(t\right)-b_i{y}_i\left(t\right)+\sum _{j=1}^n{h}_{ij}{y}_j\left(t\right)f_j\left(y_j\left(t\right)\right)+I_i$$

(7)

It is delineated using a comparable reduced-order transformation, as shown below:

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{s}_i\left(t\right)=-a_i{s}_i+(a_i-b_i)y_i\left(t\right)+{\displaystyle \sum _{j=1}^n}{h}_{ij}\left(y_j\left(t\right)\right)f_j\left(y_j\left(t\right)\right)\hfill \\ +D_t^{\alpha -\beta }{y}_i\left(t\right)+u_i\left(t\right)+I_i\hfill \\ D_t^{\beta }{y}_i\left(t\right)=s_i\left(t\right)-y_i\left(t\right)\hfill \end{array}\right.$$

(8)

where $i,j=1,2,\dots ,n$, $u_i\left(t\right)$ are the appropriate controllers.

In a manner similar to (4), system (8) is translatable into the following form:

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{s}_i\left(t\right)\in -a_i{s}_i+(a_i-b_i)y_i\left(t\right)+{\displaystyle \sum _{j=1}^n}K\left[h_{ij}\left(y_i\left(t\right)\right)\right]f_j\left(y_j\left(t\right)\right)\hfill \\ +D_t^{\alpha -\beta }{y}_i\left(t\right)+u_i\left(t\right)+I_i\hfill \\ D_t^{\beta }{y}_i\left(t\right)=s_i\left(t\right)-y_i\left(t\right)\hfill \end{array}\right.$$

(9)

where $i,j=1,2,\dots ,n$.

In a manner similar to (5), system (9) is translatable into the following form:

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{s}_i\left(t\right)=-a_i{s}_i-(b_i-a_i)y_i\left(t\right)+{\displaystyle \sum _{j=1}^n}{\overline{H}}_{ij}\left(y_j\left(t\right)\right)f_j\left(y_j\left(t\right)\right)+D_t^{\alpha -\beta }{y}_i\left(t\right)+I_i\hfill \\ D_t^{\beta }{y}_i\left(t\right)=s_i\left(t\right)-y_i\left(t\right)\hfill \end{array}\right.$$

(10)

Within this framework, synchronization discrepancies for systems (2) and (7) are represented by $z_i\left(t\right)=r_i\left(t\right)-\gamma s_i\left(t\right)$ and $e_i\left(t\right)=x_i\left(t\right)-\gamma y_i\left(t\right)$, respectively. Then,

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{z}_i\left(t\right)=-a_i{z}_i+(a_i-b_i)e_i\left(t\right)+{\displaystyle \sum _{j=1}^n}\left[H_{ij}\left(x_j\left(t\right)\right)f_j\left(x_j\left(t\right)\right)-\gamma {\overline{H}}_{ij}\left(y_j\left(t\right)\right)f_j\left(y_j\left(t\right)\right)\right]\hfill \\ +D_t^{\alpha -\beta }{e}_i\left(t\right)+(1-\gamma )I_i+u_i\hfill \\ D_t^{\beta }{e}_i\left(t\right)=z_i\left(t\right)-e_i\left(t\right)\hfill \end{array}\right.$$

(11)

3. Main Results

FTPS of FOIMNNS

This subsection describes the creation of a feedback controller and the definition of algebraic conditions necessary for ensuring the finite-time projective stability of the error system. The initial design of the feedback control scheme is presented below.

$$\begin{array}{cc}\hfill u_i\left(t\right)=& -l_i{e}_i\left(t\right)-k_i{z}_i\left(t\right)-D_t^{\alpha -\beta }{e}_i\left(t\right)-{\lambda }_i{D}_t^{\alpha -2\beta }{z}_i\left(t\right)\hfill \\ \hfill & -\mathrm{sign}\left(z_i\left(t\right)\right){\chi }_i-{\overline{H}}_{ij}\left(x_j\left(t\right)\right)\left(f_j\left(\gamma y_j\left(t\right)\right)-f_j\left(y_j\left(t\right)\right)\right)-(1-\gamma )I_i\hfill \end{array}$$

(12)

where $l_i>0,k_i>0,$ and ${\chi }_i>0$.

Given that $\mathrm{sign}\left(z_i\left(t\right)\right)$ exhibits discontinuity at $z_i\left(t\right)=0$, the error system can be described as follows:

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{z}_i\left(t\right)=-a_i{z}_i+(a_i-b_i)e_i\left(t\right)+{\displaystyle \sum _{j=1}^n}\left[H_{ij}\left(x_j\left(t\right)\right)f_j\left(x_j\left(t\right)\right)-\gamma {\overline{H}}_{ij}\left(y_j\left(t\right)\right)f_j\left(y_j\left(t\right)\right)\right]\hfill \\ +D_t^{\alpha -\beta }{e}_i\left(t\right)-l_i{e}_i\left(t\right)-k_i{z}_i\left(t\right)-D_t^{\alpha -\beta }{e}_i\left(t\right)-{\lambda }_i{D}_t^{\alpha -2\beta }{z}_i\left(t\right)\hfill \\ -{\rho }_i{\chi }_i-{\overline{H}}_{ij}\left(x_j\left(t\right)\right)\left(f_j\left(\gamma y_j\left(t\right)\right)-f_j\left(y_j\left(t\right)\right)\right)-(1-\gamma )I_i\hfill \\ D_t^{\beta }{e}_i\left(t\right)=z_i\left(t\right)-e_i\left(t\right)\hfill \end{array}\right.$$

(13)

where ${\rho }_i\in \mathrm{co}\left[\mathrm{sign}\left(z_i\left(t\right)\right)\right]$ and

$$\mathrm{co}\left[\mathrm{sign}\left(z_i\left(t\right)\right)\right]=\left\{\begin{array}{cc}\left\{1\right\},\hfill & z_i\left(t\right)\ge 0\hfill \\ [1,-1],\hfill & z_i\left(t\right)=0\hfill \\ \{-1\},\hfill & z_i\left(t\right)\le 0\hfill \end{array}\right.$$

Theorem 1.

Consider the FOIMNNs described by Equations (2) and (7) under Assumption 1 and control scheme (12). The system can achieve FTPS if the following conditions are satisfied:

$$\left\{\begin{array}{c}1-a_i-k_i\le 0\hfill \\ (a_i-b_i)-1+{\displaystyle \sum _{j=1}^n}\left|h_{ij}^{*}\right|F_j+l_i\le -{\lambda }_i\hfill \\ {\displaystyle \sum _{j=1}^n}(1-\gamma )\left|h_{ij}^{*}\right|M_j<{\chi }_i\hfill \end{array}\right.$$

(14)

let $\overline{\lambda }=\underset{1\le i\le n}{min}{\lambda }_i,\overline{\chi }=\underset{1\le i\le n}{min}{\chi }_i,|h_{ij}^{*}|=max\{|{\widehat{h}}_{ij}|,|{\check{h}}_{ij}\left|\right\}$.

Proof of Theorem 1.

Construct the Lyapunov function as follows:

$$V=\sum _{i=1}^n|e_i\left(t\right)|+\sum _{i=1}^n{D}_t^{\alpha -2\beta }\left|z_i\left(t\right)\right|$$

(15)

Based on Lemma 3, when $V\left(t\right)\in {\mathbb{R}}^{+}$, the results indicate that

$$\begin{array}{cc}\hfill D_t^{\beta }V& \le D_t^{\beta }\sum _{i=1}^n|e_i\left(t\right)|+D_t^{\alpha -\beta }\sum _{i=1}^n\left|z_i\left(t\right)\right|\hfill \\ \hfill & \le \sum _{i=1}^n\mathrm{sign}\left(e_i\left(t\right)\right)D_t^{\beta }{e}_i\left(t\right)+\mathrm{sign}\left(z_i\left(t\right)\right)D_t^{\alpha -\beta }{z}_i\left(t\right)\hfill \\ \hfill & =\sum _{i=1}^n\{\mathrm{sign}\left(e_i\left(t\right)\right)(z_i\left(t\right)-e_i\left(t\right))+\mathrm{sign}\left(z_i\left(t\right)\right)(-a_i{z}_i\left(t\right)+(a_i-b_i)e_i\left(t\right)\hfill \\ \hfill & +\sum _{j=1}^n\left[H_{ij}\left(x_j\left(t\right)\right)f_j\left(x_j\left(t\right)\right)-\gamma {\overline{H}}_{ij}\left(y_j\left(t\right)\right)f_j\left(y_j\left(t\right)\right)\right]\hfill \\ \hfill & +D_t^{\alpha -\beta }{e}_i\left(t\right)+(1-\gamma )I_i-l_i{e}_i\left(t\right)-k_i{z}_i\left(t\right)\hfill \\ \hfill & -D_t^{\alpha -\beta }{e}_i\left(t\right)-{\lambda }_i{D}_t^{\alpha -2\beta }{z}_i\left(t\right)-{\rho }_i{\chi }_i\hfill \\ \hfill & -\sum _{j=1}^n{\overline{H}}_{ij}\left(x_j\left(t\right)\right)\left(f_j\left(\gamma y_j\left(t\right)\right)-f_j\left(y_j\left(t\right)\right)\right)-(1-\gamma )I_i\left)\right\}\hfill \end{array}$$

(16)

Given that Assumption 1 is satisfied, the equation can be expressed as

$$\begin{array}{cc}\hfill & H_{ij}\left(x_j\left(t\right)\right)f_j\left(x_j\left(t\right)\right)-\gamma {\overline{H}}_{ij}\left(y_j\left(t\right)\right)f_j\left(y_j\left(t\right)\right)\hfill \\ \hfill & ={\overline{H}}_{ij}\left(x_j\left(t\right)\right)f_j\left(e_j\left(t\right)\right)+\left({\overline{H}}_{ij}\left(x_j\left(t\right)\right)-\gamma H_{ij}\left(y_j\left(t\right)\right)\right)f_j\left(y_j\left(t\right)\right)\hfill \\ \hfill & +{\overline{H}}_{ij}\left(x_j\left(t\right)\right)\left(f_j\left(\gamma y_j\left(t\right)\right)-f_j\left(y_j\left(t\right)\right)\right)\hfill \end{array}$$

(17)

where $f_j\left(e_j\left(t\right)\right)=f_j\left(x_j\left(t\right)\right)-f_j\left(\gamma y_j\left(t\right)\right)$, in which the following is the case:

$${\overline{H}}_{ij}\left(x_j\left(t\right)\right)f_j\left(e_j\left(t\right)\right)\le |h_{ij}^{*}|F_j\left|\left(e_i\left(t\right)\right)\right|$$

$$\left({\overline{H}}_{ij}\left(x_j\left(t\right)\right)-\gamma H_{ij}\left(y_j\left(t\right)\right)\right)f_j\left(y_j\left(t\right)\right)\le (1-\gamma )\left|h_{ij}^{*}\right|M_j$$

Based on the above assumptions, it follows that

$$\begin{array}{cc}\hfill D_t^{\beta }V& \le \sum _{i=1}^n\{\left|z_i\left(t\right)\right|-\left|e_i\left(t\right)\right|-a_i\left|z_i\left(t\right)\right|+(a_i-b_i)\left|e_i\left(t\right)\right|\hfill \\ \hfill & +\sum _{j=1}^n\left(|h_{ij}^{*}|F_j\left|e_i\left(t\right)\right|+(1-\gamma )\left|h_{ij}^{*}\right|M_j\right)\hfill \\ \hfill & +l_i\left|e_i\left(t\right)\right|-k_i\left|z_i\left(t\right)\right|-{\lambda }_i{D}_t^{\alpha -2\beta }\left|z_i\left(t\right)\right|-{\chi }_i\}\hfill \\ \hfill & \le \sum _{i=1}^n\{(1-a_i-k_i)\left|z_i\left(t\right)\right|+\left((a_i-b_i)-1+\sum _{j=1}^n\left|h_{ij}^{*}\right|F_j+l_i\right)\left|e_i\left(t\right)\right|\hfill \\ \hfill & +\sum _{j=1}^n(1-\gamma )\left|h_{ij}^{*}\right|M_j-{\chi }_i-{\lambda }_i{D}_t^{\alpha -2\beta }\left|z_i\left(t\right)\right|\}\hfill \end{array}$$

(18)

Given the specified conditions, it follows that

$$\begin{array}{cc}\hfill D_t^{\beta }V& =D_t^{\beta }\sum _{i=1}^n{V}_i\left(t\right)\hfill \\ \hfill & \le \sum _{i=1}^n\left(-{\lambda }_i\left[\sum _{i=1}^n|e_i\left(t\right)|+\sum _{i=1}^n{D}_t^{\alpha -2\beta }\left|z_i\left(t\right)\right|\right]+\sum _{j=1}^n(1-\gamma )\left|h_{ij}^{*}\right|M_j-{\chi }_i\right)\hfill \end{array}$$

(19)

The variable ${\eta }_i$ is defined as ${\chi }_i-{\displaystyle \sum _{j=1}^n}(1-\gamma )\left|h_{ij}^{*}\right|M_j$. Let $\overline{\eta }=\underset{1\le i\le n}{min}{\eta }_i$. Consequently, the fractional derivative of V can be expressed as follows:

$$\begin{array}{c}\hfill D_t^{\beta }V\le \sum _{i=1}^n\left(-{\lambda }_i{V}_i\left(t\right)-{\eta }_i\right)\le -\overline{\lambda }V\left(t\right)-\overline{\eta }\end{array}$$

Projective finite-time synchronization between FOIMNNs (2) and (7) can be achieved, as concluded from Lemma 1. The specific finite time is estimated by $t^{*}\le {\widehat{t}}^{*}$, and ${\widehat{t}}^{*}$ can be estimated as follows:

$${\widehat{t}}_1^{*}\le t_0+{\left({\displaystyle \frac{\Gamma (\alpha +1)v\left(t_0\right)}{\overline{\eta }}}\right)}^{{\displaystyle \frac{1}{\alpha }}}$$

   □

Remark 2.

The connection weights of FOMINNs (2) and (7) are state-dependent, categorizing these systems as state-dependent switching systems. When these weights are equal, the systems reduce to fractional-order inertial neural networks [30], making the findings of this paper relevant to their FTPS. Consequently, this study complements existing research on fractional-order inertial neural networks.

Remark 3.

The wide acknowledgment of finite-time synchronization necessity in practical applications is evident. Therefore, the finite-time synchronization results presented in this study, which are more general than those of exponential and asymptotic synchronization, are distinct from the results found in complex dynamical networks [31] and memristive neural networks [32].

Remark 4.

Projective synchronization allows for the flexible adjustment of the synchronization scale, enabling synchronization at different proportions as needed. This method enhances flexibility in managing complex systems. Unlike traditional synchronization [24,25], which demands full coherence between systems, projective synchronization allows for synchronization at predetermined ratios, thereby broadening its applicability in practical scenarios.

Theorem 2.

Consider drive-response FOIMNNs under Assumption 1. The system can achieve asymptotic projective synchronization if the following control strategy is applied:

$$\begin{array}{c}\hfill u_i\left(t\right)=-l_i{e}_i\left(t\right)-k_i{z}_i\left(t\right)-D_t^{\alpha -\beta }{e}_i\left(t\right)-\mathit{sign}\left(z_i\left(t\right)\right){\chi }_i\\ \hfill -{\overline{H}}_{ij}\left(x_j\left(t\right)\right)\left(f_j(\gamma y_j\left(t\right)-f_j\left(y_j\left(t\right)\right)\right)-(1-\gamma )I_i\end{array}$$

(20)

For the control strategy to be effective, the following conditions must be satisfied for each i:

$$\left\{\begin{array}{c}1-a_i-k_i\le 0\hfill \\ (a_i-b_i)-1+{\displaystyle \sum _{j=1}^n}\left|h_{ij}^{*}\right|F_j+l_i\le 0\hfill \\ {\displaystyle \sum _{j=1}^n}(1-\gamma )\left|h_{ij}^{*}\right|M_j={\chi }_i\hfill \end{array}\right.$$

(21)

Under these conditions, the control strategy guarantees that the drive-response FOIMNNs achieve asymptotic projective synchronization.

Proof of Theorem 2.

Combining Equation (18) with the conditions provided in Theorem 1, we can conclude that $D_t^{\beta }V<0$.

Based on Lemma 2, we can conclude that, under control scheme (20), leader system (2) and follower system (7) achieve global asymptotic projective synchronization.    □

Remark 5.

The projective synchronization’s application is broadened to include complete synchronization $(\gamma =1)$ and anti-synchronization $(\gamma =-1)$ in driver-response systems, depending on the γ values. Consequently, this analysis results in two corollaries.

Corollary 1.

The finite-time complete synchronization of the FOIMNNs, described by Equations (2) and (7), is guaranteed when the scaling factor γ is set to 1 in error system (11) and controller (12), under Assumption 1, and the sufficient condition (14) is satisfied.

Corollary 2.

The finite-time anti-synchronization of the FOIMNNs, described by Equations (2) and (7), is guaranteed when the scaling factor γ is set to $-1$ in error system (11) and controller (12), under Assumption 1, and the sufficient condition (14) is satisfied.

Remark 6.

This study discusses networks with fractional-order inertia characterized by fractional derivatives of orders α and β. This differs from the commonly seen models in the existing literature, which typically involve derivatives of orders α and $2\alpha$ [16,17]. Through employing this unique approach, this paper extends the depth and breadth of our understanding of fractional-order network systems, providing a new perspective and methodological foundation for future research.

Table 1. Overview of comparisons between the proposed method and related works.

References	System Type	Synchronization Strategy	Synchronization Type	Application Area
[24,25]	INNs	Complete Synchronization	Asymptotic Synchronization	No application mentioned in those references
[29,32]	FOMNNs	Complete Synchronization	Asymptotic Synchronization	No application mentioned in those references
[33]	FONNs	Projective Synchronization	Asymptotic Synchronization	No application mentioned in this references
[34]	NNs	Complete Synchronization	Finite-Time Synchronization	Image Encryption
This paper	FOIMNNs	Projective Synchronization	Finite-Time Synchronization	Image Encryption

provides comparisons between the proposed method and recent related works for enhanced understanding.

4. Numerical Examples for FTPS of FOIMNNs

To demonstrate the efficacy of Theorem 1, a simulation involving a three-dimensional fractional-order inertial memristive neural network (FOIMNN) model is presented:

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{r}_i\left(t\right)=-a_i{r}_i-(b_i-a_i)x_i\left(t\right)+\sum _{j=1}^n{h}_{ij}\left(x_j\left(t\right)\right)f_j\left(x_j\left(t\right)\right)+D_t^{\alpha -\beta }{x}_i\left(t\right)+I_i\hfill \\ D_t^{\beta }{x}_i\left(t\right)=r_i\left(t\right)-x_i\left(t\right)\hfill \end{array}\right.$$

where $i=1,2,3,$$a_1=2, a_2=4.0, a_3=2.8; b_1=2, b_2=3.9, b_3=2.7; I_1=0, I_2=0, I_3=0;$ and activation functions $f_1\left(x\right)=f_2\left(x\right)=f_3\left(x\right)=tanh\left(x\right)$ and $x_1\left(0\right)=1.1, x_2\left(0\right)=-1.3,x_3\left(0\right)=1.$ It is evident that the activation function meets the conditions of Assumption 1. Details of the system’s memristive connection weights are provided below:

$$\begin{array}{cc}{h}_{11}\left(x_1\left(t\right)\right)=\left\{\begin{array}{cc}0.55\hfill & |x_1\left(t\right)|\le 1\hfill \\ 0.54\hfill & |x_1\left(t\right)|>1\hfill \end{array}\right.\hfill & h_{12}\left(x_2\left(t\right)\right)=\left\{\begin{array}{cc}-0.5\hfill & |x_2\left(t\right)|\le 1\hfill \\ -0.56\hfill & |x_2\left(t\right)|>1\hfill \end{array}\right.\hfill \\ h_{13}\left(x_3\left(t\right)\right)=\left\{\begin{array}{cc}0.42\hfill & |x_3\left(t\right)|\le 1\hfill \\ 0.4\hfill & |x_3\left(t\right)|>1\hfill \end{array}\right.\hfill & h_{21}\left(x_1\left(t\right)\right)=\left\{\begin{array}{cc}0.29\hfill & |x_1\left(t\right)|\le 1\hfill \\ 0.4\hfill & |x_1\left(t\right)|>1\hfill \end{array}\right.\hfill \\ h_{22}\left(x_2\left(t\right)\right)=\left\{\begin{array}{cc}0.23\hfill & |x_2\left(t\right)|\le 1\hfill \\ 0.28\hfill & |x_2\left(t\right)|>1\hfill \end{array}\right.\hfill & h_{23}\left(x_3\left(t\right)\right)=\left\{\begin{array}{cc}-0.32\hfill & |x_3\left(t\right)|\le 1\hfill \\ -0.3\hfill & |x_3\left(t\right)|>1\hfill \end{array}\right.\hfill \\ h_{31}\left(x_1\left(t\right)\right)=\left\{\begin{array}{cc}-0.61\hfill & |x_1\left(t\right)|\le 1\hfill \\ -0.76\hfill & |x_1\left(t\right)|>1\hfill \end{array}\right.\hfill & h_{32}\left(x_2\left(t\right)\right)=\left\{\begin{array}{cc}0.8\hfill & |x_2\left(t\right)|\le 1\hfill \\ 0.78\hfill & |x_2\left(t\right)|>1\hfill \end{array}\right.\hfill \\ h_{33}\left(x_3\left(t\right)\right)=\left\{\begin{array}{cc}0.25\hfill & |x_3\left(t\right)|\le 1\hfill \\ 0.3\hfill & |x_3\left(t\right)|>1\hfill \end{array}\right.\hfill & \end{array}$$

The response system is

$$\left\{\begin{array}{c}{D}_t^{\alpha -\beta }{s}_i\left(t\right)=-a_i{s}_i+(a_i-b_i)y_i\left(t\right)+{\displaystyle \sum _{j=1}^n}{h}_{ij}\left(y_j\left(t\right)\right)f_j\left(y_j\left(t\right)\right)+D_t^{\alpha -\beta }{y}_i\left(t\right)+u_i\left(t\right)+I_i\hfill \\ D_t^{\beta }{y}_i\left(t\right)=s_i\left(t\right)-y_i\left(t\right)\hfill \end{array}\right.$$

Define $z_i\left(t\right)=r_i\left(t\right)-\gamma s_i\left(t\right)$ and $e_i\left(t\right)=x_i\left(t\right)-\gamma y_i\left(t\right)$, $i=1,2,3$, as the synchronization error system, where $0\le \beta <1$, $1\le \alpha <2\beta$. Set the parameters as follows: $\alpha =1.9$, $\beta =0.97$. And the initial conditions are $y_1\left(0\right)=2.1,y_2\left(0\right)=-2.3,$ and $y_3\left(0\right)=2.$ From inequalities (14), set the parameters as follows: $l_1=0.36$, $l_2=0.238$, $l_3=0.308$, $k_1=12$, $k_2=7$, $k_3=20$, ${\lambda }_1=0.2$, ${\lambda }_2=0.3$, ${\lambda }_3=0.1$, ${\chi }_1=3$, ${\chi }_2=2$, and ${\chi }_3=3$. Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 illustrate the state trajectories for $\gamma$ values of -1, -3, and 1, respectively. The synchronization error curves for drive system (2) and response system (7) at $\gamma =1$ are depicted in Figure 7 and Figure 8. Thus, the validation of the synchronization criterion and the correctness of Theorem 1 are confirmed.

5. Application of FTPS to Image Protection

This section applies Corollary 2 to image protection by establishing a synchronization mechanism between the drive and response systems. The drive system sequences the encryption by generating key synchronization signals based on Corollary 2, ensuring secure data transformation. The response system, using the same synchronization principle, manages the decryption by reconstructing the original image from the encrypted data.

5.1. Detailed Steps for Image Encryption

The implementation of image encryption involves four specific steps:

• Pixel-Value Extraction: Initially, each pixel of the image is extracted in terms of its RGB color space representation, i.e., the pixel values for the R, G, and B channels. Let the image be denoted as I; then, $I=\{I_R,I_G,I_B\}$, $I_R,I_G,$ and $I_B$ denote the pixel matrices corresponding to the red, green, and blue channels.
• Image Disruption Process: In using a pseudo-random sequence $S_D$ generated by the drive system, the image’s one-dimensional vector form undergoes disruption. In this step, each color channel of the image $I_R,I_G,$ and $I_B$ is unrolled into one-dimensional vectors $V_R,V_G,$ and $V_B$. These vectors are then permuted by the sequence $S_D$ to obtain the disrupted vectors $V_R^{\prime },V_G^{\prime },$ and $V_B^{\prime }$.
• Encryption Operation: The disrupted vectors $V_R^{\prime },V_G^{\prime },$ and $V_B^{\prime }$ are encrypted using diffusion sequences $S_F$ and $S_R$ also generated by the drive system, applying both forward and reverse diffusions. Specifically, for each disrupted vector $V^{\prime }$, the operation $E\left(V^{\prime }\right)=S_F\left(V^{\prime }\right)\oplus S_R\left(V^{\prime }\right)$ is performed, where ⊕ denotes the bitwise exclusive OR operation, and $S_F$ and $S_R$ are the forward and reverse diffusion sequences, respectively.
• Reassembly of Encrypted Image: Finally, the encrypted vectors $E\left(V_R^{\prime }\right),E\left(V_G^{\prime }\right),$ and $E\left(V_B^{\prime }\right)$ are rearranged and recombined to restore the two-dimensional format of the image, producing the encrypted image $I_{enc}$. In the decryption process, using sequences from the response system in reverse order for diffusion and the inverse of the disruption operation, the original image can be successfully recovered.

The core of this method lies in the design of the system’s disruption and encryption sequences, enhancing the security of the image data while maintaining the reversibility of decryption, ensuring that the image can be correctly restored under authorized access. The following Algorithm 1 outlines the steps for the image encryption process:

Algorithm 1 Image Encryption

Input: $I_R,I_G,I_B,S_D,S_F,S_R,M,N$ Output: $I_{enc}$ // Step 1: Pixel-Value Extraction $V_R=\mathrm{flatten}\left(I_R\right)$ $V_G=\mathrm{flatten}\left(I_G\right)$ $V_B=\mathrm{flatten}\left(I_B\right)$ // Step 2: Image Disruption Process for $i=1$ to $M\times N$ do    $idx=\mathrm{mod}(S_D\left(i\right)+i,M\times N)+1$    $V_R^{\prime }\left(i\right)=V_R\left(idx\right)$    $V_G^{\prime }\left(i\right)=V_G\left(idx\right)$    $V_B^{\prime }\left(i\right)=V_B\left(idx\right)$ end for // Step 3: Encryption Operation for $i=1$ to $M\times N$ do    $E\left(V_R^{\prime }\left(i\right)\right)=V_R^{\prime }\left(i\right)\oplus S_F\left(i\right)\oplus S_R\left(i\right)$    $E\left(V_G^{\prime }\left(i\right)\right)=V_G^{\prime }\left(i\right)\oplus S_F\left(i\right)\oplus S_R\left(i\right)$    $E\left(V_B^{\prime }\left(i\right)\right)=V_B^{\prime }\left(i\right)\oplus S_F\left(i\right)\oplus S_R\left(i\right)$ end for // Step 4: Reassembly of Encrypted Image $I_{en{c}_R}=\mathrm{reshape}(E\left(V_R^{\prime }\right),M,N)$ $I_{en{c}_G}=\mathrm{reshape}(E\left(V_G^{\prime }\right),M,N)$ $I_{en{c}_B}=\mathrm{reshape}(E\left(V_B^{\prime }\right),M,N)$ $I_{enc}=\{I_{en{c}_R},I_{en{c}_G},I_{en{c}_B}\}$ return $I_{enc}$

5.2. Example

This section assesses the FTS encryption method’s effectiveness in protecting images by evaluating key metrics including histogram uniformity, correlation coefficients, and information entropy. The analysis of these metrics offers a detailed insight into the method’s robustness and reliability.

5.2.1. Histogram Analysis

This section elaborates on the process of encrypting and subsequently decrypting an image (using ‘Baboon’ as an example) utilizing a specific algorithm. Figure 9a,b show the Baboon image after encryption. The image content is scrambled effectively. The effectiveness of the encryption is demonstrated by analyzing the histograms of the gray value distribution for Figure 9e,f. The histogram $H\left(v\right)$ can be expressed as

$$H\left(v\right)=\sum _{i=1}^n\delta (v_i-v)$$

where v represents the gray value, n is the total number of pixels, and $v_i$ is the gray value of pixel i. The Kronecker delta function $\delta$ counts the occurrences of each gray value v, indicating counting occurrences of each gray value v. The analysis shows consistent gray distributions, indicating that the original image information is uniformly and randomly dispersed.

Further analysis is presented in Figure 9g, which shows the distribution of gray values across each channel post encryption. The histogram shows that the gray value is fully obscured, concealing the image’s original content and protecting the information. Following the decryption, Figure 9d illustrates the complete restoration of the original Baboon image. Figure 9h shows that the gray value distribution of the decrypted image perfectly matches the original, as demonstrated by

$$H_{dec}\left(v\right)=H_{orig}\left(v\right)$$

where $H_{dec}\left(v\right)$ and $H_{orig}\left(v\right)$ are the histograms of the decrypted and original images, respectively.

This process not only validates the effectiveness of the employed encryption and decryption algorithms but also suggests the possible application of these algorithms in securing image data.

5.2.2. Correlation Coefficients

Correlation coefficients quantify the strength of a linear relationship between two variables. In the context of image encryption, the correlation of adjacent pixel values is investigated in the horizontal, vertical, and diagonal directions before and after encryption. These coefficients are calculated using the following formula:

$$r_{xy}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\displaystyle \sum _{i=1}^n}{(x_i-\overline{x})}^2{\displaystyle \sum _{i=1}^n}{(y_i-\overline{y})}^2}}}$$

where $x_i$ and $y_i$ represent adjacent pixel values in the specified direction, $\overline{x}$ and $\overline{y}$ are the mean values of these pixel pairs, and n is the total number of pixel pairs.

To visually present these data, scatter plots are employed to depict the relationship between pixel values of images before and after encryption. For unencrypted images, the scatter plot shows a high degree of linear association between adjacent pixels, as shown in Figure 10, whereas the encrypted images display a significantly reduced correlation in their scatter distribution, as shown in Figure 11, indicating a substantial decrease in correlation. The different colors in the scatter plots represent pixel values from the red, green, and blue channels: red dots for the red channel, yellow for green, and blue for blue.

This method allows us to clearly see how the encryption algorithm effectively reduces the pixel correlation, breaking the direct association between adjacent pixels in the image. These results highlight the encryption algorithm’s ability to disrupt the linear relationship between pixels, thereby improving the unpredictability of pixel values after encryption.

5.2.3. Information Entropy

Information entropy, quantifying the randomness or unpredictability in a dataset, serves as a critical metric for assessing the efficiency of encryption algorithms. In the context of image encryption, the entropy for each color channel—red, green, and blue—is computed using the designated formula:

$$H\left(X\right)=-\sum _{i=1}^{n}p\left(x_i\right){log}_{2}p\left(x_i\right)$$

where $p\left(x_i\right)$ represents the probability of occurrence of pixel value $x_i$ in the image, and n is the number of possible pixel values.

The information entropy for the original image, along with its encrypted and decrypted versions, was calculated in this study.

Table 2. Information entropy of original and encrypted images.

Images	R Channel	G Channel	B Channel
Baboon image	7.7357	7.4544	7.7736
Baboon encrypted image	7.9993	7.9991	7.9993

displays the results. An increase in information entropy across all color channels, as shown by the data, indicates that the encryption algorithm significantly enhanced the image’s randomness and security. After decryption, the information entropy returned to the same level as the original image, validating the reversibility of the encryption algorithm and ensuring that the image can be correctly restored under authorized access. These results not only demonstrate the effectiveness of the encryption algorithm in protecting image content but also suggest its potential contribution to information security and privacy protection.

These metrics affirm that the FTS encryption method provides robust protection for digital images, effectively obscuring original content and making it more challenging for unauthorized access.

5.3. Comparison

The data analyses presented in

Table 3. Correlation coefficients for RGB channels of plain and encrypted images.

Image	Channel	Plain Image Horizontal	Plain Image Vertical	Plain Image Diagonal	Encrypted Image Horizontal	Encrypted Image Vertical	Encrypted Image Diagonal
Proposed	Red	0.9319	0.8695	0.8468	−0.0012	0.0017	−0.0019
	Green	0.8963	0.8104	0.7715	0.0007	0.0021	0.0027
	Blue	0.9364	0.8863	0.8603	0.0006	0.0003	−0.0010
[34]	Red	0.9892	0.9698	0.9798	−0.0019	0.0063	0.00023
	Green	0.9827	0.9556	0.9691	−0.0030	0.0032	0.0028
	Blue	0.9576	0.9185	0.9328	0.0024	−0.0014	−0.0057
[35]	Red	0.9575	0.9444	0.9272	−0.0073	−0.0011	−0.0061
	Green	0.9769	0.9964	0.9537	0.0010	−0.0020	0.0058
	Blue	0.9539	0.9375	0.9130	−0.0013	0.0078	−0.0003
[3]	Red	0.9766	0.9869	0.9638	−0.0033	0.0119	−0.0108
	Green	0.9766	0.9881	0.9646	−0.0029	0.0113	−0.0213
	Blue	0.9535	0.9730	0.9314	−0.0040	0.0016	0.01628

and

Table 4. Performance comparison for RGB channels on Baboon image.

Algorithm	Red	Green	Blue
Proposed	7.9993	7.9991	7.9993
[34]	7.9993	7.9993	7.9993
[35]	7.9966	7.9972	7.9967
[3]	7.9993	7.9992	7.9994

illustrate that the image encryption algorithm proposed in this study significantly outperforms existing methods across several key performance metrics. The pixel correlation indices for encrypted images are particularly notable, indicating modest improvements in reducing pixel interdependencies.

Specifically, the correlation achieved by the algorithm in the horizontal direction for the red channel was −0.0012. compared to −0.0190 and −0.0033 in references [3,34], respectively. This demonstrates a correlation value closer to zero, indicating a substantial reduction in pixel dependency, thereby enhancing the security of the encryption. Additionally, for the green channel in the vertical direction, a correlation of 0.0021 was observed using the algorithm, which is lower than 0.0032 and 0.0113 reported in references [3,34], further demonstrating its improved encryption performance.

In this study, the proposed image encryption algorithm demonstrated superior performance, achieving entropy values of 7.9993, 7.9991, and 7.9993 for the red, green, and blue channels, respectively, nearly reaching the theoretical maximum entropy of 8.0000. These figures exceed the entropy values reported in reference [35], which were 7.9966 for the red channel, 7.9972 for the green channel, and 7.9967 for the blue channel. This indicates that the algorithm effectively increases the randomness of image data across all three channels, thereby enhancing image security.

Particularly in the green channel, the algorithm’s entropy value of 7.9991, which is an increase of 0.0019 over the 7.9972 reported in reference [35], might seem small but it is meaningful in the assessment of encryption security. Even minor increases can substantially raise the difficulty of cracking the encryption. Additionally, in both the red and blue channels, this algorithm’s entropy values match those of the high-performing algorithms noted in reference [3].

The differences observed in the results indicate the potential of this algorithm in reducing the predictability between image pixels, contributing to the protection of image data privacy and enhancing security. By optimizing information entropy values and reducing pixel correlation in encrypted images, the algorithm demonstrates an ability to defend against statistical analysis-based recovery attempts. The comparison of data in this study offers useful technical references and practical insights for further research on image encryption algorithms.

Remark 7.

Chaotic (unpredictable) solutions are necessary for the proposed method to be used in encryption/decryption processes. A set of parameters was chosen, causing the FOIMNNs to exhibit chaos. Without chaotic characteristics based on the chosen parameter setting, the proposed method cannot be used for image encryption/decryption.

Remark 8.

The definition of fractional-order derivatives allows them to be reduced to integer-order derivatives under certain conditions. However, the adoption of fractional orders in this study was driven by clear research needs. Current research on projective synchronization mainly focuses on integer-order systems, which present limitations in security applications. This paper introduces techniques of fractional-order chaotic systems to enhance system complexity and unpredictability, which are crucial for communication security [36]. The increased complexity enhances the system’s defense capabilities against external attacks, thereby securing the transmission of information.

6. Conclusions

In this study, the synchronization issues of FOIMNNs were investigated and their potential application in image encryption. A novel synchronization control strategy was introduced, achieving FTPS. The effectiveness of the proposed method was verified through simulations. In practical applications to image encryption, the proposed designed method showed potential in enhancing the security of image data. The results suggest that the encryption technique has the potential to protect image content and support the recovery of the original image under authorized access. While these results are preliminary, they suggest possible directions for future research in secure communications and data protection. Building on this foundation, they point to future research directions such as refining the synchronization control strategy and improving encryption efficiency.