Hidden Dynamics of a New Jerk-like System with a Smooth Memristor and Applications in Image Encryption

Abstract

Considering the dynamic characteristics of memristors, a new Jerk-like system without an equilibrium point is addressed based on a Jerk-like system, and the hidden dynamics are investigated. When changing system parameter b and fixing other parameters, the proposed system shows various hidden attractors, such as a hidden chaotic attractor (b = 5), a hidden period-1 attractor (b = 3.2), and a hidden period-2 attractor (b = 4). Furthermore, bifurcation analysis suggests that not only parameter b, but also the initial conditions of the system, have an effect on the hidden dynamics of the discussed system. The coexistence of various hidden attractors is explored and different coexistences of hidden attractors can be found for suitable system parameters. Offset boosting of different hidden attractors is discussed. It is observed that offset boosting can occur for hidden chaotic attractor, period-1 attractor, and period-2 attractor, but not for period-3 attractor and period-4 attractor. The antimonotonicity of the proposed system is debated and a full Feigenbaum remerging tree can be detected when system parameters a or b change within a certain range. On account of the complicated dynamics of the proposed system, an image encryption scheme is designed, and its encryption effectiveness is analyzed via simulation and comparison.

1. Introduction

As an exciting and fascinating phenomenon in nonlinear systems, chaos has experienced development through the decades and has achieved remarkable results since being discovered in 1963 [1]. Its application has received much attention in various fields [2,3,4,5,6,7].

Some published works show that a nonlinear system can generate two types of attractors: one is a self-excited attractor and the other is a hidden attractor [8,9]. As a new class of attractor, the hidden attractor is distinct from the traditional attractor (self-excited attractor) [10], and its attractive basin does not intersect with any unstable equilibrium point. Therefore, hidden attractors can be detected in some continuous chaotic or hyperchaotic systems with no equilibrium point or only with a stable equilibrium point. Researchers have widely researched hidden attractors and obtained many meaningful results [11,12,13,14,15,16]. In the mentioned results, the existence condition of the hidden attractor, the coexistence and transition of various hidden attractors, and localization of hidden attractors have been investigated, which enriched the research results of nonlinear dynamics.

As a famous chaotic system, the Jerk system attracted researchers’ attention. Regarding high-order Jerk Equations, high-order general Jerk circuits were designed and multi-scrolls were found in the circuit [17]. Some basic qualitative properties of a novel Jerk system were described, and complete chaos synchronization was realized when two parameters were unknown [18]. N-dimensional multi-scroll attractors have been generated in Jerk-like systems by designing nonlinear controllers [19]. A new dissipative Jerk system was presented, and bifurcation analysis revealed its multi-stability [20]. Two primary mechanisms of Jerk systems producing chaotic behavior were discovered [21]. Double-scroll chaotic attractors and multi-stability were detected in a 3D memristive Jerk system, and a scheme for controlling the system from multi-stability to monostability was designed [22]. The effects of the initial conditions of the memristor were disclosed [23]. Infinite equilibrium points were found in a 3D Jerk system disturbed by sinusoidal signal [24]. At the same time, hidden attractors of the Jerk system or Jerk-like systems were explored. Various hidden attractors in a 3D Jerk system were discovered using numerical simulations [25]. Different hidden behaviors were found in a new hyper-Jerk system [26]. Reference [27] introduced some Jerk systems with hidden chaotic behaviors. The above publications have investigated various aspects of the Jerk system and found some complicated behaviors. Some preliminary research about hidden attractors of Jerk systems was carried out.

Analysis of the existing research suggests that, although Jerk and Jerk-like systems have been explored widely and various dynamics have been detected, the considered Jerk or Jerk-like systems seldom involved memristors, which can exhibit interesting complex characteristics [28,29] and draw much attention. In particular, the hidden dynamics of Jerk or Jerk-like systems with memristors were hardly studied. The application of Jerk or Jerk-like systems has mainly focused on designing the circuit.

Motivated by the abovementioned findings, a new Jerk-like system with a smooth memristor without equilibrium is constructed, and some hidden dynamics of the proposed system are investigated. Research results will complement the dynamic theory of Jerk-like systems. On the basis of the complicated dynamics of a Jerk-like system with a smooth memristor, an image encryption method is designed, which will expand the application scope of the Jerk-like system.

Other parts of this paper are arranged as follows. A system description is given in Section 2. In Section 3, hidden dynamics of the new Jerk-like system with a smooth memristor are discussed. In Section 4, an image encryption algorithm is designed based on the chaotic series produced by the Jerk-like system with a smooth memristor, and its effectiveness is verified via experiments.

2. System Description

To construct a new system, Jerk-like system [30],

$$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=z\\ \dot{z}=-ay+bz+cxz+x^2+0.1\end{array}\right.$$

(1)

is taken into account with positive parameters $a$, $b$, $c$, and state variables $x$, $y$, $z$. $a$, $b$, $c$ are tunable parameters controlling the dynamics of system (1). By choosing different values of $a$, $b$, $c$, system (1) can produce various dynamics. System (1) involves seven terms, including only two quadratic nonlinearities. It is relatively simple and elegant. It consists of three general equations, with the third equation containing two quadratic items.

According to the characteristics of memristors, a flux-controlled memristor [31],

$$W(w)=\alpha +3\beta w^2$$

(2)

is considered, where $\alpha$ and $\beta$ are two positive memristor parameters. And then, a Jerk-like chaotic system with a smooth memristor is written as

$$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=z-d(\alpha +3\beta w^2)y\\ \dot{z}=-ay+bz+cxz+x^2+0.1\\ \dot{w}=-r(w-y)\end{array}\right.$$

(3)

where $d$ and $r$ are positive parameters. Other parameters are the same as those in system (1) and flux-controlled memristor (2).

It is obvious that Equations

$$\left\{\begin{array}{l}y=0\\ z-d(\alpha +3\beta w^2)y=0\\ -ay+bz+cxz+x^2+0.1=0\\ -r(w-y)=0\end{array}\right.$$

(4)

have no real solutions, namely, system (3) has no equilibrium point. Therefore, the attractors in following discussion of system (3) should be hidden attractors [32,33].

3. Dynamics Analysis of the New Jerk-like System with a Smooth Memristor

3.1. The Variety of Hidden Dynamics Induced by Parameter Changing

Choose initial value $(x_0,y_0,z_0,w_0)=(-1.61,1.5,-6,-0.01)$, $a=7$, $c=2$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, and for different values of $b$, the time series of system (3) can be calculated and depicted in Figure 1 and Figure 2, which indicates that various values of $b$ can make system (3) take on different hidden attractors. To further study this phenomenon, a bifurcation diagram with parameter b is depicted in Figure 3. Figure 3 confirms the results in Figure 1 and Figure 2.

Then, when we choose system parameters $a=7$, $c=2$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, and initial values $(x_0,y_0,z_0,w_0)=(-2,0,-2,0)$ and (−2,−1,−2,0), respectively, the bifurcation diagram of variable $y$ with parameter $b$ are drawn in Figure 4a,b, respectively. Figure 1, Figure 2, Figure 3 and Figure 4 indicate that parameter b has much effect on the dynamics of system (3). When fixing other parameters, the change of b can induce the diversity of the dynamic behaviors of system (3). To confirm this result, the corresponding Lyapunov exponents are calculated and given in Figure 5a,b, respectively, which coincide with the result in Figure 4. Figure 4 and Figure 5 suggest that the hidden dynamics of system (3) is not only dependent on system parameter $b$, but also relative to the initial value of the system.

3.2. Hidden Dynamics to Initial Values

In this section, the effect of initial value on the hidden dynamics of the addressed system is to be discussed.

Firstly, the effect of initial state of memristor on the hidden dynamics of system (3) is concerned. To this end, select system parameters $a=11.2$, $b=5.0$, $c=2$, $d=0.3$, $r=0.1$, $\alpha =1$, $\beta =0.1$, $x_0=0$, $y_0=0$, $z_0=0$, we change $w_0$ from −3 to 3, and the bifurcation diagram of variable $y$ is shown in Figure 6, which suggests that system (3) shows different dynamics with $w_0$ increasing from −3 to 3, such as period and chaos.

Secondly, three cases are considered: ① fix $y_0=0$, $z_0=-2$, $w_0=0$, let $x_0$ change; ② fix $x_0=-2$, $z_0=-2$, $w_0=0$, let $y_0$ change; ③ fix $x_0=-2$, $y_0=-2$, $w_0=0$, let $z_0$ change. The bifurcation diagrams of system (3) for the above three cases are depicted in Figure 7, from which we can know that when one of $x_0$, $y_0$, $z_0$ varies, system (3) exhibits different behaviors, such as chaotic attractors and period attractors.

From Figure 6 and Figure 7, we know that both the initial condition of a Jerk-like system and the initial value of a memristor have an effect on the hidden dynamics of the proposed system (3).

Based on the above, numerical simulations are performed to demonstrate the coexistence of various hidden attractors via examples. Relative results are depicted in

Table 1. Coexistence of various hidden attractors with different system parameters.

Parameters	Type of Hidden Attractors	Diagrams
$a=11.35$, $b=6.0$, $c=2$, $d=0.1$,$r=0.1$, $\alpha =0.1$, $\beta =0.3$	Coexistence of chaotic attractor (red line for initial value (−2, 0, −2, 0)) and periodic-6 attractor (blue line for initial value (−2, 1, −2, 0)).
$a=11.2$, $b=5.0$, $c=2$, $d=0.5$, $r=0.01$, $\alpha =0.1$, $\beta =0.3$	Coexistence of periodic-1 attractor (red line for initial value (−2, 0, −2, 0)) and chaotic attractor (blue line for initial value (−2, 1, −2, 0)).
$a=11.2$, $b=4.5$, $c=2$, $d=0.1$, $r=0.01$, $\alpha =0.1$, $\beta =0.3$	Coexistence of periodic-4 attractor (red line for initial value (−2, 0, −2, 0)) and periodic-1 attractor (blue line for initial value (−2, 1, −2, 0)).
$a=13$, $b=5.2$, $c=2$, $d=0.1$, $r=0.6$, $\alpha =0.1$, $\beta =0.3$	Coexistence of periodic-3 attractor (red line for initial value (−2, 0, −2, 0)) and periodic-4 attractor (blue line for initial value (−2, 1, −2, 0)).

, from which it is obvious that, by selecting various system parameters, the coexistence of different hidden attractors can be observed. This phenomenon indicates that the proposed system has complicated dynamics.

3.3. Offset Boosting of the Hidden Attractors

As we all know, offset boosting is a distinguishing property of a chaotic system. Offset boosting is an effective technique for editing an attractor or an attractor self-reproducing by changing the offset gate. It can be regarded as a useful method to take on attractor-doubling or multistability. It makes sense for exploring the complex dynamics of nonlinear systems. Results in [34,35] indicate that the change in amplitude can induce different attractors. In addition, hidden attractors were detected in some quadratic Jerk systems [36]. The coexistence of multiple attractors and boosting were found by changing parameters [37,38]. Existing results suggest that the feedback state is an effective alternative for controlling the amplitude of relative variables. Considering model (3), state variable $x$ is only involved in the third Equation. Therefore, the transformation $x\to x+k$ with constant $k$ is used to explore the offset boosting of system (3). $x$ is considered to be offset boostable. And then, system (3) can be rewritten as

$$\left\{\begin{array}{l}\dot{x}=y\\ \dot{y}=z-d(\alpha +3\beta w^2)y\\ \dot{z}=-ay+bz+c(x+k)z+{(x+k)}^2+0.1\\ \dot{w}=-r(w-y)\end{array}\right..$$

(5)

For a chaotic attractor, choose $a=7$, $b=5.0$, $c=2$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, initial value $(x(0),y(0),z(0),w(0))=(-1.61,-1.5,-6,-0.01)$, and the property of offset boosting by changing $k$ is illustrated in Figure 8.

Next, the offset boosting of the period attractors are to be discussed with initial value $(-2,0,-2,0)$. Choose $a=7$, $b=3.2$, $c=2$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, with which system (3) appears as the period-1 attractor, and the property of offset boosting by changing boosting controller $k$ is illustrated in Figure 9. Select $a=7$, $b=4$, $c=2$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, with which system (3) appears as the period-2 attractor, and the property of offset boosting by changing boosting controller $k$ is illustrated in Figure 10. Consider $a=13$, $b=5.2$, $c=2$, $d=0.1$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, with which system (3) shows the period-3 attractor, and the property of offset boosting by changing $k$ is depicted in Figure 11. Choose $a=11.2$, $b=4.5$, $c=2$, $d=0.1$, $r=0.01$, $\alpha =0.1$, $\beta =0.3$ with which system (3) shows the period-4 attractor, and the property of offset boosting by changing $k$ is depicted in Figure 12. From Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, one can know that the offset boosting of the chaotic attractor, period-1 attractor, and period-2 attractor can be provided by varying the boosting controller $k$, while offset boosting of the period-3 attractor and the period-4 attractor cannot be provided by varying the boosting controller $k$.

3.4. Antimonotonicity of a Jerk-like System with a Memristor

As we all know, the process of period-doubling bifurcation to chaos is an interesting behavior worth further study. In some nonlinear dynamical systems, reverse period-doubling sequences can be observed after period-doubling bifurcation. Periodic orbits can occur and then disappear via reverse bifurcation sequences. The phenomenon of the creation and annihilation of periodic orbits was named antimonotonicity [39]. It can be detected in system (3) by changing the value of $a$ in a certain range for several discrete values of $b$ when $c=2$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, and initial value $(x(0),y(0),z(0),w(0))=(-1.61,1.5,-6,-0.01)$. The bifurcation diagrams with controlling parameter $a$ changing are calculated and drawn in Figure 13. From Figure 13, we can know that the period-2 bubble (primary bubble) is obtained for $b=3.8$, and the period-6 bubble is observed for $b=4.7$. With $b$ increasing, more full bubbles appear until chaos occurs (full Feigenbaum remerging tree (fFrt) at $b=4.9$ and $b=5.1$). Similarly, the antimonotonicity in system (3) can also be detected with controlling parameter $b$ changing for discrete values of $a$ when $c=2$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$ and initial value $(x(0),y(0),z(0),w(0))=(-1.61,1.5,-6,-0.01)$. The bifurcation diagrams when controlling parameter b changing are depicted in Figure 14, which indicates that when $a=5$, system (3) shows a period-2 bubble (Figure 14a); when $a=6.7$, system (3) exhibits a period-6 bubble (Figure 14b); when $a=6.8$ or $a=6.9$, fFrt is also found in system (3).

4. An Image Encryption Algorithm Based on the Jerk-like System with a Smooth Memristor

At present, multimedia technology is an important means of information transmission. To avoid information leakage, various information encryption methods were proposed [40,41,42]. Because many digital images contain private information, such as confidential and personal privacy, image encryption algorithms attracted the attention of researchers [43,44,45,46,47]. Due to the complexity of a chaotic system, image encryption based on chaos theory received much attention [44,45,46,47].

4.1. Image Encryption Algorithm

According to Section 3, the new Jerk-like system with a memristor is provided with complex dynamics closely related to values of system parameters and initial conditions. According to this property, the pseudo-random sequences used in the image encryption algorithm can be generated by the proposed system. An image encryption algorithm based on the new Jerk-like system with a memristor is addressed, and the performance analysis is carried out. Suppose the size of a picture is $M\times N$, the pseudo-random sequences produced by the Jerk-like system with a memristor are denoted as $S=\left(x_j,y_j, z_j,w_j\right)$ $\left(j=1,2,3,\cdots \right)$, which can be used in the process of scrambling and diffusion of image pixels.

Suppose $\left(x_i,y_{\mathrm{i}}\right)$ and $\left(x_{\mathrm{i}+1},y_{\mathrm{i}+1}\right)$ are the positions before and after scrambling, respectively, formulas

$$\left\{\begin{array}{c}{x}_{i+1}=(x_i+y_i+abs(fix(z_j))+abs(fix(z_{j+f})))\mathrm{mod}M\\ y_{i+1}=(x_i+y_i+abs(fix(z_j))+K\sin (\frac{2\pi x_{i+1}}{N}))\mathrm{mod}N\end{array}\right.$$

(6)

are utilized in the process of pixel position scrambling with constant $K$. $abs\left(x\right)$ means the absolute value of $x$, $fix\left(x\right)$ represents a minimum integer greater than or equal to $x$. $f$ is an integer.

In the process of pixel diffusion, pixel values are modified via formula

$$v=p\oplus (c\times x_i+d\times y_i)\mathrm{mod}L$$

(7)

with $c$ and $d$ determined by

$$\left\{\begin{array}{c}c=abs\left({10}^l\right)x_j-round\left({10}^l{x}_j\right))\times {10}^3\\ d=abs\left({10}^l\right)y_j-round\left({10}^l{y}_j\right))\times {10}^3\end{array}\right.$$

(8)

where $p$ and $v$ are the pixel values before and after the diffusion, respectively. $\left(x_{\mathrm{i}},y_{\mathrm{i}}\right)$ is the position of the pixel. $L$ represents the grayscale of the pixel. $Round()$ is the rounding function. $l$ is a positive integer determined by the computer’s accuracy.

The image can be decrypted according to the inverse of the above encryption process. It should be noted that the parameter selections in the decryption progress are the same as those in the encryption process.

In following simulations, the MATLAB computing environment is used as the programming environment.

4.2. Simulations of the Encryption and Decryption

In the simulations, when choosing $a=7$, $b=5.0$, $c=2$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, and initial value $(-1.61,-1.5,-6,-0.01)$, system (3) is chaotic. $l=3.0$ and $f=20$ are selected. According to the encryption method in Section 4.1, some gray images are encrypted and decrypted (Figure 15). Figure 15 confirms the effectiveness of the encryption method based on the new Jerk-like system with a memristor. And then, encryption and decryption results for a color Lenna image are depicted in Figure 16, which means that the proposed method is also effective for color images. The time required to encrypt the images in Figure 15 and Figure 16 are listed in

Table 2. The time required to encrypt images in Figure 15 and Figure 16.

	Gray Lenna Image	Peppers	Camara Man	Color Lenna Image
Required time	34 s	33 s	32 s	35 s

Note. In Table 2, s means time unit second.

, which illustrates that the proposed encryption method do not need much time to execute the image encryption.

4.3. Performance Analysis of the Encryption Method Based on the New Jerk-like System with a Memristor

4.3.1. Size of the Key Space

According to [48], if an algorithm can resist a brute force attack, the size of the key space must be greater than $2^{100}$. In the proposed encryption method, there are 13 keys, including four initial values, seven system parameters, and two integers, $f$ and $l$. In the MATLAB computing environment, the computer has an accuracy of ${10}^{-16}$. Therefore, the key space of the addressed encryption method is ${10}^{192}$, which is far greater than $2^{100}$ and can make the cipher image effectively resist a brute force attack.

4.3.2. Statistical Feature Analysis

The histograms of Figure 15a,b are depicted in Figure 17a,b, respectively, from which we can see that the original image has prominent statistical characteristics. In contrast, the encrypted image has strong randomness. This result illustrates that the proposed encryption method can successfully hide much information about the original image, which makes it difficult to be attacked by statistical attackers.

4.3.3. Test of the Key Sensitivity

Firstly, the encryption process is sensitive to the key. To verify this result, the encryption test is performed for Figure 15a. Make one of the keys slightly change while other keys remain unchanged, and then we can observe the changes in the encrypted images. The test results are given in

Table 3. Percentage difference of the cipher image based on the new Jerk-like system with a memristor when only one key changes slightly.

	Test 1	Test 2	Test 3	Test 4
Percentage difference of cipher image	99.73%	99.82%	99.90%	99.93%

.

• Test 1. Change one initial value; for example, let $x\left(0\right)=x\left(0\right)+{10}^{-5}$.
• Test 2. Change one system parameter; for example, let $a=a+{10}^{-5}$.
• Test 3. Change parameter $l$, and let $l=l+{10}^{-5}$.
• Test 4. Change one memristor parameter $\alpha$, and let $\alpha =\alpha +{10}^{-5}$.

From Table 3, it is easy to know that, even though only one key changes slightly, the percentage difference of ciphertext images is very large and close to 100%. It means that the encryption process of the proposed method is susceptible to the key. Furthermore, comparing the percentage difference of the proposed algorithm with the traditional scrambling diffusion algorithm using the random sequences automatically generated by the computer as the pseudorandom sequence, it can be found that, using a pseudorandom sequence automatically generated by computer, the percentage difference is 99.76% when l is altered to $l+{10}^{-5}$. It suggests that the proposed image encryption algorithm after improving the key space has slightly stronger sensitivity than the traditional scrambling diffusion algorithm. Both of them have an excellent encryption effect.

Secondly, the sensitivity of the decryption process to the key is discussed.

The degree of agreement between the decrypted image and the original image is observed by slightly changing the partial key during the decryption process. With all correct keys, $x\left(0\right)=-1.61$, $y\left(0\right)=-1.5$, $z\left(0\right)=-6$, $w\left(0\right)=-0.01$, $a=7.0$, $b=5.0$, $c=2.0$, $d=0.5$, $r=0.1$, $\alpha =0.1$, $\beta =0.3$, $l=3.0$, $f=20$, the decrypted result is depicted in Figure 18a. Decryption results by changing the key $x\left(0\right)$, $a$, $l$ slightly are given in Figure 18b–d, respectively. From Figure 18, we know that even if only one key changes slightly, the decryption result is a messy image, which is meaningless. This result suggests that the decryption process of the proposed method has strong sensitivity to the key.

4.3.4. Correlation Analysis between Adjacent Pixels

The correlation coefficients between adjacent pixels in images Figure 15a,b,d,e,g,h and Figure 16a,b are calculated and listed in

Table 4. Correlation coefficient between adjacent pixels in some original images and encrypted images.

Images	Original Image			Encrypted Image
	Horizontal	Vertical	Diagonal	Horizontal	Vertical	Diagonal
Gray Lenna	0.9604	0.9846	0.9811	−0.0220	−0.0006	−0.0194
Peppers	0.9601	0.9688	0.9502	0.0035	0.0079	0.0008
Camera man	0.9589	0.9712	0.9403	0.0045	−0.0056	0.0029
Color Lenna	0.9629	0.9879	0.9838	−0.0233	−0.0002	−0.0205

, which suggests that the original images are highly correlated with adjacent pixels, while the correlation between adjacent pixels in the encrypted images are very weak. Furthermore, the results in Table 4 are very close to correlation coefficients [49]. It means that, via the proposed image encryption, the statistical characteristic of the considered images has been spread into the encrypted images with high randomness so that the encrypted images can effectively resist statistical attack.

And then, 2500 pairs of pixels are selected from the Lenna image (Figure 15a) and the encrypted image (Figure 15b), respectively. The scatter diagram about the correlation of adjacent pixels of these points are depicted in Figure 19, from which it is known that, compared with Lenna image, the relevance of the encrypted image is significantly reduced. This result is consistent with Table 4.

4.3.5. Analysis of Information Entropy

As one statistical feature, the information entropy reflects the average amount of information in an image. The information entropy of the encrypted images Figure 15b and Figure 16b can be calculated as 7.9914 and 7.9919, respectively, both of which are very close to the maximum ideal value 8 of original image. This means that the proposed encryption method maintains the integrity of the information and has strong capability of anti-entropy attack.

4.3.6. Analysis of Differential Attack

The number of pixels change rate (NPCR) and the unified average changing intensity (UACI) are two indicators to evaluate whether the encryption algorithm can resist differential attacks. In this section, some images are tested (Figure 15a,d,g and Figure 16a). The values of NPCR and UACI of the encrypted images (Figure 15b,e,h and Figure 16b) via the proposed encryption method are calculated and depicted in

Table 5. NPCR and UACI evaluations.

Test Images	NPCR	UACI
Gray Lenna	99.6194%	33.4635%
Peppers	99.4936%	33.3988%
Camera man	99.5738%	33.4406%
Color Lenna	99.5987%	33.4589%

. Comparing the result in Table 5 with Reference [48], it can be found that the gap is very small. The values of NPCR and UACI all approach 99.6093% and 33.4635%, respectively. It means that the values are satisfactory. Then, it means that the proposed image encryption method based on chaotic Jerk-like system with memristor has good ability to avoid the differential attack.

4.3.7. Analysis of the Robustness

Robustness to crop attack and noise disturbance are two important indicators to evaluate the stability of encryption algorithm.

To demonstrate the robustness of the proposed encryption algorithm. Crop attack and noise disturbance have been considered. A block of 100 × 100 pixels was removed from the encrypted image (Figure 20b) and the decrypted image can be gained as Figure 20c. Figure 20 shows that, even if part of the image information is lost, it can recover the original image very well. It suggests that the proposed encryption algorithm can resist a crop attack effectively.

Then, multiplicative noise with a variance of 0.5 and Gauss noise with a mean of zero and a variance of 0.3 were added to the encrypted image (Figure 21b). The corresponding decryption result is depicted in Figure 21c, which means that the proposed encryption algorithm can effectively resist noise disturbance.

Figure 20 and Figure 21 indicate that the proposed encryption algorithm has good robustness to information loss and noise.

5. Conclusions

Considering the complex dynamics of memristor, a new 4D Jerk-like system with a memristor is proposed. Theoretical analysis indicates that the addressed system has no equilibrium point. The following results are obtained:

(1)

System parameters can affect the types of hidden dynamics. The change in parameter can induce various hidden attractors in the new 4D Jerk-like system with a memristor, such as hidden chaotic attractor and hidden period attractors with different periods.

(2)

The coexistence of different kinds of hidden attractors can be found. By choosing appropriate system parameters, different kinds of hidden attractors can coexist, such as chaotic attractor and multi-period, period-1 attractor and chaotic attractor, multi-period attractor and period-1 attractor, multi-period attractor and multi-period attractor.

(3)

Offset boosting in system (3) is analyzed via transformation. Results suggest that, for the chaotic attractor, period-1 attractor, and period-2 attractor, offset boosting can be realized by altering the boosting controller, while offset boosting of the period-3 attractor and period-4 attractor cannot be gained by changing the boosting controller.

(4)

Antimonotonicity has also been demonstrated in the new Jerk-like system with a memristor. Namely, with parameters a or b changing, full Feigenbaum remerging tree appears.

(5)

An image encryption algorithm based on the new 4D Jerk-like system with a memristor is designed. By experiments with their analysis and some comparisons with other results, the effectiveness of the proposed encryption method is verified.