Dynamic Analysis and Application of 6D Multistable Memristive Chaotic System with Wide Range of Hyperchaotic States

Abstract

In this study, we present a novel, six-dimensional, multistable, memristive, hyperchaotic system model demonstrating two positive Lyapunov exponents. With the maximum Lyapunov exponents surpassing 21, the developed system shows pronounced hyperchaotic behavior. The dynamical behavior was analyzed through phase portraits, bifurcation diagrams, and Lyapunov exponent spectra. Parameter b was a key factor in regulating the dynamical behavior of the system, mainly affecting the strength and direction of the influence of $z_1$ on $z_2$. It was found that when the system parameter b was within a wide range of $[13,300]$, the system remained hyperchaotic throughout. Analytical establishment of multistability mechanisms was achieved through invariance analysis of the state variables under specific coordinate transformations. Furthermore, offset boosting control was realized by strategically modulating the fifth state variable, $z_5$. The FPGA-based experimental results demonstrated that attractors observed via an oscilloscope were in close agreement with numerical simulations. To validate the system’s reliability for cybersecurity applications, we designed a novel image encryption method utilizing this hyperchaotic model. The information entropy of the proposed encryption algorithm was closer to the theoretical maximum value of 8. This indicated that the system can effectively disrupt statistical patterns. Experimental outcomes confirmed that the proposed image encryption method based on the hyperchaotic system exhibits both efficiency and reliability.

1. Introduction

The origin of chaos theory can be traced back to 1963, when Lorenz [1] discovered one of the earliest chaotic systems in the world while studying air flow problems. Subsequently, research on and applications of chaos theory in the field of nonlinear systems have been widely conducted. In recent years, new chaotic systems have been widely used and studied in the fields of secure communication [2,3,4], system synchronization and control [5,6,7], nonlinear circuits [8,9,10], neural networks [11,12,13], and image encryption [14,15,16]. For three decades, researchers have conducted in-depth explorations of chaotic systems with different types of structures, like chaotic neural networks [17,18,19], discrete chaotic mapping [20,21,22], conservative chaotic systems [23,24,25], and memristive chaotic systems [26,27,28].

Hyperchaotic systems have at least two positive Lyapunov exponents, so hyperchaos is seemingly more random than chaos and oscillations are more complex [29,30,31]. Both chaotic and hyperchaotic systems can be used for secure communication, but hyperchaotic systems are more difficult to decipher than chaotic systems, so hyperchaos is more suitable for secure communication [32,33,34]. Benkouider et al. introduced an innovative, four-dimensional (4D) hyperchaotic system characterized by enhanced dynamical complexity, capable of producing diverse regimes including chaotic oscillations, hyperchaotic states, periodic patterns, and quasi-periodic dynamics through parameter modulation [35]. Zhang et al. proposed a 4D hyperchaotic system with symmetry of topologically attractive substructures in which the inverse multiplication bifurcation and multi-stability phenomena from four periods to two periods to one period are discovered [36]. Based on the 5D Euler equation, Huang et al. proposed a 5D, Hamiltonian, conserved, hyperchaotic system that can have different types of coordinate transformations and time-reversal symmetries [24]. Compared with low-dimensional chaotic systems, high-dimensional chaotic systems have higher mathematical model complexity and larger parameter sets [34]. The complexity of a mathematical model refers to its high dimensionality, multiple nonlinearities, and large number of parameters [37]. Therefore, constructing a higher dimensional hyperchaotic system is an important and challenging task. Based on the modified 3D Lorentz equations, Kopp et al. constructed a new, 6D, hyperchaotic system using the state feedback control method, which produces topologically more complex attractors with a double-winged butterfly structure [38].

In 1971, Leon O. Chua first proposed the nonlinear, passive memristor based on the relationship between magnetic flux and charge. Subsequently, memristors have been widely used in the design of chaotic circuits, and their nonlinearity and memory performance can cause memristor chaotic systems to produce a variety of chaotic attractors and exhibit complex and rich dynamic behaviors [39,40,41]. By adding memristors to a 4D, conservative, chaotic system, a novel 5D, conservative, memristive, hyperchaotic system was proposed in [23], which exhibits hyperchaos and multistability in a wide range of parameters and initial values, accompanied by transient quasi-periodic phenomena. In [42], a 5D, fractional-order, memristive, hyperchaotic system with multiple coexisting attractors was constructed by coupling the magnetron memristor with dimension expansion. In [43], a 6D, memristive, hyperchaotic system with four-wing attractors was proposed by adding magnetron memristors to the 5D, continuous, chaotic system.

In recent years, the analysis of multistable characteristics has become a research hotspot. Multistability means that under the condition of fixed system parameters, when the initial conditions change, the system will produce independent coexisting attractors or topologies of different shapes [44]. Multistability is a common phenomenon in nonlinear systems and reflects the state diversity and structure diversity of a system. Building upon the classical Jerk framework, Wang et al. developed an enhanced, multistable, chaotic system through trigonometric nonlinearity integration, exhibiting unbounded coexisting attractor generation via tangent-based bifurcation control [45]. Zhang et al. proposed a novel, memristive, multiscroll, multistable neural network in which a multistable threshold memristor is used to describe the effects of external electromagnetic radiation [46]. Yuan et al. constructed a neurodynamic, small-world architecture employing Rulkov mapping units, demonstrating initial condition-dependent multimodality that spanned synchronous regimes, chaotic itinerancy, and metastable chimera patterns [47]. Concurrently, Lai et al. engineered a non-ideal memristive oscillator with programmable, piecewise-linear state transitions, achieving topological control of scroll attractors through reel number parameterization and spontaneous decomposition into bistable, double-scroll configurations [48].

FPGAs offer high customizability, parallelism, and reconfigurability, making them ideal for rapid prototyping and efficient real-time implementation of complex chaotic systems. Mohamed Maazouz et al. implemented a chaos-based image encryption algorithm on FPGA [49]. Guohao Pan et al. observed all phenomena associated with coexisting attractors by selecting initial conditions in their FPGA-based implementation [50]. Xiangguang Sun et al. verified the practical feasibility of their dynamic information flow control framework through hardware implementation on FPGA, achieving a throughput of 500 Mbps [51].

This study constructed a 6D, memristive, multistable, hyperchaotic system (MMHS) by integrating a memristor into a 5D hyperchaotic framework. The proposed system demonstrates rich nonlinear phenomena including hyperchaos and multistability across extensive parameter ranges and initial conditions. The principal contributions are summarized as follows:

(1)

A novel, 6D, MMHS architecture was developed, with its hyperchaotic characteristics rigorously verified through the analysis of the Lyapunov exponent spectrum and the quantification of the Kaplan–Yorke dimension. Under specific parameters, the 6D MMHS exhibits either two positive Lyapunov exponents with the maximum exceeding 21 or three positive Lyapunov exponents with the maximum exceeding 4. By analyzing the Lyapunov exponents and Kaplan–Yorke dimension when the parameter b was within a wide range of $[13,300]$, its hyperchaotic characteristics over the wide range were verified.

(2)

Multistability mechanisms were analytically established through invariance analysis of state variables under specific coordinate transformations. Offset boosting control was successfully implemented via strategic modulation of the fifth state variable ($z_5$).

(3)

By discretizing the 6D MMHS and performing numerical simulation on an FPGA, the experimental results showed that the attractor on the oscilloscope closely resembled the software simulation results.

(4)

An image encryption scheme derived from the 6D MMHS was proposed, with its cryptographic security systematically evaluated through statistical and differential analyses.

From the summary discussed above, this paper is structured as follows: In Section 2, a novel, 6D MMHS is introduced, characterized by three positive Lyapunov exponents, with the maximum exceeding 21, highlighting its enhanced dynamic complexity. The fundamental dynamical properties of the system were systematically analyzed by evaluating the stability of its equilibrium points. Numerical simulations were performed to obtain the Lyapunov exponents, bifurcation diagrams, and phase portraits of the system under variations of parameters a, b, c, and d. The simulation results reveal that the system transitions gradually from periodic to hyperchaotic states as a, b, c, and d increase. Furthermore, this study demonstrates that the 6D MMHS exhibits multistability with coexisting attractors. Offset-boosting control is successfully achieved by adjusting the state variable $z_5$. Section 3 presents an implementation of the system based on a Field-Programmable Gate Array (FPGA), where attractors observed on an oscilloscope showed excellent agreement with numerical simulations. In Section 4, we propose a new image encryption method based on this hyperchaotic system. The experimental results demonstrated the effectiveness and reliability of the encryption system based on the proposed hyperchaotic system. Section 5 summarizes the results and insights from the previous discussions.

2. A 6D, Multistable, Memristive, Hyperchaotic System

2.1. Mathematical Description of the New System

A memristor is regarded as the fourth fundamental electrical element and it is a kind of nonlinear resistor. Based on the fundamental principles of memristor theory and the classification of memristors [52], a flux-controlled memristor can be determined as follows:

$$\left\{\begin{array}{c}i=W\left(\phi \right)v=\left(m+3n{\phi }^2\right)v\hfill \\ {\displaystyle \frac{d\phi }{dt}}=v\hfill \end{array}\right.$$

(1)

where $\phi$ represents the internal state of the memristor and $W\left(\phi \right)=m+3n{\phi }^2$ represents the memductance. i and v represent the output current and input voltage, respectively, while m and n are two adjustable positive parameters within the memristor. Obviously, the memristor in Formula (1) conforms to the definition of an ideal memristor [53].

Traditionally, memristive super-chaos systems have been constructed in four, five, and six dimensions by extending lower-dimensional chaotic systems with additional state variables and memristive nonlinearities. In four-dimensional systems, a memristor is typically introduced to a classical three-dimensional chaotic system, resulting in richer dynamic behaviors such as hyperchaos and multistability. Five-dimensional systems further increase complexity by adding another state variable or coupling term, often leading to more intricate attractor structures and enhanced control capabilities. Six-dimensional memristive super-chaos systems, such as the one proposed in this paper, are developed by incorporating additional feedback loops and nonlinear interactions, which significantly expand the system’s dynamic range and potential for engineering applications. These higher-dimensional extensions allow for the exploration of more complex chaotic phenomena and provide a broader platform for applications in secure communications, encryption, and random number generation.

Memristors, as nonlinear circuit elements with memory, have been widely used to enrich the dynamics of chaotic systems due to their flux-controlled or charge-controlled characteristics [54,55]. In particular, flux-controlled memristor models are often employed, where the memristor’s conductance depends on the time integral of the applied voltage. By introducing a memristor into a lower-dimensional chaotic system, it is possible to increase the system’s dimensionality and generate more complex behaviors such as multistability and hyperchaos.

In this work, we started from the five-dimensional hyperchaotic system described in Ref. [56] and extended it by incorporating the electromagnetic induction current provided by a flux-controlled memristor. Specifically, the input voltage v was identified with the state variable $z_4$ and the internal state $\phi$ of the memristor was introduced as a new state variable, $z_6$. This approach allowed the memristive effect to directly influence the system’s dynamics, leading to the construction of a novel, six-dimensional, multistable, memristive, hyperchaotic system (6D MMHS) as described below.

Therefore, by introducing the electromagnetic induction current provided by the memristor into the system described in Ref. [56], a 6D MMHS could be constructed. The input voltage v was regarded as $z_4$ and the internal state $\phi$ of the memristor was treated as a new state variable of the system. Based on the above descriptions, we propose a novel, 6D MMHS described by the following dynamical equations:

$$\left\{\begin{array}{c}{\dot{z}}_1=a(z_2-z_1)+z_2{z}_3+z_4\hfill \\ {\dot{z}}_2=z_1(b-z_3)+c{z}_4\hfill \\ {\dot{z}}_3=z_1^2+z_2{z}_1-d{z}_3\hfill \\ {\dot{z}}_4=-z_2+z_6\hfill \\ {\dot{z}}_5=z_6-z_5+kW\left(z_6\right)z_4\hfill \\ {\dot{z}}_6=-z_4\hfill \end{array}\right.$$

(2)

where $W\left(z_6\right)=m+3n{z}_6^2$. In Equation (2), $z_1$, $z_2$, $z_3$, $z_4$, $z_5$, and $z_6$ are all state variables. The coupling and control parameters are a, b, c, and d and all of them have positive values. Parameters $a,b,c,$ and d are all used to regulate the dynamical behavior of the system. They can serve as bifurcation parameters to adjust the overall dynamical characteristics of the system, which is demonstrated in the subsequent sections of this paper. The memristor strength is defined by k.

Numerical analysis under the parameterization

$$a=50,b=190,c=100,d=21,k=1,m=0.1,n=0.01$$

(3)

reveals that system (2) initialized at $Z\left(0\right)=(1,1,1,1,1,1)$ attains strong hyperchaoticity, characterized by three positive Lyapunov and MLE exponents exceeding 21.

The hyperchaotic attractor of the new, 6D MMHS is illustrated in the phase portrait shown in Figure 1. To provide a clearer view of the system’s evolution, the time series of all six state variables in the interval from 200 s to 250 s are presented in Figure 2. The phase portrait was constructed based on these time series data: specifically, by plotting one state variable against another (for example, $z_1$ versus $z_2$) at each time point within the selected interval. In this way, the phase portrait visualized the trajectory of the system in the state space, revealing the complex structure and dynamic behavior of the hyperchaotic attractor. To further analyze the frequency characteristics of the system, we computed and plotted the power spectrum density (PSD) of four representative state variables ($z_1$, $z_2$, $z_4$, and $z_5$) over the interval 200–250 s. As shown in Figure 3, all four variables exhibited broadband spectra without dominant frequency peaks, which is a typical feature of hyperchaotic systems. This result further confirms the complex and aperiodic nature of the 6D MMHS.

Figure 4a presents the Poincaré section of the system on the $z_1=0$ plane, where the intersection points are projected onto the $(z_2,z_3)$ plane. The points are densely and irregularly distributed, forming a clear fractal-like structure. This intricate pattern is a hallmark of chaotic dynamics, indicating sensitive dependence on initial conditions and the absence of periodic or quasi-periodic motion. Figure 4b shows the Poincaré section on the $z_4=0$ plane, with the intersection points projected onto the $(z_5,z_6)$ plane. Here, the points are also densely scattered, forming a cloud-like distribution without any discernible regularity or closed curves. This further confirms the presence of chaos in the system, as such a distribution is typical for chaotic attractors. Together, these Poincaré sections provide rigorous and visual evidence that the system exhibits chaotic behavior under the given parameter settings.

Under the same parameter conditions, with the total simulation time set to $T={10}^4$ s, the Lyapunov exponents of the 6D MMHS were calculated in MATLAB 2018b. The calculation followed the classical algorithm proposed by Wolf et al., which integrates the original ODE system together with its variational equations and applies the Gram–Schmidt orthonormalization procedure at regular time intervals. This method tracks the divergence of nearby trajectories in the phase space and computes the exponents as the time-averaged growth rates of infinitesimal perturbations along different directions. The algorithm operates directly on the full set of state variables and their linearized equations, so the embedding dimension is equal to the number of state variables (six in this case), and no time delay is required. Using this approach, the following Lyapunov exponents were obtained:

$$\left\{\begin{array}{cc}L{E}_1\hfill & =21.6825\hfill \\ L{E}_2\hfill & =0.1079\hfill \\ L{E}_3\hfill & =0\hfill \\ L{E}_4\hfill & =-0.0083\hfill \\ L{E}_5\hfill & =-1.0000\hfill \\ L{E}_6\hfill & =-92.7858\hfill \end{array}\right.$$

(4)

The Kaplan–Yorke dimension of the 6D MMHS was calculated as follows:

$$D_{KY}=5+{\displaystyle \frac{L{E}_1+L{E}_2+L{E}_3+L{E}_4+L{E}_5}{|L{E}_6|}}=5.224$$

(5)

From Equation (4), we deduce that the 6D MMHS is a hyperchaotic system with three positive Lyapunov exponents. Given the large value of the maximum Lyapunov exponent (MLE) in Equation (4), we conclude that the 6D MMHS exhibits strong hyperchaotic behavior, which is highly beneficial for applications of hyperchaotic systems in image encryption and secure communications. The time-varying Lyapunov exponents plot of system (2) is shown in Figure 5. As can be seen from Figure 5, the Lyapunov exponents exhibited an oscillatory state within the first 2000 s, gradually stabilized after 2000 s, and ultimately maintained a state with two positive Lyapunov exponents. Figure 5a describes the evolution of the $L{E}_2{,}_3$ and $L{E}_4$ Lyapunov exponents over time, while Figure 5b depicts the temporal changes of the $L{E}_1{,}_5$ and $L{E}_6$ Lyapunov exponents. The caption indicates that when $T={10}^4\mathrm{s}$, the values of the system’s Lyapunov exponents were as shown.

Table 1. Comparison of Lyapunov exponents and Kaplan–Yorke dimensions.

Dimension	${\mathit{LE}}_{\mathit{1}}$	${\mathit{LE}}_{\mathit{2}}$	${\mathit{LE}}_{\mathit{3}}$	${\mathit{LE}}_{\mathit{4}}$	${\mathit{LE}}_{\mathit{5}}$	${\mathit{LE}}_{\mathit{6}}$	${\mathit{D}}_{\mathit{KY}}$	Paper
6D	0.799484	0.295845	0	−0.407744	−0.659364	−14.69188	5.00192	Ref. [57]
6D	1.0529	0.6213	0.0342	0	−13.7815	−15.2637	4.12396	Ref. [58]
6D	0.0988591	0.0109865	−0.544226	−1.00557	−1.15581	−1.77091	2.20184	Ref. [59]
6D	1.0639	0.60107	0.0431	0	−13.8708	−15.1728	4.1231	Ref. [60]
6D	4.1306	0.4936	−0.0033	−3.5059	−14.8197	−99.6187	4.07523769	Ref. [43]
6D	0.653	0.326	0.000	1.558	−4.419	−12.676	4.574	Ref. [61]
6D	1.462	0.1433	0.0725	0.0449	0	−12.0700	5.1427	Ref. [62]
5D	6.053	0.028	0.000	−4.747	−46.214	/	4.029	Ref. [63]
5D	12.659	0.055	0.024	0	−67.701	/	4.189	Ref. [56]
6D	21.6825	0.1079	0	−0.0083	−1.0000	−92.7858	5.224	Section 2.1
6D	4.4553	0.1681	0.1006	0	−1.0000	−56.7186	5.066	Section 2.3.3

presents the Lyapunov exponents and Kaplan–Yorke dimension of the new system (2) in comparison with other 6D and 5D hyperchaotic systems. It can be observed that the new system (2) possesses three positive Lyapunov exponents, with its first Lyapunov exponent being significantly larger than those of other reported 6D and 5D systems. This indicates that the trajectories of the new system diverge more rapidly than those of other systems. Furthermore, the Kaplan–Yorke dimension of the new system (2) is substantially higher than those of most 6D hyperchaotic systems. These characteristics demonstrate that this novel, 6D MMHS exhibits stronger chaotic properties than many reported 6D hyperchaotic systems, making it a valuable addition to the family of hyperchaotic systems. Its distinctive features render it particularly suitable for chaos-based applications.

The 6D MMHS exhibits sustained hyperchaotic behavior across extensive parameter ranges. As demonstrated in Figure 6a,b, with fixed parameters $a=50,c=100,$$d=21,k=1,m=0.1,n=0.01$ and initial conditions $Z\left(0\right)=(1,1,1,1,1,1)$, the system maintained persistent hyperchaos for $b\in [13,300]$. The detailed quantification in Figure 6c,d reveals the following:

• Peak MLE: $L{E}_{\max }\approx 23$ within $b\in [205,240]$;
• Maximum Kaplan–Yorke dimension: $D_{KY}>5.23$.

When extending b to $(300,500)$, the system transitioned to intermittent chaos (non-persistent hyperchaos) while preserving $D_{KY}>1$. To demonstrate the hyperchaotic characteristics of the 6D MMHS within the wide parameter range of $b\in [13,300]$, Figure 7a shows the $L{E}_1$, $L{E}_2$, and $L{E}_3$ values over this interval. It can be seen that the values of $L{E}_1$ and $L{E}_2$ always remained greater than 0, indicating that the system maintained hyperchaos within this interval. In some sub-intervals, $L{E}_3$ was also greater than 0, suggesting that the system exhibited extreme hyperchaos (i.e., three positive Lyapunov exponents) in intervals where $L{E}_3>0$.

Moreover, traditional chaotic systems require parameters to be confined within specific intervals or initial values to generate chaotic behavior; otherwise, the systems may exit chaotic states and enter periodic, convergent, or divergent regimes, thereby limiting their applicability in control engineering and related domains. As demonstrated by the comparative analysis in

Table 2. Parameter range of traditional chaotic system.

Dimension	Interval Length of Chaotic State	Paper
5D	Less than 100	Ref. [64]
4D	Less than 6	Ref. [65]
4D	Less than 10	Ref. [66]
4D	Less than 50	Ref. [67]
6D	[13, 300]	This paper

, the 6D MMHSs with broader parameter adaptability exhibit superior performance in sustaining hyperchaotic characteristics under diverse operational conditions compared to conventional chaotic systems. This observation highlights their notable advantages in dynamic stability.

All numerical simulations in this work were performed using MATLAB’s built-in ode45 function, which is based on an explicit Runge–Kutta (4,5) formula, the Dormand–Prince pair. The time step and integration interval were chosen to ensure both accuracy and computational efficiency. Initial conditions and parameter values are specified in the corresponding figure captions and text. For visualization of high-dimensional dynamics, phase portraits, bifurcation diagrams, and Poincaré sections were plotted.

Figure 6. When the initial condition was $Z\left(0\right)=(1,1,1,1,1,1)$ and the parameter settings were $(a,c,d,k,m,n)=(50,100,21,1,0.1,0.01)$, the plots of Lyapunov exponents (a,c) and Kaplan–Yorke dimension varied with parameter b (b,d).

Figure 6. When the initial condition was $Z\left(0\right)=(1,1,1,1,1,1)$ and the parameter settings were $(a,c,d,k,m,n)=(50,100,21,1,0.1,0.01)$, the plots of Lyapunov exponents (a,c) and Kaplan–Yorke dimension varied with parameter b (b,d).

Figure 7. When $b\in [13,300]$, the values of $L{E}_1$, $L{E}_2$, and $L{E}_3$ were within this interval.

Figure 7. When $b\in [13,300]$, the values of $L{E}_1$, $L{E}_2$, and $L{E}_3$ were within this interval.

2.2. Equilibrium Point and Stability

The equilibrium points of the system (2) were determined by solving equations:

$$\left\{\begin{array}{c}a(z_2-z_1)+z_2{z}_3+z_4=0\hfill \\ z_1(b-z_3)+c{z}_4=0\hfill \\ z_1^2+z_2{z}_1-d{z}_3=0\hfill \\ -z_2+z_6=0\hfill \\ z_6-z_5+kW\left(z_6\right)z_4=0\hfill \\ -z_4=0\hfill \end{array}\right.$$

(6)

Clearly, $E_0=\mathbf{0}\in {\mathbf{R}}^6$ is an equilibrium point.

A simple calculation shows that there are two more equilibrium points given by

$$E_{1,2}=\left[\begin{array}{c}\pm \lambda \\ \pm {\displaystyle \frac{a}{a+b}}\lambda \\ b\\ 0\\ \pm {\displaystyle \frac{a}{a+b}}\lambda \\ \pm {\displaystyle \frac{a}{a+b}}\lambda \end{array}\right]$$

(7)

where

$$\lambda =\sqrt{{\displaystyle \frac{db(a+b)}{2a+b}}}$$

(8)

For ease of calculation, we set the system parameters as $(a,b,c,d,k,m,n)=(40,90,16,15,1,0.1,0.01)$. At this point, the system remained in a hyperchaotic state, with its Lyapunov exponents being $(L{E}_1,L{E}_2,L{E}_3,L{E}_4,L{E}_5,L{E}_6)=(12.6350,0.0538,0.0237,-0.0024,-1.0000,-67.6724)$. Under these conditions, $\lambda =\sqrt{{\displaystyle \frac{db(a+b)}{2a+b}}}=32.1302$. Also, the three equilibrium points of the system (2) were obtained as

$$\left\{\begin{array}{c}{E}_0=(0,0,0,0,0,0)\hfill \\ E_1=(32.1302,9.8862,90,0,9.8862,9.8862)\hfill \\ E_2=(-32.1302,-9.8862,90,0,-9.8862,-9.8862)\hfill \end{array}\right.$$

(9)

Based on system (2), it was not difficult to write out its Jacobian matrix as follows:

$$\mathrm{Jac}=\left[\begin{array}{cccccc}-a& a+z_3& z_2& 1& 0& 0\\ b-z_3& 0& -z_1& c& 0& 0\\ 2z_1+z_2& z_1& -d& 0& 0& 0\\ 0& -1& 0& 0& 0& 1\\ 0& 0& 0& k(3n{z}_6^2+m)& -1& 6kn{z}_4{z}_6+1\\ 0& 0& 0& -1& 0& 0\end{array}\right]$$

(10)

First, we calculated the eigenvalues of system (2) at the equilibrium point $E_0=(0,0,0,0,0,0)$. At this point, the parameters and state variables in the Jacobian matrix were all determined, and the six corresponding eigenvalues could be obtained as follows:

$$\left\{\begin{array}{c}{\lambda }_1=-1.0000\hfill \\ {\lambda }_2=-83.1884\hfill \\ {\lambda }_3=42.9844\hfill \\ {\lambda }_4=0.1020+0.9982i\hfill \\ {\lambda }_5=0.1020-0.9982i\hfill \\ {\lambda }_6=-15.0000\hfill \end{array}\right.$$

(11)

Calculations showed that the eigenvalue spectrum consisted of three negative real eigenvalues (${\lambda }_1=-1.0000$,${\lambda }_2=-83.1884$, and ${\lambda }_6=-15.0000$), one positive real eigenvalue (${\lambda }_3=42.9844$), and a pair of complex conjugate eigenvalues with positive real parts (${\lambda }_4{,}_5=0.1020\pm 0.9982i$). According to linear stability theory [68], this equilibrium point is classified as a high-dimensional saddle-focus, characterized by a 3D stable manifold governed by the contracting modes associated with the negative real eigenvalues and a 3D unstable manifold dominated by the expanding mode from the positive real eigenvalue and the spiral divergence induced by the complex eigenvalues. The positive real parts of the complex eigenvalues indicate the presence of locally unstable oscillatory dynamics near the equilibrium, while the positive real eigenvalue ${\lambda }_3$ drives rapid exponential instability along a specific direction. This configuration implies that trajectories in the vicinity of the equilibrium will transiently converge along the stable manifold while escaping via a hybrid multimodal mechanism coupling of spiral divergence with unidirectional exponential growth along the unstable manifold, consistent with phase-space bifurcation mechanisms observed in classical saddle-focus structures [69]. These linearized dynamics establish a foundation for the further investigation of global, nonlinear phenomena in the system, such as chaotic attractors or heteroclinic orbits.

Next, we calculated the eigenvalues of system (2) at the equilibrium point $E_1=(32.1302,9.8862,90,0,9.8862,9.8862)$. At this point, the system parameters in the Jacobian matrix were already determined. By substituting the calculated equilibrium point $E_1$ into the Jacobian matrix, the six corresponding eigenvalues could be obtained as follows:

$$\left\{\begin{array}{c}{\lambda }_1=-1.0000\hfill \\ {\lambda }_2=-89.0773\hfill \\ {\lambda }_3=17.0322+60.5004i\hfill \\ {\lambda }_4=17.0322-60.5004i\hfill \\ {\lambda }_5=0.0064+0.9987i\hfill \\ {\lambda }_6=0.0064-0.9987i\hfill \end{array}\right.$$

(12)

The analysis results indicated that the eigenvalue spectrum contained two negative real eigenvalues (${\lambda }_1=-1.0000$ and ${\lambda }_2=-89.0773$) and two pairs of complex conjugate eigenvalues with positive real parts: a high-frequency oscillatory mode (${\lambda }_3{,}_4=17.0322\pm 60.5004i$) and a low-frequency weakly unstable mode (${\lambda }_5{,}_6=0.0064\pm 0.9987i$). This configuration classifies the equilibrium as a high-dimensional saddle-focus, characterized by a 2D stable manifold (attracting trajectories via the negative real eigenvalues) and a 4D unstable manifold dominated by two distinct oscillatory instabilities. The dominant instability (${\lambda }_3{,}_4$) exhibits rapid exponential growth with strong oscillatory divergence ($\left|\mathrm{Im}\right(\lambda \left)\right|\gg \mathrm{Re}\left(\lambda \right)$), while the marginally positive real parts of ${\lambda }_5{,}_6$ suggest a metastable regime where trajectories transiently follow weakly damped oscillations before escaping the equilibrium neighborhood. Such spectral structure aligns with bifurcation scenarios involving multiscale chaos or intermittent bursting, as the coexistence of high- and low-frequency unstable modes often induces complex transient dynamics and heteroclinic connections in nonlinear systems [70]. These results underscore the equilibrium’s role as an organizing center for nonstationary behavior, providing a foundation for analyzing global phenomena such as chaotic attractors or transient turbulence in subsequent nonlinear studies.

After calculation, for the equilibrium point $E_2$, its corresponding eigenvalue spectrum was also obtained by Equation (12). The system admitted an exact involution symmetry under transformation $(z_1,z_2,z_3,z_4,z_5,z_6)\mapsto (-z_1,-z_2,z_3,-z_4,-z_5,-z_6)$, as verified by the invariance of the governing equations. This symmetry enforced the pairwise emergence of equilibria $E_1=(32.1302,9.8862,90,0,9.8862,9.8862)$ and $E_2=(-32.1302,-9.8862,90,0,-9.8862,-9.8862)$, which were mapped onto one another by the transformation. The Jacobian matrices at $E_1$ and $E_2$ were related by a similarity transformation $J\left(E_2\right)=P·J\left(E_1\right)·P^{-1}$, where $P=\mathrm{diag}(-1,-1,1,-1,-1,-1)$ encoded the coordinate inversion. Since similarity transformations preserve eigenvalues, we had the spectral equivalence of $E_1$ and $E_2$. Specifically, the identical sets of two negative real eigenvalues (−1.0000, −89.0773) and two pairs of complex-conjugate eigenvalues with positive real parts ($17.0322\pm 60.5004i$, $0.0064\pm 0.9987i$) arose directly from this symmetry. Such eigenvalue degeneracy ensured that both equilibria shared identical local stability properties, with 2D stable manifolds and 4D unstable manifolds. The coexistence of symmetric saddle-foci suggested the potential for heteroclinic connections between $E_1$ and $E_2$, a configuration known to underpin complex global dynamics, including chaos and transient metastability, in systems with reflectional symmetries [71,72].

In the study of six-dimensional dynamical systems, the local stability of equilibrium points is determined by the eigenvalue distribution of the Jacobian matrix. Based on the signs of the real parts and the complex nature of the eigenvalues, we established a comprehensive classification system for equilibrium points, comprising 16 topological types. In the stability distribution diagrams on the two-dimensional parameter plane $(a,b)\in [-40,40]\times [-90,90]$ (Figure 8b for the $E_1$ stability distribution; Figure 8a for the $E_0$ stability distribution), the types of equilibrium points were rigorously classified according to their topological structure as follows:

• Stable Node (SN, dark blue regions): All eigenvalues of the Jacobian matrix had negative real parts and were purely real. Trajectories in the phase space exhibited exponential convergence along a six-dimensional stable manifold without oscillation.
• Stable Focus (SF, light blue regions): All eigenvalues had negative real parts but contained complex conjugate pairs. Trajectories displayed spiral convergence with damped oscillatory characteristics.
• Unstable Node (UN, dark red regions): All eigenvalues had positive real parts and were purely real. Trajectories diverged exponentially along a six-dimensional, unstable manifold.
• Unstable Focus (UF, light red regions): All eigenvalues had positive real parts but contained complex conjugate pairs. Trajectories exhibited diverging spiral patterns with amplifying oscillations.
• Saddle Point with $dim=1$ (S1, dark green regions): Featured one eigenvalue with a positive real part (one-dimensional unstable manifold) and five eigenvalues with negative real parts (five-dimensional stable manifold), all real. Trajectories diverged along one dimension while converging along five dimensions.
• Saddle-Focus with $dim=1$ (SF1, light green regions): Contained one real eigenvalue with a positive real part (one-dimensional unstable manifold) and complex conjugate pairs with negative real parts (five-dimensional stable manifold with spiral convergence).
• Saddle Point with $dim=2$ (S2, bright yellow regions): Two real eigenvalues with positive real parts (two-dimensional unstable manifold) and four real eigenvalues with negative real parts (four-dimensional stable manifold).
• Saddle-Focus with $dim=2$ (SF2, light yellow regions): The two-dimensional unstable manifold or four-dimensional stable manifold contained complex eigenvalues, inducing spiral motion in corresponding subspaces.
• Saddle Point with $dim=3$ (S3, orange regions): Three real eigenvalues with positive real parts (three-dimensional unstable manifold) and three real eigenvalues with negative real parts (three-dimensional stable manifold).
• Saddle-Focus with $dim=3$ (SF3, light orange regions): The three-dimensional unstable manifold or stable manifold contained complex eigenvalues. This configuration could induce chaos when satisfying Shilnikov conditions.
• Saddle Point with $dim=4$ (S4, purple regions): Four real eigenvalues with positive real parts (four-dimensional unstable manifold) and two real eigenvalues with negative real parts (two-dimensional stable manifold).
• Saddle-Focus with $dim=4$ (SF4, pale purple regions): The four-dimensional unstable manifold or two-dimensional stable manifold contained complex eigenvalues, forming a high-dimensional spiral saddle structure.
• Saddle Point with $dim=5$ (S5, dark gray regions): Five real eigenvalues with positive real parts (five-dimensional unstable manifold) and one real eigenvalue with negative real part (one-dimensional stable manifold).
• Saddle-Focus with $dim=5$ (SF5, light gray regions): The five-dimensional unstable manifold contained complex eigenvalues, producing spiral divergence within strongly repulsive domains.
• Critical Point (CP, black regions): At least one eigenvalue had a real part approaching zero, corresponding to bifurcation points (e.g., Hopf bifurcation, fold bifurcation).
• Non-existent (NE, white regions): Exclusive to $E_1$, undefined when $b(a+b)(2a+b)<0$ or $2a+b=0$ (i.e., when $\lambda$ did not exist in the set of real numbers.). Under these conditions, the system possessed only the $E_0$ equilibrium point.

Due to the symmetric properties of the system, equilibrium points $E_1$ and $E_2$ shared identical eigenvalues, resulting in identical stability distribution diagrams.

Figure 8. (a) Stability classification of equilibrium point $E_0$; (b) stability classification and existence domain of non-trivial equilibrium $E_1$.

Figure 8. (a) Stability classification of equilibrium point $E_0$; (b) stability classification and existence domain of non-trivial equilibrium $E_1$.

2.3. Dynamical Behavior When System Parameters Changed

2.3.1. When a and b Changed

When parameters a and b changed, the system exhibited rich dynamic behavior. We start this discussion with a.

We investigated the dynamical characteristics of the 6D MMHS with parameter a increasing in the interval $[10,40]$ while other parameters remain fixed at $b=90$, $c=16$, $d=12$, $k=1$, $m=0.1$, and $n=0.01$. The initial condition was set to $(1,1,1,1,1,1)$. Numerical simulations revealed significant behavioral variations in system (2) under different values of a. As illustrated in the bifurcation diagram (Figure 9a) and the Lyapunov exponent spectrum (Figure 9b), the 6D MMHS exhibited periodic behavior (with near-zero MLE), chaotic behavior (with one positive Lyapunov exponent), hyperchaotic behavior (with two positive Lyapunov exponents), or extreme hyperchaotic behavior (with two positive Lyapunov exponents and exceptionally high MLE), depending on the control parameter a.

When $a\in [10,10.57]$, the state of the system was easily affected by the parameter a. At this time, the system jumped back and forth between the periodic and chaotic states. When $a\in [10.57,12.88]\cup [13.63,14.2]$, the maximum Lyapunov exponent (MLE) of the system was always less than 0, and the system was in a periodic state. When the parameter $a\in [12.88,13.63]\cup [14.2,20.4]$, the system returned to the state of jumping back and forth between the periodic and chaotic states. Moreover, if the system was in the chaotic state at this time, it had only one positive Lyapunov exponent. Figure 9c,d shows the bifurcation diagrams with parameters $a\in [15.7,15.95]$ and $a\in [16.05,16.25]$, respectively. With parameter $a\in [20.4,40]$, the system was in a hyper-chaotic state, but it jumped back and forth between the states with two positive Lyapunov exponents and three positive Lyapunov exponents.

When other parameters were fixed as $(b,c,d,k,m,n)=(90,16,12,1,0.1,0.01)$ and the initial condition of the system was still set as $(1,1,1,1,1,1)$, changing the value of a caused the system to exhibit rich, dynamic changes. When $a=12$, the system was in a periodic state, and its periodic attractor is shown in Figure 10a. The Lyapunov exponents of the system were $(L{E}_1,L{E}_2,L{E}_3,L{E}_4,L{E}_5,L{E}_6)=(0,-0.12,-0.12,-1.00,-11.88,-11.89)$. When a = 19, the system was in a chaotic state, and its chaotic attractor is shown in Figure 10b. The Lyapunov exponents of the system were $(L{E}_1,L{E}_2,L{E}_3,L{E}_4,L{E}_5,L{E}_6)=(0.012,0,-0.25,-1.00,-3.48,-27.28)$. When a = 25, the system was in a hyperchaotic state with two positive Lyapunov exponents, and its hyperchaotic attractor is shown in Figure 10c. At this time, the Lyapunov exponents of the system were $(L{E}_1,L{E}_2,L{E}_3,L{E}_4,L{E}_5,$$L{E}_6)=(10.41,0.03,0,-0.01,-1.00,-47.44)$. When a = 39.8, the system was in an extreme hyperchaotic state with three positive Lyapunov exponents, and its hyperchaotic attractor is shown in Figure 10d. At this time, the Lyapunov exponents of the system were $(L{E}_1,L{E}_2,L{E}_3,L{E}_4,L{E}_5,L{E}_6)=(11.82,0.06,0.02,0,-1.00,-63.69)$.

Next, we will discuss the system behavior when the parameter b changed.

Numerical simulations revealed that the 6D system (2) exhibited diverse dynamical behaviors as the parameter b increased within the interval $[0,90]$ while other parameters remained fixed at $a=40,c=16,d=12,k=1,m=0.1$, and $n=0.01$. As evidenced by the bifurcation analysis in Figure 11a and Lyapunov exponent in Figure 11b, system (2) underwent four distinct dynamical transitions when varying parameter b: periodic oscillations, dual-exponent hyperchaos (two positive Lyapunov exponents), triply unstable hyperchaos (three positive exponents), and equilibrium stabilization.

Concerning the parameter b within the interval [5, 20], the corresponding bifurcation diagram and Lyapunov exponent spectrum are shown in Figure 11c and Figure 11d, respectively. It can be observed that when b lay in the ranges [0, 8.5] and [10, 14.34], the Maximum Lyapunov Exponent (MLE) of the system remained below zero, indicating that the system was in a periodic state. Figure 12a demonstrates the periodic attractor when $b=4$. When b fell within [8.5, 10], the MLE alternated between positive and negative values, revealing that the system underwent intermittent transitions between chaotic and periodic states. Figure 12b displays the point attractor observed at $b=10$. For b in [14.34, 90], the system exhibited hyperchaotic behavior characterized by an MLE greater than zero and at least two positive Lyapunov exponents. The hyperchaotic attractors are respectively illustrated in Figure 12c,d: Figure 12c shows the attractor with two positive Lyapunov exponents at $b=16$, while Figure 12d presents the attractor possessing three positive Lyapunov exponents at $b=40$.

2.3.2. When c and d Changed

When parameters c and d varied individually, the system’s dynamical behavior was highly sensitive. Let us first examine the effect of parameter c.

We investigated the dynamical evolution of the system as parameter c increased within the interval [0, 30] while keeping other parameters fixed at $(a,b,d,k,m,n)=(40,90,0.8,1,0.1,0.01)$ and maintaining the initial conditions $(1,1,1,1,1,1)$. Figure 13a,b present the bifurcation diagram and the corresponding Lyapunov exponent spectrum for $c\in [0,30]$, respectively. The results revealed that the system underwent transitions between periodic, chaotic, and hyperchaotic states as c varied. Figure 14a,b depict hyperchaotic attractors in the $(z_1,z_2)$-plane at $c=1$ and $c=28$, respectively. Figure 14c shows a chaotic attractor in the $(z_1,z_2)$-plane at $c=14$, while Figure 14d displays a 3D chaotic attractor in the $z_2$-$z_1$-$z_3$ phase space at $c=14.3$. Additionally, Figure 14e,f illustrate periodic attractors in the $(z_1,z_3)$-plane at $c=11$ and in the $(z_1,z_2)$-plane at $c=13$, respectively.

Next, we turn to a discussion of parameter d.

Concerning the fixing of parameters $(a,b,c,k,m,n)=(40,90,16,1,0.1,0.01)$ and maintaining the initial system conditions $(1,1,1,1,1,1)$, the bifurcation diagram and Lyapunov exponent spectrum of the system for $d\in [0,15]$ are shown in Figure 15a and Figure 15b, respectively. When d lay in the interval [0, 1.1], the system oscillated between periodic and chaotic states. However, for $d>1.1$, the system transitioned to a hyperchaotic state. The corresponding bifurcation diagram and Lyapunov exponent spectrum for $d\in [0,1.5]$ are presented in Figure 15c,d. Figure 16a,b display phase portraits at $d=0.7$ and $d=1$, respectively, where the Maximum Lyapunov Exponent (MLE) remained below zero, indicating periodic behavior. Figure 16c,d illustrate the chaotic attractors in the $(z_1,z_3)$-plane and the 3D $z_2$-$z_1$-$z_3$ phase space at $d=0.34$. Meanwhile, Figure 16e,f demonstrate hyperchaotic attractors in the $(z_1,z_3)$-plane at $d=1.3$ and $d=1.4$, respectively.

2.3.3. Multistability and Coexistence of Attractors

Multistability or coexisting attractors represent a nonlinear phenomenon where two or more distinct attractors emerge simultaneously from different initial points [73,74,75]. Numerical simulations revealed that the new, 6D MMHS (2) exhibited rich coexisting attractor behaviors. The system showed the coexistence of two periodic attractors, two chaotic attractors, or two hyperchaotic attractors, depending on the parameter values.

The 6D MMHS (2) remained invariant under the following coordinate transformation: $(z_1,z_2,z_3,z_4,$ $z_5,z_6)\mapsto (-z_1,-z_2,z_3,-z_4,-z_5,-z_6)$. This symmetry allowed for the observation of attractor coexistence through appropriate initial point selection. This transformation was mentioned earlier. Let $Z_0=(1,1,1,1,1,1)$ and $W_0=(-1,-1,1,-1,-1,-1)$ denote two distinct initial points, with blue/red orbits representing state trajectories from $Z_0$/$W_0$, respectively.

For parameters a = 40, b = 8, c = 16, d = 12, k = 1, m = 0.1, and n = 0.01, the system exhibited two coexisting periodic attractors (Figure 17a): blue for $Z_0$; red for $W_0$.

With parameters a = 20.3, b = 90, c = 16, d = 12, k = 1, m = 0.1, and n = 0.01, the system displayed two coexisting chaotic attractors (Figure 17b) characterized by Lyapunov exponents:

$$\left\{\begin{array}{c}L{E}_1=0.2183\hfill \\ L{E}_2=0\hfill \\ L{E}_3=-0.0882\hfill \\ L{E}_4=-0.9997\hfill \\ L{E}_5=-1.0113\hfill \\ L{E}_6=-31.4155\hfill \end{array}\right.$$

(13)

Figure 17. (a) Coexisting periodic attractors in the $(z_1,z_3)$ plane; (b) coexisting chaotic attractors in the $(z_1,z_3)$ plane; (c) coexisting hyperchaotic attractors with two positive LEs in the $(z_1,z_3)$ plane; (d) coexisting hyperchaotic attractors with three positive LEs in the $(z_1,z_3)$ plane; (e) coexisting hyperchaotic attractors with three positive LEs in the $(z_1,z_2)$ plane; (f) coexisting hyperchaotic attractors with three positive LEs in the $z_1-z_2-z_3$ space.

Figure 17. (a) Coexisting periodic attractors in the $(z_1,z_3)$ plane; (b) coexisting chaotic attractors in the $(z_1,z_3)$ plane; (c) coexisting hyperchaotic attractors with two positive LEs in the $(z_1,z_3)$ plane; (d) coexisting hyperchaotic attractors with three positive LEs in the $(z_1,z_3)$ plane; (e) coexisting hyperchaotic attractors with three positive LEs in the $(z_1,z_2)$ plane; (f) coexisting hyperchaotic attractors with three positive LEs in the $z_1-z_2-z_3$ space.

When parameters a = 40, b = 15, c = 16, d = 12, k = 1, m = 0.1, and n = 0.01, the system generated two coexisting hyperchaotic attractors (Figure 17c) with Lyapunov exponents:

$$\left\{\begin{array}{c}L{E}_1=0.2496\hfill \\ L{E}_2=0.1153\hfill \\ L{E}_3=0\hfill \\ L{E}_4=-1.0024\hfill \\ L{E}_5=-1.6194\hfill \\ L{E}_6=-50.7398\hfill \end{array}\right.$$

(14)

Fixing parameters a = 40, b = 41, c = 16, d = 12, k = 1, m = 0.1, and n = 0.01, the system presented two coexisting hyperchaotic attractors (Figure 17d) with Lyapunov exponents:

$$\left\{\begin{array}{c}L{E}_1=4.4553\hfill \\ L{E}_2=0.1681\hfill \\ L{E}_3=0.1006\hfill \\ L{E}_4=0\hfill \\ L{E}_5=-1.0000\hfill \\ L{E}_6=-56.7186\hfill \end{array}\right.$$

(15)

Figure 17e and Figure 17f respectively depict the phase portraits of hyperchaotic attractors with three positive Lyapunov exponents in the $z_1-z_2$ plane and the 3D phase portraits in the $z_1-z_2-z_3$ space.

Figure 18 illustrates the bifurcation diagrams of the system with respect to parameter a for four representative values of b: (a) $b=8$, (b) $b=90$, (c) $b=15$, and (d) $b=41$, while the other parameters were fixed as $c=16$, $d=12$, $k=1$, $m=0.1$, and $n=0.01$. For each case, the bifurcation diagrams are plotted for two symmetric initial conditions: $(1,1,1,1,1,1)$ (blue) and $(-1,-1,1,-1,-1,-1)$ (red). Due to the system’s inherent symmetry, multistability emerged; however, the dynamical behaviors corresponding to these symmetric initial conditions were not identical. As shown in Figure 18, the bifurcation diagrams for the two initial conditions display distinct branches and attractor structures, further confirming the coexistence of multiple attractors and the system’s rich multistable dynamics. These results demonstrate that even under symmetric initial conditions, the system can evolve into different dynamical regimes, which is a hallmark of symmetry-induced multistability.

2.3.4. Offset Boosting Control

Offset boosting control enables amplitude manipulation through state-embedded parametric tuning, where introducing a controlled variable (p) induces spatial translation of a system’s attractor. This bidirectional displacement (along $z_5$ axes) preserves a system’s fundamental dynamic characteristics while achieving quantitative position control without altering its intrinsic stability properties.

Since the fifth state variable $z_5$ of the 6D MMHS (2) appeared exclusively in the fifth equation, this system exhibited variable-boostable hyperchaotic characteristics, with $z_5$ being fully controllable. Enhancement of $z_5$ could thus be achieved by substituting $z_5$ with $z_5+p$.

For offset-boosting control, the 6D MMHS (2) could be rewritten as

$$\left\{\begin{array}{cc}& {\dot{z}}_1=a(z_2-z_1)+z_2{z}_3+z_4\hfill \\ & {\dot{z}}_2=z_1(b-z_3)+c{z}_4\hfill \\ & {\dot{z}}_3=z_1^2+z_2{z}_1-d{z}_3\hfill \\ & {\dot{z}}_4=-z_2+z_6\hfill \\ & {\dot{z}}_5=z_6-(z_5+p)+kW\left(z_6\right)z_4\hfill \\ & {\dot{z}}_6=-z_4\hfill \end{array}\right.$$

(16)

where p serves as the offset-boosting controller.

When the parameters were set to $(a,b,c,d,k,m,n)=(40,90,16,12,1,0.1,0.01)$ and the initial state was $Z_0=(1,1,1,1,1,1)$, the offset-boosted hyperchaotic attractors of the 6D system (16) were obtained as shown in Figure 19. Figure 19a and Figure 19b respectively depict the phase portraits of hyperchaotic attractors in the $(z_2,z_5)$ plane and the phase-space $z_2-z_5-z_1$ for different values of p: negative p shifted the attractor in the positive direction, while positive p shifted it in the negative direction. Consequently, the hyperchaotic signal $z_5$ transitioned from a bipolar to a unipolar hyperchaotic signal, as demonstrated in the time-series plot of Figure 19c. This unique offset-boosting property holds significant promise for applications in secure communication and other engineering domains.

3. FPGA Implementation

Field-Programmable Gate Arrays (FPGAs) have gained increasing popularity due to their high customizability, parallel processing capabilities, and reconfigurability. These characteristics enable extremely low design and testing cycle costs for FPGA chips, making them particularly suitable for rapid prototyping and efficient implementation of highly complex dynamic chaotic systems where multiple variables interact in a highly nonlinear manner. In such scenarios, the parallel processing capacity of FPGAs facilitates more efficient computations, which is crucial for real-time concurrent processing and dynamic behavior analysis.

The FPGA hardware implementation of the 6D MMHS constitutes a significant research endeavor. In this study, for the FPGA hardware implementation, Vivado 2018.3 served as the primary development environment. The target platform was the ALINX AX7020 development board, which integrates the Xilinx XC7Z020-2CLG400I system-on-chip (SoC). This SoC features an ARM Cortex-A9 core running at 766 MHz and incorporates several key components: an image processing accelerator, internal and external memory controllers, and peripheral interfaces (Ethernet, SD/SDIO, UART), among others. Equipped with dual 40-pin (2.54 mm pitch) connectors, the AX7020 board utilized solely one row of pins in our experiment for routing chaotic signals to an oscilloscope. The post-implementation resource utilization report from Vivado 2018.3 is detailed in

Table 3. FPGA resource utilization (XC7Z020-2CLG400I).

Resource Type	Used	Available	Utilization (%)
LUT	5304	53,200	9.97
LUTRAM	376	17,400	2.16
Flip-Flop (FF)	9978	106,400	9.38
DSP48E	82	220	37.27
IO	34	125	27.20
BUFG	1	32	3.13

, demonstrating efficient hardware resource consumption. The moderate DSP utilization (37.27%) highlights the computational efficiency achieved in implementing the complex RK4 solver for the 6D system.

The numerical discretization of continuous chaotic systems frequently utilizes the classical fourth-order Runge–Kutta (RK4) algorithm. In contrast to the implicit Euler method, which possesses only first-order algebraic accuracy, the RK4 method achieves fourth-order algebraic precision, offering superior numerical solution accuracy and smaller error ranges in differential equation solving. This study implemented numerical simulations of the 6D MMHS on FPGA using the RK4 method, with its mathematical formulation described in Equation (17). The fixed-point arithmetic implementation utilized 32-bit precision (Q16.16 format) to balance computational accuracy with hardware resource constraints. The sampling frequency ($f_s$) was configured at 463 kHz, derived from the primary clock period of $108\times 20$ ns = 2.16 $\mu$s within the $Cl{k}_x$ module. To assess the impact of finite precision and discretization, the maximum absolute quantization error ($E_q$) for state variables was bounded by $E_q\le 2^{-16}\approx 1.53\times {10}^{-5}$. Phase portrait comparisons between FPGA outputs and MATLAB double-precision simulations confirmed that this error margin did not induce qualitative changes in the observed hyperchaotic attractors or multistable dynamics.

$$\left\{\begin{array}{c}{K}_1=f(x_k,y_k)\hfill \\ K_2=f\left(x_k+{\displaystyle \frac{h}{2}},y_k+{\displaystyle \frac{K_1}{2}}\right)\hfill \\ K_3=f\left(x_k+{\displaystyle \frac{h}{2}},y_k+{\displaystyle \frac{K_2}{2}}\right)\hfill \\ K_4=f(x_k+h,y_k+K_3)\hfill \\ y_{k+1}=y_k+{\displaystyle \frac{1}{6}}(K_1+2K_2+2K_3+K_4)\hfill \end{array}\right.$$

(17)

The implementation process is illustrated in Figure 20. In the $Cl{k}_x$ clock module, the clock period was configured as $108\times 20$ nanoseconds. Under stable output clock conditions, the original computation module within the $X_{reg}$ register updated results after each initial value iteration, enabling cyclic iterations to generate stable chaotic signals. Subsequently, a digital-to-analog converter (DAC) transformed the generated digital signals into analog outputs for oscilloscope visualization. Following these procedures, the Vivado 2018.3 simulation platform was utilized for synthesis and implementation to generate a bitstream file, which was programmed into the development board’s chip. Upon connecting the board to an oscilloscope, the corresponding attractor phase diagrams were successfully generated. Figure 21a–d present the circuit implementation results of the 6D MMHS phase diagrams, demonstrating robust consistency with the MATLAB simulation results shown in Figure 1a and Figure 1b, respectively. This visual congruence, maintained across multiple independent experimental runs (>10) and different initial conditions corresponding to distinct attractors, validates the robustness and reproducibility of the FPGA implementation under the chosen discretization parameters (RK4, $f_s$ = 463 kHz) and the quantization scheme (Q16.16).

4. Image Encryption

In recent years, with the rapid development of technologies such as artificial intelligence and data networks [76,77,78,79,80], people have paid more attention to their privacy and data security. Chaotic systems have received extensive attention for generating pseudo-random sequences with superior statistical properties. Therefore, chaos-based cryptographic algorithms and their applications in digital image encryption have developed into an important research branch in cryptography [81,82].

4.1. Image Encryption Process

In this section, we propose an image encryption scheme based on a 6D MMHS. In the cipher-generation phase of the proposed algorithm, the input plaintext image was denoted as P, with a size of $M\times N$. The keys were the initial values $\{x_0,y_0,z_0,w_0,u_0,v_0\}$ of the hyperchaotic system and three 8-bit random numbers $\{r_1,r_2,r_3\}$. We substituted the system initial values $\{x_0,y_0,z_0,w_0,u_0,v_0\}$ into the proposed 6D multistable memristive hyperchaotic system and, through operations, obtained six vectors $\{x_i,y_i,z_i,w_i,u_i,v_i\}$ of the required length. With the help of these six vectors, we generated three matrices, X, Y, and Z, for the subsequent encryption process. In the encryption process, we used matrix X for the first-step diffusion operation, converting the plaintext image P into matrix A. Using matrix Y to permute matrix A obtained from the first-step diffusion operation, we obtained the permuted matrix B. Using matrix Z for the second-step diffusion operation on the permuted matrix B, we obtained matrix C, which was the encrypted ciphertext image. The specific encryption process was as follows:

Step 1: Select a grayscale image with $MN$ pixels represented in binary form as the original image P to be processed.

Step 2: Use the fourth-order classical Runge–Kutta method to numerically simulate the 6D, multistable, memristive, hyperchaotic system, generating the chaotic sequence $\{x_i,y_i,z_i,w_i,u_i,v_i\}$. Generate three matrices X, Y, and Z of size $M\times N$ using Equation (18) according to the state variables x, y, z, w, u, and v.

$$\begin{array}{cc}\hfill X(i,j)& =\mathrm{floor}((x_{(i-1)\times N+j}+z_{(i-1)\times N+j}+u_{(i-1)\times N+j}+100mod1)\times {10}^{13})mod256,\hfill \\ \hfill & i=1,2,\cdots ,M,j=1,2,\cdots ,N\hfill \\ \hfill Y(i,j)& =\mathrm{floor}((y_{(i-1)\times N+j}+z_{(i-1)\times N+j}+v_{(i-1)\times N+j}+100mod1)\times {10}^{13})modK,\hfill \\ \hfill & i=1,2,\cdots ,M,j=1,2,\cdots ,N\hfill \\ \hfill Z(i,j)& =\mathrm{floor}((x_{(i-1)\times N+j}+w_{(i-1)\times N+j}+u_{(i-1)\times N+j}+100mod1)\times {10}^{13})mod256,\hfill \\ \hfill & i=1,2,\cdots ,M,j=1,2,\cdots ,N\hfill \end{array}$$

(18)

where M and N are the dimensions of the image, $K=max(M,N)$, X and Z are used for plaintext-irrelevant diffusion operations, and Y is used for plaintext-dependent permutation operations.

Step 3: This step is the first diffusion operation of the encryption algorithm. The plaintext image P is transformed into matrix A through the diffusion operation defined in Equation (19) using matrix X.

$$\begin{array}{cc}\hfill A(1,1)& =P(1,1)+X(1,1)+r_{1}mod256\hfill \\ \hfill A(1,j)& =P(1,j)+A(1,j-1)+X(1,j)mod256,j=2,3,\cdots ,N\hfill \\ \hfill A(i,1)& =P(i,1)+A(i-1,1)+X(i,1)mod256,i=2,3,\cdots ,M\hfill \\ \hfill A(i,j)& =P(i,j)+A(i,j-1)+A(i-1,j)+X(i,j)mod256,\hfill \\ \hfill & i=2,3,\cdots ,M,j=2,3,\cdots ,N\hfill \end{array}$$

(19)

The following Step 4–Step 6 describe the permutation operations of the encryption algorithm.

Step 4: Swap the positions of $A(i,j)$ and $A(m,n)$, where the position $(m,n)$ is obtained by Equation (20). The row M (columns 1 to N) and column N (rows 1 to M) of matrix A are excluded from the permutation.

$$\begin{array}{cc}\hfill m& =Y(i,j)+Y(i,N)+A(i,N)+A(M,N)modM,\hfill \\ \hfill & i=1,2,\cdots ,M-1,j=1,2,\cdots ,N-1\hfill \\ \hfill n& =Y(i,j)+Y(M,j)+A(M,j)+A(M,N)modM,\hfill \\ \hfill & i=1,2,\cdots ,M-1,j=1,2,\cdots ,N-1\hfill \end{array}$$

(20)

If $m=0$ or $n=0$ is obtained, the corresponding $A(i,j)$ does not participate in the permutation. The matrix after permutation is denoted as D.

Step 5: Permute $D(M,1\mathrm{to}N-1)$ and $D(1\mathrm{to}M,N)$. The specific permutation steps are as follows:

• Let $D(M,N)=(D(M,N)+r_2)mod256$.
• Swap the positions of $D(M,j)$ and $D(m,n)$, where m and n are obtained from Equation (21):

  $$\begin{array}{cc}\hfill m& =Y(M,j)+Y(M,N)+D(M,N)modM,\hfill \\ \hfill & j=1,2,\cdots ,N-1\hfill \\ \hfill n& =\mathrm{sum}\left(D\right(1:oM-1,j\left)\right)+Y(M,N)+D(M,N)modN,\hfill \\ \hfill & j=1,2,\cdots ,N-1\hfill \end{array}$$

  (21)
  Here, $\mathrm{sum}\left(\right)$ represents the summation of a vector. If $m=0$, $n=0$, or $n=j$ is obtained, the position of the corresponding $D(i,j)$ remains unchanged.
• Swap the positions of $D(i,N)$ and $D(m,n)$, where m and n are obtained from Equation (22):

  $$\begin{array}{cc}\hfill m& =Y(i,N)+Y(M,N)+D(M,N)modM,\hfill \\ \hfill & i=1,2,\cdots ,M-1\hfill \\ \hfill n& =\mathrm{sum}\left(D\right(i,1:oN-1\left)\right)+Y(M,N)+D(M,N)modN,\hfill \\ \hfill & i=1,2,\cdots ,M-1\hfill \end{array}$$

  (22)
  If $m=0$, $n=0$, or $m=i$ is obtained, the position of the corresponding $D(i,j)$ remains unchanged. After the above operations, matrix D is transformed into matrix E.

Step 6: Permute the position of $E(M,N)$. Swap the positions of $E(M,N)$ and $E(m,n)$, where m and n are obtained from Equation (23):

$$\begin{array}{cc}\hfill m& =\mathrm{sum}\left(Y\right(M,1:oN-1\left)\right)+\mathrm{sum}\left(E\right(M,1:oN-1\left)\right)modM\hfill \\ \hfill n& =\mathrm{sum}\left(Y\right(1:oM-1,N\left)\right)+\mathrm{sum}\left(E\right(1:oM-1,N\left)\right)modN\hfill \end{array}$$

(23)

If $m=0$ or $n=0$ is obtained, the position of $E(M,N)$ remains unchanged. After the above operations, matrix E is denoted as matrix B.

$$\begin{array}{cc}\hfill C(M,N)& =B(M,N)+Z(M,N)+r_{3}mod256\hfill \\ \hfill C(M,j)& =B(M,j)+C(M,j+1)+Z(M,j)mod256,\hfill \\ \hfill & j=N-1,N-2,\cdots ,2,1\hfill \\ \hfill C(i,N)& =B(i,N)+C(i+1,N)+Z(i,N)mod256,\hfill \\ \hfill & i=M-1,M-2,\cdots ,2,1\hfill \\ \hfill C(i,j)& =B(i,j)+C(i,j+1)+C(i+1,j)+Z(i,j)mod256,\hfill \\ \hfill & i=M-1,M-2,\cdots ,2,1,j=N-1,N-2,\cdots ,2,1\hfill \end{array}$$

(24)

Step 7: This step is the second diffusion operation of the encryption algorithm. Using matrix Z and performing the operation defined in Equation (24), we obtain the encrypted ciphertext image matrix C.

Figure 22 shows the image encryption process.

4.2. Performance and Security Analysis

4.2.1. Histogram Analysis

A histogram visualizes the spread of pixel intensities within an image. In secure image encryption, achieving a uniform histogram is a key objective as it provides strong resilience against statistical analysis-based attacks. Our evaluation employed two distinct image sets. The outcomes, presented in Figure 23, included the source images (Figure 23a,f), their encrypted counterparts (Figure 23b,g), and the pertinent histograms. The decrypted images are presented in Figure 23c,h. Figure 23d,i illustrate the histograms of the two groups of original images, while Figure 23e,j show the histograms of the encrypted images.

Visually, the encrypted images in Figure 23 exhibited flat histograms, while the histograms of the original images showed significant fluctuations. The ${\chi }^2$ statistic was commonly used to quantitatively measure the difference between them. For a grayscale image with 256 gray levels and size $M\times N$, we assumed that the frequency of pixel points for each gray value i in the histogram was $f_i$. If the distribution was uniform, then $g_i=g=MN/256$ for $i=0,1,2,\dots ,255$, and we had

$${\chi }^2=\sum _{i=0}^{255}{\displaystyle \frac{{(f_i-g)}^2}{g}}$$

(25)

This follows a ${\chi }^2$ distribution with 255 degrees of freedom, given a significance level $\alpha$ such that

$$P\left\{{\chi }^2\ge {\chi }_a^2\left(n-1\right)\right\}=\alpha$$

(26)

A commonly used significance level is $\alpha =0.05$, for which we obtain ${\chi }_{0.05}^2\left(255\right)=293.24783$. As shown in

Table 4. ${\chi }^2$ test result.

Index	Plain Image		Cipher Image
result	Baboon	Peppers	Baboon	Peppers
	$1.8735\times {10}^5$	$1.2017\times {10}^5$	235.2344	226.2598

, the test statistics of the original images were significantly larger than ${\chi }_{0.05}^2\left(255\right)$, while those of the ciphertext images were all smaller than ${\chi }_{0.05}^2\left(255\right)$. It could be concluded that under the 0.05 significance level, the histogram distribution of the ciphertext images was approximately uniform.

4.2.2. Correlation Analysis of Adjacent Pixels

Furthermore, comparing the correlation traits in plaintext images with those in their encrypted versions was imperative. Generally, adjacent pixels in plaintext images exhibit strong correlations in horizontal, vertical, positive diagonal, and negative diagonal directions, whereas adjacent pixels in ciphertext images should show no such correlations. Suppose N pairs of adjacent pixels were randomly selected from the original image and that their grayscale values were denoted as $(u_i,v_i)$, where $i=1,2,3,\dots ,N$. The correlation coefficient between vectors $\mathbf{u}=\left\{u_i\right\}$ and $\mathbf{v}=\left\{v_i\right\}$ was calculated using the following Formula (27):

$$\begin{array}{cc}\hfill & r_{xy}={\displaystyle \frac{\mathrm{cov}(u,v)}{\sqrt{D\left(u\right)}\sqrt{D\left(v\right)}}}\hfill \\ \hfill & cov(u,v)={\displaystyle \frac{1}{N}}\sum _{i=1}^N(x_i-E\left(u\right))(y_i-E\left(v\right))\hfill \\ \hfill & D\left(u\right)={\displaystyle \frac{1}{N}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{i=1}{N}}}^N{\left(u_i-E\left(u\right)\right)}^2\hfill \\ \hfill & E\left(u\right)={\displaystyle \frac{1}{N}}\sum _{i=1}{u}_i\hfill \end{array}$$

(27)

Taking the Baboon image as an example, the calculation results are shown in

Table 5. Correlation analysis of adjacent pixels.

Image	Horizontal	Vertical	Main Diagonal	Anti-Diagonal
Plain Image	0.749483	0.862469	0.740462	0.706836
Cipher image	−0.03479	−0.034369	0.005658	0.001927

. It can be seen from Table 5 that adjacent pixels in the plaintext image exhibited strong correlations, while those in the ciphertext image had correlations close to 0, indicating almost no correlation. Figure 24 visualizes adjacent pixel correlations in the plaintext and ciphertext Baboon images across four orientations: horizontal, vertical, and both positive and negative diagonals. In Figure 24, adjacent pixel pairs in the plaintext image are densely clustered along the $y=x$ line in all directions, whereas those in the ciphertext image are uniformly distributed within the rectangular region in all directions. These findings align with the results in Table 5, demonstrating the algorithm’s effective encryption performance.

4.2.3. Information Entropy

Information entropy quantifies the uncertainty within image data. Theoretically, a higher entropy value signifies greater randomness, increased information content, and reduced visual structure. The defining equation for information entropy is presented as Equation (28).

$$H=-\sum _{i=0}^{L}p\left(i\right){\log }_{2}p\left(i\right)$$

(28)

In Equation (28), L represents the number of gray levels in the image and $p\left(i\right)$ denotes the probability of occurrence of the gray value i.

Table 6. The results of information entropy.

Index	Plain Image			Cipher Image
result	Lena	Baboon	Peppers	Lena	Baboon	Peppers
	7.4451	7.3583	7.5937	7.9992	7.9994	7.9993

presents the entropy values for the plaintext and ciphertext images of Lena, Baboon, and Pepper.

The entropy value for each ciphertext image approached the theoretical maximum of 8. Taking the Baboon image as an example,

Table 7. Comparison of information entropy.

Image	Plain	Cipher	Ref. [83]	Ref. [84]	Ref. [85]
Information entropy	7.3538	7.9994	7.991	7.906	7.740

compares the encryption algorithm proposed in this paper with existing methods from the literature. The results indicate that the information entropy of the proposed encryption algorithm is closer to the theoretical maximum of 8. This demonstrates that the system can effectively disrupt the statistical patterns of a plaintext image and efficiently conceal original image information, thereby protecting an image’s privacy and security.

4.2.4. Key Sensitivity Analysis

Within image encryption, key sensitivity analysis serves as a crucial security criterion. This metric quantifies the disparity in ciphertexts generated from encrypting the same plaintext image with slightly altered keys. If the two ciphertext images exhibit a significant difference, the image cryptosystem is said to have strong key sensitivity. Conversely, if the difference is minimal, the key sensitivity of the image cryptosystem is considered poor. NPCR (Number of Pixel Change Rate) and UACI (Unified Average Changing Intensity) are prevalent metrics for assessing key sensitivity. NPCR specifically evaluates the percentage rate of altered pixels resulting from a minimal key modification, as defined by Equation (29):

$$NPCR(P_1,P_2)={\displaystyle \frac{1}{MN}}\sum _{i=1}^M\sum _{j=1}^N\left|Sign(P_1(i,j)-P_2(i,j))\right|\times 100\%$$

(29)

UACI assesses key sensitivity by computing the mean intensity variation per pixel between two ciphertext images. Its defining equation is (30)

$$UACI(P_1,P_2)={\displaystyle \frac{1}{MN}}\sum _{\mathrm{i}}^M\sum _j^N{\displaystyle \frac{\left|P_1(i,j)-P_2(i,j)\right|}{255-0}}\times 100\%$$

(30)

The key of the image encryption algorithm was $K=\{x_0,y_0,z_0,w_0,u_0,v_0,r_1,r_2,r_3\}$. By slightly perturbing the initial values with a variation of ${10}^{-13}$ and encrypting the same image again, we obtained two ciphertext images. Finally, their NPCR and UACI values were calculated. The results are shown in

Table 8. Coefficient of association.

Index	${\mathit{x}}_0$	${\mathit{y}}_0$	${\mathit{z}}_0$	${\mathit{w}}_0$	${\mathit{u}}_0$	${\mathit{v}}_0$	${\mathit{r}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	Theoretical Value
NPCR	99.611	99.6081	99.6082	99.6085	99.6118	99.6094	99.6101	99.6097	99.6107	99.6094
UACI	33.4608	33.4625	33.4657	33.469	33.4538	33.4645	33.4651	33.4624	33.4661	33.4635

. As indicated, the calculated values were close to the theoretical ones.

4.2.5. Plain Image Sensitivity Analysis

Plaintext sensitivity analysis involves encrypting two minimally distinct plaintexts with an identical key and comparing the resulting ciphertext differences. If the two ciphertext images exhibit significant differences, the image cryptosystem is said to have good plaintext sensitivity; conversely, if the differences are minimal, the system is considered to have weak plaintext sensitivity, which generally cannot resist chosen-plaintext attacks or known-plaintext attacks. Implementing stochastic perturbations at selective pixel positions produces functionally differentiable plaintexts from an original image. For example, a pixel $(i,j)$ is randomly selected from a plaintext image $P_1$, and its value $P_1(i,j)$ is changed by 1, such that the new value becomes $(P_1(i,j)+1)mod256$, resulting in a slightly different plaintext image $P_2$. As evidenced by

Table 9. Sensitivity analysis of plaintext images (Lena, Baboon, Peppers).

Index	Lena	Baboon	Pepper	Theoretical Value
NPCR	99.6106	99.6094	99.6095	99.6094
UACI	33.4567	33.4659	33.4629	33.4635

, near-optimal NPCR and UACI values substantiate the plaintext attack resilience of the 6D MMHS encryption framework.

5. Conclusions

This study established a 6D MMHS by incorporating a memristor into a 5D hyperchaotic structure, demonstrating sustained hyperchaotic behavior and multistability across wide parameter ranges and diverse initial conditions. The hyperchaotic nature of the 6D MMHS was rigorously confirmed through Lyapunov exponent spectrum analysis and Kaplan–Yorke dimension quantification, with hyperchaos maintained even when the parameter b spanned the broad interval [13, 300]. The underlying mechanisms of multistability were analytically derived via invariance analysis of state variables under coordinate transformations, providing a theoretical foundation for coexisting attractors. Strategic modulation of the fifth state variable ($z_5$) enabled effective offset boosting control, enhancing the system’s adaptability for dynamic applications. Hardware validation was achieved through FPGA-based implementation, where experimental attractors closely aligned with numerical simulations, verifying the system’s physical realizability. Furthermore, an image encryption scheme leveraging the 6D MMHS was proposed and rigorously evaluated, exhibiting cryptographic resilience through statistical uniformity, correlation coefficient minimization, and robust differential metrics (NPCR, UACI). These contributions underscore the 6D MMHS as a versatile framework for chaos-based engineering applications, with potential extensions to secure communication and nonlinear control systems. It is worth noting that, similar to other six-dimensional memristive super-chaos systems, this work calculated and analyzed key chaos statistical quantities, including phase portraits, Lyapunov exponent spectra, and bifurcation diagrams. In future studies, additional chaos-related measures such as correlation dimension, Kolmogorov–Sinai entropy, and permutation entropy could be further investigated to provide a more comprehensive understanding of the system’s complexity. The proposed 6D MMHS can also be further explored for applications such as secure wireless communication, random number generation, and nonlinear control. Optimizing the hardware implementation for higher speed and lower power consumption, as well as integrating the system into real-time embedded platforms, are promising directions. Additionally, further theoretical studies on the coexistence of attractors and parameter sensitivity, as well as extending the encryption framework to color images, video, or other multimedia data, could broaden the practical impact of this work.