Attractor Research Articles

Design and FPGA implementation of nested grid multi-scroll chaotic system

Conventional multi-scroll chaotic systems are often constrained by the number of attractors and the complexity of generation, making it challenging to meet the increasing demands of communication and computation. This paper revolves around the modified Chua’s system. By modifying its differential equation and introducing traditional nonlinear functions, such as the step function sequence and sawtooth function sequence. A nested grid multi-scroll chaotic system (NGMSCS) can be established, capable of generating nested grid multi-scroll attractors. In contrast to conventional grid multi-scroll chaotic attractors, scroll-like phenomena can be initiated outside the grid structure, thereby revealing more complex dynamic behavior and topological features. Through the theoretical design and analysis of the equilibrium point of the system and its stability, the number of saddle-focused equilibrium points of index 2 is further expanded, which can generate (2 N+2) × M attractors, and the formation mechanism is elaborated and verified in detail. In addition, the generation of an arbitrary number of equilibrium points in the y-direction is achieved by transforming the x and y variables, which can generate M×(2 N+2) attractors, increasing the complexity of the system. The system’s dynamical properties are discussed in depth via time series plots, Lyapunov exponents, Poincaré cross sections, 0–1 tests, bifurcation diagrams, and attraction basins. The existence of attractors is confirmed through numerical simulations and FPGA-based hardware experiments.

Dynamic analysis of a novel hyperchaotic system based on STM32 and application in image encryption

This paper presents a novel 4D hyperchaotic system derived from a modified 3D Lorenz chaotic system. A key aspect of this system is the presence of a single equilibrium point, and its stability is carefully analyzed. The dynamic properties, including the Lyapunov exponent spectrum, bifurcation diagram, and chaotic attractors, are investigated using MATLAB simulations. The results reveal that the system displays hyperchaotic behavior across a wide range of the parameter d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d$$\\end{document}, with its chaotic attractor transitioning through four states: hyperchaotic, chaotic, periodic, and quasi-periodic, showcasing the system's complex dynamics. Experimental validation using STM32 embedded hardware successfully reproduces these four types of chaotic attractors, confirming the theoretical predictions. The proposed hyperchaotic system is then applied to image encryption, introducing a novel encryption method. The hyperchaotic key sequence generated by this system meets 15 tests of the NIST SP800-22 standard, and further experimental validation with STM32 hardware demonstrates the algorithm's effectiveness, simplicity, non-linearity, and high security. The encrypted images and sequences are rigorously tested key space analysis, histogram similarity analysis, information entropy analysis, statistical attack analysis, differential attack analysis, key sensitivity analysis, and correlation analysis, highlighting the robustness and reliability of the proposed system. This method is versatile and can be extended to various fields, including audio and video encryption, text encryption, IoT security, financial transaction security, and medical data protection.

Parametric controllable planar multi-scroll chaotic attractors in a 3-D memristive tabu learning single neuron model

In this paper, a three-dimensional (3D) autonomous tabu learning single neuron model is proposed, which is achieved by using a sinusoidal activation function and introducing a memristor synapse. This model exhibits the remarkable capability to produce a series of planar multi-scroll chaotic attractors, and its unique feature lies in the ability to control the number of scrolls. The investigation of the planar multi-scroll chaotic attractors and its dynamical behaviors is conducted through the analysis of phase plane portraits, bifurcation diagrams, and spectral entropies. The numerical simulations unveil a compelling relationship between the number of chaotic scrolls and specific control parameters governing the model. To further validate the findings, a 3D autonomous tabu learning single neuron model is implemented on a digital hardware platform. In an effort to extend the practical significance of this research, the multi-scroll chaotic phenomenon generated by the proposed model is deployed for image encryption. The fusion of mathematical modeling, digital hardware implementation, and practical application underscores the universality and significance of the proposed single neuron model in the fields of chaotic systems and engineering applications.

Dynamics of a two-neuron hopfield neural network: Memristive synapse and autapses and impact of fractional order

There are numerous studies on Hopfield neural networks with electromagnetic induction using memristors in either autaptic or synaptic connections. In this study, we explore a novel scenario where all connections are influenced by electromagnetic induction. We investigate and compare the network’s dynamics with one memristive autapse, two memristive autapses, and a memristive synapse. The results indicate that having two memristive autapses instead of one increases the dynamical range, leading to chaotic dynamics in unequal autaptic strengths. In contrast, in the presence of the memristive synapse, chaos can emerge only in very strong synaptic strength. Using fractional-order derivatives can transform the periodic attractor of the integer-order model into a chaotic one in some parameters. Furthermore, incorporating more memristors leads to chaos at lower fractional orders.

Control, bi-stability, and preference for chaos in time-dependent vaccination campaign.

In this work, effects of constant and time-dependent vaccination rates on the Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) seasonal model are studied. Computing the Lyapunov exponent, we show that typical complex structures, such as shrimps, emerge for given combinations of a constant vaccination rate and another model parameter. In some specific cases, the constant vaccination does not act as a chaotic suppressor and chaotic bands can exist for high levels of vaccination (e.g., >0.95). Moreover, we obtain linear and non-linear relationships between one control parameter and constant vaccination to establish a disease-free solution. We also verify that the total infected number does not change whether the dynamics is chaotic or periodic. The introduction of a time-dependent vaccine is made by the inclusion of a periodic function with a defined amplitude and frequency. For this case, we investigate the effects of different amplitudes and frequencies on chaotic attractors, yielding low, medium, and high seasonality degrees of contacts. Depending on the parameters of the time-dependent vaccination function, chaotic structures can be controlled and become periodic structures. For a given set of parameters, these structures are accessed mostly via crisis and, in some cases, via period-doubling. After that, we investigate how the time-dependent vaccine acts in bi-stable dynamics when chaotic and periodic attractors coexist. We identify that this kind of vaccination acts as a control by destroying almost all the periodic basins. We explain this by the fact that chaotic attractors exhibit more desirable characteristics for epidemics than periodic ones in a bi-stable state.

Stable attractors for neural networks classification via ordinary differential equations (SA-nODE)

Abstract A novel approach for supervised classification is presented which sits at the intersection of machine learning and dynamical systems theory. At variance with other methodologies that employ ordinary differential equations for classification purposes, the untrained model is a priori constructed to accommodate for a set of pre-assigned stationary stable attractors. Classifying amounts to steer the dynamics towards one of the planted attractors, depending on the specificity of the processed item supplied as an input. Asymptotically the system will hence converge on a specific point of the explored multi-dimensional space, flagging the category of the object to be eventually classified. Working in this context, the inherent ability to perform classification, as acquired ex post by the trained model, is ultimately reflected in the shaped basin of attractions associated to each of the target stable attractors. The performance of the proposed method is here challenged against simple toy models crafted for the purpose, as well as by resorting to well established reference standards. More precisely, we achieved an accuracy of 98.06 % on the MNIST test set and 88.21% on the Fashion MNIST test set. Although this method does not reach the performance of state-of-the-art deep learning algorithms, it illustrates that continuous dynamical systems with closed analytical interaction terms can serve as high-performance classifiers.

Режимы обобщенной синхронизации грубых генераторов широкополосного сигнала при наличии частотных помех в канале связи

Possibility of the onset of generalized synchronization regimes and these applications for communication in the unidirectionally coupled wide-band generators in a presence of a frequency distortion in a communication channel is studied. There are considered three options of these generators: 1) with a robust hyperbolic chaotic attractor; 2) with a non-robust non-hyperbolic chaotic attractor; 3) with a robust strange non-chaotic attractor. Results are provided for three significant cases of frequency distortion: 1) removals in the high-frequency range; 2) removals in the low-frequency range; 3) frequency removals in the neighborhood of the main carrier frequency. Advantages of the hyperbolic and the strange non-chaotic signals for data transmission are demonstrated.

Limit Cycles of the Three-dimensional Quadratic Differential System via Hopf Bifurcation

تم في هذه الدراسة النظر في النظام التفاضلي التربيعي ثلاثي الأبعاد، حيث يصبح نقطة الأصل الإحداثيات نقطة توازن هوبف. تم دراسة وجود واستقرار الدورات النهائية التي تنبثق من نقطة هوبف. يتم حساب معاملات ليبانوف المرتبطة بنقطة هوبف باستخدام طريقة الإسقاط. أولاً، تم تحديد أربع عائلات من شروط المعلمات التي من خلالها يمكن للنظام التفاضلي التربيعي ثلاثي الأبعاد أن يظهر البعد الثالث لتشعب هوبف. تم إعطاء الدليل التحليلي لكل مجموعة من الحالات المعلمية عن طريق حساب معاملات ليبانوف، تصفير لقيمة معاملات ليبانوف الأولى و الثاني و غيرالصفري لمعاملات ليبانوف الثالثة. تم تقديم الشروط الواضحة لوجودية واستقرارية ثلاث دورات النهائية ناشئة عن كل عائلة من تشعبات هوبف. يُظهر ناتج الوجود نقطة هوبف مستقرة (غير مستقرة)، مصحوبة بظهور دورتين حديتين مستقرتين (غير مستقرة) جنبًا إلى جنب مع دورة حدية واحدة غير مستقرة (مستقرة) في جوار النقطة الأصل غير المستقر (المستقر) لنظام المعادلة التربيعية ثلاثية الأبعاد. بالإضافة إلى ذلك، يتم استخدام النتيجة لاستكشاف الدورات النهائية لنظام الجذب الفوضوي n-scroll، والذي له العديد من الاستخدامات العملية، بما في ذلك الاتصال الآمن، والتشفير، وتوليد الأرقام العشوائية، والروبوتات المتنقلة المستقلة. تم اشتقاق الشروط التي بموجبها تصبح نقطة الأصل لهذا النظام هي نقطة هوبف، ويمكن أن توجد ثلاث دورات نهائية حول نقطة هوبف. وأخيرًا، توضح العروض العددية أن النظام يخضع لتشعب هوبف فوق الحرج، مما يؤدي إلى دورتين حديتين مستقرتين وواحدة غير مستقرة. وعلاوة على ذلك، تم التحقق من كافة النتائج.

Chaotic dynamic analysis of electrical contact resistance measured in sliding current-carrying friction

This work explores the dynamic laws of electrical contact resistance (ECR) signals measured in the sliding current-carrying friction tests based on chaos theory. It is found that ECR time series have a chaotic nature, strongly verified by the positive largest Lyapunov exponent λ1. The chaos may originate from the combined friction effects on short and long timescales. Phase space analysis indicates that the ECR phase trajectory adheres to the “convergence–stability” evolution law over time, which pertains closely to energy dissipation. Hence, the chaotic attractor exists during current-carrying friction. Strikingly, wear surfaces confirm that the ECR phase trajectory evolution process corresponds to the stages of “formation–keeping” of the chaotic attractor, and also agrees with the electrical wear stages of “running-in–steady-state”. Further, the chaos quantifier variation is consistent with the phase space analysis results to identify wear transition quantitatively.

Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model

We consider a coupling of the Stommel box model and the Lorenz model, with the goal of investigating the so-called crises that are known to occur given sufficient forcing. In this context, a crisis is characterized as the destruction of a chaotic attractor under a critical forcing strength. We document the variety of chaotic attractors and crises possible in our model, focusing on the parameter region where the Lorenz model is always chaotic and where bistability exists in the Stommel box model. The chaotic saddle collisions that occur in a boundary crisis are visualized, with the chaotic saddle computed using the Saddle-Straddle Algorithm. We identify a novel sub-type of boundary crisis, namely a vanishing basin crisis. For forcing strength beyond the crisis, we demonstrate the possibility of a merging between the persisting chaotic attractor and either a chaotic transient or a ghost attractor depending on the type of boundary crisis. An investigation of the finite-time Lyapunov exponents around crisis levels of forcing reveals a convergence between two near-neutral exponents, particularly at points of a trajectory most sensitive to divergence. This points to loss of hyperbolicity associated with crisis occurrence. Finally, we generalize our findings by coupling the Stommel box model to other strange attractors and thereby show that the behaviors are quite generic and robust.

Chaotic attractor reconstruction using small reservoirs—the influence of topology

Abstract Forecasting timeseries based upon measured data is needed in a wide range of applications and has been the subject of extensive research. A particularly challenging task is the forecasting of timeseries generated by chaotic dynamics. In recent years reservoir computing has been shown to be an effective method of forecasting chaotic dynamics and reconstructing chaotic attractors from data. In this work strides are made toward smaller and lower complexity reservoirs with the goal of improved hardware implementability and more reliable production of adequate surrogate models. We show that a reservoir of uncoupled nodes more reliably produces long term timeseries predictions than more complex reservoir topologies. We then link the improved attractor reconstruction of the uncoupled reservoir with smaller spectral radii of the resulting surrogate systems. These results indicate that, the node degree plays an important role in determining whether the desired dynamics will be stable in the autonomous surrogate system which is attained via closed-loop operation of the trained reservoir. In terms of hardware implementability, uncoupled nodes would allow for greater freedom in the hardware architecture because no complex coupling setups are needed and because, for uncoupled nodes, the system response is equivalent for space and time multiplexing.

Analysis of fractional solitary wave propagation with parametric effects and qualitative analysis of the modified Korteweg-de Vries-Kadomtsev-Petviashvili equation

This study explores the fractional form of modified Korteweg-de Vries-Kadomtsev-Petviashvili equation. This equation offers the physical description of how waves propagate and explains how nonlinearity and dispersion may lead to complex and fascinating wave phenomena arising in the diversity of fields like optical fibers, fluid dynamics, plasma waves, and shallow water waves. A variety of solutions in different shapes like bright, dark, singular, and combo solitary wave solutions have been extracted. Two recently developed integration tools known as generalized Arnous method and enhanced modified extended tanh-expansion method have been applied to secure the wave structures. Moreover, the physical significance of obtained solutions is meticulously analyzed by presenting a variety of graphs that illustrate the behaviour of the solutions for specific parameter values and a comprehensive investigation into the influence of the nonlinear parameter on the propagation of the solitary wave have been observed. Further, the governing equation is discussed for the qualitative analysis by the assistance of the Galilean transformation. Chaotic behavior is investigated by introducing a perturbed term in the dynamical system and presenting various analyses, including Poincare maps, time series, 2-dimensional 3-dimensional phase portraits. Moreover, chaotic attractor and sensitivity analysis are also observed. Our findings affirm the reliability of the applied techniques and suggest its potential application in future endeavours to uncover diverse and novel soliton solutions for other nonlinear evolution equations encountered in the realms of mathematical physics and engineering.

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.

Chaos in chains: Exploring a novel supply chain model through bifurcation analysis, multi-stability and complete synchronization via backstepping control

Chaotic systems are highly sensitive to initial conditions, meaning small changes can lead to significant outcomes. This sensitivity can be leveraged to enhance the flexibility and adaptability of supply chains, allowing them to respond quickly to changing market demands and conditions. In this research work, we obtain a New Chaotic Supply Chain Model (NCSCM) by introducing a sinusoidal model uncertainty in the third differential equation of the Hamidzadeh chaotic supply chain model (2023). The sinusoidal uncertainty in the model accounts for the volatility and fluctuations of the supply chain model. We investigate the dynamic behaviors of NCSCM by performing a detailed Bifurcation Diagram (BD) analysis and Lyapunov Exponents (LEs). Next, we carry out a 0−1 test for the NCSCM to confirm the chaotic attractor of the system. We also study the multi-stability properties of the NCSCM. Finally, with the help of backstepping control, we derive new results for the complete synchronization of the novel chaotic supply chain models.

The Lorenz system as a gradient-like system

We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for 0⩽ρ⩽12 when σ = 10 and β=8/3 , significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.

Formation of human kinship structures depending on population size and cultural mutation rate

How does social complexity depend on population size and cultural transmission? Kinship structures in traditional societies provide a fundamental illustration, where cultural rules between clans determine people's marriage possibilities. Here, we propose a simple model of kinship interactions that considers kin and in-law cooperation and sexual rivalry. In this model, multiple societies compete. Societies consist of multiple families with different cultural traits and mating preferences. These values determine interactions and hence the growth rate of families and are transmitted to offspring with mutations. Through a multilevel evolutionary simulation, family traits and preferences are grouped into multiple clans with interclan mating preferences. It illustrates the emergence of kinship structures as the spontaneous formation of interdependent cultural associations. Emergent kinship structures are characterized by the cycle length of marriage exchange and the number of cycles in society. We numerically and analytically clarify their parameter dependence. The relative importance of cooperation versus rivalry determines whether attraction or repulsion exists between families. Different structures evolve as locally stable attractors. The probabilities of formation and collapse of complex structures depend on the number of families and the mutation rate, showing characteristic scaling relationships. It is now possible to explore macroscopic kinship structures based on microscopic interactions, together with their environmental dependence and the historical causality of their evolution. We propose the basic causal mechanism of the formation of typical human social structures by referring to ethnographic observations and concepts from statistical physics and multilevel evolution. Such interdisciplinary collaboration will unveil universal features in human societies.

Early warning signals of complex critical transitions in deterministic dynamics

Early Warning Signals (EWS) have generated much excitement for their potential to anticipate transitions in various systems, ranging from climate change in ecology to disease staging in medicine. EWS hold particular promise for bifurcations, a transition mechanism in which a smooth, gradual change in a control parameter of the system results in a rapid change in system dynamics. The predominant reason to expect EWS is because many bifurcations are preceded by Critical Slowing Down (CSD): if assuming the system is subject to continuous, small, Gaussian noise, the system is slower to recover from perturbations closer to the transition. However, this focus on warning signs generated by stochasticity has overshadowed warning signs which may already be found in deterministic dynamics. This is especially true for higher-dimensional systems, where more complex attractors with intrinsic dynamics such as oscillations not only become possible—they are increasingly more likely. The present study focuses on univariate and multivariate EWS in deterministic dynamics to anticipate complex critical transitions, including the period-doubling cascade to chaos, chaos-chaos transitions, and the extinction of a chaotic attractor. In a four-dimensional continuous-time Lotka–Volterra model, EWS perform well for most bifurcations, even with lower data quality. The present study highlights three reasons why EWS may still work in the absence of CSD: changing attractor morphology (size, shape, and location in phase space), shifting power spectra (amplitude and frequency), and chaotic transitional characteristics (density across attractor). More complex attractors call for different warning detection methods to utilise warning signs already contained within purely deterministic dynamics.

A periodic split attractor reconstruction method facilitates cardiovascular signal diagnoses and obstructive sleep apnea syndrome monitoring

Electrocardiogram (ECG) is a powerful tool to detect cardiovascular diseases (CVDs) and health conditions. We proposed a new method for evaluating ECG for efficient medical diagnosis in daily life. By splitting the signal according to the cardiac activity cycle, the periodic split attractor reconstruction (PSAR) method is proposed with time embedding, including three types of splitting methods to show its chaotic domain characteristics. We merged the CVDs dataset and the obstructive sleep apnea syndrome (OSAS) first-lead ECG signal dataset to validate the performance of PSAR for diagnosis and health monitoring using PSAR density maps as SE-ResNet input features. PSAR under 3 split methods showed different sensitivities for different CVDs. While in OSAS monitoring, PSAR showed good ability to recognize sleep abnormalities.

Crypto-steganographic model using chaos and coding based in deoxyribonucleic acid

Given the increase of information circulating through public channels, it is essential to create robust schemes to ensure the security of such information. The results presented here were part of the research project entitled computer security models based on mathematical tools and artificial intelligence. An algorithm focused on the encryption of images carrying steganographed texts is proposed, using chaos, artificial vision and coding based in deoxyribonucleic acid (DNA). The process consists of steganographic and cryptographic steps. In the steganographic stage, a color image was taken, the combined Canny and Sobel filters were applied to achieve its dilated edges, using Chen's chaotic attractor, the positions of the edges were selected, to hide a text in binary ASCII code using the least significant bit technique. In the encryption stage, Chen's chaotic system was used to permute the stego-image and to create a chaotic image used in the diffusion process. These two images were divided into blocks represented in DNA coding, selecting the rule to apply through the three-dimensional Logistics system, and finally applying the XOR operation by layers, obtaining a single encrypted image. To validate the proposed model, safety and performance tests were applied, obtaining comparable indicators with some current scientific references.

A secure communication scheme based on generalized modified projective synchronization of a new 4-D fractional-order hyperchaotic system

This paper focuses on the exploration of the chaotic behavior of a new 4-D fractional-order hyperchaotic system with five nonlinearities. The Adomian decomposition method is employed to solve the fractional-order hyperchaotic system. The stability of equilibrium points in this system is analyzed. Through bifurcation diagrams, Lyapunov exponent spectra, chaotic attractors, 0-1 test, C 0 complexity, and spectral entropy, the chaotic dynamics of the proposed system are investigated. Using these tools, we demonstrate the fractional-order system’s sensitivity to variation in both the derivative order and initial conditions. Moreover, a modified generalized projective synchronization is developed to implement chaos synchronization between two coupled fractional-order hyperchaotic systems. Furthermore, this work presents an application of synchronization scheme in secure communication. Numerical simulations were implemented to validate the effectiveness of the proposed secure communication scheme.