One-dimensional Maps Research Articles

Synchronization transitions in coupled q-deformed logistic maps

The concept of q-deformation has been expanded in a variety of situations including q-deformed nonlinear maps. Coupled q-deformed logistic maps with positive and negative q-diffusive coupling are investigated. The transition to state of synchronized fixed point in the thermodynamic limit for random initial conditions is studied as dynamic phase transitions. Though rare for one-dimensional maps, q-deformed maps show multistability. For weak coupling, the synchronized state may not be observed in the thermodynamic limit with random initial conditions even if it is linearly stable. Thus multistability persists. However, for large lattices with random initial conditions, five well-defined critical points are observed where the transition to a synchronized state occurs. Two of these show continuous phase transition. One of them is in the directed percolation universality class. In another case, the order parameter shows power-law decay superposed with oscillations on a logarithmic scale at the critical point and does not match with known universality classes.

[formula omitted]D-CS: A circularly shifting chaotic map generation method

Due to more complex behavior and larger number of control parameters, high-dimensional chaotic map generation methods may provide more satisfactory performance in various practical applications compared to low-dimensional counterparts. Designing n-dimensional (nD) chaotic map generation methods for generating chaotic maps in arbitrary dimensions with desired dynamics is an interesting but challenging task. In this paper, we propose a universal framework called the nD circularly shifting chaotic map generation method (nD-CS), which utilizes existing one-dimensional chaotic maps as seed maps to generate nD chaotic maps with complex and robust behaviors. Theoretical analysis proves that when specific criteria are met by its control parameters, our nD-CS can exhibit complex and robust hyperchaotic behavior in the sense of Lyapunov exponent. To demonstrate the effectiveness of our nD-CS, we first employ it to generate three examples of three-dimensional hyperchaotic maps. The results indicate high-performance indicators of these new chaotic maps. Furthermore, we compare the nD chaotic maps generated by our nD-CS to those produced by existing methods. The results demonstrate that our chaotic maps have a better overall performance.

A trio of maps: chaos on an ellipse, a shark’s fin, and Sluze’s pearls

Three one-dimensional maps are presented. The maps are analysed with a combination of Feigenbaum diagrams, the calculation of global Lyapunov exponents, and the evaluation of fixed points. It is shown that in certain cases period two orbits can be evaluated by careful observation of the form of the related pairs of equations. A range of classroom exercises are provided to enable students to investigate the maps further.

Analysis of Local Bifurcations in One-Dimensional Systems

This article delves deeper into the study of local bifurcation in maps, although only in the context of a single dimension. The various local bifurcation types common to one-dimensional maps each have their own specific conditions that must be fulfilled. An object divides into two when it bifurcates. After a bifurcation, a family of one-parameter functions retains its stable or cyclic point structure. A bifurcation in the iterative process happens when a parameter is altered, causing a qualitative change in behavior. The phenomena of local splitting can be induced by adjusting just one parameter. Both the norm form of the transcritical split and the stability of the saddle node are highlighted. As with supercritical and subcritical bifurcations, the stability of the pitchfork bifurcation is assessed, as is its norm form. Jagannath University Journal of Science, Volume 10, Number II, Dec. 2023, pp. 108-123

Surgical gestures can be used to assess surgical competence in robot-assisted surgery

To collect validity evidence for the assessment of surgical competence through the classification of general surgical gestures for a simulated robot-assisted radical prostatectomy (RARP). We used 165 video recordings of novice and experienced RARP surgeons performing three parts of the RARP procedure on the RobotiX Mentor. We annotated the surgical tasks with different surgical gestures: dissection, hemostatic control, application of clips, needle handling, and suturing. The gestures were analyzed using idle time (periods with minimal instrument movements) and active time (whenever a surgical gesture was annotated). The distribution of surgical gestures was described using a one-dimensional heat map, snail tracks. All surgeons had a similar percentage of idle time but novices had longer phases of idle time (mean time: 21 vs. 15 s, p < 0.001). Novices used a higher total number of surgical gestures (number of phases: 45 vs. 35, p < 0.001) and each phase was longer compared with those of the experienced surgeons (mean time: 10 vs. 8 s, p < 0.001). There was a different pattern of gestures between novices and experienced surgeons as seen by a different distribution of the phases. General surgical gestures can be used to assess surgical competence in simulated RARP and can be displayed as a visual tool to show how performance is improving. The established pass/fail level may be used to ensure the competence of the residents before proceeding with supervised real-life surgery. The next step is to investigate if the developed tool can optimize automated feedback during simulator training.

A relative investigation of one-dimensional chaotic maps intended for light-weight cryptography in smart grid

Technology is quite important in today's contemporary society, especially when it comes to information transmission. The free and open interchange of data via networks is essential for the development of communication technologies. As a result, it is essential to ensure the security of transmitted data, especially digital data like readings from smart meters in smart grids. As a result, the study of chaos-based cryptography has received a great deal of interest in the realm of data security. Due to its favourable characteristics, including ergodicity, pseudo-randomness, non-periodicity, and sensitivity to beginning circumstances, chaotic maps have become a popular choice for data encryption. Researchers have concentrated on factors like the highest Lyapunov exponent and sensitivity to beginning circumstances while evaluating these maps. The MATLAB platform has been used to run simulations that produce bifurcation diagrams and calculate Lyapunov exponents in order to analyse and compare various chaotic maps. Further research is required since these experiments have shown considerable improvements in the chaotic behaviour of one-dimensional maps over time. There are two key areas covered by the review. On the basis of their bifurcation diagrams and Lyapunov exponents, it compares newly presented maps in the first place. It then looks at the techniques used to combine one-dimensional chaotic maps. The work shows the potential of chaotic maps for cryptography applications and emphasises the growing significance of data security in the digital age.

Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions

UDC 517.9 Motivated by the recent work by Ma and Magal [Proc. Amer. Math. Soc. (2021); https://doi.org/10.1090/proc/15629] on the global stability property of the Gurtin–MacCamy's population model, we consider a family of scalar nonlinear convolution equations with unimodal nonlinearities. In particular, we relate the Ivanov and Sharkovsky analysis of singularly perturbed delay differential equations in [https://doi.org/10.1007/978-3-642-61243-5_5] with the asymptotic behavior of solutions of the Gurtin–MacCamy's system. According the classification proposed in [https://doi.org/10.1007/978-3-642-61243-5_5], we can distinguish three fundamental kinds of continuous solutions of our equations, namely, solutions of the asymptotically constant type, relaxation type and turbulent type. We present various conditions assuring that all solutions belong to the first of these three classes. In the setting of unimodal convolution equations, these conditions suggest a generalized version of the famous Wright's conjecture.

IMAGE ENCRYPTION TECHNIQUE BASED ON A BINARY COMBINATION OF MULTIPLE CHAOTIC MAPS AND DNA SEQUENCE OPERATIONS

The huge advance of digital communication and networks has led to enormous storage and transmission of information over public networks. Nevertheless, the assurance of information security remains incomplete across these unsecured networks. Digital images are the primary means for sharing information over open networks. Consequently, the confidentiality of digital images during storage and transmission has become a crucial concern, particularly when sharing sensitive information. Image encryption has emerged as a solution to this problem. This paper presents an image encryption technique based on multiple one-dimensional chaotic maps and DNA coding. The method employs three one-dimensional chaotic maps, including the Logistic map, Tent map, and piecewise map, multiple times to produce 18 random sequences with different initial values and parameters. SHA-512 hash function is used to indicate the initial values of chaotic maps. For encrypting images, the binary elements from various sequences of chaotic maps are amalgamated to alter the pixel intensities of the image in the diffusion process. Dynamic DNA coding is performed through a random selection of DNA rules and operations (XOR, XNOR, and Addition) to each pixel in the image. The technique is enforced using circular rotations which are applied randomly to each key. The proposed technique is evaluated using many standard images. Different performance metrics have been measured. The empirical findings illustrate the security and resilience of the suggested method and its ability to resist statistical and differential attacks.

A Monte Carlo approach to understanding the impacts of initial-condition uncertainty, model uncertainty, and simulation variability on the predictability of chaotic systems: Perspectives from the one-dimensional logistic map.

The predictability of the logistic map is investigated for the joint impact of initial-condition (IC) and model uncertainty (bias + random variability) as well as simulation variability. To this end, Monte Carlo simulations are carried out where IC bias is varied in a wide range of 10-15-10-3, and, similarly, model bias is introduced in comparable range. It is found that while the predictability limit of the logistic map can be continuously extended by reducing IC bias, the introduction of the model bias imposes an upper limit to the predictability limit beyond which further reductions in IC bias do not lead to an extension in the predictability limit, effectively restricting the feasible joint space spanned by the IC-model biases. It is further observed that imposing a lower limit to the allowed variability among ensemble solutions (so as to prevent the ensemble variability from collapse) results in a similar constraint in the joint IC-model-bias space; but this correspondence breaks down when the imposed variability limit is too high (∼x>0.7 for the logistic map). Finally, although increasing the IC random variability in an ensemble is found to consistently extend the allowed predictability limit of the logistic map, the same is not observed for model parameter random variability. In contrast, while low levels of model parameter variability have no impact on the allowed predictability limit, there appears to be a threshold at which an abrupt transition occurs toward a distinctly lower predictability limit.

BCLM: a novel chaotic map for designing cryptography-based security mechanism for IEEE C37.118.2 PMU communication in smart grid

Abstract Today, creating a smart grid that is resistant to cyberattacks is a subject of utmost significance. One of the components of the smart grid that is most susceptible to a cyber-attack is the phasor measuring unit (PMU). The reason is that PMU employs IEEE C37.118.2 communication standards, which specify the structure and sequencing of data packets but offer no security measures. Users must implement the security techniques to ensure the protection of PMU data. Additionally, PMU communicates via a public wide-area network, raising the risk to security. In addition, PMU is a crucial component of the smart grid, enabling different crucial choices for the reliable functioning of the smart grid to be made using its data. This research suggests a chaos-based data encryption solution to close the knowledge gap and reduce the confidentiality assault on PMU data. For this, a brand-new boost converter logarithmic map (BCLM), a one-dimensional (1D) chaotic map, has been presented. The research demonstrates how the suggested chaotic map has better chaotic qualities than conventional chaotic maps. The pseudorandom generator is the chaotic BCLM system. The PMU data are encrypted using the random sequence produced by the BCLM chaotic system. The suggested chaotic map is not computationally demanding, making it simple to implement in a PMU device with limited resources.

Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models

We demonstrate a basic technique for simplifying time-periodic competition models, which is based on the utilization of periodic delta functions as population growth rates. We show that the Poincare mapping splits into a sequence of one-dimensional mappings. The study of the corresponding stable equilibria allows us to make conclusions concerning the coexistence and selection of the family of competitors. In particular, in “all vs. all” systems, for one of the populations to dominate, it is enough to surpass the others with a certain margin, and the correspondent stock constant does not depend on the number of competitors. We present paradoxical examples, where (1) a low-productive population can displace a highly productive one, (2) the displacement is non-transitive, (3) the coexistence is non-transitive. We also show how the delta functions can be utilized for the analysis of a “predator–prey” system.

Enhancing logistic chaotic map for improved cryptographic security in random number generation

The use of chaotic maps for cryptography is advantageous due to their desirable characteristics, including complexity, unpredictability, sensitivity to initial values and control parameters, and ergodicity. However, most chaos-based systems have limitations that make them impractical, such as low efficiency, insufficient security, and small keyspace. To overcome these challenges, this paper proposes an enhanced logistic map (ELM), a one-dimensional digital chaotic map based on the logistic map and anterior perturbation method. The ELM map leverages perturbations of chaotic states, leading to enhanced randomness, data distribution, and low correlation, with improved security properties. The ELM map is simple yet capable of generating large chaotic behavior. The findings show that the ELM map is highly secure and can be used in various cryptographic applications. A new pseudorandom number generator (PRNG) is also proposed based on a combination of ELM chaotic maps. The resulting PRNG is highly dependent on the chaotic properties of the maps and the combination rule. Statistical analysis shows that the proposed method is statistically secure, highlighting the potential of the ELM map for other cryptographic applications.

Lyapunov exponents and extensivity of strongly coupled chaotic maps in regular graphs

In Thermodynamics and Statistical Physics, a system’s property is extensive when it grows with the system size. When it happens, the system can be decomposed into separate components, which has been done in many systems with weakly interacting components, such as for various gas models. Similarly, Ruelle conjectured 40 years ago that the Lyapunov exponents (LEs) of some sufficiently large chaotic systems are extensive, which led to study the extensivity properties of chaotic systems with strong interactions. Because of the complexities in these systems, most results achieved so far are restricted to numerical simulations. Here, we derive closed-form expressions for the LEs and entropy rate of coupled maps in finite- and infinite-sized regular graphs, according to the coupling strength, map’s chaoticity, and graph’s spectral properties. We show that this type of system has either 4 or 5 cases for the LEs, depending on the graph’s extreme Laplacian eigenvalues. These cases represent qualitatively different collective behaviours emerging in parameter space, including chaotic synchronisation (N−1 negative LEs) and incoherent chaos (N positive LEs). From the entropy rate, we show that the ring and complete graphs (nearest-neighbour and all-to-all couplings, respectively) are extensive in all parameter regions outside the chaotic synchronisation region. Although our derivations are restricted to one-dimensional maps with constant positive derivative (i.e., chaotic), our approach can be used to find LE and entropy rates for other regular graphs (such as for cyclic graphs) or be the basis for tackling small world graphs via perturbative methods.

Chaotic Phenomenal and One-dimensional Logistic Map

Introducing the basic properties of logistic map. The paper includes the proofing of the stability of fixed points in different domain of and determining the property of period doubling of Logistic map, describing Chaos phenomenon in logistic map by showing the stability of fixed points in different intervals of , researching 3-period limit cycle in the Logistic Map bifurcation diagram, showing a special characteristic of the logistic map on the region on the bifurcation diagram that exist 3-period limit cycles, collecting the different value of at one, two, four, eight, sixteen and thirty-two pitchfork points from the Logistic Map Bifurcation diagram to calculate Feigenbaum constant and proving the Feigenbaum can also be used to determine the pitchfork bifurcation speed of logistic map when r is between zero to negative two. Showing the basic properties of the Mandelbrot Set. Showing the result of computer simulation in real part to the Mandelbrot Set to determine the relationship between the Logistic Map bifurcation diagram and Mandelbrot Set.

Period addition cascade in a model of neuron-glial interaction

We analyze a system of four differential equations that describe the dynamics of a neuron-glial network using the mean field approximation [1, 2]: (τ=–E+α ln(1+e1/α(JuxE+I0)), =(1–x)/τD–uxE,(1) =U(y)–u/τF+U(y)(1–u)E, =–y/τy+βσ(x), where E(t) is the average activity, x(t) is the fraction of available neurotransmitter released into the synaptic gap with a probability u(t); y(t) is the fraction of the gliatransmitter released by the astrocyte. Sigmoidal functions U(y) and σ(x) correspond to changes in the base probability level u(t) during the release of the gliatransmitter and activation of astrocytes during neurotransmitter release, respectively. The input inhibitory current corresponds to a bifurcation parameter with a negative value of I0 0, while the remaining parameters have positive and fixed values. The rest of the parameters are positive and fixed. For a detailed description of the model, including information on the type of functions and parameter values, refer to works [1, 2]. For a constant value of U(x)=const, the first three equations in system (1) represent the Tsodyks–Markram model, which explains the short-term synaptic plasticity phenomenon [1]. The model was enhanced with a fourth equation for y in [2], incorporating the influence of astrocytes via the concept of a tripartite synapse [3]. Model (1) illustrates a wide range of dynamic behavior including quiescence, regular tonic activity, and chaotic bursting activity. These behaviors correspond to various sets in the phase space, such as stable equilibrium states, limit cycles of period 1, limit cycles of any period n∈N, and chaotic attractors. Changing the I0 l parameter causes sets to bifurcate, resulting in the loss of stability of certain attractors and the emergence of others, leading to a shift in the dynamic regime. Therefore, in terms of dynamics, the conditions for bifurcation and the characteristics of the newly formed attractors are crucial. In this presentation, we have obtained a series of numerical bifurcations in system (1) that correspond to the shift from tonic activity to burst activity, resulting in subsequent modifications to the bursts. Specifically, our findings demonstrate that an increase in the number of spikes per burst is determined by a period adding cascade where the aforementioned limit cycle of period n becomes unstable, allowing for a previously established stable cycle of period n+1 to occupy the position of the “main” attractor. This process culminates in the vanishing of the orbit with an endless period due to the saddle-node bifurcation of cycles, followed by the creation of a dependable cycle with a period of 1. The main properties of the cascade were reproduced in our model one-dimensional piecewise-smooth map z¯=1−z6, forz0,μ−1−μ(z-1)6, forz0, where z∈R1, μ is a bifurcational parameter. The map’s results suggest that an increase in current I0 i in model (1) may lead to the emergence and disappearance of quasi-strange attractors (quasi-attractors), implying chaotic behavior in connection with burst variation.

An innovative orthogonal matrix based on nonlinear chaotic system for compressive sensing

Compressed sensing exploits the signal’s sparsity by non-uniform sampling to achieve high-quality signal reconstruction at low sampling rates. This work aims to show the efficient performance of chaotic binary orthogonal matrices (CBOM) in compressed sensing. The nonlinear, high dimensional, irregular, and high complexity properties of chaotic systems can provide more diverse and efficient ways of sampling and reconstructing signals. The CBOM construction method proposed in this paper is divided into two steps, in the first step, the real-valued sequence of a one-dimensional chaotic map is binarised using the proposed Threshold-Matching Symbol Algorithm (TMSA) to obtain a chaotic binary sequence (CBS). The i.i.d properties of the CBS were proved using the Perron–Frobenius operator and the properties of the joint probability. The binarized CBS conditionally preserves the pseudo-random of chaotic sequences, as evidenced by the derivation of the well-distribution measure and the k-order correlation measure. In the second step, the binarised sequence CBS was split and orthogonalized to construct CBOM, which satisfies low storage and correlation. We prove that CBOM obeys the Restricted Isometric Condition (RIP). The orthogonalization of the matrix will further reduce matrix column correlation and improve the quality of the reconstruction. Numerical simulation results show that the proposed matrix has considerable sampling efficiency, comparable to Gaussian and partial Hadamard matrices, close to the theoretical limit. Meanwhile, the generation and reconstruction time of the proposed matrix is smaller than other matrices. Our framework covers partial one-dimensional chaotic maps, including Chebyshev, Tent, Logistic, and so on. We can easily apply this paradigm to various fields.

Secure key real-time update and dynamic DNA encryption for CO-OFDM-PON based on a hybrid 5-D chaos.

A double key (DK) real-time update and hybrid five-dimensional (5-D) hyperchaotic deoxyribonucleic acid (DNA) dynamic encryption scheme is proposed, which can ensure the security in the orthogonal frequency division multiplexing passive optical network (OFDM-PON). Chaotic sequences for DNA dynamic encryption are produced using a four-dimensional (4-D) hyperchaotic Lü system and a one-dimensional (1-D) logistic map. In this scheme, the DK consists of an external key set, which is stored locally, and an internal key, which is associated with the plaintext and external key. In addition, a pilot cluster is used as the carrier of key transmission and key embedding is achieved by converting key to phase information of the pilot. To verify the feasibility of the scheme, a simulation validation is performed on a 46.5Gb/s 16 quadrature amplitude modulation (QAM) coherent OFDM-PON system transmitted over an 80 km transmission distance. The results show that the proposed scheme can improve the security performance of OFDM-PON at a low OSNR cost of 0.3 dB and the key space is expanded to (8.514 × 10102)S. When the correlation redundancy (CR) G⩾7, the 0 bit error rate (BER) of key can be achieved and the key can be updated and distributed in real-time without occupying additional secure channels.

Synchronization and firing patterns of coupled one-dimensional neuron maps

Synchronization and firing patterns of coupled one-dimensional neuron maps

Exploiting one-dimensional exponential Chebyshev chaotic map and matching embedding for visually meaningful image encryption

As a result of their prominent initial value sensitivity, pseudo-randomness, and unpredictability nature, chaotic maps are widely used in multimedia security. However, the majority of chaotic maps now in use have issues like narrow parameter scope, poor traversability, and insufficient chaotic performance. To address these issues, a one-dimensional exponential Chebyshev (1-DEC) chaotic map is constructed in this paper which performs satisfactorily. Through the use of a bifurcation diagram, Lyapunov exponents, sample entropy, and permutation entropy, the chaotic dynamics of the 1-DEC are thoroughly examined. Subsequently, a visually meaningful image encryption (VMIE) scheme combined with matching embedding based on 1-DEC was presented. Specifically, a cyclic shift confusion method across rows and columns (CSCARC) is proposed to eliminate the correlation of adjacent pixels. Then, an adaptive embedding method based on image energy and dynamic matching (AEIEDM) is proposed to fuse the privacy information into carrier images. In addition, secret image share and authentication information is introduced in this scheme, which improves disaster-tolerance performance and implements the distributed management and storage of data. Subscribers have the capabilities of authentication, management, and data integrity checking. Ultimately, the simulation results and analysis demonstrate that the proposed scheme possesses satisfactory performance in terms of security and robustness, and the visual quality of the steganographic images was above 50.9 dB.

Classification and rapid non-destructive quality evaluation of different processed products of Cyperus rotundus based on near-infrared spectroscopy combined with deep learning

The quality of traditional Chinese medicine is very important for human health, but the traditional quality control method is very tedious, which leads to the substandard quality of many traditional Chinese medicine. In order to solve the problem of time-consuming and laborious traditional quality control methods, this study takes traditional Chinese medicine Cyperus rotundus as an example, a comprehensive strategy of near-infrared (NIR) spectroscopy combined with One-dimensional convolutional neural network (1D-CNN) and chaotic map dung beetle optimization (CDBO) algorithm combined with BP neural network (BPNN) is proposed. This strategy has the advantages of fast and non-destructive. It can not only qualitatively distinguish Cyperus rotundus and various processed products, but also quantitatively predict two bioactive components. In classification, 1D-CNN successfully distinguished four kinds of processed products of Cyperus rotundus with 100 % accuracy. Quantitatively, a CDBO algorithm is proposed to optimize the performance of the BPNN quantitative model of two terpenoids, and compared with the BP, whale optimization algorithm (WOA)-BP, sparrow optimization algorithm (SSA)-BP, grey wolf optimization (GWO)-BP and particle swarm optimization (PSO)-BP models. The results show that the CDBO-BPNN model has the smallest error and has a significant advantage in predicting the content of active components in different processed products. To sum up, it is feasible to use near infrared spectroscopy to quickly evaluate the effect of processing methods on the quality of Cyperus rotundus, which provides a meaningful reference for the quality control of traditional Chinese medicine with many other processing methods.