A Hidden Chaotic Attractor with an Independent Amplitude-Frequency Controller

School of Electronic Information Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, China Jiangsu Collaborative Innovation Center of Atmospheric Environment and Equipment Technology (CICAEET), Nanjing University of Information Science & Technology, Nanjing 210044, China Department of Computer Engineering, Faculty of Engineering, Hitit University, Corum 19030, Turkey


Introduction
Since chaos was first modeled by Edward Norton Lorenz in 1963 [1], great attention has been attracted for the reason that in such deterministic systems, chaos refers to the presence of seemingly random irregular motion [2]. In fact, chaos is a universal phenomenon in nonlinear systems and an inherent property of nonlinear dynamic systems. Chaos is a fundamental concept in nonlinear systems, and it is frequently used to characterize phenomena including bifurcation and periodic motion [3,4]. It is found that a chaotic system will display bifurcation under specific parameters and that periodic and aperiodic motion can become entangled. Many famous 3-D chaotic systems have been proposed, including Arneodo systems, Sprott systems, Chen systems, Lv-Chen systems, Cai systems, and T systems [5][6][7][8]. A chaotic circuit is a significant research topic and an important representation of chaos. Chua proposed the first chaotic circuit in 1984 [9][10][11][12], trying to bridge the gap between chaos theory and chaotic circuits. People have performed significant research on chaos theory and produced numerous novel breakthroughs in recent years as a result of the wide application of chaos [13][14][15][16][17]. For information encryption, chaotic sequence signals have a significant application value. Chaos has been extensively applied for many aspects due to the intricate relationship between chaos and cryptography, and significant advancement has been performed in this area. Because of the irregularities and unpredictability of chaotic signals, chaos and corresponding fundamental systems have received a lot of attention [18][19][20][21][22][23][24]. A high-dimensional system's chaotic dynamics seem to be more complex, and sometimes, they show hyperchaos for more than two positive Lyapunov exponents [25,26]. erefore, it has been proven that chaotic signals can contribute to enhancing the security of chaos-based communication as well as digital encryption. Furthermore, a hidden chaotic attractor is an important phenomenon found ten years ago. e hidden oscillation has received a lot of attention because of its potential threat and possible applications. Finding hidden chaotic attractors in nonlinear dynamical systems has become a major issue in nonlinear dynamical system research [27][28][29]. Many hidden attractors have been found in memristive systems [30] and hyperchaotic systems [31]. However, there is not much attention on hidden attractors with amplitude-frequency control. Recently, Wang et al. [32] studied the amplitude control and encryption application of chaotic signal, and they found that modular circuit cells with systematically configured parameters are useful for implementing multipiecewise Chua's diode. Wang et al. [33,34] also studied the hidden oscillations in Chua's circuit and modified Sprott-A systems, where all the basins of attraction are not intersected with any equilibrium point indicating hidden attractor.
In this paper, a chaotic system with a line of equilibrium points is focused on, where the attractor stands in the region with negative y, and thus, the basin of attraction does not intersect with all equilibrium points indicating hidden attractor. Furthermore, the amplitude and frequency of hidden oscillation can be controlled by a single knob. Circuit implementation shows the convenience of amplitude-frequency control of the chaotic signal. Image encryption proves the merits of this system. e remaining of this paper is organized as follows. In Section 2, the system model is elaborated. In Section 3, the dynamics of the system are analyzed. In Section 4, the system is implemented in a simulated circuit. Finally, a chaotic system is applied to image encryption. e conclusions are presented in the last section.

A Novel Hidden Attractor
In reference [35], the 3-dimensional chaotic system is articulated as e system state variables are x, y, and z, while the constants a, b, and c are the real coefficients, with dots representing a derivative of time t. When a � 0.9, b � 1, c � 1, and IC � (2, −2, 2), the system (1) exhibits chaos in the region with negative y, as shown in Figure 1. ere are six terms in this new system, with two nonlinear items. e Lyapunov exponents are L 1 � 0.1314, L 2 � 0, and L 3 � −0.8453.

Dynamical Analysis
e equilibrium points of the system can be calculated as where a � 0.9, b � 1, and c � 1. Solving this equation, the dynamical system (1) has three nontrivial equilibrium points in (2) which is independent of the value of the parameters a, b, and c.
As |J − λI| � 0, the characteristic equation is In the Jacobian matrix of system (1), the equilibrium point E is defined as e eigenvalues of the matrix at the equilibrium point can be determined as follows: e eigenvalues of the system are obtained while a � 0.9, b � 1, and c � 1. e obtained all eigenvalues are given in Equation (7). e equilibrium E 1 (0, −c, 0) is a saddle-focus point of index-2; therefore, this equilibrium point E 1 is unstable, and we can see in E 2 that λ 1 is a positive real number, and λ 2 and λ 3 are a pair of complex conjugate 2 Complexity eigenvalues with a negative real number. e equilibrium E 2 (0, c/ab − 1, bc/ab − 1) is a saddle-focus point of index-1; therefore, this equilibrium point E 2 is also unstable.

Dissipativity Analysis.
Inference µ is a region in the horizontal surface A 3 , and V(t) is set to be the volume of µ(t).
Here, we obtain erefore, the dissipativity of the proposed chaotic system is where F is the 3-dimensional chaotic system, and b and z are the real parameters. e above equation is rewritten as erefore, we can obtain V(t) � e c V(0); if ∇ × F < 0, then system (1) is dissipative and the state of the system is bounded by the state of the system (when a � 0.9, b � 1, and c � 1).

Amplitude and Frequent Control.
In system (1), parameter c is a single knob for amplitude and frequency control. Let x ⟶ mx, y ⟶ my, z ⟶ mz, t ⟶ t/m (m > 0); then, system (1) turns to be indicating that the parameter c can control the amplitude and frequency of all variables x, y, and z, as shown in Figure 2. In Figure 3(a), when the linear coefficient c is increased, system (1) keeps the chaotic state of all the time and the chaotic area continue to increase. e rescaled amplitude and frequency can also be proved by Lyapunov exponents, as shown in Figure 3(b).

Circuit Implementation
Circuit verification is also an essential step in the implementation of the proposed chaotic system to ensure its correctness. Meanwhile, the key issue in the circuit is how to implement a circuit expression for a 3D chaotic system by converting and transforming it into a realistic circuit using electronic devices such as capacitors and resistors. To authenticate the efficiency of our proposed 3-dimensional chaotic system, the circuit implementation is designed and simulated with the NI multisim circuit simulation software, and the simulation results are detailed in this section. For this circuit construction, it is transformed to be A chaotic system is defined as a system that has several causes and multiplication, addition, and differentiation, and differentiation exists in the system equations, and a realistic expression of this system is obtained utilizing a summation, an integrator, and a transformer. e relevant circuit implementation is depicted in Figure 4 based on the above explanations. e state variables x, y, and z in the system (1) correspond to the state voltages of the capacitors C 1 , C 1 , and C 3 in the simplified circuit, and the corresponding circuit components can be selected as follows: LM741CN is selected as an operational amplifier; there is a general time scale 1000 for better displaying in the oscilloscope. Figure 5 shows the phase trajectory of the system (1) in the analog oscilloscope. e area of the phase track will change with the organization of the resistance R 1 , as shown in Figure 5(d).

Application in Image Encryption
e chaotic system with a linear equilibrium in this section has a higher level of unpredictability, a larger keyspace, and a higher level of complexity, all of which make the encryption stronger secure in concept.

Encryption Application with a Chaotic System.
e control parameters and initial conditions of the system are (Table 1) μ � 3.9999 and x 0 � 0.6209, respectively, and a standard color image is chosen for testing, as illustrated in Figure 6. e new chaotic system parameters are a � 0.9, b � 1, and c � 1.
e initial conditions are set as X 0 � 0.7361, Y 0 � 0.4663, Z 0 � 0.1501, and U 0 � 0.7653. Table 2 describes the selected keys that were chosen. M 1 and N 1 are the zeroing parameters used throughout encryption; k 1 is the average gray level of G in the original image, and the original image k 2 is the average gray level of B. Figure 6(b) shows the encrypted image after simulation. e image after encryption is chaotic and fundamentally different from the image before processing, as it can be observed. e properly decrypted image, which can be seen in Figure 6(c), is identical to the original image ( Table 2).

Security Analysis.
Secure analysis is the most essential and fundamental requirement for an encrypted system. In general, a chaotic image encryption algorithm requires a substantial keyspace, reversible encryption and decryption, strong antiattack characteristics, and the other performances, and then the performance of chaos-based encryption is the next to determine by using the following eligibility requirements: keyspace analysis, histogram analysis, information entropy analysis, and correlation analysis.

Keyspace Analysis.
e keyspace can approach (10 16 ) 10 � 10 160 , satisfying the security level of the keyspace (greater than 2 128 ) using a 64-bit processor, floating-point precision up to 10 − 16 . e modified system can be observed to be more resistant to the attacker's comprehensive attack. e decrypted image cannot be acquired accurately when the algorithm key is changed slightly, as shown in Figure 7. As a result, the new system has a high level of key sensitivity. e images used throughout the investigations are 256 × 256 conventional images with a 2 3 � 8-bit grayscale.

Histogram Analysis.
e distribution of pixel intensity values within an image is represented by a histogram. It can be utilized to defend against statistical attacks. e frequency distribution of each gray level pixel can be visually displayed using a gray histogram, which is a statistical analytical method. e histogram of the image becomes smooth even after encryption, compared to the fluctuation before encryption, thus preventing the attacker from accessing the original image information through statistical analysis, resulting in information leakage, and ensuring information.

Information Analysis for Entropy.
e image's information entropy can be used to evaluate the level of uncertainty and randomness in the image distribution of its pixel gray value. Typically, the higher the image's entropy, the more consistent the image's gray distribution. In a grayscale image, each pixel is coded in 8 bits. As a result, an image's maximum entropy value is 8. e information entropy of the encrypted image should be near 8, which indicates the best amount of uncertainty, owing to a decent encryption process. To compute the information entropy, many users use the method as follows: where p(x i ) represents the probability of the occurrence of the gray value x i and H indicates the gray level of the image. eoretically, for a completely random digital image with a grayscale of 256 has an evenly distributed pixel value in [0, 255], then p(x i ) � 1/256 (i ∈∈[0, 255]), and the estimated information entropy is 8 bits. If the image is encrypted, the closer the image's information entropy is near 8, the better the encryption features.

Correlation Statistical Analysis.
On the one hand, the correlation level of adjacent pixels is larger when the correlation coefficient degree of adjacent pixels is higher. On the other hand, the lower the correlation, the smaller the coefficient. As a result, calculating the correlation coefficient can be justified the algorithm's security. e lower the correlation and the advanced security, the smaller the coefficient. e correlation coefficients were calculated from the three channels (R, G, and B) with three directions: horizontal, vertical, and diagonal, to measure the correlation between the original image and adjacent pixels of the ciphertext image; N pairs of adjacent pixels were selected from the image, and the correlation coefficients were calculated from the three channels (R, G, B) with three directions: horizontal, vertical, and diagonal. To equivalence the autocorrelation of an unadorned and encrypted image, we have calculated the correlation coefficient r of each pair of pixels by using the following formula: where cov(X, Y) represents the correlation and autocorrelation function, X and Y are the grayscale values of two adjacent pixels in the image, and N denotes the sample. E is the expected value operator, and D(X) represents the variance of the variable x. e values of r XY lie in the range Key         Here, we can see from Table 3 that the correlation coefficients of the original image are all very close to 1, whereas the correlation coefficients of the encrypted image are all very close to 0, indicating that the encrypted image's pixel point distribution is very highly discrete.
If we compare Tables 3 and 4 with each other, we can see some different points. In Table 3   99.07%, and diagonal 98.54%, and in Table 4, the original correlation coefficient values are average horizontal 96.72%, vertical 98.17%, and diagonal 95.18%, that is, the original points in Table 3 are closer to 1 and the encrypted points are  also closer to 0 than Table 4. So, Table 3 indicates that the encrypted points of distribution are highly discrete. e correlation and autocorrelation plots for each image are shown in Figures 9-11. In Table 3, horizontal, vertical, and diagonal directions provide their correlation and autocorrelation coefficients. Table 3 also includes significant data from references for comparison.

Conclusion
e single nonbifurcation parameter of the system can effectively modify the amplitude and frequency of the demonstrated three-dimensional chaotic system. Numerical simulation and circuit experiment based on multisim agree to each other by proving the phenomenon. As a typical application, the property of image encryption is exhaustively analyzed. With the chaotic signal from the new system, a color image is well encrypted and decrypted in the keyspace. Histogram and correlation of adjacent pixels are used for showing the high encryption performance.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon reasonable request.