Two-dimensional Coupled Map Lattices Research Articles

Spatiotemporal Patterns in a Space–Time Discrete SIRS Epidemic Model with Self- and Cross-Diffusion

In this work, a two-dimensional Coupled Map Lattice (CML) method is applied to a continuous Reaction–Diffusion (RD) SIRS epidemic model. The model involves a complex, nonlinear incidence rate and the spatially heterogeneous self- and cross-diffusion coefficients. The CML method applied here results in a space–time discrete counterpart of the model. The basic reproduction number is derived, and the local stability of the fixed points of the discrete model is established. The two typical bifurcations of codimension 1, such as the flip and the Neimark–Sacker (NS) bifurcations, are investigated by utilizing the center manifold theory and the normal form theorem. Furthermore, the codimension-2 fold–flip bifurcation about the endemic fixed point is discussed in detail. Turing instability criterion of the cross-diffusion epidemic model is established by utilizing the CML method under periodic boundary conditions. Finally, the numerical simulations clarify both the effectiveness of all theoretical findings and the effects of time step size and cross-diffusion on the spatiotemporal patterns. The main contribution of this work is the discovery of some interesting Turing and non-Turing patterns which may be induced by flip–Turing instability, NS–Turing instability or fold–flip–Turing instability. Concretely speaking, spiral patterns are observed near the flip bifurcation threshold value and ultimately evolve into an irregular defect pattern. Stripe pattern, labyrinth pattern and circular pattern are observed near the NS bifurcation threshold value. The disorderly interwoven ribbon pattern, mosaic pattern and some irregular oscillatory patterns are observed near the fold–flip bifurcation point. Our results greatly help in predicting the long-term behavior of the space–time discrete SIRS epidemic model under study.

An efficient image encryption scheme integrating chaotic keystream generator with S-box and triangular block scrambling

A novel image encryption scheme has been developed to effectively meet the growing need for image security and privacy. This scheme combines a chaotic keystream generator with S-box and triangular block scrambling to provide efficient encryption. The keystream generator consists of a two-dimensional coupled map lattice (2DCML), linear interval transformation (LIT) and an S-box. By using a 2DCML, two initial chaotic sequences are generated and used to construct an S-box. Then, a final chaotic sequence is obtained after undergoing LIT and S-box processing. The proposed algorithm applies two rounds of scrambling operations and two rounds of substitutions. During the scrambling processes, global scrambling and triangular block scrambling are combined. Diffusion operations use an S-box while incorporating a key policy linked with image content to enhance resistance against chosen-plaintext attacks. The simulation outcomes, coupled with the security analysis, substantiate the efficacy and robust encryption performance of the proposed image encryption scheme.

Pattern dynamics analysis of a time-space discrete FitzHugh-Nagumo (FHN) model based on coupled map lattices

This paper investigates the dynamics of a discrete FitzHugh-Nagumo (FHN) model with self-diffusion on two-dimensional coupled map lattices. The primary objective is to analyze the complex dynamics of neuronal systems in a discrete setting. Through the application of central manifold and normal form analysis, it has been demonstrated that the system is capable of undergoing Neimark-Sacker and flip bifurcations even in the absence of diffusion. Additionally, when influenced by diffusion, the system can manifest pure Turing instability, Neimark-Sacker-Turing instability, and Flip-Turing instability. In the numerical simulation section, the path from bifurcation to chaos is explored by calculating the maximum Lyapunov exponent and drawing the bifurcation diagram. The interconversion between various Turing instabilities is simulated by varying the values of the time step and the self-diffusion coefficient. This study contributes to a deeper understanding of the complexity of neural network systems.

Complex patterns in a space–time discrete mathematical model of antibiotic resistance in hospitals with self-diffusion

In this work, we extend an antibiotic resistance mathematical model in hospitals spatially on a two-dimensional coupled map lattice. The complex dynamics of the resultant space–time discrete model are investigated. An invasion reproduction number Rar and two control parameters Rsc and Rrc for sensitive bacteria and resistant bacteria, respectively, are defined. The three numbers play an important role in the existence and stability of fixed points. According to the normal form theorem and the center manifold theory, we show that the discrete model exhibits both the flip and the Turing bifurcations. We prove that the existence of the interior fixed point excludes the possibility of the occurrence of a Neimark–Sacker bifurcation. Numerical simulations, including phase diagrams, Lyapunov exponents, bifurcation diagrams, and complex patterns, are enrolled to verify the analyses. The complex patterns induced here are either a result of the pure Turing instability, or the flip instability (non-Turing), while no Turing-flip patterns are formed. The results of the current study show that the space–time discrete version of the model of antibiotic resistance in hospitals captures complex and rich nonlinear dynamics and adds to our understanding of the complicated pattern formation of the model.

Design and Analysis of Hash Function Based on Two-dimensional Integer Chaotic Map

<p>Combining two-dimensional coupled map lattice and dynamic integer tent map, a hash function construction scheme with variable length output is proposed. The dynamic integer tent map is used as the nonlinear function of two-dimensional coupled map lattice, and dynamic parameters are added to the two-dimensional coupled map lattice, the bit logic decision function of the dynamic integer tent map is used to control the change of the dynamic parameters. At the same time, the dynamic parameters are evolved by the cyclic displacement mode, which enhances the correlation between the plaintext difference and the parameters. The test results show that this hash function has strong security, simple implementation, and can be used as an ideal algorithm to replace the traditional hash function.</p> <p> </p>

Spatiotemporal chaos in two-dimensional dynamic coupled map lattices system based on elementary cellular automata

Spatiotemporal chaos in two-dimensional dynamic coupled map lattices system based on elementary cellular automata

Performance of the 2D Coupled Map Lattice Model and Its Application in Image Encryption

The two‐dimensional coupled map lattice (2D CML) model has been extensively employed as the basis component for designing various schemes in the cryptography system due to its complicated chaotic dynamic behavior. In this study, we analyze the chaotic characteristics of the 2D CML model, such as the Lyapunov exponent (LE), synchronization stability, bifurcation, and ergodicity. We then show that the chaotic sequences generated by the 2D CML model are random according to the NIST testing. Furthermore, we propose an image encryption scheme based on the 2D CML model and Singular Value Decomposition (SVD). In our scheme, the SVD method is used to reduce the image storage, and the Red, Green, and Blue channels of a color image will be encrypted through confusion and diffusion. The simulation results, as well as the results of the comparison with other schemes, demonstrate that our scheme possesses outstanding statistics, excellent encryption performance, and high security. It has great potential for ensuring the security of digital images in real applications.

A Novel Compressive Image Encryption with an Improved 2D Coupled Map Lattice Model

The digital image, as the critical component of information transmission and storage, has been widely used in the fields of big data, cloud and frog computing, Internet of things, and so on. Due to large amounts of private information in the digital image, the image protection is fairly essential, and the designing of the encryption image scheme has become a hot issue in recent years. In this paper, to resolve the shortcoming that the probability density distribution (PDD) of the chaotic sequences generated in the original two-dimensional coupled map lattice (2D CML) model is uneven, we firstly proposed an improved 2D CML model according to adding the offsets for each node after every iteration of the original model, which possesses much better chaotic performance than the original one, and also its chaotic sequences become uniform. Based on the improved 2D CML model, we designed a compressive image encryption scheme. Under the condition of different keys, the uniform chaotic sequences generated by the improved 2D CML model are utilized for compressing, confusing, and diffusing, respectively. Meanwhile, the message authentication code (MAC) is employed for guaranteeing that the encryption image be integration. Finally, theoretical analysis and simulation tests both demonstrate that the proposed image encryption scheme owns outstanding statistical, well encryption performance, and high security. It has great potential for ensuring the digital image security in application.

A new image encryption algorithm based on\xa0the OF-LSTMS and chaotic sequences

In this paper, a novel image encryption algorithm based on the Once Forward Long Short Term Memory Structure (OF-LSTMS) and the Two-Dimensional Coupled Map Lattice (2DCML) fractional-order chaotic system is proposed. The original image is divided into several image blocks, each of which is input into the OF-LSTMS as a pixel sub-sequence. According to the chaotic sequences generated by the 2DCML fractional-order chaotic system, the parameters of the input gate, output gate and memory unit of the OF-LSTMS are initialized, and the pixel positions are changed at the same time of changing the pixel values, achieving the synchronization of permutation and diffusion operations, which greatly improves the efficiency of image encryption and reduces the time consumption. In addition the 2DCML fractional-order chaotic system has better chaotic ergodicity and the values of chaotic sequences are larger than the traditional chaotic system. Therefore, it is very suitable to image encryption. Many simulation results show that the proposed scheme has higher security and efficiency comparing with previous schemes.

A stream cipher algorithm based on 2D coupled map lattice and partitioned cellular automata

The two-dimensional coupled map lattice (2D CML) is a spatiotemporal chaotic model with complex dynamic behavior and has high potential for designing stream cipher. We propose an offset 2D CML model by adding different offsets for each lattice. The offset 2D CML model has better chaotic properties, such as larger Lyapunov exponent (LE) and more uniform chaotic sequences, than the original 2D CML model, which provides a good basis for constructing stream cipher. We combine the offset 2D CML model with the partitioned cellular automata (PCA) and propose a stream cipher algorithm. In our algorithm, the PCA is used to control the extraction of pseudo-random number from the offset 2D CML model, which effectively hides the orbit information of system and enhances the difficulty of attacking chaotic sequences. Moreover, some fast nonlinear transform operations are specially introduced into our algorithm to further improve the complexity and the running speed. Theory analysis and simulation test both confirm that the proposed stream cipher algorithm has excellent statistical performance, high security and computational efficiency. It has great potential for guaranteeing data security in the Internet.

A new image encryption algorithm based on two-dimensional spatiotemporal chaotic system

In this paper, a new image encryption algorithm based on the two-dimensional spatiotemporal chaotic system is proposed.This system mixes linear neighborhood coupling and the nonlinear chaotic map coupling of lattices, and it has more cryptographic features in dynamics than the system of coupled map lattices does. The two-dimensional coupled map lattices (2DCML) system is only a special case in this chaotic system. In addition, bit-level permutation is employed to strengthen security of the cryptosystem. Simulations have been carried out, and the results demonstrate that the proposed algorithm has properties of large key space, high sensitivity to key, strong resisting attack. So, it is more secure and effective algorithm for encryption of digital images.

Dependence of Initial Value on Pattern Formation for a Logistic Coupled Map Lattice.

The logistic coupled map lattices (LCML) have been widely investigated as well as their pattern dynamics. The patterns formation may depend on not only fluctuations of system parameters, but variation of the initial conditions. However, the mathematical discussion is quite few for the effect of initial values so far. The present paper is concerned with the pattern formation for a two-dimensional Logistic coupled map lattice, where any initial value can be linear expressed by corresponding eigenvectors, and patterns formation can be determined by selecting the corresponding eigenvectors. A set of simulations are conducted whose results demonstrate the fact. The method utilized in the present paper could be applied to other discrete systems as well.

THE INTERPLAY OF UNCORRELATED NOISE AND SMALL WORLD EFFECT ON PATTERN FORMATION IN TWO-DIMENSIONAL COUPLED MAP LATTICES

By using a neuron-like map model to denote the generic dynamics of excitable systems, Gaussian-noise-induced pattern formation in the two-dimensional coupled map lattices with nearest-neighbor coupling and shortcut links has been studied. Given the appropriate initial values and parameter regions, with all nodes concerned, the functions of δ(n), χ and ℜ are introduced to analyze the evolution of pattern formation. It is found that there exists a critical εc beyond which the stable rotating spiral wave will appear. After introducing the Gaussian noise for the homogeneous ε region, different spatiotemporal stable patterns will be achieved. Additionally, the importance of the parameter I on the coupling strength C is discussed.

Spatio-temporal dynamics and quantification of daisyworld in two-dimensional coupled map lattices

We spatially extend the daisyworld model on a two-dimensional toroidal coupled map lattice (CML – a generalisation of cellular automata). We investigated whether this tightly coupled system of local nonlinear dynamics with bi-directional life-environment feedback can generate a specific kind of behaviour, characterised by global stability coexisting with local instability. We introduce appropriate metrics to measure the spatio-temporal dynamics of the daisyworld system. Specifically, we evaluate spatial autocorrelation using Moran's I, and local and global temporal fluctuation through the permutation entropy and the temporal standard deviation. We categorise a range of different behaviours that can arise in such scenarios, and relate them through a parameter analysis.

Turing instability for a two-dimensional Logistic coupled map lattice

In this Letter, stability analysis is applied to a two-dimensional Logistic coupled map lattice with the periodic boundary conditions. The conditions of Turing instability are obtained, and various patterns can be exhibited by numerical simulations in the Turing instability region. For example, space–time periodic structures, periodic or quasiperiodic traveling wave solutions, stationary wave solutions, spiral waves, and spatiotemporal chaos, etc. have been observed. In particular, the different pattern structures have also been observed for same parameters and different initial values. That is, pattern structures also depend on the initial values. The similar patterns have also been seen in relevant references. However, the present Letter owes to pattern formation via diffusion-driven instabilities because the system is stable in the absence of diffusion.

Anti-synchronization of nonlinear-coupled spatiotemporal chaotic systems

The anti-synchronization of two nonlinear-coupled spatiotemporal chaotic systems is discussed. A special nonlinear-coupled term is constructed through suitably separating the spatiotemporal chaotic systems to linear and nonlinear terms, and anti-synchronization of two two-dimensional coupled map lattices is realized. The nonlinear-coupled method is further generalized to anti-synchronize the complex networks composed of two-dimensional coupled map lattices. The artificial simulation results show that this method is still effective.

One-way hash function construction based on 2D coupled map lattices

An algorithm for constructing one-way hash function based on spatiotemporal chaos is proposed. A two-dimensional coupled map lattices (2D CML) with parameters leading to the largest Lyapunov exponent is employed. The state of the 2D CML is dynamically determined by its previous state and the message bit at the corresponding positions. The hash value is obtained by a linear transform on the final state of the 2D CML. Theoretical analysis and computer simulation indicate that our algorithm has good statistical properties, strong collision resistance and high flexibility. It is practical and reliable, with high potential to be adopted as a strong hash function for providing data integrity.

INHERENT GLOBAL STABILIZATION OF UNSTABLE LOCAL BEHAVIOR IN COUPLED MAP LATTICES

The behavior of two-dimensional coupled map lattices is studied with respect to the global stabilization of unstable local fixed points without external control. It is numerically shown under which circumstances such inherent global stabilization can be achieved for both synchronous and asynchronous updating. Two necessary conditions for inherent global stabilization are derived analytically.

Characterizing spontaneous irregular behavior in coupled map lattices

Two-dimensional coupled map lattices display, in a specific parameter range, a stable phase (quasi-) periodic in both space and time. With small changes to the model parameters, this stable phase develops spontaneous eruptions of non-periodic behavior. Although this behavior itself appears irregular, it can be characterized in a systematic fashion. In particular, parameter-independent features of the spontaneous eruptions may allow useful empirical characterizations of other phenomena that are intrinsically hard to predict and reproduce. Specific features of the distributions of lifetimes and emergence rates of irregular states display such parameter-independent properties.

Local dimension and finite time prediction in coupled map lattices

Forecasting, for obvious reasons, often become the most important goal to be achieved. For spatially extended systems (e.g. atmospheric system) where the local nonlinearities lead to the most unpredictable chaotic evolution, it is highly desirable to have a simple diagnostic tool to identify regions of predictable behaviour. In this paper, we discuss the use of the bred vector (BV) dimension, a recently introduced statistics, to identify the regimes where a finite time forecast is feasible. Using the tools from dynamical systems theory and Bayesian modelling, we show the finite time predictability in two-dimensional coupled map lattices in the regions of low BV dimension.