Hidden Attractors in a New 4D Memristor-Based Hyperchaotic System: Dynamical Analysis, Circuit Design, Synchronization, and Its Applications

Abstract

This paper presents a novel four-dimensional (4D) memristive system, notable for its simplicity and unique dynamic behaviors. Comprising only seven terms and devoid of equilibrium points, this system is capable of generating hidden attractors. A comprehensive analysis of its dynamic properties is conducted, including Lyapunov exponents, Kaplan–Yorke dimensions, temporal diagrams and phase portraits, multistability, and offset boosting. Numerical simulations over a sufficiently long time interval show that the Lyapunov function remains bounded, thereby confirming the dissipative nature and global stability of the newly proposed 4D hyperchaotic system. The theoretical model is further validated through electronic simulations of the chaotic system using Multisim software. Additionally, the paper explores synchronization between two identical 4D hyperchaotic systems. The proposed system, despite its structural simplicity, exhibits intricate chaotic dynamics, making it suitable for various practical applications. In particular, a method for chaotic encryption and decryption of an information signal was developed and validated through numerical testing. Based on the method of complete synchronization of chaotic systems, the possibility of detecting a weak signal is demonstrated. The proposed system is also implemented on an Arduino UNO to demonstrate its practical applicability for real-time chaotic signal generation and image encryption.

1. Introduction

Recent advances in chaos theory have significantly expanded its scope of applications, making it an integral tool in diverse engineering and scientific domains. Its influence spans secure communications [1], cryptographic systems [2,3], medicine [4], neural and genetic networks [5,6], and memristive systems [7], among other fields.

Lorenz’s discovery of a three-dimensional (3D) chaotic system [8] has sparked extensive exploration into the behavior and properties of various chaotic systems. To exhibit chaotic behavior, a dynamic system must include at least five terms. Systems at this threshold of simplicity are notably rarer than those with six or more terms. Their algebraic elegance not only makes them attractive but also minimizes the number of electronic components required for circuit implementation. Sprott pioneered the introduction of several five-term chaotic systems [9], followed by contributions from Van der Schrier and Maas [10], Elwakil et al. [11], Toncharoen and Srisuchinwong [12], and Munmuangsaen and Srisuchinwong [13]. However, despite these advances, the past decade has seen only a modest number of novel five-term chaotic attractors proposed [14,15,16]. For example, Chang and Kim [14] introduced a chaotic five-term system with one multiplier, one quadratic term, and one absolute value term. Another study by Tamba et al. [15] presented a new 3D five-term chaotic system distinguished by hyperbolic sinusoidal nonlinearity and the absence of equilibrium points. Furthermore, Gokyildirim et al. [16] described a new five-term 3D chaotic flow characterized by cubic nonlinearity, a linear term, a constant term, and two other nonlinear components.

However, chaos researchers have increasingly shifted their focus towards dynamic systems with dimensions greater than three, particularly hyperchaotic systems, which display more intricate and complex dynamics compared to standard chaotic systems. The defining feature of hyperchaotic systems is the presence of multiple positive Lyapunov exponents (LEs), indicating a higher degree of unpredictability. Specifically, a four-dimensional (4D) autonomous hyperchaotic system is characterized by two positive Lyapunov exponents, one zero exponent, and one negative exponent [17]. In this context, there is a growing trend to develop new chaotic and hyperchaotic models that address the following objectives:

(1)

Ensuring the presence of multiple positive Lyapunov exponents;

(2)

Minimizing the number of terms in the system for simplicity;

(3)

Maximizing the Kaplan–Yorke dimension to achieve higher system complexity.

Chua’s groundbreaking discovery of the memristor [18]—a device that couples electric charge and magnetic flux, effectively functioning as a resistor with memory, has proven instrumental in tackling these challenges, enabling the design of more efficient and sophisticated chaotic and hyperchaotic systems. Numerous chaotic and hyperchaotic systems that incorporate memristors as nonlinear elements have been developed, demonstrating the versatility of memristors in generating complex dynamics. In addition, various equivalent circuits for modeling memristor emulators have been extensively explored in the literature [19].

In this work, we focus specifically on 4D chaotic systems that integrate memristors, leveraging their unique properties to enhance system behavior and complexity. Pham et al. [20] introduced a 4D memristive system comprising 12 terms, 5 of which are nonlinear. This system is particularly notable for its lack of equilibrium points and its ability to exhibit periodic, chaotic, and hyperchaotic dynamics within specific parameter ranges. Rajagopal et al. [21] proposed another 4D memristive system where the equilibrium structure is dependent on a control parameter, allowing for either no equilibrium or an equilibrium line. By varying this parameter, the system transitions between chaotic and hyperchaotic behavior. This system includes 11 terms, 5 of which are non-linear. Singh et al. [22] presented a simpler 4D chaotic memristor-based system without equilibria, featuring only 9 terms, including two nonlinear ones. An even more compact design was introduced by Tamba et al. [23], who developed a 4D two-scroll chaotic memristive system with just 7 terms and three nonlinearities. This system exhibits diverse and complex dynamics, including offset boosting, remerging period-doubling bifurcations, and hidden extreme multi-stability. Irfan [24] proposed a 4D hyperchaotic hyperjerk memristive system characterized by a line equilibrium and containing seven algebraic terms with a single non-linearity. Remarkably, this system incorporates an intrinsic memristive nonlinearity, an inherent property of the memristor itself, making it a highly streamlined and elegant design.

This study presents a streamlined 4D seven-term system derived from a 3D five-term chaotic system incorporating a memristor. For ease of comparison,

Table 1. Comparison of 4D memristive chaotic systems based on their main characteristics.

Reference	Terms	Attractor	Equilibria	No. of NL Terms
Pham et al. [20]	12	Chaotic/hyperchaotic	None	5
Rajagopal et al. [21]	11	Chaotic/hyperchaotic	None or line	5
Singh et al. [22]	9	Chaotic	None	2
Tamba et al. [23]	7	Two-scroll chaos	None	3
Irfan [24]	7	Hyperchaotic	Line equilibrium	1
This work	7	Hyperchaotic	None	3

presents the main characteristics of various 4D memristive chaotic systems, including the newly proposed one.

The proposed hyperchaotic system, notable for its absence of equilibrium points and its ability to generate hidden attractors, features only three non-linear terms. This approach highlights the potential to construct new hyperchaotic dynamic systems from simpler 3D five-term chaotic systems, paving the way for the development of electronic chaos generators with increased complexity and functionality.

This manuscript is organized as follows. The introduction provides a concise overview of the current state of research and the problem addressed in this study. Section 2 details the derivation of a novel 4D hyperchaotic dynamical system that incorporates a memristor. Section 3 analyzes the dynamic properties of the proposed non-linear 4D system. This includes examining its fixed points, computing the Lyapunov spectrum and dimension, and investigating phenomena such as multistability and offset-boosting control. Section 4 focuses on implementing an electronic circuit for a hyperchaotic chaos generator using the Multisim environment. The circuit’s functionality is validated by comparing simulation results with those obtained through the Mathematica environment. Section 5 explores the numerical analysis of synchronization between two identical 4D hyperchaotic systems. Synchronization is achieved using the adaptive control method, as described in [25,26,27,28]. In Section 6, we apply adaptive synchronization of chaotic signals to secure communication. Section 7 covers the identification of weak signals using a new 4D hyperchaotic system. A method for detecting weak signals based on chaotic system synchronization is presented. The signal perturbs the master system, causing a loss of synchronization with the slave. The resulting synchronization error reveals the presence of a weak signal. In Section 8, the practical implementation of the newly introduced four-dimensional memristive hyperchaotic system is successfully achieved using the cost-effective and readily available Arduino UNO microcontroller. Conclusions summarize the key findings and contributions of this study.

2. A New 4D Memristive Dynamical System: Derivation and Key Properties

The primary objective in designing a new 4D dynamical system incorporating a memristor is to develop a model with minimal terms while still exhibiting complex behaviors, such as hidden attractors and hyperchaos. This section introduces a novel 4D memristive two-wing chaotic system consisting of only seven terms. This system is based on the model proposed in [13] and is defined as follows:    

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_1}{dt}}& =a(-x_1+x_2),\hfill \\ \hfill {\displaystyle \frac{d{x}_2}{dt}}& =-x_1{x}_3,\hfill \\ \hfill {\displaystyle \frac{d{x}_3}{dt}}& =x_1{x}_2-b.\hfill \end{array}\right.$$

(1)

Here, a and b are positive constant parameters. The system (1) exhibits highly complex chaotic behavior when $a=5, b=90$, and the initial conditions (ICs) are set to $x_1\left(0\right)=5, x_2\left(0\right)=10, x_3\left(0\right)=5$. In this case, the Lyapunov exponents for the system (1) are $L{E}_1=1.4913, L{E}_2=0.0026, L{E}_3=-6.4939$. Furthermore, the Lyapunov dimension $D_L$ of the system (1) is fractional, with a value of $D_L=2.2301$ [13].

By adding a fourth state variable and connecting it to the original system through a memristor, the system (1) expands to a 4D system, producing hyperchaotic behavior. We use the absolute memristor model for simplicity, namely Bao’s magnetically controlled memristor [29], which is defined by the following equations:

$$\left\{\begin{array}{c}{i}_m=W\left(\phi \right)u_m,\hfill \\ {\displaystyle \frac{d\phi }{dt}}=u_m,\hfill \\ W\left(\phi \right)=\alpha +\beta \left|\phi \right|.\hfill \end{array}\right.$$

(2)

The symbols $u_m$, $i_m$, and $\phi$ in equation (2) represent the input, output, and state variables of the memory device, respectively. Whereas $\alpha$ and $\beta$ are constant coefficients, set to $\alpha =1$ and $\beta =0.5$, the function $\phi$ represents the magnetic flux. An origin-crossing smooth quadratic nonlinear characteristic curve is visible in the graph of the system (2). The memristor is powered by a sinusoidal AC voltage source, $u_m$, which can be written as $u_m=Asin\left(2\pi ft\right)$, where A and f stand for the amplitude and frequency of the external signal, respectively.

Figure 1 illustrates the memristor circuit in the simulation results, which show how it reacts to sinusoidal excitation with alternating current. A closed curve, more precisely, a hysteresis loop that passes through the origin, is formed by the memristor’s current-voltage characteristic. Interestingly, the area of the hysteresis loop shrinks with increasing frequency f, but expands with increasing amplitude A. This behavior is consistent with the fundamental properties of memristors.

A new set of 4D nonlinear dynamical equations based on memristor is produced by integrating the expressions (2) into the system of nonlinear dynamic equations in (1), as illustrated below:   

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_1}{dt}}& =a{x}_2-(a+\alpha )x_1-\beta \left|x_4\right|x_1,\hfill \\ \hfill {\displaystyle \frac{d{x}_2}{dt}}& =-x_1{x}_3,\hfill \\ \hfill {\displaystyle \frac{d{x}_3}{dt}}& =x_1{x}_2-b,\hfill \\ \hfill {\displaystyle \frac{d{x}_4}{dt}}& =x_1.\hfill \end{array}\right.$$

(3)

In this formulation, we replace the flux notation $\phi$ with a new dynamic variable $x_4$. The system (3) consists of seven terms, including four constant parameters and three nonlinear terms. In particular, it represents the minimum number of terms necessary to generate hyperchaotic dynamics in a four-dimensional autonomous system. Such a 4D system with only seven terms is uncommon in the literature, underscoring a key advantage of the proposed system (3).

3. Dynamical Analysis

In this section, we consider some of the basic dynamic characteristics of the new 4D dynamic system.

3.1. Symmetry and Dissipative

The system (3) is easily verifiable to be symmetric with respect to the $x_3$-axis and to remain invariant under the following coordinate transformation $\mathbf{T}$:

$$\mathbf{T}:(x_1,x_2,x_3,x_4)\to (-x_1,-x_2,x_3,-x_4).$$

(4)

In order to further describe the behavior of the system, the divergence or trace of the matrix of the system (3) is calculated as

$$Tr\left(f\left(x_i\right)\right)=\nabla ·f\left(x_i\right)=\sum _{i=1}^4{\displaystyle \frac{\partial {\dot{x}}_i}{\partial x_i}}=-(a+\alpha +\beta |x_4\left|\right).$$

(5)

Consequently, for all positive values of the parameters, the system (3) is dissipative.

3.2. Equilibrium Points

The suggested 4D system has no points of equilibrium. The equilibrium states of a dynamic system (3) are found by setting all ${\dot{x}}_i=0$ in the system (3) and solving the system equations.

$$\left\{\begin{array}{cc}\hfill 0& =a{\tilde{x}}_2-(a+\alpha ){\tilde{x}}_1-\beta \left|{\tilde{x}}_4\right|{\tilde{x}}_1,\hfill \\ \hfill 0& =-{\tilde{x}}_1{\tilde{x}}_3,\hfill \\ \hfill 0& ={\tilde{x}}_1{\tilde{x}}_2-b,\hfill \\ \hfill 0& ={\tilde{x}}_1.\hfill \end{array}\right.$$

(6)

From the fourth equation, we find that ${\tilde{x}}_1=0$. Substituting this value into the third equation yields a contradictory result, $-b=0$, indicating that the system has no equilibrium points. Consequently, all attractors generated by the system (3) are hidden attractors. The system’s attractors are regarded as hidden since they are not directly related to any of the system’s equilibrium points, including stable nodes, saddles, or foci. In particular, the attractors in this system are located in areas of phase space that lack equilibrium points, which makes it more difficult to detect them. For these attractors, standard methods, such as the Shil’nikov theorem [30], that frequently depend on perturbing trajectories close to equilibrium points are ineffective.

3.3. Lyapunov Exponents and Kaplan–Yorke Dimension

The system’s behavior depends on variations in multiple parameters, making multiparameter analysis a complex process. Therefore, it is preferable to study the system by varying a single parameter as a control parameter. The evolution of the system for varying values of the control parameter can be analyzed through the Lyapunov exponent spectrum, providing insights into its dynamic behavior and stability characteristics.

A deeper understanding of a dynamic system’s behavior under changes in a control parameter can be achieved by examining its Lyapunov exponents (LEs). These exponents represent the average exponential rates at which nearby trajectories in phase space diverge or converge. Analyzing the Lyapunov exponents provides insight into the system’s stability and the presence of chaotic behavior. A positive Lyapunov exponent (LE) indicates instability or chaos in the system. A negative LE indicates asymptotic stability, which may correspond to convergence toward an equilibrium point or a stable periodic orbit. Thus, by examining the signs of the LEs, we can classify the system’s behavior as quasi-regular (2-torus), chaotic, or hyperchaotic. We computed all Lyapunov exponents using a as a control parameter, with other parameters fixed at $\alpha =1, \beta =0.1525, b=90$, and the initial conditions given by (7):

$$x_1\left(0\right)=5,x_2\left(0\right)=10,x_3\left(0\right)=5,x_4\left(0\right)=5.$$

(7)

The dynamical behaviors (3) are classified as shown in

Table 2. Lyapunov exponents for different values of the parameter a.

a	$\mathbf{Lyapunov}\mathbf{Exponents}$$({\mathit{LE}}_1,{\mathit{LE}}_2,{\mathit{LE}}_3,{\mathit{LE}}_4)$	$\mathbf{Signs}$	$\mathbf{Behavior}$
$0.7$	$(0.1743,\mathbf{0}.\mathbf{0003},-0.17889,-6.3915)$	$(+,0,-,-)$	$\mathrm{Chaotic}$
2	$(0.2270,\mathbf{0}.\mathbf{0067},-0.3792,-7.7904)$	$(+,0,-,-)$	$\mathrm{Chaotic}$
$2.6$	$(0.8428,\mathbf{0}.\mathbf{0170},-\mathbf{0}.\mathbf{0161},-6.9004)$	$(+,0,0,-)$	$\mathrm{Chaotic}$$2-\mathrm{torus}$
3	$(1.0087,0.0328,-\mathbf{0}.\mathbf{0049},-7.7900)$	$(+,+,0,-)$	$\mathrm{Hyperchaotic}$
5	$(0.8859,0.0556,-\mathbf{0}.\mathbf{0071},-10.8456)$	$(+,+,0,-)$	$\mathrm{Hyperchaotic}$
$5.1$	$(1.1164,\mathbf{0}.\mathbf{0087},-\mathbf{0}.\mathbf{0229},-10.5894)$	$(+,0,0,-)$	$\mathrm{Chaotic}$$2-\mathrm{torus}$
10	$(0.1065,-\mathbf{0}.\mathbf{0227},-1.2504,-15.2400)$	$(+,0,-,-)$	$\mathrm{Chaotic}$

.

From Table 2 we see examples of hyperchaos in system (3) for the parameter values $a=3$ and $a=5$, respectively. The dissipative nature of the system (3) is confirmed by the fact that the sum of the six Lyapunov exponents is negative: $L_1+L_2+L_3+L_4+L_5+L_6<0$. To evaluate the complexity of the attractors in the new hyperchaotic system (3), we calculate the Lyapunov dimension (or Kaplan–Yorke) for the parameter values $a=3$ and $a=5$ as follows:

$$D_{KY}{|}_{a=3}=\xi +{\displaystyle \frac{1}{|L{E}_{\xi +1}|}}\sum _{i=1}^{\xi }L{E}_i=3+{\displaystyle \frac{1.0366}{7.79}}\approx 3.133,$$

$$D_{KY}{|}_{a=5}=\xi +{\displaystyle \frac{1}{|L{E}_{\xi +1}|}}\sum _{i=1}^{\xi }L{E}_i=3+{\displaystyle \frac{0.9344}{10.8456}}\approx 3.086,$$

(8)

where $\xi$ is determined from the conditions

$$\begin{gathered} \mathrm{for}a=3:\sum _{i=1}^{\xi }L{E}_i>0\Rightarrow \sum _{i=1}^{3}L{E}_i=1.0366,\sum _{i=1}^{\xi +1}L{E}_i=-6.7534<0, \\ \mathrm{for}a=5:\sum _{i=1}^{\xi }L{E}_i>0\Rightarrow \sum _{i=1}^{3}L{E}_i=0.9344,\sum _{i=1}^{\xi +1}L{E}_i=-9.9112<0. \end{gathered}$$

(9)

Here, $\xi$ represents the count of the first nonnegative Lyapunov exponents in the spectrum. The dimension $D_{KY}$ of (8) is fractional. We found that the Kaplan-Yorke dimension (8) is significantly higher than that of the chaotic system (1), indicating greater complexity in the dynamics of the system (3).

The dynamics of the Lyapunov exponents associated with hyperchaotic behavior for $a=5$ is illustrated in Figure 2.

The bifurcation diagram and the Lyapunov exponent spectrum concerning variations in the parameter a are depicted in Figure 3. These results indicate that system (3) exhibits chaotic and hyperchaotic regimes over a broad interval $a\in (0.1,20)$.

3.4. Temporal Diagrams and Phase Portraits

Time diagrams provide a visual representation of how the state variables of a system evolve over time, effectively revealing oscillatory patterns, transient dynamics, or steady-state behavior. We plotted the temporal evolution of the state components for the hyperchaotic system (3), as illustrated in Figure 4.

The time series of the variables $x_1, x_2, x_3, x_4$ exhibit an aperiodic structure, a hallmark feature of chaotic systems. An interesting behavior of the state variable $x_4$ is observed during computer simulations. The simulation time for the $x_4$ coordinate was extended to $t=$ 10,000. As shown in Figure 4d, the amplitude of $x_4$ increases to approximately $t=700$. Beyond this point ($t>700$), the growth ceases and $x_4$ exhibits chaotic fluctuations within the range of roughly 114 to 118.

Phase portraits offer a geometric representation of the trajectories of a dynamical system in its state space, providing a clear visualization of chaotic attractors and their structure. Figure 5 also presents the solutions of Equation (3) with initial conditions (7), depicted as phase portraits of hidden attractors in the planes $x_1{x}_2$, $x_1{x}_3$, $x_2{x}_3$, $x_1{x}_4$, $x_2{x}_4$, and $x_3{x}_4$. The analysis of the Poincaré section concerning the plane $x_3=5$ shows that the trajectory densely fills a subset of the $(x_1,x_2)$-plane, indicating the presence of chaotic dynamics (see Figure 4f). The scattered structure of points rules out periodic or quasiperiodic motion and supports the hypothesis of sensitive dependence on initial conditions. As illustrated in Figure 5, the dynamic variables $x_i$ ($i=1,2,3,4$) exceed the range supported by the power supply capabilities of the operational amplifiers. To address this limitation, the variables $x_i$ in the dynamical system (3) are rescaled as follows: $x_1=5X_1, x_2=10X_2, x_3=10X_3, x_4=20X_4$. After applying this transformation, the hyperchaotic system (3) is reformulated as follows:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{X}_1}{dt}}& =10X_2-6X_1-3.05\left|X_4\right|X_1,\hfill \\ \hfill {\displaystyle \frac{d{X}_2}{dt}}& =-5X_1{X}_3,\hfill \\ \hfill {\displaystyle \frac{d{X}_3}{dt}}& =5X_1{X}_2-9,\hfill \\ \hfill {\displaystyle \frac{d{X}_4}{dt}}& =0.25X_1.\hfill \end{array}\right.$$

(10)

For this system, the initial conditions (7) are similarly transformed as follows:

$$X_1\left(0\right)=1,X_2\left(0\right)=1,X_3\left(0\right)=0.5,X_4\left(0\right)=0.25.$$

(11)

Thus, the transformed system (10) can be utilized to design an analog chaos generator circuit.

3.5. Stability Analysis via the Lyapunov Function

The considered hyperchaotic system does not possess any equilibrium points, as the algebraic conditions for stationarity cannot be satisfied due to the nonlinear structure, particularly the non-smooth term $|X_4|X_1$ in the first equation. In such systems, classical local linearization techniques become inapplicable, and stability analysis must rely on alternative global methods. A common approach involves the use of a Lyapunov function to investigate the boundedness and dissipativity of the system. Let us consider the following Lyapunov function:

$$V\left(X\right)={\displaystyle \frac{1}{2}}(X_1^2+X_2^2+X_3^2+X_4^2),$$

(12)

which is a standard positive-definite energy-like function. Taking the time derivative along the trajectories of the system yields the following.

$${\displaystyle \frac{dV}{dt}}=10X_1{X}_2-6X_1^2-3.05\left|X_4\right|X_1^2-9X_3+0.25X_1{X}_4.$$

(13)

This expression is not sign-definite, but the presence of dominant negative quadratic terms $-6X_1^2$ and $-3.05|X_4|X_1^2$, together with a constant dissipative sink term $-9X_3$, suggests that the energy function $V\left(X\right)$ is dissipative on average. Although the remaining terms can be positive, they are linear and cannot offset the unbounded growth of the quadratic terms. We perform an upper bound estimation for the derivative of the Lyapunov function $V\left(t\right)$ and show that it is bounded, which implies that the function $V\left(t\right)$ itself remains bounded as well. We consider the absolute value of the derivative.

$$\left|{\displaystyle \frac{dV}{dt}}\right|\le |10X_1{X}_2|+6X_1^2+3.05|X_4|X_1^2+9|X_3|+0.25|X_1\left|\right|X_4|.$$

(14)

Assume that the trajectory is confined within a bounded region:

$$|X_i\left(t\right)|\le R\mathrm{for}\mathrm{all}i=1,2,3,4.$$

(15)

Then we obtain the following upper bound:

$$\left|{\displaystyle \frac{dV}{dt}}\right|\le 10R^2+6R^2+3.05R·R^2+9R+0.25R^2=:\mathrm{\Phi }\left(R\right).$$

(16)

If the system starts in a region such that $X_i\left(0\right)\in [-1.5,1.5]$, we may take $R\approx 2$, which yields: $\mathrm{\Phi }\left(2\right)=107.4$. Therefore, the derivative of the Lyapunov function is bounded in magnitude by

$$\left|{\displaystyle \frac{dV}{dt}}\right|\le 107.4.$$

(17)

Thus, the derivative ${\displaystyle \frac{dV}{dt}}$ is bounded and the Lyapunov function $V\left(t\right)$ remains confined within a finite range. This supports the conclusion that the system is globally bounded and dissipative.

These theoretical findings are confirmed by numerical simulations (see Figure 6), where the Lyapunov function remains within finite bounds over a long time interval. This provides further evidence for the existence of a hidden attractor and supports the classification of the system as hyperchaotic, yet globally stable in the Lyapunov sense.

3.6. Sensitivity Analysis

In practical applications, understanding sensitivity limits becomes especially critical when modeling complex systems, ranging from climate models and biological populations to cryptographic algorithms and neural networks. The presence of two positive Lyapunov exponents in hyperchaotic systems significantly amplifies the sensitivity to initial conditions compared to conventional chaotic systems. This occurs because of an accelerated loss of predictability, as the forecasting time horizon diminishes much more rapidly. While for a system with one positive exponent $L{E}_1$ the characteristic predictability time is approximately $1/L{E}_1$, for a hyperchaotic system with two exponents $L{E}_1$ and $L{E}_2$ the effective predictability time may be on the order of $1/(L{E}_1+L{E}_2)$ or even shorter, depending on the geometry of the phase space. This characteristic implies that infinitesimal perturbations in initial conditions can generate substantially divergent evolutionary pathways over time, making long-term forecasting fundamentally unreliable. Figure 7 illustrates the temporal evolution of the trajectories for the proposed hyperchaotic system (10) under two marginally different initial conditions. The trajectory corresponding to the nominal initial condition $X_1\left(0\right)=1$ is depicted in blue, while the trajectory with the perturbed initial condition $X_1\left(0\right)=1.0001$ is shown in red. As evident in Figure 7, a marked divergence in the trajectory emerges after $t\ge 8\mathrm{s}$, demonstrating that the proposed system exhibits a marked sensitivity to the initial conditions.

3.7. Testing the Randomness of Chaotic Sequences

The randomness of chaotic sequences generated by the hyperchaotic system (10) with initial conditions (11) is evaluated using the NIST SP800-22 statistical test suite, implemented via a standard Python library.

Table 3. Results of the NIST SP800-22 randomness tests for a 10,000-bit chaotic sequence.

No.	Test Name	p-Value	Result
1	Frequency (Monobit) Test	1.00000	PASS
2	Block Frequency Test	0.96718	PASS
3	Runs Test	0.13887	PASS
4	Cumulative Sums Test (Forward)	0.55996	PASS
5	Cumulative Sums Test (Backward)	0.55996	PASS
6	Discrete Fourier Transform (Spectral) Test	0.14203	PASS
7	Approximate Entropy Test	0.88952	PASS
8	Serial Test	0.72385	PASS
9	Longest Run of Ones in a Block Test	0.40256	PASS
10	Rank Test	0.37439	PASS
11	Non-overlapping Template Matching Test	0.29135	PASS
12	Overlapping Template Matching Test	0.96518	PASS
13	Universal Statistical Test (Maurer’s Test)	0.94858	PASS
14	Linear Complexity Test	0.27652	PASS
15	Random Excursions (Variant) Test:
	State −4:	0.85648	PASS
	State −3:	0.91479	PASS
	State −2:	0.58058	PASS
	State −1:	0.33856	PASS
	State 1:	0.86431	PASS
	State 2:	0.55385	PASS
	State 3:	0.55109	PASS
	State 4:	0.78617	PASS

presents the results of all 15 core tests applied to a 10,000-bit sequence produced by system (10). According to the NIST criteria, a sequence passes a test if the corresponding p-value exceeds 0.01, indicating that the sequence exhibits sufficient randomness for that statistical property. If all tests are passed, the sequence is considered to possess good overall randomness characteristics [31].

3.8. Multistability and Offset Boosting Control

Multistability describes the phenomenon where two or more attractors coexist for the same set of system parameters but differ according to the initial conditions. The new hyperchaotic system (10) demonstrates this behavior, exhibiting coexisting attractors under varying initial conditions.

Table 4. The system (10) demonstrates multistability when using different ICs.

$\mathbf{Planes}$	$\mathbf{Initial}$$\mathbf{Conditions}$	$\mathbf{Color}$	$\mathbf{Sign}$$\mathbf{of}$$\mathit{LEs}$
$X_3{X}_1$	$(1,1,0.5,0.25)$	$\mathrm{red}$	$(+,+,0,-)$$hyperchaotic$
$X_3{X}_1$	$(-1,-1,0.5,-0.25)$	$\mathrm{blue}$	$(+,+,0,-)$$hyperchaotic$
$X_3{X}_2$	$(1,1,0.5,0.25)$	$\mathrm{red}$	$(+,+,0,-)$$hyperchaotic$
$X_3{X}_2$	$(1.05,1.05,-0.5,-0.25)$	$\mathrm{blue}$	$(+,+,0,-)$$hyperchaotic$
$X_1{X}_2$	$(1,1,0.5,0.25)$	$\mathrm{red}$	$(+,+,0,-)$$hyperchaotic$
$X_1{X}_2$	$(-1.05,-1.05,0.4,-0.35)$	$\mathrm{blue}$	$(+,+,0,-)$$hyperchaotic$

provides data for two different attractors obtained by solving system (10) with the same control parameters but different initial conditions. The behavior of these attractors, as detailed in Table 4, is visually depicted in Figure 8. To provide a more compelling illustration of multistability and the coexistence of attractors observed in the phase portraits of Figure 8, we employed the Poincaré section method. Among several sections tested, the plane $X_2=0.5$ with projection on the $(X_1,X_4)$ plane was found to be the most informative. This choice is justified by the fact that $X_2$ plays a crucial role in generating chaotic nonlinearity through the interaction term $X_1{X}_3$, as well as in the evolution equation for ${\dot{X}}_3$. Figure 9 shows the resulting Poincaré sections: the attractors corresponding to different initial conditions (listed in Table 4) are colored blue, while those generated from the same initial conditions are colored red. The clear separation of red and blue patterns highlights the coexistence of multiple attractors and demonstrates the pronounced sensitivity of the system to initial conditions.

Figure 9 shows the resulting Poincar’e sections: the attractors corresponding to different initial conditions (listed in Table 4) are colored blue, while those generated from the same initial conditions are colored red. The clear separation of red and blue patterns highlights the coexistence of multiple attractors and demonstrates the pronounced sensitivity of the system to initial conditions.

The offset boost control method proves to be a powerful tool for hyperchaotic systems, allowing attractors to be flexibly shifted in a desired direction by introducing an offset. This approach is particularly useful in engineering applications [32]. By adding a constant offset to specific system variables, chaotic signals can be manipulated across the phase space. In the equations of system (10), the state variable $X_3$ appears exclusively in the second equation, making it an ideal candidate for control. By replacing $X_3$ with $X_3+k$, where k is a constant, we can directly influence this variable. As shown in Figure 10, for a positive value of $k=5$, the attractor (green) moves in the negative direction, and for a negative value of $k=-5$, the attractor (red) moves in the positive direction. The attractor highlighted in blue corresponds to $k=0$, which means that there is no change. The size of the attractor along the coordinate $X_3$ does not change for any constant value of k, as follows from the third equation of the system (10). However, along the coordinate $X_2$, the attractor changes size. This is especially noticeable for the red attractor, since for negative k, the $X_2$ coordinate grows according to the second equation of the system (10). As demonstrated on the left side of Figure 10, adjusting the bias gain control k transitions the signal $X_3$ from bipolar to unipolar form. Furthermore, modifying k moves the attractor along the $X_3$ axis, as shown on the right side of Figure 10.

4. Electronic Circuit Implementation of the New Hyperchaotic System

In this section, we realize the theoretical model of the new hyperchaotic system (10) through the implementation of electronic circuits. By applying Kirchhoff’s laws, the electrical analog of system (10) is formulated as follows:

$$\left\{\begin{array}{cc}\hfill C_1{\displaystyle \frac{d{U}_1}{d\tau }}& ={\displaystyle \frac{U_2}{R_{12}}}-{\displaystyle \frac{U_1}{R_{11}}}-{\displaystyle \frac{|U_4|U_1}{R_{13}·K}},\hfill \\ \hfill C_2{\displaystyle \frac{d{U}_2}{d\tau }}& =-{\displaystyle \frac{U_1{U}_3}{R_{21}·K}},\hfill \\ \hfill C_3{\displaystyle \frac{d{U}_3}{d\tau }}& ={\displaystyle \frac{U_1{U}_2}{R_{31}·K}}-{\displaystyle \frac{{\tilde{V}}_b}{R_{32}}},\hfill \\ \hfill C_4{\displaystyle \frac{d{U}_4}{d\tau }}& ={\displaystyle \frac{U_1}{R_{41}}},\hfill \end{array}\right.$$

(18)

where ${\tilde{V}}_b$ is a stable DC voltage source to implement the constant (=9) in a system (10), $R_{ij}$ are resistors $(i,j)=1,2,3,4$, $U_i\left(\tau \right)$ are voltage values, $C_i$ are capacitors and K is a scaling coefficient for the multiplier. We choose the normalized resistor as $R_0=100$ $\mathrm{k}\mathrm{\Omega }$ and the normalized capacitor as $C_0=1$ $\mathrm{nF}$. Then the time constant is equal to $t_0=R_0{C}_0={10}^{-4}$ s. We rescale the state variables of the system (18) as follows $U_1=U_0{\tilde{X}}_1, U_2=U_0{\tilde{X}}_2, U_3=U_0{\tilde{X}}_3, U_4=U_0{\tilde{X}}_4$, $K=U_0{K}^{\prime }$, and $\tau =t_{0}t$. The Equation (18) can then be rewritten in a dimensionless form. By substituting $R_0$, $C_1=C_2=C_3=C_4=C_0$, and $K^{\prime }=10$ into (18) and comparing the numerical results with the output voltages of the system (10), the resistor values are determined as follows:  

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{\tilde{X}}_1}{dt}}& ={\displaystyle \frac{100\mathrm{k}}{R_2}}{\tilde{X}}_2-\left({\displaystyle \frac{100\mathrm{k}}{R_1}}+{\displaystyle \frac{100\mathrm{k}}{R_3·10}}\left|{\tilde{X}}_4\right|\right){\tilde{X}}_1,\hfill \\ \hfill {\displaystyle \frac{d{\tilde{X}}_2}{dt}}& =-{\displaystyle \frac{100\mathrm{k}}{R_4·10}}{\tilde{X}}_1{\tilde{X}}_3,\hfill \\ \hfill {\displaystyle \frac{d{\tilde{X}}_3}{dt}}& ={\displaystyle \frac{100\mathrm{k}}{R_5·10}}{\tilde{X}}_1{\tilde{X}}_2-{\displaystyle \frac{100\mathrm{k}}{R_6}}{V}_b,\hfill \\ \hfill {\displaystyle \frac{d{\tilde{X}}_4}{dt}}& ={\displaystyle \frac{100\mathrm{k}}{R_7}}{\tilde{X}}_1,\hfill \end{array}\right.$$

(19)

where

$$R_1=16.667 \mathrm{k}\mathrm{\Omega },R_2=10 \mathrm{k}\mathrm{\Omega },R_3=3.278 \mathrm{k}\mathrm{\Omega },R_4=R_5=2 \mathrm{k}\mathrm{\Omega },R_6=11.11 \mathrm{k}\mathrm{\Omega },R_7=400 \mathrm{k}\mathrm{\Omega }.$$

The analog circuit modules implementing the equations of system (19) are shown in Figure 11.

These circuits incorporate standard components, including resistors (R), capacitors (C), diodes $D1$–$D2$ (1N4001), multipliers $M_1$–$M_3$ (AD633), operational amplifiers $A1$–$A16$ (TL084ACN), and a supply voltage of $\pm 15$ V. The constant 9 is achieved using a constant voltage source $V_b=1$ V. To implement the absolute value function $|·|$, a standard electronic circuit [33] is used, as shown in Figure 12.

The phase portraits in Figure 13 demonstrate a striking similarity between the Mathematica simulation results (Figure 5) and those obtained from Multisim (Figure 13).

5. Control and Synchronization Mechanisms for a New Hyperchaotic System

Developing a new chaotic generator based on 4D nonlinear dynamical equations requires studying its synchronization capabilities to ensure practical application. This section explores the application of the adaptive control method [25,26,27,28] to synchronize two identical 4D hyperchaotic systems. Adaptive control of chaotic system synchronization provides robustness to parameter uncertainties and external disturbances, does not require precise knowledge of the master (or drive) system model, and enables effective synchronization even in high-dimensional and nonlinear systems. This makes the method particularly valuable for practical applications, including encryption, weak signal detection, and so on. The goal of complete synchronization is to use the master system’s output to control the slave (or response) system, ensuring that the slave system’s output asymptotically follows the master system’s output over time. Active feedback control is utilized when the system parameters are known and measurable, whereas adaptive feedback control is employed when the system parameters are uncertain or unknown.

5.1. Adaptive Control

This subsection introduces an adaptive controller to stabilize the novel 4D hyperchaotic system with unknown parameters. The novel controlled 4D chaotic system is expressed as follows:  

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_1}{dt}}& =a{x}_2-(a+\alpha )x_1-\beta \left|x_4\right|x_1+u_1,\hfill \\ \hfill {\displaystyle \frac{d{x}_2}{dt}}& =-x_1{x}_3+u_2,\hfill \\ \hfill {\displaystyle \frac{d{x}_3}{dt}}& =x_1{x}_2-b+u_3,\hfill \\ \hfill {\displaystyle \frac{d{x}_4}{dt}}& =x_1+u_4,\hfill \end{array}\right.$$

(20)

where $u_1, u_2, u_3, u_4$ are the adaptive control inputs to be designed using estimates $\widehat{a}\left(t\right), \widehat{b}\left(t\right), \widehat{\alpha }\left(t\right), \widehat{\beta }\left(t\right)$ for the unknown parameters $a, b, \alpha , \beta$, respectively. The adaptive feedback control law is defined as follows:

$$\left\{\begin{array}{c}{u}_1=-\widehat{a}\left(t\right)x_2+(\widehat{a}\left(t\right)+\widehat{\alpha }\left(t\right))x_1+\widehat{\beta }\left(t\right)\left|x_4\right|x_1-{\kappa }_1{x}_1,\hfill \\ u_2=x_1{x}_3-{\kappa }_2{x}_2,\hfill \\ u_3=-x_1{x}_2+\widehat{b}\left(t\right)-{\kappa }_3{x}_3,\hfill \\ u_4=-x_1-{\kappa }_4{x}_4.\hfill \end{array}\right.$$

(21)

The closed-loop control system is obtained by substituting (21) into (20), resulting in the following:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_1}{dt}}& =(a-\widehat{a}\left(t\right))x_2-(a-\widehat{a}\left(t\right)+\alpha -\widehat{\alpha }\left(t\right))x_1-(\beta -\widehat{\beta }\left(t\right))\left|x_4\right|x_1-{\kappa }_1{x}_1,\hfill \\ \hfill {\displaystyle \frac{d{x}_2}{dt}}& =-{\kappa }_2{x}_2,\hfill \\ \hfill {\displaystyle \frac{d{x}_3}{dt}}& =-(b-\widehat{b}\left(t\right))-{\kappa }_3{x}_3,\hfill \\ \hfill {\displaystyle \frac{d{x}_4}{dt}}& =-{\kappa }_4{x}_4.\hfill \end{array}\right.$$

(22)

To simplify (22), the parameter estimation error is expressed as:

$${\xi }_a\left(t\right)=a-\widehat{a}\left(t\right),{\xi }_b\left(t\right)=b-\widehat{b}\left(t\right){\xi }_{\alpha }\left(t\right)=\alpha -\widehat{\alpha }\left(t\right),{\xi }_{\beta }\left(t\right)=\beta -\widehat{\beta }\left(t\right).$$

(23)

By incorporating (23), the closed-loop system (22) takes the following form:  

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_1}{dt}}& ={\xi }_a{x}_2-({\xi }_a+{\xi }_{\alpha })x_1-{\xi }_{\beta }\left|x_4\right|x_1-{\kappa }_1{x}_1,\hfill \\ \hfill {\displaystyle \frac{d{x}_2}{dt}}& =-{\kappa }_2{x}_2,\hfill \\ \hfill {\displaystyle \frac{d{x}_3}{dt}}& =-{\xi }_b-{\kappa }_3{x}_3,\hfill \\ \hfill {\displaystyle \frac{d{x}_4}{dt}}& =-{\kappa }_4{x}_4.\hfill \end{array}\right.$$

(24)

Next, we introduce a Lyapunov function defined as:

$$\mathrm{\Lambda }={\displaystyle \frac{1}{2}}\left(\sum _i{x}_i^2+\sum _j{\xi }_j^2\right),i=(1,2,3,4)j=(a,b,\alpha ,\beta ),$$

(25)

which is a positive function of ${\mathbb{R}}^4$. Differentiating (25) concerning t, we obtain

$${\displaystyle \frac{d\mathrm{\Lambda }}{dt}}=-\sum _{i=1}^4{\kappa }_i{x}_i^2+{\xi }_a(x_1{x}_2-x_1^2-\dot{\widehat{a}})+{\xi }_b(-x_3-\dot{\widehat{b}})+{\xi }_{\alpha }(-x_1^2-\dot{\widehat{\alpha }})+{\xi }_{\beta }(-|x_4|x_1^2-\dot{\widehat{\beta }})$$

(26)

From (26), the update law for the parameter estimates is derived as:

$$\dot{\widehat{a}}=x_1{x}_2-x_1^2,\dot{\widehat{b}}=-x_3,\dot{\widehat{\alpha }}=-x_1^2,\dot{\widehat{\beta }}=-\left|x_4\right|x_1^2,$$

(27)

where $(·)$ on the symbol indicates the differentiation with respect to time t.

Theorem 1.

If $k_i$ (i = 1, 2, 3, 4) are positive gain constants, the adaptive control law (21) and the parameter update law (27) stabilize the 4D system (3) with unknown system parameters globally and exponentially for all initial conditions.

Proof. 

Substituting the parameter update law (27) into (26), we obtain the time derivative of $\mathrm{\Lambda }$ as

$${\displaystyle \frac{d\mathrm{\Lambda }}{dt}}=-{\kappa }_1{x}_1^2-{\kappa }_1{x}_2^2-{\kappa }_3{x}_3^2-{\kappa }_4{x}_4^2.$$

(28)

Then $d\mathrm{\Lambda }/dt$ is a negative semidefinite function on $R^4$. Integrating (28) from 0 to t, we get

$$\kappa {\int }_0^t{∥x\left(\tau \right)∥}^{2}d\tau \le \mathrm{\Lambda }\left(0\right)-\mathrm{\Lambda }\left(t\right),$$

(29)

where $\kappa =\min \left\{{\kappa }_1,{\kappa }_2,{\kappa }_3,{\kappa }_4\right\}$. Applying Barbalat’s lemma [34], it can be concluded that $x\left(t\right)\to 0$ exponentially as $t\to \infty$ for all initial conditions $x\left(0\right)\in {\mathbb{R}}^4$. This completes the proof.

We numerically solve systems (3) and (27) under the application of the adaptive control law (21). The parameter values for the 4D hyperchaotic system (3) are set as follows:

$$a=5,b=90,\alpha =1,\beta =0.1525.$$

The control gains are chosen as

$${\kappa }_1=3,{\kappa }_2=0.8,{\kappa }_3=2,{\kappa }_4=1.$$

The initial values of the hyperchaotic system (20) are taken as

$$x_1\left(0\right)=5,x_2\left(0\right)=10,x_3\left(0\right)=x_4\left(0\right)=5.$$

Also, as initial conditions for parameter estimates

$$\widehat{a}\left(0\right)=4.99,\widehat{b}\left(0\right)=89.1,\alpha \left(0\right)=1.05,\beta \left(0\right)=0.305.$$

   □

Figure 14 illustrates the exponential convergence of the controlled states of the 4D hyperchaotic system (20).

5.2. Adaptive Synchronization

In this subsection, we consider the adaptive synchronization of two identical 4D systems with unknown system parameters. The system (3) serves as the drive system, while the following system is taken as the response (or slave) system:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{y}_1}{dt}}& =a{y}_2-(a+\alpha )y_1-\beta \left|y_4\right|y_1+u_1,\hfill \\ \hfill {\displaystyle \frac{d{y}_2}{dt}}& =-y_1{y}_3+u_2,\hfill \\ \hfill {\displaystyle \frac{d{y}_3}{dt}}& =y_1{y}_2-b+u_3,\hfill \\ \hfill {\displaystyle \frac{d{y}_4}{dt}}& =y_1+u_4.\hfill \end{array}\right.$$

(30)

Here, $y_1,y_2,y_3,y_4$ represent the signal variables of the slave system, and $u_1,u_2,u_3,u_4$ are the controllers designed to achieve global chaos synchronization between systems (3) and (30). The synchronization error between identical chaotic systems is defined as $e_i\left(t\right)=y_i\left(t\right)-x_i\left(t\right)$, for $i=1,2,3,4$. The synchronization error dynamics are expressed as

$$\left\{\begin{array}{c}{\dot{e}}_1=a{e}_2-(a+\alpha )e_1-\beta \left(\right|y_4|y_1-|x_4\left|x_1\right)+u_1,\hfill \\ {\dot{e}}_2=-(y_1{y}_3-x_1{x}_3)+u_2,\hfill \\ {\dot{e}}_3=(y_1{y}_2-x_1{x}_2)+u_3,\hfill \\ {\dot{e}}_4=e_1+u_4.\hfill \end{array}\right.$$

(31)

The adaptive control law is formulated as follows.

$$\left\{\begin{array}{c}{u}_1=-\widehat{a}\left(t\right)e_2+(\widehat{a}\left(t\right)+\widehat{\alpha }\left(t\right))e_1+\widehat{\beta }\left(t\right)\left(\right|y_4|y_1-|x_4\left|x_1\right)-k_1{e}_1,\hfill \\ u_2=(y_1{y}_3-x_1{x}_3)-k_2{e}_2,\hfill \\ u_3=-(y_1{y}_2-x_1{x}_2)-k_3{e}_3-(b-\widehat{b}\left(t\right)),\hfill \\ u_4=-e_1-k_4{e}_4,\hfill \end{array}\right.$$

(32)

where $k_1,k_2,k_3,k_4$ are positive constants that control the synchronization speed; $\widehat{a}\left(t\right)$, $\widehat{b}\left(t\right)$, $\widehat{\alpha }\left(t\right)$, $\widehat{\beta }\left(t\right)$ are estimates of unknown parameters $a,b,\alpha ,\beta$, respectively. The parameter estimation errors ${\xi }_a,{\xi }_b,{\xi }_{\alpha },{\xi }_{\beta }$ are defined from (23). Substituting (32) into the error dynamics (31), we obtain

$$\left\{\begin{array}{c}{\dot{e}}_1={\xi }_a{e}_2-({\xi }_a+{\xi }_{\alpha })e_1-{\xi }_{\beta }\left(\right|y_4|y_1-|x_4\left|x_1\right)-k_1{x}_1,\hfill \\ {\dot{e}}_2=-k_2{e}_2,\hfill \\ {\dot{e}}_3=-{\xi }_b-k_3{e}_3,\hfill \\ {\dot{e}}_4=-k_4{e}_4.\hfill \end{array}\right.$$

(33)

Now, consider the Lyapunov function defined by

$$\mathrm{\Lambda }={\displaystyle \frac{1}{2}}\left(\sum _i{e}_i^2+\sum _j{\xi }_j^2\right),i=(1,2,3,4);j=(a,b,\alpha ,\beta ).$$

(34)

Here $\mathrm{\Lambda }$ is a quadratic positive definite function and differentiating it with respect to t, we find

$${\displaystyle \frac{d\mathrm{\Lambda }}{dt}}=-{\sum }_{i=1}^4{k}_i{e}_i^2+{\xi }_a(e_1{e}_2-e_1^2-\dot{\widehat{a}})+{\xi }_b(-e_3-\dot{\widehat{b}})+{\xi }_{\alpha }(-e_1^2-\dot{\widehat{\alpha }})+{\xi }_{\beta }(-(|y_4|y_1-|x_4|x_1)e_1-\dot{\widehat{\beta }})$$

(35)

We define the parameter update law as follows.

$$\dot{\widehat{a}}=e_1{e}_2-e_1^2,\dot{\widehat{b}}=-e_3,\dot{\widehat{\alpha }}=-e_1^2,\dot{\widehat{\beta }}=-\left(\right|y_4|y_1-|x_4\left|x_1\right)e_1.$$

(36)

Theorem 2.

The adaptive control law (32) and the parameter update law (36) synchronize identical 4D hyperchaotic systems (3) and (20) with unknown system parameters globally and exponentially for all initial conditions, provided that $k_1,k_2,k_3,k_4$ are positive gain constants.

Proof. 

According to the Lyapunov stability theory, it follows that if $\mathrm{\Lambda }$ is a positive definite function and $d\mathrm{\Lambda }/dt$ is a negative semidefinite function. Substituting the parameter update law (36) into (35), we obtain the time derivative of $\mathrm{\Lambda }$ as:

$${\displaystyle \frac{d\mathrm{\Lambda }}{dt}}=-k_1{e}_1^2-k_2{e}_2^2-k_3{e}_3^2-k_4{e}_4^2.$$

(37)

Integrating (37) from 0 to t, we get

$$k{\int }_0^t{∥e\left(\tau \right)∥}^{2}d\tau \le \mathrm{\Lambda }\left(0\right)-\mathrm{\Lambda }\left(t\right),$$

(38)

where $k=\min \left\{k_1,k_2,k_3,k_4\right\}$. By also applying Barbalat’s lemma [34], we conclude that $e\left(t\right)\to 0$ exponentially as $t\to \infty$ for all initial conditions $e\left(0\right)\in {\mathbb{R}}^4$. This concludes the proof. The results of adaptive synchronization are validated using Maple software. For numerical simulations, the parameter values of the master system (3) and the slave system (30) are set to match the hyperchaotic case:

$$a=5,b=90,\alpha =1,\beta =0.1525.$$

The gain constants are chosen as:

$$k_1=3,k_2=0.8,k_3=0.8,k_4=1.$$

The initial conditions for the drive system (3) are as specified in (7). The initial conditions for the response system (30) are chosen as:

$$y_1\left(0\right)=-5,y_2\left(0\right)=-10,y_3\left(0\right)=6,y_4\left(0\right)=-5.$$

The initial values for the parameter estimates are given as:

$$\widehat{a}\left(0\right)=4.99,\widehat{b}\left(0\right)=89.1,\widehat{\alpha }\left(0\right)=1.05,\widehat{\beta }\left(0\right)=0.305.$$

   □

Figure 15 and Figure 16 illustrate the results of synchronization.

Figure 15 shows the exponential convergence of synchronization errors $e_1,e_2,e_3,e_4$ to zero over time, and the timing diagrams are shown in Figure 16.

6. Adaptive Control Method for Chaotic Masking and Decoding Useful Signals

The hyperchaotic system (10) exhibits pronounced sensitivity to initial conditions (see Figure 6) and parameter variations, which makes it highly effective in masking chaotic signals. In this approach, the information signal $S\left(t\right)$ is embedded in one of the state variables of the system, effectively concealing it within the chaotic dynamics. The receiver, lacking knowledge of the exact system parameters, employs adaptive synchronization to recover the system dynamics and extract the encrypted signal simultaneously.

We consider an adaptive control method for synchronizing two identical 4D hyperchaotic systems initialized with different initial conditions and uncertain or unknown system parameters. A sinusoidal signal $S\left(t\right)=3sin\left(0.2t\right)$ is selected as the information signal, while the chaotic state variable $X_1$ serves as the carrier signal for the encryption of information. The encrypted signal $M\left(t\right)$ is mathematically expressed as the superposition of the information and carrier signals: $M\left(t\right)=S\left(t\right)+X_1\left(t\right)$. Consequently, the transmission equations can be derived from (10) as follows:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{X}_1}{dt}}& =2a{X}_2-(a+\alpha )M\left(t\right)-20\beta \left|X_4\right|X_1,\hfill \\ \hfill {\displaystyle \frac{d{X}_2}{dt}}& =-5X_1{X}_3,\hfill \\ \hfill {\displaystyle \frac{d{X}_3}{dt}}& =5X_1{X}_2-9,\hfill \\ \hfill {\displaystyle \frac{d{X}_4}{dt}}& =0.25M\left(t\right).\hfill \end{array}\right.$$

(39)

In practice, the receiver observes only a highly fluctuating chaotically masked signal, from which it is impossible to extract $s\left(t\right)$ without prior knowledge of the dynamics and parameters of the system. The chaotic nature of the carrier signal effectively conceals the information content, rendering conventional signal processing techniques ineffective.

Therefore, for successful signal decoding, the state variable $Y_1$ in the receiver must be synchronized with the transmitter variable $X_1$. Once synchronization is achieved, the information signal can be recovered as:

$$S_1\left(t\right)=M\left(t\right)-Y_1\left(t\right)=S\left(t\right)+X_1\left(t\right)-Y_1\left(t\right)=S\left(t\right)-e_1\left(t\right),$$

where $S_1\left(t\right)$ represents the recovered information signal, $e_1\left(t\right)$ is an encryption error. Consequently, the equations governing the receiving system are derived from (20) as

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{Y}_1}{dt}}& =2a{Y}_2-(a+\alpha )M_1\left(t\right)-20\beta \left|Y_4\right|Y_1+U_1,\hfill \\ \hfill {\displaystyle \frac{d{Y}_2}{dt}}& =-5X_1{X}_3+U_2,\hfill \\ \hfill {\displaystyle \frac{d{Y}_3}{dt}}& =5X_1{X}_2-9+U_3,\hfill \\ \hfill {\displaystyle \frac{d{Y}_4}{dt}}& =0.25M_1\left(t\right)+U_4,\hfill \end{array}\right.$$

(40)

where $M_1\left(t\right)=S\left(t\right)+Y_1\left(t\right)$. It is evident that as the error $e_1$ approaches zero, successful signal recovery becomes achievable. $U_1,U_2,U_3,U_4$ are rescaled adaptive control inputs (21) to be designed using estimates $\widehat{a}\left(t\right),\widehat{b}\left(t\right),\widehat{\alpha }\left(t\right),\widehat{\beta }\left(t\right)$ for unknown parameters $a,b,\alpha ,\beta$, respectively.

To numerically verify secure communication, we utilized the nonlinear Equations (39) and (40) and applied the Runge–Kutta–Fehlberg (rkf45) method in the Maple computing environment. The initial conditions for the drive system (39) were set as follows:

$$X_1\left(0\right)=1,X_2\left(0\right)=1,X_3\left(0\right)=0.5,X_4\left(0\right)=0.25,$$

(41)

and the response system was initialized with:

$$Y_1\left(0\right)=-1,Y_2\left(0\right)=-1,Y_3\left(0\right)=0.6,Y_4\left(0\right)=-0.25.$$

(42)

Figure 17 presents the simulation results for the secure communication scheme. The original information signal $S\left(t\right)$ (Figure 17a) transforms into a chaotically masked state after encryption, as shown in Figure 17b, effectively concealing the content of the information and ensuring secure transmission. Following the decryption process, the information signal is successfully restored to its original form, as shown in Figure 17c. Figure 17d illustrates the waveform of the reconstruction error $e\left(t\right)=S\left(t\right)-S_1\left(t\right)$, which exhibits characteristics similar to the synchronization error observed in the adaptive control method discussed in the previous section (see Figure 16). In particular, the reconstruction error decreases exponentially and converges to zero after approximately $t=3.5$ s, indicating successful synchronization and signal recovery. The transient period corresponds to the time required for the adaptive controller to estimate the unknown system parameters and achieve synchronization between the transmitter and receiver dynamics.

These results demonstrate that the proposed adaptive control synchronization method effectively performs information encryption and decryption within acceptable error bounds, validating its potential for secure chaotic communication applications.

7. Application of Complete Synchronization of Hyperchaotic Systems to Detect Weak Signals

The application of chaotic synchronization methods for weak signal detection began with the foundational work of Pecora and Carroll [35], who demonstrated that specific subsystems of nonlinear chaotic systems can be synchronized by coupling them through standard signals [36]. Haykin and Li [37] advanced this field by presenting a novel method for detecting signals in ‘noise’, based on the assumption that the ‘noise’ is chaotic with at least one positive Lyapunov exponent, and employing neural networks for implementation [38]. Practical implementations were demonstrated in studies [39] using Chua’s circuit, where researchers developed a method to detect weak sinusoidal signals in intense noise using two synchronization systems to estimate different signal parameters [40]. An alternative approach [41] employing the Duffing oscillator revealed that this system exhibits high sensitivity to detect weak periodic signals while maintaining robustness against interference from signals at other carrier frequencies. In these seminal works [35,37,39,41], the theoretical and practical foundations for applying nonlinear dynamical systems to weak signal detection problems were established.

The weak signal detection mechanism is based on the fact that a weak external signal slightly disrupts the chaotic dynamics of the master system. While the slave system tries to synchronize with the master, this disturbance prevents perfect synchronization, resulting in a measurable synchronization error. By monitoring this error, in particular its amplitude and spectral characteristics, it becomes possible to detect and characterize the weak signal. We consider the detection of a weak periodic signal using two coupled hyperchaotic systems (10) in the master-slave configuration. The following set of nonlinear differential equations governs the master system:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{X}_1}{dt}}& =10X_2-6X_1-3.05\left|X_4\right|X_1+\epsilon sin\left(\omega t\right),\hfill \\ \hfill {\displaystyle \frac{d{X}_2}{dt}}& =-5X_1{X}_3,\hfill \\ \hfill {\displaystyle \frac{d{X}_3}{dt}}& =5X_1{X}_2-9,\hfill \\ \hfill {\displaystyle \frac{d{X}_4}{dt}}& =0.25X_1,\hfill \end{array}\right.$$

(43)

where $\epsilon$ and $\omega$ define the amplitude and frequency of the weak periodic signal injected. The slave system receives the output of the master system through linear diffusive coupling:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{d{Y}_1}{dt}}& =10Y_2-6Y_1-3.05\left|Y_4\right|Y_1+k(X_1-Y_1),\hfill \\ \hfill {\displaystyle \frac{d{Y}_2}{dt}}& =-5Y_1{Y}_3+k(X_2-Y_2),\hfill \\ \hfill {\displaystyle \frac{d{Y}_3}{dt}}& =5Y_1{Y}_2-9+k(X_3-Y_3),\hfill \\ \hfill {\displaystyle \frac{d{Y}_4}{dt}}& =0.25Y_1+k(X_4-Y_4),\hfill \end{array}\right.$$

(44)

where k is the coupling coefficient. The synchronization error $e_1\left(t\right)=X_1\left(t\right)-Y_1\left(t\right)$ remains small when $\epsilon =0$, but increases significantly as soon as a weak signal is introduced. This property is exploited for signal detection, enabling the identification of low-amplitude signals through deviations from the synchronized regime.

7.1. Numerical Results

The system of master-slave differential Equations (43) and (44) was numerically integrated using the Runge–Kutta method with a fixed time step over the interval $t\in [0,100]$. Both systems are initialized with the same initial conditions:

$$X_1\left(0\right)=Y_1\left(0\right)=1,X_2\left(0\right)=Y_2\left(0\right)=1,X_3\left(0\right)=Y_3\left(0\right)=0.5,X_4\left(0\right)=Y_4\left(0\right)=0.25.$$

The weak external signal applied to the master system was of the form:

$$s\left(t\right)=\epsilon sin\left(\omega t\right),ϵ=0.01,\omega =0.55,k=1.35.$$

Simulation results show that, in the absence of the signal $\epsilon =0$, the slave system completely synchronizes with the master and all synchronization errors converge to zero, as shown in the upper part of Figure 18. However, with the weak signal active, a deviation from perfect synchronization becomes evident, as shown in the lower part of Figure 18. The presence of a weak signal results in an apparent deviation from complete synchronization, which facilitates its detection.

7.2. Filtering the Synchronization Error near Frequency $\omega$

Figure 18 shows the evolution in time of the synchronization error $e\left(t\right)=X_1\left(t\right)-Y_1\left(t\right)$. During an initial short time interval, complete synchronization is observed with $e\left(t\right)\approx 0$, indicating that the slave system successfully tracks the master system. However, as time passes, chaotic desynchronization bursts appear, resulting in intermittent spikes in the error signal. To investigate whether a weak periodic signal is embedded in the error dynamics, numerical bandpass filtering was applied. The filtering results are presented in Figure 19. The error signal was filtered in a narrow frequency band centered on the known external frequency $\omega$ to isolate the corresponding harmonic component from the chaotic broadband background.

Figure 20 shows the filtered synchronization error, where a quasi-sinusoidal component becomes visible.

To further analyze its temporal structure, the amplitude envelope was computed using the Hilbert transform, as shown in Figure 20. The red dashed curve represents the slowly varying envelope, which captures amplitude modulations induced by the chaotic background. This modulation illustrates how the weak periodic forcing interacts with the underlying chaotic dynamics, resulting in a non-uniform response.

This behavior is typical in synchronization-based signal detection and does not imply failure. In contrast, the emergence of a modulated sinusoidal component in the filtered error confirms the successful detection of the weak signal hidden within the chaotic system.

7.3. Spectral Detection of the Weak Signal

To assess the presence and extractability of the weak periodic signal, two spectral analyses were performed: one based on the raw synchronization error $e\left(t\right)=X_1\left(t\right)-Y_1\left(t\right)$ and the other on the bandpass-filtered version of this signal. Figure 21 presents the amplitude spectrum of the full error signal $e\left(t\right)$. The spectrum exhibits a broadband chaotic structure, characteristic of the underlying nonlinear dynamics. Despite this complexity, a distinct and narrow peak is observed at the driving frequency $\omega =0.55$ Hz, indicating that the weak periodic input leaves a detectable spectral trace even when embedded in chaotic fluctuations.

In contrast, Figure 22 shows the spectrum of the error signal after narrowband filtering centered at $\omega$. Here, the spectral energy is sharply concentrated in a narrow range around the expected frequency, while the chaotic background is strongly suppressed. This confirms that the weak periodic component has been effectively extracted from the broadband error dynamics.

This comparison demonstrates the effectiveness of synchronization-based detection followed by frequency-selective filtering in identifying and isolating weak periodic signals buried in chaotic dynamics.

8. Practical Implementation of a 4D Memristor-Based Hyperchaotic System on Arduino UNO

This section demonstrates the practical feasibility of implementing a newly proposed four-dimensional memristive hyperchaotic system (10) on a widely available and inexpensive Arduino UNO microcontroller.

8.1. Chaotic Dynamic Based on Arduino-Generated Chaotic Sequences

To verify the feasibility of low-cost hardware for real-time nonlinear system simulation, system (10) was implemented on an Arduino UNO microcontroller using the explicit Euler method with time step $\Delta t=0.01$ and initial conditions (11). The experimental setup consisted of the Arduino Uno-based system connected to a computer, with the testing procedures documented in Figure 23.

The state variables were computed iteratively and transmitted data over UART (Universal Asynchronous Receiver/Transmitter) at 115,200 baud. Figure 24 presents a screenshot that shows the communication between the microcontroller and the computer.

The resulting data stream was acquired and processed in MATLAB (R2017a), where phase portraits were reconstructed in projection planes $(X_1,X_2)$, $(X_1,X_3)$ and $(X_2,X_3)$, as shown in Figure 25.

In particular, the trajectories obtained from the microcontroller closely matched those produced by symbolic-numerical integration in Mathematica (see Figure 5), as well as those generated in circuit-level simulation in Multisim (see Figure 12). This shows that even simple embedded systems can accurately capture and reproduce complex non-linear dynamics, enabling real-time analysis and validation of mathematical models through both software and hardware platforms.

8.2. Image Encryption and Decryption Using Arduino-Generated Chaotic Sequences

To demonstrate secure image processing using hardware-generated chaotic sequences, we developed a two-part implementation involving an Arduino UNO and MATLAB. The Arduino executes the numerical integration of the 4D chaotic system (10) using fourth-order functions.

The Runge–Kutta method (RK4) to generate a chaotic sequence in real time with initial conditions specified in (11). The chaotic data stream is transmitted via the serial port and is used in MATLAB to perform encryption and decryption of grayscale images. The encryption process includes two operations: a chaotic permutation of pixels and a bitwise XOR with the chaotic sequence. Decryption applies the inverse operations to recover the original image. The initial conditions for the system are set according to (11). The Arduino code (Algorithm 1) employs the Runge–Kutta 4th-order method for numerical integration, whereas the MATLAB code (Algorithm 2) performs image processing and visualization.

Algorithm 1: Chaos generation on Arduino (see system (10), initial conditions (11))

Input: Initial state ${\mathbf{x}}_0$, step size h, iterations N      Output: Chaotic bytes

Algorithm 2: Image encryption and decryption in MATLAB

Input: Grayscale image I, chaotic byte stream $\mathbf{c}$    Output: Encrypted image E, decrypted image $\widehat{I}$ 1 Flatten I into vector $\mathbf{v}$ 2 Receive chaotic sequence $\mathbf{c}$ of same length 3 Generate permutation $\pi$ by sorting $\mathbf{c}$ 4 Apply permutation: ${\mathbf{v}}^{\prime }=\mathbf{v}\left(\pi \right)$ 5 Encrypt: $\mathbf{e}={\mathbf{v}}^{\prime }\oplus \mathbf{c}$ 6 Decrypt: ${\mathbf{v}}^{″}=\mathbf{e}\oplus \mathbf{c}$ 7 Recover original order: $\widehat{\mathbf{v}}={\mathbf{v}}^{″}\left({\pi }^{-1}\right)$ 8 Reshape into image $\widehat{I}$

The simulation results using the grayscale images are shown in Figure 26. The perfect reconstruction quality is confirmed by the PSNR value of ∞ dB, indicating that there is no distortion between the original and decrypted images.

9. Conclusions

In this paper, we introduced a novel four-dimensional memristor-based dynamical system by integrating an absolute memristor, as proposed by Bao [29], with the Munmuangsaen equations of five terms [13]. The proposed system satisfies the established criteria for generating hyperchaos: (a) it is dissipative, (b) it operates in a four-dimensional phase space, and (c) it incorporates three nonlinear terms. In particular, this hyperchaotic system comprises only seven terms and lacks equilibrium points, which may give rise to hidden attractors.

Characterized by two positive Lyapunov exponents, the system is classified as hyperchaotic, with a Kaplan–Yorke dimension ($D_{KY}=3.133$) underscoring its high complexity. The bounded evolution of the Lyapunov function over extended simulations provides clear evidence of the dissipativity and global asymptotic stability of the constructed 4D hyperchaotic system. Numerical simulations reveal the phenomenon of multistability, resulting in the coexistence of hyperchaotic attractors under varying initial conditions. Furthermore, the system’s offset-boosting control behavior maintains constant Lyapunov exponent values, demonstrating that its hyperchaotic nature persists across different booster parameter settings.

The electronic circuit simulation of the proposed 4D system, implemented in Multisim 14, shows excellent agreement with the results obtained from Mathematica.

This study also presents new results for the application of the adaptive control method in synchronizing two identical 4D hyperchaotic systems with unknown parameters. The adaptive controllers were designed systematically using a feedback control strategy to ensure synchronization. The effectiveness of the proposed method was rigorously validated through Lyapunov stability theory. Numerical simulations demonstrated that all system state variables achieve synchronization, with synchronization errors converging to zero in a short time.

Using an adaptive control approach, a simple and effective scheme for chaotic masking and recovery of a harmonic signal was successfully demonstrated. The analysis confirms that complete synchronization of chaotic systems provides a practical approach for detecting weak periodic signals embedded in nonlinear dynamics. Injecting a weak signal into the master system disrupts synchronization, resulting in a measurable error in synchronization.

This sensitivity enables reliable identification of weak inputs, making the method suitable for a range of engineering applications, including sensor systems, covert communications, and signal monitoring in noisy environments.

The use of the Arduino UNO microcontroller demonstrates the practical feasibility of implementing real-time simulations of complex hyperchaotic systems on low-cost hardware. Moreover, its integration with MATLAB for image encryption and decryption confirms its potential in lightweight, hardware-based cryptographic applications.