Nonlinear Dynamics in the Coupled Fractional-Order Memristor Chaotic System and Its Application in Image Encryption

. Tis work presents two forms of coupled fractional-order memristor chaotic systems. Te existence and uniqueness of solutions are studied. Moreover, the range of parameters and time span at which the proposed two models exhibit continuous dependence on initial conditions are examined. Te unique equilibrium point for each system is found, and the corresponding stability analysis is carried out. Te regions of stability in the space of parameters are obtained, whereas numerical simulations are employed to confrm theoretical results. Te bifurcation diagrams, in addition to Lyapunov exponents, are utilized to examine the efects of key parameters in two models. A chaos-based encryption scheme is presented as an application to utilize complicated chaotic behaviors in coupled circuits.


Introduction
Several phenomena that are subject to spatio-temporal development have been encountered in various felds of application, such as physics, engineering, biology, economy, and chemistry [1,2].To better study and interpret emerging nonlinear phenomena, the theories of dynamical systems and chaos are helpful tools that achieve these goals.Te applications of dynamical systems and chaos have been involved in many disciplines such as electronic circuits [3,4], chaos and synchronization [5][6][7][8], mathematical biology [1], image encryption [9], secure communications [10], cryptography [11], and neuroscience research [12].
Nonlinear electronic circuits can be considered a very useful practical tool for studying nonlinear phenomena and chaos.Tis line of research has attracted considerable interest, especially due to the works of L.O.Chua from the mid-1960s.Chua proposed a resistor with two terminals having a piecewise-continuous voltage-current characteristic referred to as Chua's diode, and Chua's circuit is the name of the resulting circuit [13].Since then, several developments have been introduced in the feld of nonlinear circuits.For instance, Shinriki et al. inserted a nonlinear circuit called the modifed Van der Pol oscillator (MVPO) [14].King and Gaito derived a nonlinear circuit based on the MVPO circuit and called this circuit an autonomous Van der Pol-Dufng (ADVP) oscillator [15].A modifcation to the ADVP circuit was conducted through adding a resistor in parallel with the inductor in the baseline ADVP circuit [16].In the modifed ADVP, the dynamics of the original ADVP oscillator are confned in a small range of the circuit's parameter.
Chua theoretically forecasted the memristor as a fundamental fourth circuit component in 1971.Te memristor is a nonlinear two-terminal component, in which the induced magnetic fux is the function of an electric charge that passes through the device [17,18].Te memristor has taken its place along with other conventional circuit components, which include the resistor, capacitor, and inductor, when Williams et al. [18] fabricated the frst solid-state implementation of memristor circuit components.Tere are many important applications of memristors such as learning networks, ultradense nonvolatile memories, high-frequency oscillators, and secure communications applications employing nanoscale memristor-based circuits [18][19][20][21].
Te multistability analysis of a novel memristor-based chaotic circuit has been introduced in [22] along with circuit implementations and application in image encryption.Examples of chaotic dynamics in biological systems and chaosbased image encryption schemes can be found in [23][24][25] and references therein.
Recently, the applications of fractional calculus in mathematical modeling of dynamical systems have attracted increasing attention.Several theoretical and experimental studies have indicated that fractional-order derivatives are more adequate and can provide accurate mathematical models of intricate systems with memory, in contrast to classical integer-order models [26].Fractional-order differential equations have been verifed as the more appropriate modeling tool for several real problems in many felds of science, engineering [26], biological systems [27,28], economic systems [29,30], circuit theory [31,32], and many more.Te chaotic behavior has also been found in many fractional-order systems [33], such as fractional Lu systems [34] and fractional Lorenz systems [35].From an application point of view, chaotic systems are widely used to design algorithms dedicated to image encryption [36][37][38][39].Indeed, fractional-order chaotic systems have good advantages in terms of pseudorandomness and ergodicity and are extremely sensitive to initial conditions which are very appropriate for image encryption [40].Synchronization of fractional chaotic systems and its applications in encryption of images have been presented in [41].In addition, the improper fractional-order laser chaotic system and its application for image encryption are considered in [42].
Fractional-order diferential equations are known to better model natural, engineering, and physical systems having memory characteristics.On the other hand, the fourth basic circuit element, i.e., the memristor, relates the values of the fux across its terminals with the past values (history) of the current passed through it.In other words, the memristor can memorize the history of its state.So it is more adequate to employ fractional-order diferential equations with memristor-based circuits.Te other point which motivates this work is to examine the more general case where coupled fractional-order chaotic memristor-based circuits are established.Indeed, there are some interesting questions which arise and need to be investigated, for example, what are the infuences of the fractional order and type of coupling on the dynamical characteristics and equilibrium points and their stability in the coupled system.Finally, it is also essential to explore which potential applications can be established based on the analytical/theoretical results obtained in the study.Te paper is an attempt to explore the answers to these questions.Due to the distinct characteristics of memristors, they are crucial elements in chaotic circuits with very small sizes and low-power consumption.Terefore, the present work establishes theoretical/numerical frameworks for practical studies which implement memristors in efcient image encryption hardware.Te proposed image encryption method, on the other hand, creates chaotic sequences which depend on tiny changes in both secret keys and input plain images.Tis implies that the proposed scheme can resist known-plaintext attacks, known-ciphertext attacks, and diferential attacks, while it can be realizable in small, very fast, and low-powerconsumed appliances.
With the rapid development of the Internet and wireless communication systems, images and real-time videos are considered among the most important carriers of information.So ensuring the security of transmission, storage and access to digital images has become a very active topic for researchers.From a mathematical point of view, the fascinating features of chaotic dynamics attracted the attention of mathematicians, engineers, and computer scientists.More specifcally, chaos has the characteristics of random-like behavior, unpredictability, ergodicity, sensitive dependence on parameters and initial values, and the ability to produce very complicated behaviors from relatively simple-structure systems, which render chaotic systems an ideal choice in the feld of image encryption.Chaotic systems also have the advantages of being easily realizable on feldprogrammable gate arrays (FPGAs), digital signal processors (DSPs), or microcontrollers.In addition, they can be included in ultrafast secure communication systems which utilize chaotic semiconductor lasers or ring fber lasers.Some of the recent developments in the feld of chaos-based cryptography can be seen in [43][44][45][46][47][48][49].
In this work, two forms of coupled fractional-order memristor chaotic systems along with the study of the existence and uniqueness of the proposed model's solution and conditions of continuous dependence on initial conditions are presented in Section 2. Te stability analysis of the unique equilibrium point for each system is presented in Section 3. Te regions of stability in the space of parameters are obtained, and numerical simulations are employed to verify the theoretical results.In Section 4, the bifurcation diagrams and Lyapunov exponents are employed to inspect the efects of key parameters in the two systems.Finally, a chaos-based encryption scheme is presented in Section 5 as an application to utilize complicated chaotic behaviors in the coupled circuit.

The Proposed Coupled Fractional-Order System
In this section, two forms of coupling are introduced for the proposed coupled fractional-order memristor-based chaotic system, namely, partial coupling and complete coupling cases.First, some key defnitions and properties of fractional calculus have been reviewed.
Memristors can be classifed as fux-controlled or charge-controlled memristors.In fux-controlled memristors, the charge q is related to the fux ϕ of the memristor by the relation where G is a fux-dependent function.By diferentiating the above equation w.r.t.time, it yields where i(t) denotes the current that passes through the memristor, v(t) is the voltage across its terminals, and W(ϕ) refers to the incremental memductance function which describes the change rate of the charge with fux.In particular, Similarly, the second type of memristors is the chargecontrolled memristor which can be described by the following equations: In this work, the fux-controlled memristor is used.
For the proposed partial coupling case, the system is described as For the complete coupling case, the system is proposed in the form Systems ( 7) and ( 8) are coupled systems of two fractional-order chaotic memristor circuits.Te circuit is illustrated in Figure 1 and consists of two resistors, two capacitors, one inductor, and the memristor element.Te generalization to the fractional-order case is introduced, and the diferent cases of couplings are examined from theoretical/numerical viewpoints along with image encryption applications.
2.1.Existence and Uniqueness.Te proposed system (7) can be expressed in the following form: Te solution of this initial value problem is therefore obtained by Te existence of this solution is examined in the region Π × J where ω � x, y, z, w and i � 1, 2, 3, 4. Te equivalence between the above integral (10) and original system (7) is utilized as follows.
Te right hand side of (10) is referred to as Ω(X), and hence, we obtain

Mathematical Problems in Engineering
Te following inequality can be obtained after some calculations: Here, the supremum norm is employed for the class C 1 of diferentiable continuous functions on J. Now, we get the following theorem which states the suffcient condition for existence and uniqueness of system (7).
then the solution of system (3) exists, and it is unique on Π × J.
Proof.For Λ < 1, the mapping X � Ψ(X) is a contraction mapping, and the theorem follows immediately from the Banach fxed-point theorem.
On the other side, the second proposed system ( 8) is expressed as Te solution of the initial value problem is written as Te existence of the solution is examined in the region Π × J where ω � x, y, z, w and i � 1, 2, 3, 4. Te equivalence between the above integral (15) and original system (8) is employed as follows.
Te right hand side of ( 15) is referred to as Φ(X), and hence, we obtain Te following inequality can be obtained after some calculations:  Mathematical Problems in Engineering Now, we get the following theorem which states the sufcient condition for existence and uniqueness of (8).□ then the solution of system ( 8) exists, and it is unique on Π × J.
Proof.Te proof of Teorem 1 is extended to the case of the complete coupling case in Teorem 2. (7) and ( 8) Theorem 3. Assume that the conditions of Teorem 1 (Teorem 2) are satisfed, then the solution of system (7) (system ( 8)) exhibits continuous dependence on initial conditions; i.e., for every ϱ > 0, there exists κ > 0 such that for two initial conditions satisfying

Continuous Dependence of Initial Conditions for the Solutions of Coupled Systems
Proof.Consider any two solutions of system (7) (system ( 8)) which evolve from two close initial conditions X 01 and X 02 in the way that 0 Assume that the conditions of Teorem 1 (Teorem 2) are satisfed, thus In view of these two equations, it is obvious that where 0 < Λ < 1 as stated above.

Stability of the Equilibrium Point
Te unique equilibrium point of the proposed two systems (7) and ( 8) is X * � (0, 0, 0, 0, 0, 0, 0, 0).Te interesting point here is that the original uncoupled 4D system has a line of equilibria, i.e., an infnite number of nonisolated equilibrium points.Partial and complete couplings are found to eliminate these lines of equilibria in the way that only the origin equilibrium point persists.
Te Jacobian matrix of the fractional-order model ( 7) at X * is evaluated as follows: whereas the Jacobian matrix of the fractional-order model ( 8) at X * is evaluated as follows: For system (7), the characteristic equation of J 1 is obtained as follows: where

Mathematical Problems in Engineering
On the other side, for system (8), the characteristic equation of J 2 is obtained as follows: where Numerical investigation of stability regions in the space of parameters is a necessary step due to the very complicated exact forms of stability conditions of equilibrium points.It is known that the arguments of eigenvalues of the Jacobian matrix should satisfy to ensure the local asymptotic stability of the equilibrium point.
Stability regions for the equilibrium point of the proposed system (7)  For the suggested system (8), stability regions for the equilibrium point of the system are illustrated in Figure 5  Numerical simulations are used to verify these obtained regions.In Figure 8, the time series plots and phase portraits for system (7) are shown at the following values of parameters ϵ � 0.05, ξ � 0.7, δ � 0.5, α � 7.82, β � 7.813, c � 2, and σ � 5 and illustrate the asymptotic stability of the origin equilibrium point.Also, in Figure 9, the time series plots and phase portraits for system (48 point of the proposed system (8)) are shown at the following values of parameters ϵ � 0.3, ξ � 0.9, δ � 0.6, α � 1, β � 1.3, c � 2.8, and σ � 0.03 and depict the asymptotic stability of the origin equilibrium point.
We fx α 1 � 1.4 and the varying parameter β in the range [1, 10.5], as depicted in Figure 11(a).For 1 ≤ β ≤ 3.7 there is 2-period doubling, and then, the chaotic behavior emerges.We fx β � 5; then, we examine the efect of the parameter α 2 through the range [0.1, 12.5], as shown in Figure 11(b).For  Mathematical Problems in Engineering 0.1 ≤ α 2 ≤ 3.8, there is 3-period doubling, and for 3.8 ≤ α 2 ≤ 12.5, the system exhibits chaotic behavior.We fx α 2 � 6.5; then, we change the value of ϵ ∈ [0, 3], as displayed in Figure 12.It is clear that, in the range [0, 1.2], there is quasi-periodic behavior, and then, the system goes to chaos and back to 2-period doubling in [2.6, 3].Te parameter ϵ is fxed with the value ϵ � 1.5; then, we change the value of parameters δ ∈ [0, 0.99] and δ ∈ [0, 10], as depicted in Figures 13(a) and 13(b), respectively.In the range of 0 ≤ δ ≤ 0.3, the system exhibits chaotic behavior, and then, 2period doubling behavior appears.For the parameter σ in the range from σ � 0.5 to σ � 4, we fx δ � 0.6, and the bifurcation diagram and the corresponding Lyapunov exponent plot are shown in Figures 14(a   We fx � 3, and then, we change the value of the fractional order ξ ∈ [0.75, 1], as displayed in Figure 15.For 0.75 ≤ ξ ≤ 0.865, there is 2-period doubling, and then, the system goes to quasi-periodic and back to chaos.Te phase portrait at two diferent values for the fractional order ξ is depicted in Figures 16 and 17 to verify the bifurcation diagram.

Image Encryption and the Decryption Algorithm
We are motivated by coupled fractional-order memristor chaotic systems and their sensitivity to initial conditions to utilize the generated randomness time series from the coupled system in an image encryption application.Te algorithm fundamentally consists of two parts: original image pixel scrambling with the pixel value remains unchanged and image difusion based on the chaotic time series from the coupled fractional-order memristor chaotic systems.

Encryption
Steps.Te steps of the proposed algorithm are illustrated as follows: (1) Te original image is put as a matrix B M×N with dimensions M × N. (2) Te value of the constant B c is set as a perturbation value that relies only on the original image, and it is defned as where B(i, j) is the positions of original image pixels.We use the image constant B c as a perturbation value for one of the system parameters, for example, σ or α 1 , to make the original image contribute to evaluating the secret key and the scrambling process.(3) Te chaotic time series are obtained as For the secret key, we use the mod function between x 1 (i) and 256, where the range [0, 256] indicates image pixel values.Also, we use the mod function between x 2 (i), y 2 (i) and M and N, respectively, to get a new position for the pixel value image matrix B for the scrambling process.(5) Te repositioned image matrix B sc is obtained after using newRow and newCol as new rows and columns for pixel positions without altering pixel values.(6) Te secret key is transformed to the matrix using sKey � reshape(secretKey, M, N). ( Step 7. To obtain the encrypted image B en , the bitwise XOR operation between sKey and B sc is applied:

Decryption Steps.
To decode the image, we reverse the encryption steps as follows: Step 1.To obtain the decrypted image B de , the bitwise XOR operation between B en and sKey is applied: 14 Mathematical Problems in Engineering B de (i, j) � B en (i, j) ⊕ sKey(i, j). ( Step 2. Te positions of the pixels in the image are inverted to get the original image using newRow and newCol sequences.

Security Analysis.
Te encryption algorithm of images is applied based on the coupled fractional-order memristor chaotic systems.Four test images, cameraman, Lena, bird, and baboon, with a size of 256 × 256 are used in the experiment.Te perturbation value B c of the cameraman, Lena, bird, and baboon images in the algorithm through (32) is 2.7641 × 10 − 8 , 2.8891 × 10 − 8 , 1.6707 × 10 − 8 , and 1.5513 × 10 − 8 , respectively.Te proposed encryption algorithm is implemented on the four test images, as illustrated in Figure 18 which depicts the original, scrambled, encrypted, and decrypted images for the presented algorithm.
To confrm the efectiveness of the proposed algorithm, statistical and randomness tests have been carried out such as key space, histogram analysis, information entropy, correlation coefcients of adjacent pixels, key sensitivity, and cropping attacks.Te key space refers to the key size range, which can determine the security of the encryption algorithm which is resistant to brute force attacks.In this paper, the secret key is composed of the double precision based on the chaotic coupled system parameters α 1 , α 2 , β, ϵ, δ, σ, ξ, B c and initial conditions (x 1 (0), y 1 (0), z 1 (0), w 1 (0), x 2 (0), y 2 (0), z 2 (0), w 2 (0)).Te digit numbers are to be at least 10 − 15 in each parameter.Te proposed algorithm has a key space value of 10 240 which is compared to the recommended key space that should be at least 2 100 [61].Consequently, the large key space of the proposed algorithm provides resistance to various types of attacks.

Histogram Analysis.
Te histogram is an indicator of statistics in the image, which refects the total number of pixels for each value in the image.To deny an adversary from obtaining statistical information, the histogram of the encrypted image should be uniformly distributed.For the proposed algorithm, the histogram of the original and encrypted images is shown in Figure 19, demonstrating that the encrypted image has a uniform histogram regardless of the original image; hence, it can resist statistical analysis attacks.
Moreover, the histogram variance can be used to quantify the histogram.Te smaller the variance of the encrypted image, the more uniform the distribution.Te histogram variances for the original and encrypted images are shown in Table 1.In all the cases, the signifcant reduction of the variance can be observed compared with the original image.Te reduction for the cameraman, Lena, bird and baboon images is verifed with values of approximately 99%.

Information Entropy.
Information entropy is a tool used to measure the randomness of pixels in an image.Te values of entropy are 7.9978, 7.9971, 7.9974, and 7.9973 for cameraman, Lena, bird, and baboon images, respectively.It is clear that the mentioned values approach the ideal value ≈8, which means that the encrypted image has high randomness and is less feasible to show information for the encryption scheme.In Table 2, the entropy of the proposed algorithm is compared with the similar literature work.

Correlation Analysis.
Te correlation analysis is used to represent the degree of association between adjacent pixels, and the correlation coefcient is close to 1 in the original image.Te calculation method of the correlation coefcient between adjacent pixels is as follows: where cov(u, v) is the covariance function, E(u) � 1/N  N i�1 u i , and σ u � 1/N  N i�1 (u i − E(u) 2 .Te correlation analysis of the cameraman, Lena, bird, and baboon original and encrypted images in three directions is evaluated and shown in Table 3.In addition, the correlations of the bird image before and after encryption in each direction are depicted in Figure 20.It is clear that the correlation coefcient of the encrypted image is close to zero in all directions.Terefore, the encryption process

Key Sensitivity Analysis.
Te encryption algorithm should be very sensitive to its secret key.Terefore, a slight change in the key will lead to a complete change in the encryption result.Hence, if the secret key changes slightly, the decrypted image will be diferent from the original image.To measure the sensitivity of the encryption key, NPCR (pixel change rate) and UACI (pixel average change intensity) have been used.NPCR and UACI are evaluated by the following equations [65]: We verifed the sensitivity of secret key values associated with x 1 (0) and y 1 (0) by adding 10 − 12 in the initial conditions.Te values of NPCR and UACI between the encrypted image and the original image are evaluated in Table 4.It can be confrmed that the proposed algorithm has a good sensitivity to the encryption key.Mathematical Problems in Engineering 5.3.6.Cropping Attack.To investigate the immunity of the encryption algorithm to some information loss or tampered information, the cropping attack is utilized.In the proposed algorithm, we can still obtain an identifable image by decryption when an internal block of the encrypted image with a dimension of 256 × 75 is exchanged with a black block in

Conclusion
In this study, two coupling schemes for fractional-order memristor-based chaotic circuits are introduced.It is observed that the line of nonisolated equilibrium points in uncoupled circuits becomes a unique equilibrium point in the two coupled cases.Te regions of stability in the space of parameters are depicted for each coupled system.Te variety of nonlinear dynamics exhibited by the coupled circuits is examined via bifurcation diagrams, Lyapunov exponents, and phase portraits.A suggested chaos-based encryption scheme is presented.Security tests and analysis are carried out to confrm reliability of the encryption scheme.Tis work is limited to the case of two coupled fractionalorder chaotic circuits.Te work can be extended to the more realistic and general cases of coupled memristor-based circuits.In particular, diferent topologies for networks of coupled circuits can be considered in future studies.For example, it is interesting to explore the nonlinear dynamics and chaos synchronization in ring, starhub, tree, or hierarchical networks of fractional memristor-based chaotic circuits.Moreover, new chaos-based encryption schemes can be examined in diferent confgurations of fractional circuit networks.Mathematical Problems in Engineering
) and 14(b), respectively.Te fgures show that, in the interval [0.5, 1.75], the value of the maximal Lyapunov exponent is very close to
2, . . ., k, and k � M × N; after eliminating the transient values, the required size k is obtained from the solutions of the chaotic system.(4) Te standard values for the secret key and new pixel positions are obtained as follows: secretKey i � mod floor x 1 (i) × 10 15  , 256  , newRow i � mod floor x 2 (i) × 10 15  , M   + 1, newCol i � mod floor y 2 (i) × 10 15  , N   + 1.

Figure 18 :
Figure 18: Te original, shufed, encrypted, and decrypted images in frst, second, third, and fourth columns, respectively, for cameraman, Lena, bird, and baboon images with a size of 256 × 256.

Figure 19 :
Figure 19: Histograms for cameraman, Lena, bird, and baboon images in the frst, second, and third rows, respectively.First column: original images; second column: encrypted images.

Figure 20 :
Figure 20: Te correlation of the bird original image in the frst column and the encrypted image in the second column for the three directions.

Figure 21 :
Figure 21: Te frst and second rows represent the encrypted images and the respective decrypted images as a result of the cropping attack, respectively.

Table 1 :
Te histogram variances for original and encrypted images for cameraman, Lena, bird, and baboon images with their reduction.

Table 2 :
Comparison of the entropy of the proposed algorithm with the literature.

Table 3 :
Te correlation coefcients of the four images in diferent directions.