Startseite Asymmetrical novel hyperchaotic system with two exponential functions and an application to image encryption
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Asymmetrical novel hyperchaotic system with two exponential functions and an application to image encryption

  • Ali A. Shukur EMAIL logo , Mohanad A. AlFallooji und Viet-Thanh Pham
Veröffentlicht/Copyright: 2. Februar 2024
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Abstract

In this article, asymmetrical novel system with two exponential functions, which can show hyperchaotic behavior, has been proposed. Although new system possesses only one unstable equilibrium. The dynamical behaviors of such system are discovered by computing the Lyapunov exponents and bifurcation diagram. Furthermore, the synchronization of the proposed system are also presented by an adaptive synchronization approach of two identical hyperchaotic systems. An application to image encryption has been obtained.

1 Introduction and formulation of the system

In the last two decades, a large number of chaotic systems have been studied with their application in weather forecasting [1], telecommunication [2], biological modeling [3], and so on. Chaotic systems can be classified into three categories by physical, dynamical, and algebraic features according to the number of dimensions, the number of wings, and number of equilibrium points, respectively. After Lorenz’s discovery (1964 [4]), researchers have attempted to provide chaotic systems with unique peculiarities (physical and dynamical) such as biological model of Rössler that contain only one nonlinear term [5], electronic circuit of Chua that exhibit two scroll chaotic behavior [6], and simplification of the Lorenz system by the Chen and Lu system [7,8]. On the other side, chaotic systems, with special algebraic structure were provided, such as Wei’s system that has no equilibria [9] and Wang’s system that has only one stable equilibrium [10]. In the late 1970s, Rössler proposed a very interested choatic system with more then one positive Lypunouv exponent, later was called hyperchoatic [11]. After that, hyperchaotic systems got the interest of researchers in different areas, and many of them have been introduced, especially, 4D hyperchaotic Lorenz- type system [12]. Note that such kind of systems with unusual peculiarities can have neither heteroclinic orbit nor homoclinic orbit, and thus, the Shilnikov method [13] may not help to verify the chaos. Thus, hyperchaotic systems are more complicated.

From a computational point of view, a hidden attractor was observed in some kinds of chaotic and hyperchaotic systems such as systems with no equilibria. To have better understanding to hidden attractor, we refer to the studies by Kuznetsov [14] and Leonov and Kuznetsov [15].

In this article, a novel hyperchaotic system with two exponential functions is proposed. This article is organized as follows: Sections 2 introduces the theoretical model of the system and studies some of it is fundamental properties. In Section 3, the ultimate boundedness of the proposed system has been obtained. In Section 4, the possibility of the synchronization scheme of the proposed systems is studied. In Section 5, an application to image encryption was processed.

2 The proposed new model

Zhouchao and Qigui [9] presented a very special kind of chaotic system with the exponential function such as:

(1) x ˙ = α ( y x ) ; y ˙ = β y + k x z ; z ˙ = d e x y ,

where α , β , c , and d are system parameters and x z , exp ( x y ) are the nonlinear terms. System (1) shows complex dynamics when α = 0.78 , β = 0.5 , c = 0.37 , and d = 2.5 . Besides, system (1) has two equilibria, and due to this fact, one can easily check that (1) is not topologically equivalent to the original Lorenz or any Lorenz like systems. In the study by Pham et al. [16], the following 4D hyperchaotic system was constructed:

(2) x ˙ = α ( y x ) ; y ˙ = β y + k x z + c w ; z ˙ = d e x y ; w ˙ = m x .

Pham et al. [16] reported that system (2) has no equilibrium and it can be classified as a chaotic system with the hidden strange attractor where a basin of attraction does not contain neighborhoods of the equilibrium points.

On the basis of the idea of system (2), we introduce asymmetrical four-dimensional hyperchaotic system with two exponential functions such as follows:

(3) x ˙ = α ( y x ) + m w ; y ˙ = β y + k x z ; z ˙ = d x y x w e x w ; w ˙ = r 1 + e y ,

where 1 1 + e y is a sigmoid function; x , y , and z are state variables; and α , β , m , d , and r , are real constant parameters of systems (3) to control.

In what follows, we test the most important peculiarities of complex dynamical systems such as: disspativity, stability, Lyapunov exponents (LEs), and bifurcation diagram. First, system (3) is disspative for β < 0 with the divergence

V = x ˙ x + y ˙ y + z ˙ z + w ˙ w = α + β .

For example, considering α = 1 and β = 2 , we have V = 2 . Thus, the considered systems are dissipative. The exponential contraction rate is calculated as follows:

d V d t = e 2 t .

Therefore, each volume containing the system trajectory shrinks to zero as t and system orbits are ultimately bounded. Thereby, the existence of attractor of system (3) is proved.

To set the stability, the following algebraic equations must be held

(4) α ( y x ) + m w = 0 , β y + x z = 0 , d x y x w exp ( x w ) = 0 r 1 + exp ( y ) = 0 .

From (4) follows that the system (3) have no equilibrium points and this similar to system (2) which means that system (3) is hidden strange attractor in contrast to system (2), one can easily observe that systems (3) is asymmetrical.

System (3) shows complex dynamics for α = 0.7 , β = 0.1 , k = 1 , d = 6.5 , r = 0.5 , and m = 2.9 with the initial condition ( 1.2 , 1.3 , 1.5 , 1.5 ) . In this case, the LEs of the system are found to be L 1 = 0.237154 , L 2 = 0.024746 , L 3 = 0.000309 , and L 4 = 1.030919 and are shown in Figure 1(a). The bifurcation diagrams show the local maxima of the variable z ( t ) when changing the value of r , β , and d , which is shown in Figure 1(b)–(d). The dynamics of (3) is shown in Figure 2.

Figure 1 
               (a) Lyapunov exponents of the system (3). (b), (c) and (d) Bifurcation diagrams of (3) when varying parameter 
                     
                        
                        
                           r
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                        r,\beta 
                     
                  , and 
                     
                        
                        
                           d
                        
                        d
                     
                  , respectively.
Figure 1

(a) Lyapunov exponents of the system (3). (b), (c) and (d) Bifurcation diagrams of (3) when varying parameter r , β , and d , respectively.

Figure 2 
               (a) 
                     
                        
                        
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                   plane. (b) 
                     
                        
                        
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Figure 2

(a) x - y plane. (b) y - z plane. (c) z - w plane. (d) x - y - w plane.

3 Ultimate boundedness

Chaotic systems are bounded, and estimating the boundedness is one of the most important and difficult tasks. The bounds of chaotic systems play a very significant role in chaos control and chaos synchronization [17,18]. Here, we discuss the boundedness property of system (3).

Theorem 3.1

Suppose that the parameters α , β , m , d , and r of system (3) are positives. Then, all the system orbits, including chaotic orbits, are trapped in a bounded region.

Proof

Construct the following Lyapunov function:

V ( x , y , z , w ) = 1 2 [ x 2 + y 2 + z 2 + w 2 ] .

Along the orbits of (3), we have

V ˙ ( x , y , z , w ) = α x y α x 2 + m w x + β y 2 + x y z + d z z x w e x w + r w x 1 + e y = m w 2 α α x 2 + α x 2 β β y 2 + z + d x w e x w 2 2 + x w + r 2 ( 1 + e y ) 2 + m w 2 α 2 α x 2 β 2 z 2 d x w e x w 2 2 ( x w ) 2 r 2 ( 1 + e y ) 2 .

Let D 0 > 0 be sufficiently large, so that for all ( x , y , z , w ) satisfying V ( x , y , z , w ) = D for D > D 0 with the condition

m w 2 α α x 2 + α x 2 β 2 + z 2 + d x w e x w 2 2 + ( x w ) 2 + r 2 ( 1 + e y ) 2 > z + d x w e x w 2 2 + x w + r 2 ( 1 + e y ) 2 + m w 2 α 2 .

Consequently, we have the surface { ( x , y , z , w ) V ( x , y , z , w ) = D } and V ˙ ( x , y , z , w ) < 0 , which implies that the set { ( x , y , z , w ) V ( x , y , z , w ) D } is confined a trapping region of all solutions of system (3).

4 Adaptive control of the proposed chaotic system

Let consider the driver system defined as follows:

(5) x ˙ = α ( x 2 x 1 ) + m x 4 ; y ˙ = β x 2 + x 1 x 3 ; z ˙ = d x 1 x 2 x 1 x 4 e x 1 x 4 ; w ˙ = r x 1 1 + e x 2 .

Here, we consider the adaptive synchronization of identical novel chaotic systems with unknown parameters. The response system is presented as follows:

(6) x ˙ = α ( x 2 x 1 ) + m x 4 + u 1 ; y ˙ = β x 2 + x 1 x 3 + u 2 ; z ˙ = d x 1 x 2 x 1 x 4 e x 1 x 4 + u 3 ; w ˙ = r x 1 1 + e x 2 + u 4 ,

where x 1 , x 2 , x 3 , and x 4 are the states, α , β , m , d , and r are unknown system parameters, and U = [ u 1 , u 2 , u 3 , u 4 ] T is the adaptive controller to be determined.

We consider the adaptive controller defined by

(7) u 1 = ε α ( t ) x 2 + ε α ( t ) x 1 ε m ( t ) x 4 k 1 x 1 ; u 2 = ε β ( t ) x 2 + x 1 x 3 k 2 x 2 ; u 3 = ε d ( t ) + x 1 x 2 + x 1 x 4 e x 1 x 4 k 3 x 3 ; u 4 = ε r ( t ) x 1 1 + e x 2 k 4 x 4 ,

where ε α , ε β , ε m , ε d , and ε r are estimates of α , β , m , d , and r , respectively, and k 1 , k 2 , k 3 , k 4 > 0 . By substituting (7) into (6), we obtain the closed-loop system:

(8) x 1 ˙ = [ α ε α ( t ) ] x 2 + [ α ε α ( t ) ] x 1 [ m ε m ( t ) ] x 4 k 1 x 1 ; x 2 ˙ = [ β ε β ( t ) ] x 2 k 2 x 2 ; x 3 ˙ = [ d ε d ( t ) ] + x 1 x 2 + x 1 x 4 e x 1 x 4 k 3 x 3 ; x 4 ˙ = [ r ε r ( t ) ] x 1 1 + e x 2 k 4 x 4 ;

with parameter estimation errors defined as follows:

(9) ε 2 ( t ) = [ α ε α ( t ) ] ; ε 3 ( t ) = [ m ε m ( t ) ] ; ε 4 ( t ) = [ β ε β ( t ) ] ; ε 5 ( t ) = [ d ε d ( t ) ] ; ε 6 ( t ) = [ r ε r ( t ) ] .

Next, we reduce (8) to

(10) x 1 ˙ = ε 2 x 2 + ε 2 x 1 ε 3 x 4 k 1 x 1 ; x 2 ˙ = ε 4 x 2 k 2 x 2 ; x 3 ˙ = ε 5 k 3 x 3 ; x 4 ˙ = ε 6 x 1 1 + e x 2 k 4 x 4 .

Theorem 4.1

If the controller are chosen as (7) and update laws of parameters are given by

(11) ε 2 ˙ ( t ) = x 1 x 2 x 1 2 ; ε 3 ˙ ( t ) = x 1 x 4 ; ε 4 ˙ ( t ) = x 2 2 ; ε 5 ˙ ( t ) = x 3 ; ε 6 ˙ ( t ) = x 1 x 2 1 + e x 2 .

Then the synchronization between the driver system (5) and the response system (6) is achieved if k 1 , k 2 , k 3 , and k 4 are positive constants.

Proof

We consider the Lyapunov function defined by

V ( x 1 , x 2 , x 3 , x 4 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 ) = 1 2 ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + ε 2 2 + ε 3 2 + ε 4 2 + ε 5 2 + ε 6 2 ) .

Taking time derivative of the aforementioned function along the trajectories of (11), we have

(12) V ˙ = k 1 x 1 2 k 2 x 2 2 k 3 x 3 2 k 4 x 4 2 ,

which is a negative definite function for k 1 , k 2 , k 3 , k 4 > 0 . Thus, due to the Lyapunov stability theory, we obtain that ε 2 ( t ) 0 , ε 3 ( t ) 0 , ε 4 ( t ) 0 , ε 5 ( t ) 0 , ε 6 ( t ) 0 exponentially when t . In other words, the synchronization occurred between the driver system (5) and the response system (6).

5 An application to image encryption

5.1 Encryption procedure

In this section, we study the exploit of the proposed system in theory of image cryptosystem. The scheme of the presented image cryptosystem is shown in Figure 3(a). In particular, the following steps need to be done:

  1. Calculate the following value:

    X = ( P + M N ) ( M N + 2 23 ) ,

    and then updates the value of X using the formula:

    X ( i ) = mod ( X ( i 1 ) * 1 e K , 1 ) ,

    where M N is the total number of pixels in the image, P is the sum of the pixel values, K is the system’s dimensions, and i = 2 , , K .

  2. Solve the system (3) using the initial values from the previous step.

  3. Sort the output of the differential equations the system (3) and store the sorting indices.

  4. Rearrange the original image after convert it to vector R of pixel values according to the order of the sorted indices form the previous step.

  5. Reshape the vector of pixel values into a matrix with M rows and N columns and store it as matrix R , then extract 2 × 2 block of elements from matrix R after that store it in matrix C x . In particular, the two blocks are chosen based on their position in the image by iterating over every second row (for i = 1 : 2 : M ) and every second column (for j = 1 : 2 : N ) of the image, and after selecting a 2 × 2 block of pixels centered at that position.

  6. The matrix multiplication C x * A is performed, and the obtained matrix is stored in the corresponding block of C 2 × 2 = C x * A , which represents the encrypted image, where A is an arbitrary 2 × 2 matrix, which we called secret.

Figure 3 
                  (a) The scheme of the encryption algorithm; Airplane as an example and (b) the scheme of the encryption algorithm; Airplane as an example.
Figure 3

(a) The scheme of the encryption algorithm; Airplane as an example and (b) the scheme of the encryption algorithm; Airplane as an example.

5.2 Decryption procedure

The decryption is usually a reverse of encryption process. The decryption scheme corresponding to the proposed encryption approach is displayed in Figure 3(b).

After convert, the encrypted image to a double-precision floating-point array P and obtaining the size of it ( M * N ) , then according to the number of rounds that is used in encryption and by going from the last round to the first round, we apply a linear transformation to each 2 × 2 block of pixels in the encrypted image using the adjoint of matrix A that is used in encryption. Then we sort the elements of the transformed image using the indices in the key for that round, flatten the transformed image into a single vector, and sort the vector using the same indices. After that we reshape the sorted vector into a matrix with the same size as the original image and apply the modulo operation elementwise to ensure that all values are within the range 0–255. After all the rounds have been completed, the decrypted image converted to an unsigned 8-bit integer. For more information about the recent encryption, methods based on chaotic and as well hyperchaotic system, we refer the readers to the previous studies [2124].

6 Performance and security analysis

6.1 NPCR and UACI tests

Its well known that in cryptosystem, minor modifications are possible to the plain image, and to test the sensitivity, there are two measures, shortly denoted by UACI and NPCR. The equations for computing the NPCR and UACI are as follows, respectively:

NPCR = 1 M N i = 1 M j = 1 N dist ( i , j )

and

UACI = 1 255 × M N i = 1 M j = 1 N C 1 ( i , j ) C 2 ( i , j ) ,

where M and N are the rows and columns in the image, respectively. dist ( i , j ) is the difference between C 1 and C 2 , given by the following equation.

dist ( i , j ) = 0 , if C 1 ( i , j ) = C 2 ( i , j ) ; 1 if C 1 ( i , j ) C 2 ( i , j ) .

Table 1 displays the comparative results of UACI and NPCR.

Table 1

NPCR and UACI

PNCR UACI
99.6100 33.4674 Ref. [19]
99.2000 31.9249 Ref. [20]
99.5830 26.3126 Ref. [21]
99.5706 26.3570 Proposed system

6.2 Statistical analysis

Here, we will check the following statistical indices:

  1. Correlation analysis. By N p , we denote the neighboring pixels. In the usual images, N p values are very close to each other, and this means that connected pixels are highly correlated in the original images. For specialist, highly correlated feature can be used to break the cipher. In particular, in the cipher image, N p must be highly uncorrelated. The correlation coefficient between any two pixels is given in the following equation:

    C a b = i = 1 K t ( α i { α } ) ( β i { β } ) i = 1 K t ( α i { α } ) 2 i = 1 K t ( β i { β } ) 2 ,

    where { . } indicates the expected values of the random variables, α i and β i be the grayscale values of N p and K t is the total number of pixels taken for the calculation. In Figure 4(a) and (b), we show the correlation between the original image and the encrypted/decrypted image, respectively. The correlation coefficient for the original vs encrypted image is 0.002197 , while the correlation coefficient for the original vs decrypted image is 1. These results indicate that the encryption process has successfully scrambled the pixels of the original image, making it difficult to infer any information about the original image from the encrypted image. In contrast, the decryption process has successfully restored the original image from the encrypted image.

  2. Histogram and Chi-square test. To visual description to the distribution of pixel intensities in the image, we need to consider the histogram of an image, that gives a graphic view.

Note that original images usually have non-uniform histograms because pixel intensities are limited within some range. Its well known that this property can be used by cryptanalysts to intercept the cipher using histogram-based attacks. Thus, a secure encryption should produce cipher images with uniform histograms. Figures 5 and 6 show two grayscale images, namely Airplane and Peppers, along with their histograms with depicts the encrypted analogs of the original images along with their histograms. It can be observed that the original images exhibit non-uniform histograms while encrypted images have highly uniform distributions of pixel intensities. The Chi-square test by the following equation:

X 2 = i = 0 255 ( o i g ) 2 g ,

where o i indicates the frequency of occurrence of a particular pixel value and g = ( M N ) 256 . In the experiment process, the distribution is considered to be uniform when the Chi-square test is found to be more than a significance level μ ( μ [ 0 , 1 ] ) , and this way, the null hypothesis is accepted. Table 2 shows the success rate of the Chi-square test of different encryption schemes.

  1. Entropy information. One of the most important measure in dynamical systems theory is entropy, and the statistical test for calculating randomness in a sequence is defined as follows:

    H ( X ) = i = 1 K s p ( x i ) log ( p ( x i ) ) ,

    where X is he source, p ( x k ) is the probability of the element x k , and K c is the number of different elements generated by X . The ideal of entropy is to obtain when all pixel levels appear with an equal probability showing that all pixel uniformly distributed. In Table 3, we show different entropies considered in the literature.

Figure 4 
                  Distribution of connected pair pixels in image of airplane. (a) vertically and (b) diagonally.
Figure 4

Distribution of connected pair pixels in image of airplane. (a) vertically and (b) diagonally.

Figure 5 
                  (a) Original images, (b) its histogram, (c) ciphered image, and (d) its histogram.
Figure 5

(a) Original images, (b) its histogram, (c) ciphered image, and (d) its histogram.

Figure 6 
                  (a) Original images, (b) its histogram, (c) ciphered image, and (d) its histogram.
Figure 6

(a) Original images, (b) its histogram, (c) ciphered image, and (d) its histogram.

Table 2

X 2 test

Ref. [19] Ref. [20] Ref. [21] Proposed system
0.5113 0.3661 0.28155 0.25156
Table 3

Entropy information

Ref. [19] Ref. [20] Ref. [21] Proposed system
7.9993 7.98615 7.9970 7.9973

6.3 Keyspace analysis

For a good encryption process, the keyspace must be large enough to withstand bruteforce assaults. The suggested algorithm’s key consists of the starting values x , y , z , w as well as the parameters α , β , d , m , and r . Also, the operations of the proposed encryption algorithm are governed by parameters α , β , d , m , r and matrix A . In particular, the algorithm iterates for a specified number of rounds to enhance security. The keys in this algorithm are not explicitly generated; rather, they are derived from the parameters α , β , d , m , r , and A , which means that these parameters play a crucial role in determining the behavior of the algorithm and, consequently, the resulting encryption. The choice of values for these parameters significantly impacts the resulting ciphertext. For instance, larger values may lead to more complex transformations, potentially increasing the size of the keyspace. In addition, the matrix A introduces another layer of complexity, as its elements influence the linear transformation applied to the image data. As a result, when an attacker attempts to brute-force the key, they must account for the influence of A along with the other parameters. This increases the complexity of the encryption process and enlarges the keyspace, making it even more challenging for an attacker to deduce the correct key through trial and error.

6.4 Occlusion attack

It is well known that cipher images will loss data when it transferred through a communication channel. Completely or partially, the lost data can affect the decryption process. To measure the strength of the presented cryptosystem, the occlusion attack test is applied to the cipher images. Testing is conducted using 1/16, 1/8, 1 4 , and 1/2 data losses. We can compare Figure 7(a)--(d), which show the encrypted images with above mentioned losses, and Figure 7(e)–(h), which show analogous ciphered images.

Figure 7 
                  Occlusion attacks result.
Figure 7

Occlusion attacks result.

Visually, we can observe that most of the information is retained even after information of half of the encrypted image’s information is lost.

6.5 Noise attack

Particularly, while images goes through media channels that are often subjected to the noise. Obviously, this noise can affect the quality of the decrypted image if the corresponding cipher image is subjected to it. By contaminating the cipher image with salt and pepper nose where densities 0.005 , 0.05 , 0.1 , and 0.3 . The analogous decrypted images are shown if Figure 8(a)–(d).

Figure 8 
                  Salt and pepper noise. (a) 0.005, (b) 0.05, (c) 0.100, and (d) 0.300.
Figure 8

Salt and pepper noise. (a) 0.005, (b) 0.05, (c) 0.100, and (d) 0.300.

From the aforementioned figures, one can see that these images are noisy but readable.

7 Discussion

In this article, the complex dynamics of a four dimensional hyperchaotic system involved in two exponential terms was investigated. Ultimate bound of this system was obtained. Adaptive control of the new system was obtained. As an application to the proposed system, the proposed cryptosystem consists of several stages: first, some initial values are calculated by applying proposed equation in Section 5.1, where its parameters depending on the dimensions of the image to be encrypted, the initial values that were calculated previously were used to solve a system of differential equations in (3), and after that, the output indices of the differential equations will be the initial values that are used to rearrange the pixels of the input image. The resulting image is rebuilt in 2 matrices. Each matrix multiplies a secret matrix of dimension 2 × 2 . Also, the resulting matrix is converted into unit 8, which represents the encrypted image. At the end, throughout this study, we observed that the proposed system involved to sigmond function is making the system weaker. In particular, the other dynamics of the introduced system of this article are expected to be further studied.

  1. Funding information: This research received no external funding.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: Viet-Thanh Pham who is the co-author of this article, is a current Editorial Board member of Nonlinear Engineering. This fact did not affect the peer-review process. The authors declare no other conflict of interest.

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Received: 2023-01-15
Revised: 2023-11-06
Accepted: 2023-11-19
Published Online: 2024-02-02

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Editorial
  2. Focus on NLENG 2023 Volume 12 Issue 1
  3. Research Articles
  4. Seismic vulnerability signal analysis of low tower cable-stayed bridges method based on convolutional attention network
  5. Robust passivity-based nonlinear controller design for bilateral teleoperation system under variable time delay and variable load disturbance
  6. A physically consistent AI-based SPH emulator for computational fluid dynamics
  7. Asymmetrical novel hyperchaotic system with two exponential functions and an application to image encryption
  8. A novel framework for effective structural vulnerability assessment of tubular structures using machine learning algorithms (GA and ANN) for hybrid simulations
  9. Flow and irreversible mechanism of pure and hybridized non-Newtonian nanofluids through elastic surfaces with melting effects
  10. Stability analysis of the corruption dynamics under fractional-order interventions
  11. Solutions of certain initial-boundary value problems via a new extended Laplace transform
  12. Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method
  13. Fractional-order lead networks to avoid limit cycle in control loops with dead zone and plant servo system
  14. Modeling anomalous transport in fractal porous media: A study of fractional diffusion PDEs using numerical method
  15. Analysis of nonlinear dynamics of RC slabs under blast loads: A hybrid machine learning approach
  16. On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations
  17. Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation
  18. Influence of damage on large displacement buckling analysis of beams
  19. Approximate numerical procedures for the Navier–Stokes system through the generalized method of lines
  20. Mathematical analysis of a combustible viscoelastic material in a cylindrical channel taking into account induced electric field: A spectral approach
  21. A new operational matrix method to solve nonlinear fractional differential equations
  22. New solutions for the generalized q-deformed wave equation with q-translation symmetry
  23. Optimize the corrosion behaviour and mechanical properties of AISI 316 stainless steel under heat treatment and previous cold working
  24. Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena
  25. Investigation of the lubrication performance of a marine diesel engine crankshaft using a thermo-electrohydrodynamic model
  26. Modeling credit risk with mixed fractional Brownian motion: An application to barrier options
  27. Method of feature extraction of abnormal communication signal in network based on nonlinear technology
  28. An innovative binocular vision-based method for displacement measurement in membrane structures
  29. An analysis of exponential kernel fractional difference operator for delta positivity
  30. Novel analytic solutions of strain wave model in micro-structured solids
  31. Conditions for the existence of soliton solutions: An analysis of coefficients in the generalized Wu–Zhang system and generalized Sawada–Kotera model
  32. Scale-3 Haar wavelet-based method of fractal-fractional differential equations with power law kernel and exponential decay kernel
  33. Non-linear influences of track dynamic irregularities on vertical levelling loss of heavy-haul railway track geometry under cyclic loadings
  34. Fast analysis approach for instability problems of thin shells utilizing ANNs and a Bayesian regularization back-propagation algorithm
  35. Validity and error analysis of calculating matrix exponential function and vector product
  36. Optimizing execution time and cost while scheduling scientific workflow in edge data center with fault tolerance awareness
  37. Estimating the dynamics of the drinking epidemic model with control interventions: A sensitivity analysis
  38. Online and offline physical education quality assessment based on mobile edge computing
  39. Discovering optical solutions to a nonlinear Schrödinger equation and its bifurcation and chaos analysis
  40. New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation
  41. Study of weakly nonlinear double-diffusive magneto-convection with throughflow under concentration modulation
  42. Variable sampling time discrete sliding mode control for a flapping wing micro air vehicle using flapping frequency as the control input
  43. Error analysis of arbitrarily high-order stepping schemes for fractional integro-differential equations with weakly singular kernels
  44. Solitary and periodic pattern solutions for time-fractional generalized nonlinear Schrödinger equation
  45. An unconditionally stable numerical scheme for solving nonlinear Fisher equation
  46. Effect of modulated boundary on heat and mass transport of Walter-B viscoelastic fluid saturated in porous medium
  47. Analysis of heat mass transfer in a squeezed Carreau nanofluid flow due to a sensor surface with variable thermal conductivity
  48. Navigating waves: Advancing ocean dynamics through the nonlinear Schrödinger equation
  49. Experimental and numerical investigations into torsional-flexural behaviours of railway composite sleepers and bearers
  50. Novel dynamics of the fractional KFG equation through the unified and unified solver schemes with stability and multistability analysis
  51. Analysis of the magnetohydrodynamic effects on non-Newtonian fluid flow in an inclined non-uniform channel under long-wavelength, low-Reynolds number conditions
  52. Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems
  53. Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation
  54. Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering
  55. Solution of third-order nonlinear integro-differential equations with parallel computing for intelligent IoT and wireless networks using the Haar wavelet method
  56. Mathematical modeling and computational analysis of hepatitis B virus transmission using the higher-order Galerkin scheme
  57. Mathematical model based on nonlinear differential equations and its control algorithm
  58. Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation
  59. Space–time variable-order carbon nanotube model using modified Atangana–Baleanu–Caputo derivative
  60. Minimal universal laser network model: Synchronization, extreme events, and multistability
  61. Valuation of forward start option with mean reverting stock model for uncertain markets
  62. Geometric nonlinear analysis based on the generalized displacement control method and orthogonal iteration
  63. Fuzzy neural network with backpropagation for fuzzy quadratic programming problems and portfolio optimization problems
  64. B-spline curve theory: An overview and applications in real life
  65. Nonlinearity modeling for online estimation of industrial cooling fan speed subject to model uncertainties and state-dependent measurement noise
  66. Quantitative analysis and modeling of ride sharing behavior based on internet of vehicles
  67. Review Article
  68. Bond performance of recycled coarse aggregate concrete with rebar under freeze–thaw environment: A review
  69. Retraction
  70. Retraction of “Convolutional neural network for UAV image processing and navigation in tree plantations based on deep learning”
  71. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part II
  72. Improved nonlinear model predictive control with inequality constraints using particle filtering for nonlinear and highly coupled dynamical systems
  73. Anti-control of Hopf bifurcation for a chaotic system
  74. Special Issue: Decision and Control in Nonlinear Systems - Part I
  75. Addressing target loss and actuator saturation in visual servoing of multirotors: A nonrecursive augmented dynamics control approach
  76. Collaborative control of multi-manipulator systems in intelligent manufacturing based on event-triggered and adaptive strategy
  77. Greenhouse monitoring system integrating NB-IOT technology and a cloud service framework
  78. Special Issue: Unleashing the Power of AI and ML in Dynamical System Research
  79. Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme
  80. Special Issue: Nonlinear Analysis and Design of Communication Networks for IoT Applications - Part I
  81. Research on the role of multi-sensor system information fusion in improving hardware control accuracy of intelligent system
  82. Advanced integration of IoT and AI algorithms for comprehensive smart meter data analysis in smart grids
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