Attractor Research Articles

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.

Global history, the emergence of chaos and inducing sustainability in networks of socio-ecological systems.

In this study, we propose a simplified model of a socio-environmental system that accounts for population, resources, and wealth, with a quadratic population contribution in the resource extraction term. Given its structure, an analytical treatment of attractors and bifurcations is possible. In particular, a Hopf bifurcation from a stable fixed point to a limit cycle emerges above a critical value of the extraction rate parameter. The stable fixed-point attractor can be interpreted as a sustainable regime, and a large-amplitude limit cycle as an unsustainable regime. The model is generalized to multiple interacting systems, with chaotic dynamics emerging for small non-uniformities in the interaction matrix. In contrast to systems where a specific parameter choice or high dimensionality is necessary for chaos to emerge, chaotic dynamics here appears as a generic feature of the system. In addition, we show that diffusion can stabilize networks of sustainable and unsustainable societies, and thus, interconnection could be a way of increasing resilience in global networked systems. Overall, the multi-systems model provides a timescale of predictability (300-1000 years) for societal dynamics comparable to results from other studies, while indicating that the emergent dynamics of networks of interacting societies over longer time spans is likely chaotic and hence unpredictable.

Hidden Dynamics of a New Jerk-like System with a Smooth Memristor and Applications in Image Encryption

Considering the dynamic characteristics of memristors, a new Jerk-like system without an equilibrium point is addressed based on a Jerk-like system, and the hidden dynamics are investigated. When changing system parameter b and fixing other parameters, the proposed system shows various hidden attractors, such as a hidden chaotic attractor (b = 5), a hidden period-1 attractor (b = 3.2), and a hidden period-2 attractor (b = 4). Furthermore, bifurcation analysis suggests that not only parameter b, but also the initial conditions of the system, have an effect on the hidden dynamics of the discussed system. The coexistence of various hidden attractors is explored and different coexistences of hidden attractors can be found for suitable system parameters. Offset boosting of different hidden attractors is discussed. It is observed that offset boosting can occur for hidden chaotic attractor, period-1 attractor, and period-2 attractor, but not for period-3 attractor and period-4 attractor. The antimonotonicity of the proposed system is debated and a full Feigenbaum remerging tree can be detected when system parameters a or b change within a certain range. On account of the complicated dynamics of the proposed system, an image encryption scheme is designed, and its encryption effectiveness is analyzed via simulation and comparison.

Dynamics, circuit design, feedback control of a new hyperchaotic system and its application in audio encryption

In this study, a 4D hyperchaotic system is constructed based on the foundation of a 3D Lü chaotic system. The newly devised hyperchaotic system possesses a sole equilibrium point, showcasing a simplified system structure that reduces complexity. This simplification offers a clearer opportunity for in-depth analysis of dynamic behaviors in the realm of scientific research. The proposed hyperchaotic system undergoes an in-depth examination of its dynamical characteristics, including chaotic attractors, equilibrium point stability, Lyapunov exponents’ spectrum, and bifurcation diagram. Numerical analysis results reveal that the attractor of this hyperchaotic system exhibits highly complex, non-periodic, and fractal structural dynamics. Its motion demonstrates extreme sensitivity and randomness, even within a wide range of variations in parameter d, affirming its hyperchaotic properties with two positive Lyapunov exponents. Hyperchaotic bifurcation diagrams typically exhibit highly intricate structures, such as fractals, branches, and period doubling characteristics, signifying that even minor parameter adjustments can lead to significant changes in system behavior, presenting diversity and unpredictability. Subsequently, to further investigate the practical utility of this hyperchaotic system, a linear feedback control strategy is implemented. Through linear feedback control, the hyperchaotic system is stabilized at its unique equilibrium point. Experimental validation is conducted using both computer software simulation Matlab, electronic circuit simulation Multisim, and embedded hardware STM32. The results of these experiments consistently align, providing theoretical support for the application of this hyperchaotic system in practical domains. Finally, leveraging the hyperchaotic keys generated by this hyperchaotic system, audio encryption is achieved using a cross-XOR algorithm, which is then realized on the embedded hardware platform STM32. The results show that the audio encryption scheme based on the hyperchaotic system is feasible, and the method is simple to implement, has nonlinear characteristics and certain algorithm complexity, which can be applied to audio encryption, image encryption, video encryption, and more.

Fractal functions associated with Reich contractions: an approximation of chaotic attractors

Fractal functions associated with Reich contractions: an approximation of chaotic attractors

Multistability, period-adding, and fractality in a plasma oscillator

In this paper, we report on a periodically driven plasma oscillator modeled by a six-parameter nonhomogeneous second-order ordinary differential equation. We fix four of these parameters, and investigate the dynamics of this system by varying the other two, namely, the amplitude A and the angular frequency ω of the driving. In other words, we investigate the (ω,A) parameter plane, where the dynamical behavior of each point was characterized by the magnitude of the largest Lyapunov exponent. Then, we show that this parameter plane reveals the occurrence of the multistability phenomenon in the system. Properly generated bifurcation diagrams confirm this finding. Basins of attraction of coexisting periodic and chaotic attractors in the phase-space are presented. We also report on the organization of periodicity and chaos in the (ω,A) parameter plane. Typical periodic structures were detected embedded in a chaotic region, namely, the cuspidal, the non-cuspidal, and the shrimp-like. At a certain location on the parameter plane, the organization of the shrimp-like periodic structures resembles a fractal, since the same shape is seen when we look through different scales. Elsewhere these same structures appear organized in a period-adding sequence.