Attractor Research Articles

A Hybrid Algorithms Based on the Aizawa Attractor and Rabbit-Lightweight Cipher for Image Encryption

Social media and networks rely heavily on images. Those images should be distributed in a private manner. Image encryption is therefore one of the most crucial components of cyber security. In the present study, an effective image encryption technique is developed that combines the Rabbit Algorithm, a simple algorithm, with the Attractor of Aizawa, a chaotic map. The lightweight encryption algorithm (Rabbit Algorithm), which is a 3D dynamic system, is made more secure by the Attractor of Aizawa. The process separates color images into blocks by first dividing them into bands of red, green, and blue (RGB). The presented approach generates multiple keys, or sequences, based on the initial parameters and conditions, which are then utilized for encrypting each one of the blocks. The peak signal-to-noise ratio (PSNR), Mean Squared Error (MSE), and structural similarity index measure (SSIM) tests are among performance tests that are used to assess the effectiveness of the approach. With the help of National Institute of Standards and Technology (NIST), the produced keys have been checked for randomness. Additionally, research is being done into how long decryption and encryption take. The encryption approach is particularly secure because of the chaotic attractor's unpredictability. The correlation coefficient, Shannon entropy, and a histogram are used to analyse the encryption technique's performance. The findings demonstrate that the suggested encryption approach dependably operates since it makes use of a mechanism to strengthen security against the attack of known or selected plain-text. The outcomes also demonstrated that both decoding and encoding using the suggested encoding approach require less time. For instance, images 3's decoding time is 0.002000, whereas images 3's encoding time is 0.001200.

Hamilton energy of a complex chaotic system and offset boosting

The complex differential system can be obtained by introducing complex variable in the real differential system. Complex variables can be decomposed into real component and imaginary component, which makes the complex differential systems have more complex dynamic behaviors. Complex chaotic system is used in secure communications to increase the security of cryptographic systems. In this study, we designed a complex differential system by incorporating a complex variable into a 3D differential system. Dynamics of this complex differential system are investigated by applying typical nonlinear analysis tools. Furthermore, Hamilton energy function for complex differential system is obtained based on Helmholtz’s theorem. The values of Hamilton energy with different oscillations of complex differential system are calculated. In addition, offset boosting control for the complex chaotic signal is realized by adding a constant to variable of complex system. Simulation shows that the position of the chaotic attractor in phase space can be flexibly shifted by applying the offset parameter.

Attractor-repeller construction of Shintani domains for totally complex quartic fields

The units of a number field k act naturally on the real vector space k⊗QR, and so on open subsets of (k⊗QR)⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k, whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for totally complex number fields has been published. Here we give a quick way to compute Shintani domains for totally complex quartic number fields.To construct these domains we exploit the existence of a forward attractor and backward repeller set for this action. We prove that the number of polyhedral cones needed for the Shintani domain is bounded by an absolute constant, a fact previously known only for cubic or quadratic fields. We also show that our algorithm for finding a Shintani domain runs in polynomial time and we give a table describing the results of running the algorithm on more than 168,000 fields.

A New 3D Chaotic Attractor in Gene Regulatory Network

This paper introduces a new 3D chaotic attractor in a gene regulatory network. The proposed model has eighteen parameters. Formulas for characteristic numbers of critical points for three-dimensional systems were considered. We show that the three equilibrium points of the new chaotic 3D system are unstable and deduce that the three-dimensional system exhibits chaotic behavior. The possible outcomes of this 3D model were compared with the results of the Chua circuit. The bifurcation structures of the proposed 3D system are investigated numerically, showing periodic solutions and chaotic solutions. Lyapunov exponents and Kaplan-Yorke dimension are calculated. For calculations, the Wolfram Mathematica is used.

Chaotic neural networks with complex-valued weight and its applications in electrocardiogram classification

As neural networks are easy to converge to local minimum, the ergodicity of chaotic system is helpful to tackle this problem. Besides, the real parts and imaginary parts in complex-valued chaotic systems are independent, which increase the ergodic property and unpredictability of the chaotic signals. Therefore, we propose a new chaotic neural network with complex-valued weight for electrocardiogram classification. Firstly, a complex Logistic chaotic map is put forward, and its bifurcation diagram, Lyapunov index, and chaotic attractors are analyzed. Secondly, based on the ergodicity of complex Logistic chaotic map and a novel neuron function, the learning algorithm including complex-valued weight iteration for the chaotic neural network is proposed. Finally, the MIT-BIH data-base is used to verify the proposed method. The chaotic neural network with real Logistic map and other classification methods are also adopted for comparison. The results show that our chaotic neural network has a certain improvement in the accuracy of electrocardiogram classification.

Pseudo and true singularly degenerate heteroclinic cycles of a new 3D cubic Lorenz-like system

In contrast to the coexistence of infinitely many pseudo and true singularly degenerate heteroclinic cycles with nearby two-scroll hyperchaotic Lorenz-like attractors coined in four-dimensional systems, this note reports the ones with nearby bifurcated chaotic attractors in a new 3D cubic Lorenz-like system. Except for the rabbit head/boat shape two-scroll attractors, there exist globally exponentially asymptotically stable parabolic type equilibria and a pair of heteroclinic orbits. In addition, numerical simulations also validate the correctness of the theoretical analysis.

Limits to predictability of the asymptotic state of the Atlantic Meridional Overturning Circulation in a conceptual climate model

Anticipating critical transitions in the Earth system is of great societal relevance, yet there may be intrinsic limitations to their predictability. For instance, the asymptotic state of a dynamical system possessing multiple chaotic attractors depends sensitively on the initial condition in the proximity of a fractal basin boundary. Here, we approach the problem of final-state sensitivity of the Atlantic Meridional Overturning Circulation (AMOC) using a conceptual climate model, composed of a slow bistable ocean coupled to a fast chaotic atmosphere. First, we explore the occurrence of long chaotic transients in the monostable regime, which can mask a loss of stability near bifurcations. In the bistable regime, we explicitly construct the chaotic saddle using the edge tracking technique. We quantify the final-state sensitivity through the maximum Lyapunov exponent and the lifetime of the saddle and find that the system exhibits a fractal basin boundary with almost full phase space dimension, implying vanishing predictability of the second kind near the basin boundary. Our results demonstrate the usefulness of studying non-attracting chaotic sets in the context of predicting climatic tipping points, and provide guidance for the interpretation of critical transitions in higher-dimensional climate models.

Scroll-growing/controlling chaotic attractors in cyclic Hopfield neural networks via memristive bridging

In this paper, a memristor-based neural network is proposed, which is implemented by two tri-neuron resistive-cyclic Hopfield neural networks (RC-HNNs) via memristive bridging. The memristor-bridged network has a line equilibrium set composed of infinitely many index-2 saddle-foci, but it can produce multi-scroll chaotic attractors contrary to Shil’nikov’s criterion. Complex bifurcation behaviors, scroll-growing chaotic attractors over time, and homogeneous coexisting attractors are revealed by numerical methods. Further, a scroll-control scheme is designed and scroll-controlling chaotic attractors are demonstrated numerically. The results show that the memristor-bridged network can not only generate scroll-growing chaotic attractors over time, but also produce scroll-controlling chaotic attractors by limiting the dynamic range of the internal state of the bridging memristor. Finally, an analog electronic circuit is designed for the memristor-bridged network, and PSIM circuit simulations are used to verify the numerical simulations.

Reservoir computing with higher-order interactive coupled pendulums.

The reservoir computing approach utilizes a time series of measurements as input to a high-dimensional dynamical system known as a reservoir. However, the approach relies on sampling a random matrix to define its underlying reservoir layer, which leads to numerous hyperparameters that need to be optimized. Here, we propose a nonlocally coupled pendulum model with higher-order interactions as a novel reservoir, which requires no random underlying matrices and fewer hyperparameters. We use Bayesian optimization to explore the hyperparameter space within a minimal number of iterations and train the coupled pendulums model to reproduce the chaotic attractors, which simplifies complicated hyperparameter optimization. We illustrate the effectiveness of our technique with the Lorenz system and the Hindmarsh-Rose neuronal model, and we calculate the Pearson correlation coefficients between time series and the Hausdorff metrics in the phase space. We demonstrate the contribution of higher-order interactions by analyzing the interaction between different reservoir configurations and prediction performance, as well as computations of the largest Lyapunov exponents. The chimera state is found as the most effective dynamical regime for prediction. The findings, where we present a new reservoir structure, offer potential applications in the design of high-performance modeling of dynamics in physical systems.

A lightweight chaos-based encryption scheme for IoT healthcare systems

The Internet of Things (IoT) has transformed the healthcare industry by enabling new services and capabilities through connected devices and sensors. These devices, such as smartwatches and fitness tracker bands, can monitor various health metrics such as heart rate, blood pressure, and sleep quality. However, the security of these systems is a critical concern, as the unauthorized disclosure or access of healthcare information could have serious consequences for patients. This information can include personal and medical data, diagnostic and treatment information, as well as private locations. Lightweight encryption schemes are commonly used in IoT systems to protect this sensitive data. These schemes are designed to be fast and efficient, allowing them to encrypt and decrypt data in real-time, which is important for systems with limited computing power or storage capacity. In the last decade, there has been a significant increase in research and development in chaos cryptography. Due to chaotic systems’ unpredictable and sensitive nature, they can provide robust cryptographic schemes for data transmission between embedded devices. This makes it difficult for attackers to intercept and manipulate the data. Therefore, this work proposes a chaos-based lightweight encryption scheme for IoT healthcare systems with a primary application in the encryption of wearable devices. The proposed encryption scheme is based on a 2D 4-scroll chaotic attractor system, uniquely characterized for this work. The scheme is tested on an ARM-based microcontroller for encrypting PPG (Photoplethysmogram) biosignal data. The obtained results show that the chaos-based lightweight encryption scheme effectively improves the security of healthcare IoT systems while maintaining the real-time flow of data.

Combining intransitive and higher-order effects in a coupled oscillator framework: A case study of an ant community.

A growing body of literature recognizes that pairwise species interactions are not necessarily an appropriate metaphorical molecule of community ecology. Two examples are intransitive competition and nonlinear higher-order effects. While these two processes have been discussed extensively, the explicit analysis of how the two of them behave when simultaneously part of the same dynamic system has not yet been explored theoretically. A concrete situation exists on coffee farms in Puerto Rico in which three ant species form an intransitive competitive triplet, and that triplet is strongly influenced, nonlinearly, by a fly parasitoid that modifies the competitive ability of one of the species. Using this arrangement as a template, we explore the dynamical consequences with a simple ordinary differential equation (ODE) model. Results are complicated and include alternative periodic and chaotic attractors. The qualitative structures of those complications, however, may be approximately retrieved from the basic natural history of the system.

Viscous effect in the late time evolution of phantom universe

We investigate the cosmological implications of a phantom dark energy model with bulk viscosity. We explore this model as a possible way to resolve the big rip singularity problem that plagues the phantom models. We use the latest type Ia supernova and Hubble parameter data to constrain the model parameters and find that the data favor a significant bulk viscosity over a non-constant potential term for the phantom field. We perform a dynamical analysis of the model and show that the only stable and physical attractor corresponds to a phantom-dominated era with a total equation of state that can be greater than -1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-1$$\\end{document} due to the viscosity. We also study the general effect of viscosity on the phantom field and the late time evolution of the universe. We apply the statefinder diagnostic to the model and find that it approaches a nearby fixed point asymptotically, indicating that the universe can escape the big rip singularity with the presence of bulk viscosity. We conclude that bulk viscosity can play an important role in affecting the late-time behavior as well as alleviating the singularity problem of the phantom universe.

Online real-time learning of dynamical systems from noisy streaming data

Recent advancements in sensing and communication facilitate obtaining high-frequency real-time data from various physical systems like power networks, climate systems, biological networks, etc. However, since the data are recorded by physical sensors, it is natural that the obtained data is corrupted by measurement noise. In this paper, we present a novel algorithm for online real-time learning of dynamical systems from noisy time-series data, which employs the Robust Koopman operator framework to mitigate the effect of measurement noise. The proposed algorithm has three main advantages: (a) it allows for online real-time monitoring of a dynamical system; (b) it obtains a linear representation of the underlying dynamical system, thus enabling the user to use linear systems theory for analysis and control of the system; (c) it is computationally fast and less intensive than the popular extended dynamic mode decomposition (EDMD) algorithm. We illustrate the efficiency of the proposed algorithm by applying it to identify the Van der Pol oscillator, the chaotic attractor of the Henon map, the IEEE 68 bus system, and a ring network of Van der Pol oscillators.

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.

Complex dynamics of a four-species food-web model: An analysis through Beddington-DeAngelis functional response in the presence of additional food

Abstract The four-dimensional food-web system consisting of two prey species for a generalist middle predator and a top predator is proposed and investigated. The middle predator is predating over both the prey species with a modified Holling type-II functional response. The food-web model is effectively formulated, exhibits bounded behavior, and displays dissipative dynamics. The proposed model’s essential dynamical features are studied regarding local stability. We investigated the four species’ survival and established their persistence criteria. In the proposed model, a transcritical bifurcation occurs at the axial equilibrium point. The numerical simulations reveal the persistence of a chaotic attractor or stable focus. The conclusion is that increasing the food available to the middle predator may make it possible to manage and mitigate the chaos within the food chain.

On two-parameter bifurcation and analog circuit implementation of a Chameleon chaotic system

In this paper, the two-parameter space bifurcation of a three-dimensional Chameleon system is investigated. It is called Chameleon since the type and the number of the system equilibrium are adjustable for different parameter configurations. Aided by the computation analysis, the graphic structures of two-parameter bifurcation of the Chameleon system are characterized for the first time. With different two-parameter configurations, the bifurcation evolution shows that various self-excited and hidden attractors exist. In addition, numerical demonstration of the two-dimensional slice through the attraction basin space is presented. The results show that the basin of attraction of the typical hidden chaotic attractor does not associated with the origin, which makes the attractor difficult to be numerically localized and experimentally observed. To solve the problem, offset boost scheme is adopted to control the basin of attraction and make it touch the origin, which allows to coin the hidden attractor via configuring zero initial value and making it feasible in experimental observation. Finally, the analog circuit-assisted experiment validated the feasibility of the scheme.

Scale-Free Chaos in the 2D Harmonically Confined Vicsek Model.

Animal motion and flocking are ubiquitous nonequilibrium phenomena that are often studied within active matter. In examples such as insect swarms, macroscopic quantities exhibit power laws with measurable critical exponents and ideas from phase transitions and statistical mechanics have been explored to explain them. The widely used Vicsek model with periodic boundary conditions has an ordering phase transition but the corresponding homogeneous ordered or disordered phases are different from observations of natural swarms. If a harmonic potential (instead of a periodic box) is used to confine particles, then the numerical simulations of the Vicsek model display periodic, quasiperiodic, and chaotic attractors. The latter are scale-free on critical curves that produce power laws and critical exponents. Here, we investigate the scale-free chaos phase transition in two space dimensions. We show that the shape of the chaotic swarm on the critical curve reflects the split between the core and the vapor of insects observed in midge swarms and that the dynamic correlation function collapses only for a finite interval of small scaled times. We explain the algorithms used to calculate the largest Lyapunov exponents, the static and dynamic critical exponents, and compare them to those of the three-dimensional model.

A Novel Conservative Chaotic Circuit with a Cubic Root

A new conservative chaotic system with a cubic root is constructed in this paper. The system has no equilibria. An offset parameter is introduced into the system. By modulating different offset parameters and corresponding initial conditions, multiple coexisting conservative chaotic attractors can be generated consequently. By replacing the offset parameter with an appropriate sign function, multiscroll conservative chaotic attractors can be generated effectively by connecting multiple coexisting chaotic attractors based on offset boosting. Moreover, the number of scrolls tend to be infinite. Finally, a novel chaotic circuit with a cubic root is designed successfully by connecting two multipliers in opposite directions. The hardware implementation of the chaotic circuit is also completed to further verify the existence of conservative chaos and the consistency with the numerical simulations.

Bifurcation analysis of complex switching oscillations in a Kerr microring resonator.

Microresonators are micron-scale optical systems that confine light using total internal reflection. These optical systems have gained interest in the past two decades due to their compact sizes, unprecedented measurement capabilities, and widespread applications. The increasingly high finesse (or Q factor) of such resonators means that nonlinear effects are unavoidable even for low power, making them attractive for nonlinear applications, including optical comb generation and second harmonic generation. In addition, light in these nonlinear resonators may exhibit chaotic behavior across wide parameter regions. Hence, it is necessary to understand how, where, and what types of such chaotic dynamics occur before they can be used in practical devices. We study here the underlying mathematical model that describes the interactions between the complex-valued electrical fields of two optical beams in a single-mode resonator with symmetric pumping. Recently, it was shown that this model exhibits a wide range of fascinating behaviors, including bistability, symmetry breaking, chaos, and self-switching oscillations. We employ here a dynamical system approach to perform a comprehensive theoretical study that allows us to identify, delimit, and explain the parameter regions where different behaviors can be observed. Specifically, we present a two-parameter bifurcation diagram that shows how (global) bifurcations organize the observable dynamics. Prominent features are curves of Shilnikov homoclinic bifurcations, which act as gluing bifurcations of pairs of periodic orbits or chaotic attractors, and a Belyakov transition point (where the stability of the homoclinic orbit changes). In this way, we identify and map out distinctive transitions between different kinds of chaotic self-switching behavior in this optical device.

Solution and dynamics analysis of fractal-fractional multi-scroll Chen chaotic system based on Adomain decomposition method

In recent years, newly proposed fractal-fractional calculus operators have been used to predict the chaotic behavior of some attractors, and the new operators can capture self-similarity in the studied chaotic attractors. The results of varying with fractal order and fractional order are interesting, however, the dynamics are barely analyzed in these studies, and the law of fractal order and fractional order on the system is not clear. In this paper, the fractal-fractional differential operators with inverse operator property are redefined in the Caputo sense, and a new numerical scheme is proposed based on the sixth-order Adomain decomposition method. Then the dynamics of the new multi-scroll Chen chaotic system and its three degradation modes are analyzed completely in the fractal-fractional sense. The influence of the change of fractal order and fractional order on the chaotic system is revealed. Finally, a new physical phenomenon of the fractal-fractional chaotic system is captured by observing the coexistence of multi-scroll and two-scroll chaotic attractors of the new system through a wood-grain-like attractor basin, as well as transient chaos, chaotic jumps, and complex state transition behaviors. It is shown that fractal-fractional multi-scroll Chen system has complex and specific dynamical behaviors.