Attractor Research Articles

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.

Opposition to Synchronization of Bistable State in Motif Configuration of Rössler Chaotic Oscillator Systems

This paper presents the study of the opposition to the synchronization of bistable chaotic oscillator systems in basic motif configurations. The following configurations were analyzed: Driver-response oscillator systems coupling, two driver oscillator systems to one response oscillator, and a three-oscillator systems ring unidirectional configuration. The study was conducted using the differential equations representing the piecewise linear Rössler-like electronic circuits; the initial conditions were changed to achieve a bistable characteristic Homoclinic H-type or Rössler R-type attractor. Analyzing a sweep of the initial conditions, the basin attractor was obtained. It can be observed that each system has a preferred Homoclinic chaotic attractor with any perturbation or change in initial conditions. A similarity analysis based on the coupling factor was also performed and found that the system has a preferentially Homoclinic chaotic attractor.

Quantifying strange property of attractors in quasiperiodically forced systems

Quasiperiodically forced systems are an important class of dynamical systems exhibiting quasiperiodic, strange nonchaotic, and chaotic attractors. A major concern is the identification of the parameter range in which each one of the attractors is present. In this work, based on the phase sensitivity proposed by Pikovsky and Feudel, we define a measure to quantitatively distinguish quasiperiodic attractors, strange nonchaotic attractors, and chaotic attractors. Particularly, we can determine the boundary points of these three attractors in parameter space. The reliability of this measure is verified in smooth and non-smooth systems.

Fractional-order-induced symmetric multi-scroll chaotic attractors and double bubble bifurcations in a memristive coupled Hopfield neural networks

The multi-state memristor is a type of memristor capable of memorizing multiple conductance states. This special feature is successfully used to mimic the behavior of neural synapses and simulate the influences of electromagnetic induction and radiation in artificial neural networks such as Hopfield neural networks. This paper introduces and examines a novel fractional-order Hopfield neural network model consisting of two non-identical sub-Hopfield neural networks coupled by a multi-stable flux controlled memristor (MCFHNN). The pinched hysteresis loops of the multi-stable flux controlled memristor are analyzed via numerical simulations. The equilibrium points of MCFHNN model and their stability are investigated. The introduced MCFHNN model is solved by using Adomian Decomposition Method (ADM). By using some examination methods such as spectral entropy complexity, bifurcation diagrams, phase portraits, maximum Lyapunov exponent and basins of attraction, we analyze the dynamic behaviors of MCFHNN model associated with coupling strength, fractional-order and initial states. Numerical results demonstrate that the proposed MCFHNN model is capable to develop abundant and complicated dynamics including, chaos, multiple transient transition behaviors, double bubble bifurcations, initial-boosted behavior and symmetric multi-scroll chaotic attractors. It is found that the complexity results are in agreement with those obtained from the bifurcation diagrams. This proves that the complexity can well reflect the dynamic behaviors of MCFHNN model. In order to support theoretical and numerical results, the designed MCFHNN model is implemented with the help of an Arduino-Due microcontroller board based on the Atmel SAM3X8E ARM Cortex-M3 processor. The results are in very good agreement with those obtained from computational simulations.

Cascade tri-neuron hopfield neural network: Dynamical analysis and analog circuit implementation

This paper studies a cascade tri-neuron Hopfield neural network (CTN-HNN) with no connection between the first neuron and the third neuron. Such incompletely connected neuronal structure may be part of complex neural networks with negligible inputs, and may be present in abnormal neural networks with some neurological diseases. It was proved that the stable point, limit cycle, single-scroll chaotic attractor and double-scroll chaotic attractor were observed in the network by adjusting the synaptic weight from the second neuron to itself. To further dynamic analysis, we first demonstrate that this three-dimensional (3D) autonomous nonlinear dynamical system is symmetric about the origin and bounded ultimately, and has one unstable saddle-focus zero point and two symmetric nonzero points with different stability types. Afterwards, taking two synaptic weights as adjustable parameters, the numerical analysis reveals that the CTN-HNN exhibits coexisting chaotic and periodic attractors, coexisting bursting firings and long-term transient chaos by employing multiple numerical methods. Finally, an analog printed circuit is built using one main circuit module and three activation function modules, thus effectively demonstrating the complex dynamics of the network.

Correlation dimension of high-dimensional and high-definition experimental time series

The correlation dimension (CD) is a nonlinear measure of the complexity of invariant sets. First introduced for describing low-dimensional chaotic attractors, it has been later extended to the analysis of experimental electroencephalographic (EEG), magnetoencephalographic (MEG), and local field potential (LFP) recordings. However, its direct application to high-dimensional (dozens of signals) and high-definition (kHz sampling rate) 2HD data revealed a controversy in the results. We show that the need for an exponentially long data sample is the main difficulty in dealing with 2HD data. Then, we provide a novel method for estimating CD that enables orders of magnitude reduction of the required sample size. The approach decomposes raw data into statistically independent components and estimates the CD for each of them separately. In addition, the method allows ongoing insights into the interplay between the complexity of the contributing components, which can be related to different anatomical pathways and brain regions. The latter opens new approaches to a deeper interpretation of experimental data. Finally, we illustrate the method with synthetic data and LFPs recorded in the hippocampus of a rat.

Design and Implementation of Grid-Wing Hidden Chaotic Attractors With Only Stable Equilibria

Design and Implementation of Grid-Wing Hidden Chaotic Attractors With Only Stable Equilibria

A Triple-Memristor Hopfield Neural Network With Space Multistructure Attractors and Space Initial-Offset Behaviors

Memristors have recently demonstrated great promise in constructing memristive neural networks with complex dynamics. This paper proposes a memristive Hopfield neural network with three memristive coupling synaptic weights. The complex dynamical behaviors of the triple-memristor Hopfield neural network (TM-HNN), which have never been observed in previous Hopfield-type neural networks, include space multi-structure chaotic attractors and space initial-offset coexisting behaviors. Bifurcation diagrams, Lyapunov exponents, phase portraits, Poincaré maps, and basins of attraction are used to reveal and examine the specific dynamics. Theoretical analysis and numerical simulation show that the number of space multi-structure attractors can be adjusted by changing the control parameters of the memristors, and the position of space coexisting attractors can be changed by switching the initial states of the memristors. Extreme multistability emerges as a result of the TM-HNN’s unique dynamical behaviors, making it more suitable for applications based on chaos. Moreover, a digital hardware platform is developed and the space multi-structure attractors as well as the space coexisting attractors are experimentally demonstrated. Finally, we design a pseudo-random number generator to explore the potential application of the proposed TM-HNN.

DYNAMICS OF EVOLUTION OF CHIALVO SYSTEM: STUDY OF CHAOS AND COMPLEXITY

Chialvo map modelling a neuron has been investigated numerically as it exhbit generically some of the intriguing/ complex features of several excitable biological systems. Bifurcation diagram obtained for this map shows pattern of repeated period-doubling route to chaos. Appearance of complicated periodic windows within bifurcation suggests presence of complexity within the map. Regular and chaotic attractors are drawn here in different parameter space. Lyapunov exponents describing the stability of the systems have been computed for different cases, clearly revealing complexity involved dynamics. Numerical simulation have been extended to calculate topological entropy as a measure of complexity and presented through graphics. Topological entropy graphs showing significant increase within certain parameter range. Correlation dimension of certain chaotic attractors also obtained.

Lifetimes of metastable windy states in two-dimensional Rayleigh–Bénard convection with stress-free boundaries

Two-dimensional horizontally periodic Rayleigh–Bénard convection between stress-free boundaries displays two distinct types of states, depending on the initial conditions. Roll states are composed of pairs of counter-rotating convection rolls. Windy states are dominated by strong horizontal wind (also called zonal flow) that is vertically sheared, precludes convection rolls and suppresses heat transport. Windy states occur only when the Rayleigh number $Ra$ is sufficiently above the onset of convection. At intermediate $Ra$ values, windy states can be induced by suitable initial conditions, but they undergo a transition to roll states after finite lifetimes. At larger $Ra$ values, where windy states have been observed for the full duration of simulations, it is unknown whether they represent chaotic attractors or only metastable states that would eventually undergo a transition to roll states. We study this question using direct numerical simulations of a fluid with a Prandtl number of 10 in a layer whose horizontal period is eight times its height. At each of seven $Ra$ values between $9\times 10^6$ and $2.25\times 10^7$ we have carried out 200 or more simulations, all from initial conditions leading to windy convection with finite lifetimes. The lifetime statistics at each $Ra$ indicate a memoryless process with survival probability decreasing exponentially in time. The mean lifetimes grow with $Ra$ approximately as $Ra^4$ . This analysis provides no $Ra$ value at which windy convection becomes stable; it might remain metastable at larger $Ra$ with extremely long lifetimes.

Mathematical and Numerical Investigations for a Cholera Dynamics With a Seasonal Environment

We propose a mathematical model for the vibrio cholerae spread under the influence of a seasonal environment with two routes of infection. We proved the existence of a unique bounded positive solution, and that the system admits a global attractor set. The basic reproduction number R0 was calculated for both cases, the fixed and seasonal environment which permits to characterise both, the extinction and the persistence of the disease. We proved that the phage-free equilibrium point is globally asymptotically stable if R0≤1, while the disease will be persist if R0>1. Finally, extensive numerical simulations are given to confirm the theoretical findings.

Multi-ring julia fractal chaotic systems with separated scrolls and nested scrolls

Chaos and fractals are two branches of nonlinear science. However, few scholars have combined chaos and fractals for generating chaotic attractors with different topologies. The main goal of this paper is to combine fractals and chaos to generate separated multi-scroll attractors and nested multi-scroll attractors, as well as to compensate for the lack of randomness and complexity of general chaotic systems. Combining chaotic attractors with two different fractal processes, a class of multi-scroll chaotic attractors with different behaviors are generated, including multi-ring attractors, separated multi-ring attractors, and nested multi-ring attractors. The experiments of spectral entropy (SE) complexity indicates that the proposed multi-ring fractal systems have higher complexity than other chaotic systems. Furthermore, from the perspective of engineering applications, the random performance of one of the multi-ring fractal systems is tested by the NIST test suite. The results demonstrate that the multi-ring fractal system has promising randomness and is well-suited for image encryption applications. Finally, the numerical simulation results are well illustrated by the micro-controller unit (MCU) hardware experiments, which reveal the feasibility of the designed multi-ring attractors, separated multi-ring attractors, and nested attractors.

Coexisting multi-period and chaotic attractor in fully connected system via adaptive multi-body interaction control

Coexisting multi-period and chaotic attractor in fully connected system via adaptive multi-body interaction control

A novel financial system with one stable and two unstable equilibrium points: Dynamics, coexisting attractors, complexity analysis and synchronization using integral sliding mode control

In the world of economics, financial risk is a significant issue that concerns countries and all economic agents. It refers to the potential for experiencing losses due to unpredictable fluctuations in financial or investment activities caused by endogenous factors. Chaos theory provides an explanation for the complexity observed in economic time series, as it supports an endogenous view. This article proposed an enhanced chaos-based financial risk system with one stable and two unstable equilibrium points. The existence of stable and unstable equilibrium points shows that the proposed system has both hidden and self-excited chaotic attractors. By analyzing its dynamic behavior, the study unveils the transition from stability to chaos in the improved financial risk system, while also describing how the complex dynamics of the system affect changes in financial market risk. In addition, this study explores the fundamental dynamic behavior of the financial risk system, including stability analysis, Lyapunov exponent, bifurcation analysis, coexisting attractors, 0–1 test, algorithm and spectral entropy (SE). Finally, with the help of integral sliding mode control (ISMC), we discover the outcomes for the asymptotic synchronization of two chaotic financial systems that are constructed as master and slave systems. Simulations are provided to accompany and support the findings from this research.

High-dimensional memristive neural network and its application in commercial data encryption communication

With the development of economic globalization, demands for digital business data sharing among different subsidiary companies are constantly emerging. How to achieve secure transmission of commercial data in the Internet of Things (IoT) is a challenging task. In this paper, we propose a memristive neural network-based method for encrypted communication of commercial data. First, a high-dimensional memristive Hopfield neural network (MMHNN) model with three memristive synapses and seven neurons is proposed and its complex dynamical behaviors are deeply analyzed. The results of theoretical analysis and numerical simulations show that the proposed MMHNN can generate quasi-periodic, periodic, chaotic and hyperchaotic attractors, as well as some controllable 1-direction (1D), 2D, 3D hyperchaotic multi-scroll attractors and other types of parametric regulatory dynamics and initial value regulation dynamics. In addition, an analog equivalent circuit for the MMHNN is constructed, and the circuit simulations experimental results demonstrate the correctness and feasibility of the multi-scroll attractors phenomena. Finally, a novel MMHNN-based encryption scheme for IoT commercial data images is proposed by applying hyperchaotic spatial multi-scroll attractors, and the performance analysis shows that the proposed system achieves good encryption performances, especially in terms of keyspace, information entropy, correlation and differential attacks. The hardware experiments further verify the effectiveness and superiority of the proposed commercial data encryption scheme based on the MMHNN.

Foreword to the special issue of Journal of Difference Equations and Applications on ‘Lozi, Hénon, and other chaotic attractors, theory and applications’

This foreword presents the 32 articles of the special issue devoted to the Lozi, Hénon and other chaotic attractors proposed by 78 authors from 20 countries (Algeria, Brazil, Canada, China, Czech Republic, France, Germany, India, Italy, Jamaica, Japan, Lebanon, New Zealand, Portugal, Russia, Saudi Arabia, Spain, Ukraine, Tunisia, USA). These original research papers consider chaotic attractors, both in their theoretical and applied aspects.