Complex Chaotic Behavior Research Articles

A low-time-consumption image encryption combining 2D parametric Pascal matrix chaotic system and elementary operation

The rapid development of the big data era has resulted in traditional image encryption algorithms consuming more time in handling the huge amount of data. The consumption of time cost needs to be reduced while ensuring the security of encryption algorithms. With this in mind, the paper proposes a low-time-consumption image encryption (LTC-IE) combining 2D parametric Pascal matrix chaotic system (2D-PPMCS) and elementary operation. First, the 2D-PPMCS with robustness and complex chaotic behavior is adopted. Second, the SHA-256 hash values are applied to the chaotic sequences generated by 2D-PPMCS, which are processed and applied to image permutation and diffusion encryption. In the permutation stage, the pixel matrix is permutation encrypted based on the permutation matrix generated from the chaotic sequences. For diffusion encryption, elementary operations are utilized to construct the model, such as exclusive or, modulo, and arithmetic operations (addition, subtraction, multiplication, and division). After analyzing the security experiments, the LTC-IE algorithm ensures security and robustness while reducing the time cost consumption.

A Four-Dimensional No-Equilibrium Chaotic System with Multi-Scroll Chaotic Hidden Attractors and its Application in Image Encryption

Abstract In recent years, constructing hidden attractors with multi-scroll has become a key discussion point in the research and application fields of chaos science. In this paper, with the existing four-dimensional (4D) chaotic system as the base, a new four-dimensional chaotic system featuring significant characteristics of multi-scroll hidden attractors is constructed by adding a nonlinear function. Comprehensive studies including theoretical analyses and numerical simulations have been carried out on the dynamic properties of the new chaotic system, and all the results show that this system exhibits extremely complex chaotic behaviours and excellent unpredictability, which has great value in image encryption. Therefore, an image encryption scheme based on the new chaotic system is proposed, which cleverly integrates the new scrambling algorithm based on parity coordinate transformation and the new rotational diffusion algorithm. And the effectiveness of this encryption algorithm has been thoroughly analyzed and tested. The results based on the experiments show that this encryption algorithm exhibits significant advantages in performance, which can greatly enhance the security of images during encryption and transmission.

Design and Analysis of a Novel Fractional-Order System with Hidden Dynamics, Hyperchaotic Behavior and Multi-Scroll Attractors

The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system’s order is extended beyond integer values, providing a richer dynamic behavior. The system’s hidden dynamics are revealed through detailed numerical simulations and theoretical analysis, demonstrating complex attractors and bifurcations. The hyperchaotic nature of the system is verified through Lyapunov exponents and phase portraits, showing multiple positive exponents that indicate a higher degree of unpredictability and complexity. Additionally, the system’s multi-scroll attractors are analyzed, showcasing their potential for secure communication and encryption applications. The fractional-order approach enhances the system’s flexibility and adaptability, making it suitable for a wide range of practical uses, including secure data transmission, image encryption, and complex signal processing. Finally, based on the proposed fractional-order system, we designed a simple and efficient medical image encryption scheme and analyzed its security performance. Experimental results validate the theoretical findings, confirming the system’s robustness and effectiveness in generating complex chaotic behaviors.

An image encryption scheme without additional key transmission based on an N-dimensional closed-loop coupled triangular wave model

Currently, many image encryption schemes either require additional secure channels for key transmission or employ a key that exhibits a lack of correlation with the plaintext, thereby reducing the security of the algorithm. This study proposes a novel image encryption scheme to address this issue. A new N-dimensional closed-loop coupled triangular wave model is developed to overcome the limitations of the sine function in chaotic systems while reducing the difficulty of the chaotic system design. Experimental analysis demonstrates that the generated chaotic systems exhibit complex chaotic behaviors and outperform other chaotic systems designed using the sine function. Specifically, we utilize difference extension technology to embed key information into the plaintext image and design a scrambling and diffusion algorithm based on Latin rectangles to encrypt the remaining pixel matrix without key information. Finally, the pixel matrix containing the key information is concealed to obtain the final ciphertext image. Simulation analysis and security evaluation demonstrate that the proposed algorithm outperforms the encryption performance and security of several advanced image encryption algorithms.

Intermittent-like synchronization and desynchronization phenomena in a Colpitts network model.

This study investigates the dynamics of a modified Colpitts oscillator, exhibiting complex periodic and chaotic behaviors. Our research explores the dynamics and synchronization of coupled chaotic Colpitts oscillators, crucial for understanding their potential applications and behaviors. The main discovery is the emergence of a phase in which the systems achieve either complete synchronization or desynchronization. This behavior depends on the values of the coupling parameter. The subsequent challenge involves understanding how the coupling parameter influences the emergence of this synchronization phenomenon.

A lightweight symmetric image encryption cryptosystem in wavelet domain based on an improved sine map

In the era of big data, the number of images transmitted over the public channel increases exponentially. As a result, it is crucial to devise the efficient and highly secure encryption method to safeguard the sensitive image. In this paper, an improved sine map (ISM) possessing a larger chaotic region, more complex chaotic behavior and greater unpredictability is proposed and extensively tested. Drawing upon the strengths of ISM, we introduce a lightweight symmetric image encryption cryptosystem in wavelet domain (WDLIC). The WDLIC employs selective encryption to strike a satisfactory balance between security and speed. Initially, only the low-frequency-low-frequency component is chosen to encrypt utilizing classic permutation and diffusion. Then leveraging the statistical properties in wavelet domain, Gaussianization operation which opens the minds of encrypting image information in wavelet domain is first proposed and employed to all sub-bands. Simulations and theoretical analysis demonstrate the high speed and the remarkable effectiveness of WDLIC.

Image encryption algorithm based on ElGamal cryptography and selective random diffusion

Some current image encryption schemes are independent of the plaintext, leading to a vulnerability to the chosen-plaintext attack under symmetric structure. To address this issue, an asymmetric image encryption algorithm with plaintext correlation is suggested, after analyzing various image encryption schemes using chaotic systems. First, a three-dimensional New Logistic-Sine Map (NewLSM) is designed by coupling Logistic map and Sine map, considering them as seed maps. Analyses prove that the NewLSM has a wider continuous chaotic interval and more complex chaotic behavior than seed maps. Secondly, to enhance the keystream associated with plaintext, a new key acquisition model is constructed, i.e., NewMKG, by combining the hash function SHA-3 with the public key ElGamal cryptography. Then, employing the NewLSM and the NewMKG, a new image encryption algorithm with asymmetric structure is presented using classical framework of confusion-diffusion. In particular, a new diffusion method is proposed after confusion, namely, selective random diffusion (NewSRD). Moreover, experiments and analyses indicate that a good performance can be reached for the proposed algorithm. For example, the value of information entropy for the obtained cipher image is close to eight.

Reliability and robustness of oscillations in some slow-fast chaotic systems.

A variety of nonlinear models of biological systems generate complex chaotic behaviors that contrast with biological homeostasis, the observation that many biological systems prove remarkably robust in the face of changing external or internal conditions. Motivated by the subtle dynamics of cell activity in a crustacean central pattern generator (CPG), this paper proposes a refinement of the notion of chaos that reconciles homeostasis and chaos in systems with multiple timescales. We show that systems displaying relaxation cycles while going through chaotic attractors generate chaotic dynamics that are regular at macroscopic timescales and are, thus, consistent with physiological function. We further show that this relative regularity may break down through global bifurcations of chaotic attractors such as crises, beyond which the system may also generate erratic activity at slow timescales. We analyze these phenomena in detail in the chaotic Rulkov map, a classical neuron model known to exhibit a variety of chaotic spike patterns. This leads us to propose that the passage of slow relaxation cycles through a chaotic attractor crisis is a robust, general mechanism for the transition between such dynamics. We validate this numerically in three other models: a simple model of the crustacean CPG neural network, a discrete cubic map, and a continuous flow.

A Simple Chaotic Model With Complex Chaotic Behaviors and Its Hardware Implementation

Recently, due to the initial sensitivity, ergodicity, boundedness, and unpredictability of chaos, it has been widely used in the fields of secure communication, signal processing, chaotic synchronization, etc. Moreover, due to the low complexity of hardware implementation, widespread attention to one-dimensional (1D) chaotic maps has been paid from researchers. However, some existing 1D maps have simple chaotic behaviors and discrete chaotic intervals, which have significantly affected the practical application. To address these shortcomings, this paper proposes a 1D chaotic model based on the absolute value function and the cosine function called the cosine-coupled chaotic model (CCCM). It can use existing 1D chaotic maps as seed maps to construct a series of 1D chaotic maps with continuous chaotic intervals and complex chaotic behaviors. To verify the effectiveness of the CCCM, we first generate three new chaotic maps using the CCCM, and then their chaotic properties are analyzed by using various measures including Lyapunov exponent (LE), sample entropy (SE), correlation dimension (CD), 0-1 test and sensitivity. The experimental results confirm that the new chaotic maps have wider chaotic intervals and more complex chaotic behaviors than the corresponding seed maps and some other advanced chaotic maps. Then we use a field programmable gate array (FPGA) as the hardware platform for simple hardware implementations of the new chaotic maps. And a new pseudo-random number generator (PRNG) is introduced to verify practical applications of the new maps. Finally, experimental results show that the new chaotic maps can be implemented on FPGA and the proposed PRNG has strong random properties.

Regular and Chaotic Oscillations in a Geostrophic Flow with Vertical Shear

In the framework of a two-level quasi-geostrophic model, the stability of flow with a constant vertical shear is investigated. Analytical expressions for the increment of perturbation growth in linear stability theory were obtained. The Galerkin method with three basic Fourier harmonics was used to describe the nonlinear dynamics of perturbations. A nonlinear system of ordinary differential equations is formulated for amplitudes of Fourier harmonics. It is shown that in the absence of bottom friction all solutions of the system describe periodic mode of nonlinear oscillations or vascillations. The situation changes fundamentally in the model with bottom friction. In this case, for a wide range of parameter values, the system solutions exhibit complex chaotic behavior. Thus, chaos or turbulence emerges for large-scale motions.

Hydrodynamics of an odd active surfer in a chiral fluid

We theoretically and computationally study the low-Reynolds-number hydrodynamics of a linear active microswimmer surfing on a compressible thin fluid layer characterized by an odd viscosity. Since the underlying three-dimensional fluid is assumed to be very thin compared to any lateral size of the fluid layer, the model is effectively two-dimensional. In the limit of small odd viscosity compared to the even viscosities of the fluid layer, we obtain analytical expressions for the self-induced flow field, which includes non-reciprocal components due to the odd viscosity. On this basis, we fully analyze the behavior of a single linear swimmer, finding that it follows a circular path, the radius of which is, to leading order, inversely proportional to the magnitude of the odd viscosity. In addition, we show that a pair of swimmers exhibits a wealth of two-body dynamics that depends on the initial relative orientation angles as well as on the propulsion mechanism adopted by each swimmer. In particular, the pusher–pusher and pusher–puller-type swimmer pairs exhibit a generic spiral motion, while the puller–puller pair is found to either co-rotate in the steady state along a circular trajectory or exhibit a more complex chaotic behavior resulting from the interplay between hydrodynamic and steric interactions. Our theoretical predictions may pave the way toward a better understanding of active transport in active chiral fluids with odd viscosity, and may find potential applications in the quantitative microrheological characterization of odd-viscous fluids.

An Image Encryption Transmission Scheme Based on a Polynomial Chaotic Map.

Most existing chaotic systems have many drawbacks in engineering applications, such as the discontinuous range of chaotic parameters, weak chaotic properties, uneven chaotic sequence outputs, and dynamic degradation. Therefore, based on the above, this paper proposes a new method for the design of a three-dimensional chaotic map. One can obtain the desired number of positive Lyapunov exponents, and can also obtain the desired value of positive Lyapunov exponents. Simulation results show that the proposed system has complex chaotic behavior and high complexity. Finally, the method is implemented into an image encryption transmission scheme and experimental results show that the proposed image encryption scheme can resist brute force attacks, correlation attacks, and differential attacks, so it has a higher security.

Image encryption algorithm based on 2D‐CLICM chaotic system

AbstractImage encryption algorithm based on a two‐dimensional Chebyshev‐Logistic‐Infinite Collapse Map (2D‐CLICM) is proposed. A new two‐dimensional chaotic map is designed, and numerical studies show that 2D‐CLICM has a larger chaotic region, better randomness and more complex chaotic behavior. An image encryption algorithm (CLICM‐IE) is developed based on the 2D‐CLICM. The randomly generated key is used to generate more random chaotic sequences and the security is greatly improved. To obtain lower computation and time complexity, the whole process performs row encryption for each image pixel. The Arnold transformation preprocesses the image and then the encryption of image pixels is accomplished by controlling the random confusion and diffusion parts with the sequence generated by 2D‐CLICM. The results show that the CLICM‐IE algorithm is deeply random and unpredictable, and resists a variety of attacks. These results can provide good basis and practical application for the chaos generation and image encryption.

An Improved Absolute-Cosine Chaotification Model and Its Simple Application in PRNG

Chaotic systems are widely used in various aspects such as information security, signal processing, and synchronous control. The structural complexity and the chaotic behavior of chaotic systems are two significant factors affecting their practical applications. In this paper, we propose a universal two-dimensional (2D) absolute-cosine chaotic model (ACCM). The 2D-ACCM is composed of a nonlinear bounded cosine function and an absolute value function. It can construct new chaotic maps with simple structures and complex chaotic behaviors on the basis of existing chaotic systems. To verify the effectiveness of the proposed system, we first choose two existing one-dimensional (1D) chaotic maps and one existing 2D chaotic map as the seed maps of the 2D-ACCM to generate two new maps, respectively. The results of chaotic behavior analysis show that these two new maps have more complex chaotic behavior and wider chaotic ranges than seed maps and some advanced chaotic maps. Then a hardware experiment platform based on a field-programmable gate array (FPGA) is used for the hardware implementation of the new maps. Finally, a simple chaos-based pseudo-random number generator (PRNG) is introduced to show the practical application. The experimental results show that the new maps can be easily implemented on the FPGA and the chaos-PRNGs can generate pseudo-random numbers with excellent randomness.

Magneto-Aero-Elastic Superharmonic and Subharmonic Resonances, Bifurcations and Chaos of Conductive Spinning Circular Plates

This paper focuses on the magneto-aero-elastic nonlinear resonances, bifurcations and chaos for superharmonic and subharmonic responses of conductive circular plates spinning in air and magnetic fields. The electromagnetic forces acting on in-plane are presented by using electromagnetic field theory, and a rotational dissipative damping aerodynamic model is employed to characterize the aerodynamic forces acting on the plates. Considering the effects of the geometric nonlinearity and centrifugal tensioning, the magneto-aero-elastic governing equations of spinning conductive circular plates are derived by the Hamiltonian principle. Based on the mode shapes with Bessel functions, the transverse nonlinear vibration differential equations of spinning plates with clamped-edge are derived by the Galerkin method. Amplitude-frequency equations for subharmonic and superharmonic responses are obtained by using the approach of multiple scales, respectively. Numerical calculations indicate the influences of important parameters (e.g. magnetic field intensity, excitation amplitude and rotational speed) on amplitude frequency characteristics. Moreover, it is demonstrated by the numerical results that system responses are sensitive to the variation of these important parameters, where the complex intermittency, period-doubling and chaotic dynamical behaviors exist.

A Novel Hyperchaotic 2D-SFCF with Simple Structure and Its Application in Image Encryption.

In this paper, a novel image encryption algorithm is proposed based on hyperchaotic two-dimensional sin-fractional-cos-fractional (2D-SFCF), called sin-fractional-cos-fractional image-encryption (SFCF-IE). The 2D-SFCF is constructed from two one-dimensional cosine fractional (1-DCFs), and it has a more complex chaotic behavior with a larger parameter space than one-dimensional chaotic systems. Compared with the two-dimensional (2D) chaotic system, the 2D-SFCF has a simple structure, and the parameter space in the chaotic state is continuous, which is beneficial to generating the keystream in the cryptosystem. Therefore, in the novel image encryption algorithm, we use the 2D-SFCF to generate the keystream of the cryptosystem. The encryption algorithm is a process of scrambling and diffusion. Different from common diffusion methods, the diffusion starting position of the SFCF-IE is randomly generated, enhancing the algorithm’s security. Simulation experiments show that the image encrypted by this algorithm has better distribution characteristics and can resist common attack methods.

Multi-Image Compression-Encryption Algorithm Based on Compressed Sensing and Optical Encryption.

In order to achieve large-capacity, fast and secure image transmission, a multi-image compression–encryption algorithm based on two-dimensional compressed sensing (2D CS) and optical encryption is proposed in this paper. Firstly, the paper uses compressed sensing to compress and encrypt multiple images simultaneously, and design a new structured measurement matrix. Subsequently, double random phase encoding based on the multi-parameter fractional quaternion Fourier transform is used to encrypt the multiple images for secondary encryption, which improves the security performance of the images. Moreover, a fractional-order chaotic system with more complex chaotic behavior is constructed for image compression and encryption. Experimental results show that the algorithm has strong robustness and security.

Dynamic analysis and image encryption application of a sinusoidal-polynomial composite chaotic system

Due to complex properties of ergodicity, non-periodic ability and sensitivity to initial states, chaotic systems are widely used in cryptography. In this paper, we propose a sinusoidal-polynomial composite chaotic system (SPCCS), and prove that it satisfies Devaney’s definition of chaos: the sensitivity to initial conditions, topological transitivity and density of periodic points. The experimental results show that SPCCS has better unpredictability and more complex chaotic behavior than the classical chaotic maps. Furthermore, we provide a new image encryption algorithm combining pixel segmentation operation, block chaotic matrix confusing operation and pixel diffusion operation with the proposed SPCCS. The detailed simulation results verify superiority and effectiveness of the proposed image encryption algorithm.

Nonlinear characteristics of a multi-degree-of-freedom wind turbine’s gear transmission system involving friction

In this study, a 42-degree-of-freedom (42-DOF) translation–torsion coupling dynamic model of the wind turbine’s compound gear transmission system considering time-varying meshing friction, time-varying meshing stiffness, meshing damping, meshing error and backlash is proposed. Considering the different meshing between internal and external teeth of planetary gear, the time-varying meshing stiffness is calculated by using the cantilever beam theory. An improved meshing friction model takes into account the mixed elastohydrodynamic lubrication to calculate the time-varying meshing friction. The bifurcation diagram of the excitation frequency is utilized to analyze the bifurcation and chaos characteristics of the system. Meanwhile, the nonlinear dynamic characteristics of the gear system are identified by using the time domain diagrams, phase diagrams, Poincare maps and amplitude–frequency spectrums of the gear system. The results show that the system has complex bifurcation and chaotic behaviors including periodic, quasi-periodic, chaotic motion. The bifurcation characteristics of the system become complicated and the chaotic region increases considering the effects of friction in the high-frequency region.

A Fast Image Encryption Algorithm Based on Improved 6-D Hyper-Chaotic System

Aiming at the security and efficiency of image transmission, a fast image encryption algorithm based on an improved 6-D chaotic system is proposed. Firstly, we design a hyper-chaotic system with more complex chaotic behavior, analyze the Lyapunov exponential spectrum, chaotic attractor and randomness of the system, and generate random sequences through randomness enhancement operation. Secondly, image preprocessing is used to select pixels from the original image to form a thumbnail, the size of the key space can be changed by adjusting the thumbnail. Thirdly, the hash value of the original image is used as the initial values of the hyper-chaotic system to realize the uniqueness of the key. The row encryption matrix and column encryption matrix are generated according to the maximum and minimum values of row and column pixels in the thumbnail. These two encryption matrices are composed of the full arrangement of random sequences, which refers to the random combination of random sequences in a certain order. Before the encryption, the Arnold transformation is performed on the original image and then the cipher image is obtained by row encryption and column encryption respectively. The experimental results illustrate that the proposed algorithm has excellent security performance, robustness and the speed of encryption and decryption is very fast.