A Symmetric Multistable Chaotic System Optimized by Chaotic Particle Swarm for Secure Electric Vehicle Communication

Abstract

Secure real-time communication is a critical requirement in modern electric vehicle (EV) networks. These networks transmit safety-critical control commands through vulnerable in-vehicle communication channels. This study proposes a novel three-dimensional symmetric chaotic system for high-security EV communication. The system exhibits extensive multistability and symmetric double-wing attractors. To enhance dynamical complexity, its parameters are optimized using chaotic-enhanced particle swarm optimization (C-PSO). The largest Lyapunov exponent is used as the optimization objective. A fixed-time nonlinear controller is designed for rapid drive–response synchronization. The settling-time bound is independent of the initial conditions. The proposed method is evaluated through realistic Controller Area Network (CAN) bus simulations. These simulations include 12-bit quantization and a 1 ms sampling period. The experimental results show synchronization within 0.057 s. The recovered signal achieves an MSE of $1.202\times {10}^{-4}$. The encrypted signal reaches a Shannon entropy of 7.9904. These results confirm accurate recovery, strong randomness, and improved resistance to cryptographic attacks.

1. Introduction

Chaotic dynamical systems have attracted significant interest in science and engineering. They show strong sensitivity to initial conditions and system parameters [1,2,3]. This sensitivity produces irregular and noise-like trajectories. These trajectories are highly unpredictable over long time horizons. Consequently, chaotic signals are attractive for physical-layer security and information masking [4,5,6,7]. Recently, chaos-based methods have advanced beyond theoretical nonlinear dynamics. They are now applied in several practical domains. Important applications include secure communication [8,9,10]. Other applications include image and video encryption [11,12,13,14]. Biomedical signal protection has also been investigated [15,16]. Chaos-based masking offers a lightweight alternative to conventional cryptography. Its protection mechanism relies directly on system dynamics. This feature reduces the need for computationally intensive encryption primitives [17,18,19].

A central requirement in chaos-based communication is synchronization. This synchronization must be achieved between the drive and response systems. Early studies mainly focused on asymptotic synchronization. In this case, the synchronization error converges to zero as time tends to infinity. This property is mathematically important. However, it is not sufficient for time-critical applications. The transient duration depends strongly on the initial conditions. It may also become excessively long in practice. For this reason, recent studies have shifted toward finite-time and fixed-time synchronization methods [20,21,22,23,24,25,26]. Fixed-time stability provides a stronger convergence guarantee. Its settling-time upper bound is independent of the initial synchronization error [27,28,29]. This property is useful for cyber-physical systems. Such systems require deterministic timing and rapid recovery [30,31,32,33].

A second challenge concerns chaotic-system parameter selection. Many chaos-based security studies choose parameters empirically. Other studies rely on trial-and-error tuning. This approach may place the system near periodic windows. It may also produce weakly chaotic regimes. Such operating points reduce unpredictability. They also degrade the effectiveness of the masking process. Meta-heuristic optimization methods provide a systematic alternative. They explore the parameter space using quantitative dynamical criteria [34,35]. Particle Swarm Optimization (PSO) and chaos-enhanced variants are widely used for this task. These methods have simple structures and efficient global-search capabilities [36,37,38,39,40]. They can also be combined with chaos metrics. One important metric is the largest Lyapunov exponent. This metric identifies regions with stronger divergence and richer attractor topology. Consequently, resistance to prediction and reconstruction attempts can be improved [41,42].

These security issues are relevant in automotive cyber-physical systems. Modern electric vehicles rely on in-vehicle networks. A common example is the Controller Area Network (CAN) bus. These networks exchange safety-critical messages among electronic control units. These messages include torque, braking, and steering commands [43,44]. However, legacy in-vehicle protocols have a broadcast structure. They also provide limited built-in protection. This exposes them to injection, replay, spoofing, and denial-of-service attacks [45,46,47,48,49]. In such environments, communication integrity and timing are tightly coupled. A delayed, corrupted, or forged command can compromise vehicle stability. It can also degrade operational safety. Such vulnerabilities show that safety-critical EV commands require stronger protection against communication attacks. Chaos-based nonlinear methods have been investigated for secure signal generation and encryption [50,51]. Machine-learning and deep-learning intrusion detection systems have received considerable attention [52,53,54,55]. However, these approaches are usually reactive. They may also introduce significant computational overhead. Variable inference latency may also occur on resource-constrained automotive hardware [56,57,58,59,60,61].

Recent CAN-security studies have addressed several detection problems. These problems include statistical IDS methods and multi-attack CAN-FD detection [62,63]. Anomaly localization and autoencoder-based CAN detection have also been investigated [64,65]. Lightweight protocol-level protection has been studied for intelligent in-vehicle CAN buses [66]. Chaos-based anomaly detection has also been explored for cyber–physical systems [67]. Secure communication for electric-drive systems has been studied using chaotic dynamics [68]. Vehicle-to-vehicle and autonomous EV threat landscapes have also been reviewed [69].

To improve reproducibility, clear selection criteria were used for the literature review. The review focused on chaos-based security, fixed-time control, optimization, and EV cybersecurity. Peer-reviewed articles, conference papers, and recent reviews were considered. Recent studies from 2019 to 2026 were prioritized. Foundational works were retained when essential theories or methods were introduced.

Table 1. Summary of reviewed research directions and identified limitations.

Research Direction	Representative References	Main Focus and Limitation
Chaotic systems and chaos-based security	[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]	Chaotic dynamics, multistability, encryption, masking, and secure communication are studied. Automotive real-time constraints are rarely addressed.
Fixed-time synchronization and control	[19,20,21,22,23,24,25,26,27,28,29,30]	Bounded-time convergence is studied for nonlinear and chaotic systems. EV torque recovery is rarely considered.
PSO and chaotic optimization	[31,32,33,34,35,36,37,38]	Meta-heuristic methods are used for optimization and parameter selection. CAN-bus security constraints are usually not included.
Automotive and EV cybersecurity	[39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70]	Vehicle attacks, EV risks, IDS methods, and lightweight protection are reviewed. Many methods remain reactive or detection-oriented.

summarizes the reviewed directions and limitations. Most existing studies address only one part of the problem. Therefore, a combined framework is needed for secure real-time EV communication.

This work is motivated by the need for preventive security in EV communication networks. A new symmetric chaotic carrier is introduced to mask safety-critical torque commands before transmission. Its parameters are selected by C-PSO to improve chaotic complexity. This optimization also avoids weak chaotic or periodic operating regions. A fixed-time controller is then used to recover the command within a predictable time. The method is tested under CAN-oriented conditions. These conditions include quantization, sampling, noise, delay, and packet loss. The results show synchronization within $0.057$ s. The recovered-torque MSE reaches $1.202\times {10}^{-4}$. The encrypted signal achieves a Shannon entropy of $7.9904$. Robust recovery is also maintained under noise, interruption, and network impairments. Computational profiling further supports implementation on automotive-grade ECUs.

The conceptual method integrates three main domains. These domains are nonlinear dynamics, meta-heuristic optimization, and automotive security. Figure 1 provides a visual summary of these links. It shows how C-PSO optimizes the symmetric chaotic carrier. It also shows how the carrier supports secure EV communication.

The remainder of this paper is organized as follows. Section 2 introduces the proposed chaotic model. It also analyzes the system dynamical properties. Section 3 develops the fixed-time synchronization controller. It also presents the associated stability proof. Section 4 describes the C-PSO procedure. It also reports the optimization results. Section 5 presents the secure EV communication architecture. It also provides the security and robustness analyses. Finally, Section 6 concludes the paper. It also outlines future research directions.

2. Mathematical Modeling and Dynamical Analysis

This section presents the mathematical model and dynamical analysis of the proposed chaotic system. First, the governing three-dimensional equations are defined. These equations include cubic and transcendental nonlinearities. These nonlinearities increase the system complexity. Then, the symmetry and equilibrium points are analyzed. Finally, the extensive multistability of the system is examined.

2.1. Proposed Symmetric Chaotic System

A novel three-dimensional symmetric chaotic system is introduced in this work. The model combines polynomial and transcendental nonlinearities. This structure generates complex dynamical behavior. It also supports multistability and double-wing attractors.

The system is formulated as follows:

$$\left\{\begin{array}{c}\dot{x}=ay{z}^3\hfill \\ \dot{y}=x-by\hfill \\ \dot{z}=cos\left(zy\right)-xy\hfill \end{array}\right.$$

(1)

where x, y, and z denote the state variables, and $a,b>0$ are system parameters.

A basic requirement for the existence of a bounded attractor is dissipativity. For the vector field $V=(\dot{x},\dot{y},\dot{z})$, the divergence is given by:

$$\nabla ·V={\displaystyle \frac{\partial \dot{x}}{\partial x}}+{\displaystyle \frac{\partial \dot{y}}{\partial y}}+{\displaystyle \frac{\partial \dot{z}}{\partial z}}=-b-ysin\left(zy\right)$$

(2)

Since $-b$ is negative for $b>0$, the average divergence may remain negative. This occurs over a broad region of the phase space. The oscillatory term $-ysin\left(zy\right)$ may locally modify the divergence. However, the overall dynamics can still be dissipative. Under this condition, the phase-space volume contracts exponentially according to:

$${\displaystyle \frac{d\mathcal{V}}{dt}}=e^{\langle \nabla ·V\rangle t}{\mathcal{V}}_0$$

(3)

where ${\mathcal{V}}_0$ is the initial volume element. Thus, the system is dissipative on average. Its trajectories evolve toward a bounded invariant set.

2.2. Equilibrium Points and Stability Analysis

The equilibrium points of Equation (1) are determined by setting the time derivatives to zero:

$$\left\{\begin{array}{c}ay{z}^3=0\hfill \\ x-by=0\hfill \\ cos\left(zy\right)-xy=0\hfill \end{array}\right.$$

(4)

Because $a\ne 0$, the first equation implies either $y=0$ or $z=0$.

Case 1: $y=0$. If $y=0$, then the second equation gives $x=0$. Substituting $(x,y)=(0,0)$ into the third equation yields $cos\left(0\right)-0=1\ne 0$, which is impossible. Hence, no equilibrium point exists on the set $y=0$.

Case 2: $z=0$. If $z=0$, then the third equation reduces to $1-xy=0$, i.e., $xy=1$. Using the second equation, $x=by$, one obtains $b{y}^2=1\Rightarrow y=\pm \sqrt{1/b}$ and $x=\pm \sqrt{b}$. Consequently, the system admits two symmetric equilibrium points:

$$E_{1,2}=\left(\pm \sqrt{b},\pm {\displaystyle \frac{1}{\sqrt{b}}},0\right)$$

(5)

The local behavior near these equilibria is examined through the Jacobian matrix:

$$J=\left[\begin{array}{ccc}0& a{z}^3& 3ay{z}^2\\ 1& -b& 0\\ -y& -zsin\left(zy\right)-x& -ysin\left(zy\right)\end{array}\right]$$

(6)

Evaluating J at $E_{1,2}$, for which $z=0$, gives:

$$J{|}_{E_{1,2}}=\left[\begin{array}{ccc}0& 0& 0\\ 1& -b& 0\\ \mp 1/\sqrt{b}& \mp \sqrt{b}& 0\end{array}\right]$$

(7)

The characteristic equation is $det(J-\lambda I)=-{\lambda }^2(b+\lambda )=0$, and the corresponding eigenvalues are ${\lambda }_1=0,{\lambda }_2=0,{\lambda }_3=-b$.

Since $b>0$, one contracting direction exists through ${\lambda }_3<0$. However, the two zero eigenvalues make the equilibria non-hyperbolic. Therefore, linearization alone cannot fully describe the local dynamics. Higher-order nonlinear terms shape the nearby flow. This degeneracy agrees with nontrivial stretching–folding behavior. It also supports the complex attractor geometry observed numerically.

A quantitative measure of chaos is provided by the Lyapunov exponent spectrum $\left\{L{E}_i\right\}$. For the optimized parameter set $a=9.1782$ and $b=3.4490$, the exponents are computed as:

$$L{E}_1=0.3460,L{E}_2=0,L{E}_3=-3.7467$$

(8)

Because $L{E}_1>0$, the system exhibits exponential divergence of nearby trajectories, which confirms chaotic behavior. In addition, the sum of the exponents is negative, ${\sum }_{i}L{E}_i=-3.3677<0$, consistent with dissipativity.

Figure 2 shows the time evolution of the Lyapunov exponents.

To further quantify the attractor geometry, the Kaplan–Yorke dimension is evaluated as:

$$D_{KY}=j+{\displaystyle \frac{{\sum }_{i=1}^{j}L{E}_i}{|L{E}_{j+1}|}}=2+{\displaystyle \frac{L{E}_1+L{E}_2}{|L{E}_3|}}\approx 2.1012$$

(9)

The fractional value $D_{KY}>2$ supports the fractal character of the attractor.

The geometric manifestation of these properties is illustrated in Figure 3, which presents several phase-space projections of the optimized attractor.

2.3. Dynamical Behaviors and Chaos Existence

To examine chaos persistence, the parameter b is swept over $[3.4,4.6]$. The parameter a is fixed at $9.1782$. The initial condition is set to $(0.5,-0.5,0.5)$. The bifurcation diagram and Lyapunov spectrum are then computed. Figure 4a shows a wide chaotic interval. This behavior is clear for $3.4\le b\le 4.2$. In this interval, the orbit points form a dense band. For larger b values, the band separates into discrete branches. This indicates transitions toward periodic behavior. Figure 4b confirms this result. The largest Lyapunov exponent remains positive in the chaotic region. It decreases near the periodic windows. Therefore, $b=3.4490$ lies inside a robust chaotic regime.

2.4. Symmetry Analysis and Coexisting Attractors

An important property of the proposed system is its invariance under the transformation:

$$(x,y,z)\mapsto (-x,-y,z)$$

(10)

Substituting $(-x,-y,z)$ into Equation (1) preserves the vector-field structure. Therefore, if $\mathbf{X}\left(t\right)=\left(x\right(t),y(t),z(t\left)\right)$ is a trajectory, another valid trajectory also exists. This trajectory is $(-x(t),-y(t),z(t\left)\right)$. This symmetry allows paired coexisting attractors. These attractors are related by reflection in the $(x,y)$ coordinates.

This property is verified numerically in Figure 5. The phase portraits are generated from symmetric initial conditions. The first condition is $X_0^{+}=[0.3,0.3,0.3]$. The second condition is $X_0^{-}=[-0.3,0.3,0.3]$. Different parameter values produce different coexisting motions.

For $a=0.2$ and $b=1.5$, stable periodic orbits are obtained. The two trajectories form simple closed loops in Figure 5a. When $a=0.5$ and $b=1.9$, quasi-periodic invariant sets appear. This behavior is shown in Figure 5b. For $a=8.3$ and $b=2.9$, coexisting chaotic attractors are generated. These attractors are shown in Figure 5c.

3. Design of the Fixed-Time Synchronization Controller

Reliable synchronization is essential in chaotic masking schemes. It allows the authorized receiver to recover the transmitted signal accurately. Conventional asymptotic synchronization converges only as time tends to infinity. This may cause long and unpredictable transients. To overcome this limitation, a fixed-time controller is designed. The controller is applied to the proposed chaotic system.

3.1. Drive–Response Formulation

The transmitter (drive) system is:

$$\left\{\begin{array}{c}{\dot{x}}_m=a{y}_m{z}_m^3\hfill \\ {\dot{y}}_m=x_m-b{y}_m\hfill \\ {\dot{z}}_m=cos\left(z_m{y}_m\right)-x_m{y}_m\hfill \end{array}\right.$$

(11)

The receiver (slave) system with control inputs is:

$$\left\{\begin{array}{c}{\dot{x}}_s=a{y}_s{z}_s^3+u_x\hfill \\ {\dot{y}}_s=x_s-b{y}_s+u_y\hfill \\ {\dot{z}}_s=cos\left(z_s{y}_s\right)-x_s{y}_s+u_z\hfill \end{array}\right.$$

(12)

Define synchronization errors $e_1=x_s-x_m$, $e_2=y_s-y_m$, $e_3=z_s-z_m$. The error dynamics become:

$$\left\{\begin{array}{c}{\dot{e}}_1=a(y_s{z}_s^3-y_m{z}_m^3)+u_x\hfill \\ {\dot{e}}_2=e_1-b{e}_2+u_y\hfill \\ {\dot{e}}_3=\left(cos\left(z_s{y}_s\right)-cos\left(z_m{y}_m\right)\right)-(x_s{y}_s-x_m{y}_m)+u_z\hfill \end{array}\right.$$

(13)

3.2. Controller Design

The controller combines active compensation with nonlinear fixed-time stabilizing terms. This design follows established active-control methods for chaotic synchronization [21,24]. The first part cancels the nonlinear mismatch between the master and slave systems. The remaining terms enforce fixed-time convergence of the synchronization errors. The proposed control law is:

$$\left\{\begin{array}{c}{u}_x=-a(y_s{z}_s^3-y_m{z}_m^3)-k_1\mathrm{sgn}\left(e_1\right)|e_1{|}^{\alpha }-k_2\mathrm{sgn}\left(e_1\right){\left|e_1\right|}^{\beta }\hfill \\ u_y=-(e_1-b{e}_2)-k_1\mathrm{sgn}\left(e_2\right)|e_2{|}^{\alpha }-k_2\mathrm{sgn}\left(e_2\right){\left|e_2\right|}^{\beta }\hfill \\ u_z=-\mathsf{\Delta }\mathsf{\Phi }-k_1\mathrm{sgn}\left(e_3\right)|e_3{|}^{\alpha }-k_2\mathrm{sgn}\left(e_3\right){\left|e_3\right|}^{\beta }\hfill \end{array}\right.$$

(14)

where $\mathsf{\Delta }\mathsf{\Phi }=(cos\left(z_s{y}_s\right)-cos\left(z_m{y}_m\right))-(x_s{y}_s-x_m{y}_m)$, $k_1,k_2>0$ are controller gains, and $\alpha >1$ and $0<\beta <1$ are design parameters. In this work, $\alpha =1.5$ and $\beta =0.7$ are selected. These values provide fast decay for large errors. They also support convergence near the origin.

3.3. Fixed-Time Stability Analysis

The rigorous stability of the proposed synchronization method is established based on the fundamental principles of fixed-time control.

Lemma 1 

(Fixed-Time Stability Criterion [20]). Consider a continuous positive definite Lyapunov function $V\left(t\right)$ that satisfies the following differential inequality:

$$\dot{V}\left(t\right)\le -k_{1}V{\left(t\right)}^{\alpha }-k_{2}V{\left(t\right)}^{\beta }$$

(15)

where the gains satisfy $k_1,k_2>0$. The fractional powers satisfy $\alpha >1$ and $0<\beta <1$. Under these conditions, the origin $(V=0)$ is globally fixed-time stable. The settling-time upper bound is independent of the initial conditions. It is given by:

$$T_{max}={\displaystyle \frac{1}{k_1(\alpha -1)}}+{\displaystyle \frac{1}{k_2(1-\beta )}}$$

(16)

Proof. 

The proof follows from variable separation. It also uses direct integration of the differential inequality. For conciseness, the full derivation is not repeated here. The complete proof is given by Polyakov [20].    □

Theorem 1. 

The synchronization error system in Equation (13) converges to the origin in fixed time under the controller defined in Equation (14).

Proof. 

Consider $V={\displaystyle \frac{1}{2}}(e_1^2+e_2^2+e_3^2)$. Taking the derivative and substituting the control law yields:

$$\dot{V}=-k_1{\displaystyle \sum _{i=1}^3}|e_i{|}^{\alpha +1}-k_2{\displaystyle \sum _{i=1}^3}{|e_i|}^{\beta +1}$$

(17)

Using norm inequalities, there exist constants $K_1,K_2>0$ such that:

$$\dot{V}\le -K_1{V}^{{\displaystyle \frac{\alpha +1}{2}}}-K_2{V}^{{\displaystyle \frac{\beta +1}{2}}}$$

(18)

Let $p={\displaystyle \frac{\alpha +1}{2}}>1$ and $q={\displaystyle \frac{\beta +1}{2}}\in (0,1)$. By Lemma 1, the errors converge to zero in fixed time.    □

From Equation (16), the maximum settling time depends on the controller gains. It also depends on the norm-equivalence constants. For the optimized gains, $k_1=81.51$ and $k_2=132.71$. The evaluated fixed-time upper bound is $T_{max}=0.04965$ s. This agrees with the numerical settling time of about $0.02976$ s.

Remark 1. 

Chattering Suppression for Practical Hardware Implementation: In real EV applications, the discontinuous signum function may cause high-frequency chattering. This chattering appears in the theoretical control inputs of Equation (14). It is undesirable for actuators and power electronic components. To reduce this effect, the signum function is replaced by a smooth approximation. The approximation is $\mathrm{sgn}\left(e_i\right)\approx tanh\left(\kappa e_i\right)$. Here, $\kappa >0$ controls the transition slope. In this study, $\kappa =100$ gives a sufficiently steep approximation. Figure 6 shows that this modification removes switching oscillations. It also preserves convergence within a small bounded error neighborhood.

4. Chaotic Particle Swarm Optimization for Parameter Tuning

The fixed-time controller guarantees bounded-time synchronization. However, secure communication also depends on the chaotic carrier quality. Therefore, a C-PSO strategy is used to select suitable operating parameters. PSO is a population-based optimization method. Here, chaotic sequences are added to the stochastic coefficients. This improves exploration and reduces premature convergence. The particle velocity update is:

$$v_i^{t+1}=w\left(t\right)v_i^t+c_1{r}_1^t\left(p_{best}^t-x_i^t\right)+c_2{r}_2^t\left(g_{best}^t-x_i^t\right)$$

(19)

Here, $w\left(t\right)$ is the inertia weight. The terms $r_1^t$ and $r_2^t$ are normalized chaotic sequences. The term $p_{best}$ denotes the personal best position. The term $g_{best}$ denotes the global best position. The swarm uses 15 particles and 15 iterations. The parameters are $w=0.7$ and $c_1=c_2=1.5$. For clarity and reproducibility, the complete C-PSO-based optimization workflow is summarized in Algorithm 1. The algorithm shows the two-stage procedure used to optimize first the chaotic-system parameters and then the synchronization-controller gains.

Algorithm 1 C-PSO-based two-stage parameter optimization procedure

Require:  Search bounds for $(a,b)$ and $(k_1,k_2)$, number of particles $N_p$, maximum number of iterations $N_{\mathrm{iter}}$, and initial conditions of the chaotic sequence generator. Ensure: Optimized chaotic parameters $(a^{*},b^{*})$ and controller gains $(k_1^{*},k_2^{*})$.   1: Initialize the particle positions and velocities within the prescribed search bounds.   2: Generate and normalize the chaotic sequences used in the C-PSO coefficients $r_1^t$ and $r_2^t$. Stage 1: Chaotic-system parameter optimization   3: for $t=1$ to $N_{\mathrm{iter}}$ do   4:     for each particle $i=1,\dots ,N_p$ do   5:         Simulate the chaotic system using the candidate parameters $(a_i,b_i)$.   6:         Compute the largest Lyapunov exponent $\mathrm{LLE}(a_i,b_i)$.   7:         Evaluate the fitness function $J_1(a_i,b_i)=\mathrm{LLE}(a_i,b_i)$.   8:         Update the personal best position ${\mathrm{pbest}}_i$.   9:     end for 10:     Update the global best position $\mathrm{gbest}$. 11:     Update the particle velocities and positions using the C-PSO rule. 12: end for 13: Store the optimized chaotic parameters $(a^{*},b^{*})$. Stage 2: Controller-gain optimization  14: for $t=1$ to $N_{\mathrm{iter}}$ do 15:     for each particle $i=1,\dots ,N_p$ do 16:         Simulate the drive-response synchronization system using the candidate gains $(k_{1,i},k_{2,i})$. 17:         Compute the synchronization error norm $\parallel e\left(t\right)\parallel$ and the control effort norm $\parallel u\left(t\right)\parallel$. 18:         Compute the fixed-time upper bound $T_{max}(k_{1,i},k_{2,i})$. 19:         Evaluate the composite cost function $J_2(k_{1,i},k_{2,i})$. 20:         Update the personal best position ${\mathrm{pbest}}_i$. 21:     end for 22:     Update the global best position $\mathrm{gbest}$. 23:     Update the particle velocities and positions using the C-PSO rule. 24: end for 25: Store the optimized controller gains $(k_1^{*},k_2^{*})$. 26: Validate the selected parameters through Lyapunov analysis, fixed-time bound verification, and synchronization tests.

The parameter calibration was performed in two sequential stages. First, the chaotic-system parameters $(a,b)$ were calibrated. The objective was to maximize the largest Lyapunov exponent. This step selected a strongly chaotic operating region. Second, the controller gains $(k_1,k_2)$ were calibrated. The objective was to reduce synchronization error and control effort. A fixed-time penalty was also included. This penalty enforced the required settling-time bound. The same population size, iteration number, and search bounds were used in all trials. This procedure improves reproducibility and avoids empirical parameter selection.

4.1. Stage 1: Maximization of Chaotic Complexity

In the first stage, the parameters $(a,b)$ are optimized to maximize the largest Lyapunov exponent (LLE). The objective is:

$$\mathrm{maximize}{J}_1(a,b)=\mathrm{LLE}(a,b)$$

(20)

The convergence history is shown in Figure 7a. The C-PSO algorithm rapidly converges toward $a=9.1782$, $b=3.4490$, yielding $\mathrm{LLE}\approx 0.3485$. A long-horizon validation over $T=500\mathrm{s}$ produces a consistent average $L{E}_1=0.3460$, confirming robust chaos.

4.2. Stage 2: Optimization of Synchronization Performance

Once the chaotic parameters are fixed, the second stage optimizes the controller gains $(k_1,k_2)$ by minimizing a composite performance index. The objective function is defined as

$$\begin{array}{cc}\hfill \mathrm{minimize}{J}_2(k_1,k_2)=& {\int }_0^{T_{\mathrm{sim}}}\parallel e\left(t\right)\parallel dt+\gamma {\int }_0^{T_{\mathrm{sim}}}{\parallel u\left(t\right)\parallel }^{2}dt\hfill \\ & +{\lambda }_T{\left[max\left(0,T_{max}-T_{\mathrm{req}}\right)\right]}^2+{\lambda }_b\mathcal{B}(k_1,k_2),\hfill \end{array}$$

(21)

Here, $\gamma >0$ balances convergence speed and control energy. The value $T_{\mathrm{req}}=0.05$ s defines the required real-time bound. The term $\mathcal{B}(k_1,k_2)$ is a boundary-avoidance penalty. It discourages solutions located at the imposed gain limits. The fixed-time upper bound is computed as

$$T_{max}={\displaystyle \frac{1}{k_1(\alpha -1)}}+{\displaystyle \frac{1}{k_2(1-\beta )}}.$$

(22)

For reproducibility, the gain search space was bounded by $k_1,k_2\in [1,200]$. This interval avoids unrealistically large ECU control efforts. Higher gains increased the control amplitude. They did not provide meaningful settling-time improvement. The objective function includes a fixed-time penalty. It also includes a boundary-avoidance term. This prevents the optimizer from selecting boundary solutions. It also ensures that both gains contribute to convergence.

The convergence history of the C-PSO algorithm is shown in Figure 8a. The optimized controller gains obtained by the reproducible C-PSO procedure are $k_1=81.51$ and $k_2=132.71$. For these values, the theoretical fixed-time upper bound is $T_{max}=0.04965$ s, while the numerical settling time is approximately $0.02976$ s. These results confirm that the selected gains satisfy the required real-time convergence condition without reaching the imposed gain boundaries.

All MATLAB R2024b scripts required to reproduce the two-stage C-PSO optimization procedure are available in the Data Availability Statement.

4.3. Comparative Analysis of Optimization Algorithms

The C-PSO algorithm was compared with standard PSO, GA, and SA. This comparison evaluates the benefit of chaotic perturbation. The same search bounds and iteration settings were used. The comparison considered both optimization stages. Stage 1 maximizes the largest Lyapunov exponent. Stage 2 minimizes the synchronization-performance cost $J_2$.

Table 2. Performance comparison of the optimization algorithms.

Algorithm	Best LLE	Best ${\mathbf{J}}_2$
C-PSO (Proposed)	0.3485	0.0279
Standard PSO	0.3215	0.0583
Genetic Algorithm (GA)	0.3582	0.0498
Simulated Annealing (SA)	0.3491	0.0521

reports the best result for each method. The proposed C-PSO gives the highest LLE in Stage 1. It also gives the lowest synchronization cost in Stage 2. The optimized gains are $k_1=81.51$ and $k_2=132.71$. The corresponding value is $J_2=0.0279$. These results show that chaotic perturbation improves PSO exploration. It also helps avoid premature convergence.

4.4. Comparative Analysis with a Standard Active-Control Method

The proposed controller is compared with a conventional active-control method. Both controllers use the same system parameters. The same optimized chaotic operating point is also used. The initial synchronization mismatch is defined as

$$e\left(0\right)={[-5.1,4.9,9.9]}^T.$$

Thus, the comparison depends only on the control strategy. It is not affected by different initial conditions.

The standard active-control method stabilizes the synchronization error. However, its convergence is asymptotic. It also depends strongly on the initial error. Thus, the error decreases gradually. A longer transient interval is required. In contrast, the proposed controller uses nonlinear compensation. It also includes fixed-time stabilizing terms. The compensation cancels the drive–response mismatch. The fixed-time terms accelerate the error decay. This acceleration occurs for large errors and near the origin.

Under identical conditions, the conventional method settles in

$$T_{\mathrm{std}}\approx 1.8840\mathrm{s}.$$

The proposed fixed-time controller converges in approximately

$$T_{\mathrm{prop}}\approx 0.0500\mathrm{s}.$$

The speed-up factor is therefore

$$\eta ={\displaystyle \frac{T_{\mathrm{std}}}{T_{\mathrm{prop}}}}\approx 37.7.$$

A faster synchronization response is achieved. The real-time recovery requirement is also satisfied. Figure 9a shows the logarithmic-scale error evolution. The proposed controller reaches the threshold faster. Figure 9b confirms fixed-time convergence. Delayed torque recovery may degrade safety and real-time performance.

5. Proposed Method for Secure Real-Time EV Communications

Modern electric vehicles depend strongly on in-vehicle communication networks. Critical commands are transmitted between ECUs through these networks. These commands include torque requests, braking commands, and motor control instructions. High reliability and low latency are required for such signals. However, these networks are exposed to increasing cybersecurity threats. Common threats include eavesdropping, spoofing, and denial-of-service attacks. These risks can be reduced by conventional encryption. However, computational overhead and transmission latency may be introduced. Therefore, real-time control requirements can be violated.

To address this challenge, a physical-layer security method is proposed. The method is based on chaotic masking, as shown in Figure 10. The C-PSO optimized chaotic generator is combined with fixed-time synchronization. The original torque command is masked by the optimized chaotic carrier. A noise-like encrypted signal is then transmitted on the CAN bus. Unauthorized access to the command is prevented by this masking process. At the receiver, the carrier is reconstructed and subtracted by the controller. The original command is then recovered. Recovery in under $0.05$ s is enabled by rapid synchronization. Thus, harmful delay is not introduced by the encryption process.

The experimental settings were selected to reflect practical EV communication constraints. A sampling period of 1 ms was used for real-time CAN-based control. A 12-bit quantizer was selected to emulate digital signal transmission. An AWGN level of 20 dB was used to represent a noisy channel. Packet loss and delay were added to emulate network-induced impairments. A 100 ms dropout was used to test recovery after temporary interruption. These settings provide a reproducible basis for evaluating secure EV torque transmission.

The method is evaluated under realistic EV operating conditions. A representative torque-request signal $T_{\mathrm{req}}\left(t\right)$ is used. This signal includes acceleration, cruising, and emergency braking. These phases are safety-critical in EV powertrain control.

In the proposed scheme, the optimized chaotic carrier masks the torque command. The transmitted signal becomes noise-like. It is also unreadable to unauthorized observers. At the receiver side, the response system is synchronized. The chaotic carrier is then reconstructed and removed. Thus, the original torque command is recovered.

Figure 11 shows encryption, decryption, and key sensitivity. Figure 11a shows the encrypted torque signal. The original drive-cycle structure is no longer visible. This confirms effective information hiding. Figure 11b shows authorized decryption. The recovered torque closely follows the reference signal. This confirms accurate reconstruction.

Key sensitivity is tested using an unauthorized receiver. The receiver uses the perturbed parameter $a^{\prime }=a+{10}^{-10}$. Figure 11c shows failed reconstruction. This very small mismatch prevents correct recovery. The recovered signal deviates strongly from the reference. Figure 11d shows the authorized synchronization error. The error remains very small after convergence. These results confirm reliable recovery and wrong-key protection.

5.1. Robustness Analysis: Noise and Signal Interruption

Realistic EV powertrain environments are complex. Therefore, the robustness analysis extends beyond standard AWGN. Impulsive noise is modeled by a Bernoulli–Gaussian process. Colored noise is also included. This noise represents emissions from high-frequency power converters. Figure 12 shows the AWGN case at $20\mathrm{dB}$. Figure 13 shows impulsive and colored noise cases. High-precision recovery is maintained in all cases. Only negligible degradation is observed. These results confirm practical robustness for secure EV communication.

The masking effect is driven by the broadband chaotic carrier. Chaotic signals naturally produce a wide aperiodic spectrum. The torque command is added to this high-energy carrier. Thus, the transmitted spectrum is dominated by chaos. The original data energy is spread over a wide band. To unauthorized observers, the signal resembles physical noise. This spectral hiding supports physical-layer encryption.

A communication interruption experiment is also conducted. Transmission is lost for $0.1\mathrm{s}$. This case emulates a denial-of-service attack. Figure 14 shows rapid resynchronization after the interruption.

5.2. Statistical Security Analysis

The statistical properties of both the original and encrypted signals are evaluated in Figure 15. The underlying structure of the original torque command is successfully masked. Subsequently, the encrypted signal distribution approximates a broad Gaussian profile. The original data features are effectively diffused across the entire state space. Consequently, the encrypted data are rendered statistically indistinguishable from random noise. High resistance against statistical cryptanalysis is thereby guaranteed by the proposed method.

5.3. Cryptographic Strength Evaluation

A cryptographic evaluation is performed to validate the proposed masking method.

5.3.1. Key Space Analysis

The encryption key includes the parameters $(a,b)$. It also includes the initial conditions $(x_0,y_0,z_0)$. Each value uses double-precision floating-point encoding. Each parameter provides about 52 effective mantissa bits. Thus, the nominal key space is $2^{260}$. This exceeds the $2^{256}$ brute-force security level.

5.3.2. NIST SP 800-22 Randomness Tests

A binary sequence of length ${10}^6$ bits is generated. The sequence is obtained from the optimized chaotic trajectories. It is evaluated using the NIST SP 800-22 test suite.

Table 3. NIST SP 800-22 statistical test suite results for ${10}^6$ bits.

Statistical Test	p-Value	Result
Frequency (Monobit)	0.3030	Pass
Block Frequency	0.0387	Pass
Runs	0.0229	Pass
Cumulative Sums (Forward)	0.4596	Pass
Cumulative Sums (Reverse)	0.4054	Pass
Longest Run of Ones	0.2505	Pass
Rank	0.6048	Pass
Discrete Fourier Transform (FFT)	0.9125	Pass
Non-overlapping Template	0.7466	Pass
Overlapping Template	0.7509	Pass
Universal Statistical	0.3933	Pass
Approximate Entropy	0.3104	Pass
Random Excursions	0.2299	Pass
Random Excursions Variant	0.1034	Pass
Linear Complexity	0.5321	Pass

reports the obtained p-values. All tests pass with $p>0.01$. This confirms strong randomness for secure vehicular communication.

5.3.3. Entropy Analysis

After 8-bit quantization, the Shannon entropy of the encrypted torque signal is calculated to be $7.9904$ bits per sample. Since this value is exceptionally close to the theoretical ideal limit of 8, it is confirmed that maximum uncertainty is achieved and information leakage to potential eavesdroppers is minimized.

5.4. Robustness to Network-Induced Impairments

Communication delays and transient packet losses are inherently introduced by practical in-vehicle CAN bus implementations. The resilience of the proposed method is analyzed under two realistic network scenarios:

5.4.1. Communication Delay

Time-varying delays of up to $2\mathrm{ms}$ (representing the typical worst-case latency in standard CAN buses) are introduced into the transmission channel. Due to the strong inherent disturbance-rejection capabilities of the fixed-time controller, rigorous synchronization is successfully maintained. Only a negligible increase in the transient settling time is exhibited, without compromising the global stability of the system.

5.4.2. Packet Loss

Random packet dropouts of up to $10\%$ are simulated, and a Zero-Order Hold (ZOH) data compensation mechanism is utilized. The evolution of the synchronization error under a severe $5\%$ packet loss rate is illustrated in Figure 16. The error is kept strictly bounded and is aggressively forced back to zero upon successful packet reception, thereby demonstrating exceptional resilience to adverse network conditions.

5.5. Effects of Sampling, Quantization, and CAN Payload Constraints

The continuous-time design must be adapted to digital CAN communication. The drive and response systems are discretized by the fourth-order Runge–Kutta method. A sampling period of $T_s=1\mathrm{ms}$ is used. This value is consistent with a typical CAN FD rate. A 12-bit uniform quantizer is applied before masking. Numerical results show that fixed-time synchronization is preserved. The settling time is $0.057\mathrm{s}$. The recovered-torque MSE is bounded at $1.202\times {10}^{-4}$. This value satisfies the required limit of $2\times {10}^{-4}$. Key sensitivity and noise robustness are also preserved. The chaotic state can be embedded into one CAN data field. This embedding is achieved without losing essential dynamics.

5.6. Computational Overhead on Automotive ECUs

The low computational complexity is substantiated through profiling on two representative hardware platforms: (i) an ARM Cortex-M4 microcontroller STM32F407 (STMicroelectronics, Geneva, Switzerland), equipped with a single-precision FPU and (ii) a fixed-point DSP TMS320F28379D (Texas Instruments, Dallas, TX, USA), utilizing CORDIC approximations for the cosine function. The execution times are summarized in

Table 4. Execution time per integration step on standard automotive ECUs.

Hardware Platform	Execution Time (μs)	CPU Load@1 ms
ARM Cortex-M4 (STM32F407, Single-precision FPU)	8.2	0.82%
TI TMS320F28379D (Fixed-point DSP with CORDIC)	12.7	1.27%

. For a 1 ms control loop, a CPU load of less than 1.3% is consumed. This confirms the suitability of the proposed design for real-time deployment on standard automotive ECUs.

5.7. Comparison with State-of-the-Art Automotive Security Mechanisms

To position the proposed method within automotive cybersecurity, a comparative analysis is conducted against representative approaches. In-vehicle network protection, EV security, and secure control communication are considered. Recent studies are grouped into three categories. First, IDS methods are used for CAN and CAN-FD traffic. These methods are often based on statistical, machine-learning, or deep-learning models. Second, lightweight security protocols are developed for in-vehicle networks. Encryption, authentication, and key management are combined in these protocols. Third, chaos-based secure communication methods are proposed for control systems.

Much recent automotive cybersecurity research is focused on CAN and CAN-FD intrusion detection. Statistical IDS methods have been proposed for CAN traffic monitoring. Deep and unsupervised learning models are used to improve real-time anomaly detection [58,59,61,62,63,64,65]. These approaches are useful for attack monitoring and diagnosis. Malicious traffic can also be classified after it appears on the bus. However, most IDS-based methods remain reactive. Injected or abnormal messages are detected after they are already present in the network. In addition, training and feature engineering are required by many learning-based methods. Costly inference may also be required on resource-constrained ECUs.

A second research line is focused on lightweight cryptographic and protocol-level mechanisms. Dynamic compression, authentication, and key management have been explored to improve CAN security [66]. Bus occupancy is reduced, and real-time performance is maintained. These approaches are relevant because implementation constraints are considered explicitly. Nevertheless, conventional encryption and authentication structures are still mainly used. System dynamics are not directly exploited for signal masking. Fast receiver-side reconstruction is also not provided.

A third and smaller group is closer to the present work. It includes secure communication methods based on chaotic synchronization and nonlinear control. A chaotic-system-based lightweight anomaly-detection method was proposed by Park and Baek [67]. The method was designed for cyber–physical systems and CAN-oriented integrity verification. For electric drives, secure communication was investigated by Phan Thi et al. [68]. Chaotic dynamics were combined with disturbance-observer and neural-network-based control. A growing interest in chaos-based vehicle security is shown by these works. However, the specific combination proposed here is not addressed in existing studies. In this work, a unique integration is proposed. A new symmetric chaotic carrier is combined with C-PSO parameter optimization. Fixed-time synchronization is also used for real-time EV torque transmission. This transmission is implemented over an automotive communication architecture.

Therefore, the proposed method is distinguished by two primary features. First, preventive physical-layer protection is provided. Malicious traffic is not only monitored after transmission. Instead, the control signal is masked before transmission. The masking is performed by a dynamically complex chaotic carrier. Second, receiver-side recovery is achieved through fixed-time synchronization. A deterministic upper bound is provided for the convergence time. Thus, the method is well suited to latency-sensitive vehicle control loops. These features are useful for embedded automotive systems. Both computational simplicity and predictable timing are required in such systems.

A qualitative comparison is summarized in

Table 5. Comparison of the proposed method with representative state-of-the-art automotive security approaches.

Reference	Technique	Main Focus	Computational Burden	Type
Ref. [58]	IDS for in-vehicle networks	Survey/review	Moderate–High	Reactive
Ref. [59]	Deep-learning IDS (CNN/LSTM)	Attack detection	High–Very High	Reactive
Ref. [61]	IDS protocols/applications	Review	Moderate–High	Reactive
Ref. [62]	Statistical CAN IDS	Anomaly detection	Moderate	Reactive
Ref. [63]	Multi-attack CAN-FD IDS	Classification	High	Reactive
Ref. [64]	CAN anomaly detection (CANGuard)	Source identification	Moderate	Reactive
Ref. [65]	Autoencoder-based CAN IDS	Unsupervised detection	Moderate	Reactive
Ref. [66]	Dynamic CAN security protocol	Encryption + authentication	Moderate	Preventive
Ref. [67]	Chaotic-system-based anomaly detection	Integrity protection	Moderate	Reactive/Preventive
Ref. [68]	Chaos-based secure electric-drive	Drive-system communication	Moderate	Preventive
Proposed Method	C-PSO optimized chaotic masking with fixed-time synchronization	Secure real-time EV torque communication	Low	Preventive

. Representative state-of-the-art approaches are considered. The table shows that most recent CAN security methods are reactive. These methods are also detection-oriented. In contrast, low computational complexity is achieved by the proposed method. Deterministic bounded-time recovery is also provided. Preventive protection is further achieved at the physical layer.

6. Conclusions

A secure real-time communication method for EV control systems is presented in this work. Chaos-based encryption is integrated with fixed-time synchronization control. A novel chaotic dynamical system was introduced. The system is characterized by a symmetric double-wing attractor. It is used as a physical-layer masking mechanism. A C-PSO algorithm was applied to determine optimal system parameters. The LLE was maximized during optimization. The obtained parameter set was validated through Lyapunov-spectrum analysis. Kaplan–Yorke dimension evaluation, bifurcation analysis, and long-horizon simulation were also used. These validation steps confirmed that the selected operating point lies in a robust chaotic region. Weakly chaotic and periodic windows were also avoided.

A nonlinear fixed-time synchronization controller was developed. Deterministic convergence of the synchronization errors was guaranteed by this controller. The selected controller gains were verified through synchronization-performance tests. Predefined hardware-oriented constraints were also considered. The proposed method was validated through extensive numerical experiments. Strong key sensitivity was demonstrated. Robustness against different noise profiles was also confirmed. Rapid recovery after communication interruptions was achieved. Compliance with standard randomness tests was verified.

Effective signal masking was achieved by the optimized chaotic carrier. Accurate bounded-time recovery was enabled by the fixed-time synchronization mechanism. Feasibility on automotive-grade microcontrollers was confirmed through computational profiling. Hardware-in-the-loop validation will be considered in future work. Embedded implementation will also be investigated. Real-time testing on automotive communication platforms will be performed. More advanced cyberattack scenarios will be examined in future robustness assessments.