Chaos Attractors Research Articles

Dynamic Analysis, Image Encryption Application and FPGA Implementation of a Discrete Memristor-Coupled Neural Network

This paper presents a novel discrete memristor model that incorporates exponential and absolute value functions. A discrete coupled memristor neural network model is constructed based on this memristor design. The periodic and chaotic regions of the discrete neural network model are determined using bifurcation and Lyapunov exponent spectrum. Furthermore, by varying the initial values of the discrete memristive neural network, we observe the coexistence of chaos and periodic attractors, as well as periodic attractors. Additionally, an application to color image encryption based on the discrete system model is given. Security analysis is conducted in the aspects of key space, histogram analysis, correlation analysis, sensitivity analysis, Peak Signal-to-Noise Ratio (PSNR), and information entropy analysis. The analysis results show that the algorithm has a key space size of [Formula: see text], and the information entropy of baboon graph is 7.9993, which is very close to the ideal value of 8. It shows that the image encryption algorithm is feasible and effective. Finally, the implementation of the discrete memristive neural network model is realized using Field Programmable Gate Array (FPGA). The experimental implementation is conducted using the Verilog language on the Vivado 2018.3 platform, and the obtained results align with the numerical simulation results obtained through MATLAB.

Offset boosting in a discrete system.

Offset boosting plays an important role in chaos application in electronic engineering. A direct variable substitution typically will destroy the dynamics of a discrete map even though the initial condition is well considered. The internal fundamental reason is that the left-hand side of a discrete system does not have the dimension of variable differentiation (DVD) like the one of a continuous system. When the key property of DVD is completely preserved, the offset boosting based on a parameter or the initial condition can be reasonably achieved like in a differential system. Consequently, by the initial condition-oriented offset boosting, flexible multistability like attractor self-reproducing or attractor doubling can be further realized. A circuit experiment is completed for the verification of reliable offset boosting. The systematic exploration of offset boosting in a map will cast a new light on chaos regulation and attractor transportation in a discrete map. As a simple case, a two-dimensional Hénon map is taken as the example demonstrating the achievement of offset boosting via the parameter or initial condition.

BOUALLEGUE EQUATION SYSTEM AND ITS APPLICATIONS IN CHAOS THEORY

This paper presents a new contribution in chaos theory using Bouallegue Equation System, this system is built of three blocks, These blocks are connected in cascade, the first block is a system of chaotic attractor, the second block is a system of fractal processes and the third block is a system of neural networks. In this work, we have generated a novel class of hyper chaotic attractors by coupling with neuron containing multi dendrites, each dendrite has its activation function. The activation function contains four parameters form, position of dendrite, parameter of equilibrium and polarity. Each dendrite can take four behaviors by configuration the parameter n, its structure contains four variables (n,p,q,χ), so we call this capacity or this potential of activation function, a Variable Structure Model of Neuron (VSMN, for short). The impact of activation function in chaotic attractor generates a new classes of hyper chaos attractors with fluctuation, cutting, folding, amplifying in attractors. Simulations have demonstrated the validity and feasibility of the proposed method. Two results of applications using Bouallegue Equation System in fractal theory are also given to show the similarities in medicine illustrating the fractal chromosome and the fractal lung, and in biology displaying a fractal flower.

Capturing chaos: a multidisciplinary approach to nonlinear population dynamics

The hypothesis of chaotic population dynamics was proposed in ecology by Robert May in the mid-1970s. At that time the idea was controversial, and it remains a fascinating and unsettled issue today. We report the results of a 20-year laboratory research programme that continued in the tradition of the pioneering ecologist Thomas Park using the Tribolium flour beetle model. We present biological evidence of complex population dynamics – including bifurcations, chaos, saddle nodes, phase switching, resonance effects, and multiple attractors – by using a low-dimensional difference equation model for Tribolium together with carefully designed, conducted, and statistically analysed experiments. The model, parameterized with data, also explains the results of historical Tribolium experiments, such as the classical competition studies of Thomas Park and his colleagues. Our research programme has inspired other studies using the Tribolium mathematical and laboratory model. This work was conducted by a multidisciplinary team, which included Jim Cushing.

Design of memristor hyperchaotic circuit with burst oscillation and infinite attractor coexistence and its application

A 4D hyperchaotic system with symmetry based on memristors is proposed and the chaotic properties of the circuit are verified by phase diagrams, Poincaré cross sections, bifurcation, Lyapunov exponents, and 0–1 tests. In addition, the system is found to have rich dynamical behaviors such as transient chaos, intermittent chaos, offset enhancement, burst oscillation, and infinite attractor coexistence. The complexity comparison analysis reveals that this hyperchaotic system has good symmetry as well as the initial values of the system that are more suitable for image encryption. Circuit simulation of the system using simulation software proved the usability of the system. A field programmable gate array (FPGA) is used to implement the actual digital circuit of the system. The final FPGA results agree with the numerical simulation and prove the practical value of the constructed memristor hyperchaotic system. Finally, chaotic circuits are applied to DNA image encryption. It is discovered to have a very large key space, strong key sensitivity, outstanding anti-cropping, noise immunity and high security performance. Thus, the proposed hyperchaotic system has good advantages for application in the field of image encryption.

The Parallel Chaotification Map and Its Application

A universal modular plus parallel (MPP) method is proposed to construct enhanced chaotic maps, including one-dimensional MPP chaotic map (1D-MPPCM) and high-dimensional MPPCM (HD-MPPCM). It is theoretically proved that 1D-MPPCM model can significantly increase the Lyapunov exponent (LE) and parameter range of seed chaotic maps. To further increase the system dimension, the HD-MPPCM model is established through the close-loop parallel coupling mechanism. Based on several typical seed chaotic maps, some new parallel chaotic maps are obtained by self-parallel and hybrid-parallel, and their dynamics are analyzed by phase diagram, LEs, permutation entropy (PE) complexity and statistic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi^2$</tex-math> </inline-formula> . The simulation results show that the proposed maps have large maximum Lyapunov exponent (MLE), PE complexity, and uniform distribution. In particular, HD-MPPCMs have some interesting characteristics, such as full positive LEs, hyperchaotic behavior, global chaos, and full attractor distribution, which are the potential model for engineering applications. To further verify the practicability, the proposed maps are implemented on DSP platform, and applied to pseudo-random number generator (PRNG).

Chaos and multi-layer attractors in asymmetric neural networks coupled with discrete fractional memristor

This article introduces a novel model of asymmetric neural networks combined with fractional difference memristors, which has both theoretical and practical implications in the rapidly evolving field of computational intelligence. The proposed model includes two types of fractional difference memristor elements: one with hyperbolic tangent memductance and the other with periodic memductance and memristor state described by sine functions. The authenticity of the constructed memristor is confirmed through fingerprint verification. The research extensively investigates the dynamics of a coupled neural network model, analyzing its stability at equilibrium states, studying bifurcation diagrams, and calculating the largest Lyapunov exponents. The results suggest that when incorporating sine memristors, the model demonstrates coexisting state variables depending on the initial conditions, revealing the emergence of multi-layer attractors. The article further demonstrates how the memristor state shifts through numerical simulations with varying memductance values. Notably, the study emphasizes the crucial role of memductance (synaptic weight) in determining the complex dynamical characteristics of neural network systems. To support the analytical results and demonstrate the chaotic response of state variables, the article includes appropriate numerical simulations. These simulations effectively validate the presented findings and provide concrete evidence of the system’s chaotic behavior.

A nonlinear perturbed coupled system with an application to chaos attractor

In this paper, a general system of quadratically perturbed system of modified fractional differential equations (FDEs) is considered for the solution existence, solution uniqueness, stability results, numerical scheme and computational applications. The presumed perturbed system is more general and several preexisting problems become its special cases. Fixed point results are applied for the theoretical results. The Lagrange’s polynomial is used to approximate the nonlinear system and a useful numerical scheme is established. For an application, a complex system of chaotic attractor is given and is computational studied via some graphical presentation.

Coexistence behavior of asymmetric attractors in hyperbolic-type memristive Hopfield neural network and its application in image encryption

The neuron model has been widely employed in neural-morphic computing systems and chaotic circuits. This study aims to develop a novel circuit simulation of a three-neuron Hopfield neural network (HNN) with coupled hyperbolic memristors through the modification of a single coupling connection weight. The bistable mode of the hyperbolic memristive HNN (mHNN), characterized by the coexistence of asymmetric chaos and periodic attractors, is effectively demonstrated through the utilization of conventional nonlinear analysis techniques. These techniques include bifurcation diagrams, two-parameter maximum Lyapunov exponent plots, local attractor basins, and phase trajectory diagrams. Moreover, an encryption technique for color images is devised by leveraging the mHNN model and asymmetric structural attractors. This method demonstrates significant benefits in correlation, information entropy, and resistance to differential attacks, providing strong evidence for its effectiveness in encryption. Additionally, an improved modular circuit design method is employed to create the analog equivalent circuit of the memristive HNN. The correctness of the circuit design is confirmed through Multisim simulations, which align with numerical simulations conducted in Matlab.

Simulation and dynamical analysis of a chaotic chameleon system designed for an electronic circuit

The moment when stability moves to instability and order moves to disorder constitutes a chaotic systems; such phenomena are characterized sensitively on the basis of initial conditions. In this manuscript, a fractal–fractionalized chaotic chameleon system is developed to portray random chaos and strange attractors. The mathematical modeling of the chaotic chameleon system is established through the Caputo–Fabrizio fractal–fractional differential operator versus the Atangana–Baleanu fractal–fractional differential operator. The fractal–fractional differential operators suggest random chaos and strange attractors with hidden oscillations and self-excitation. The limiting cases of fractal–fractional differential operators are invoked on the chaotic chameleon system, including variation of the fractal domain by fixing the fractional domain, variation of the fractional domain by fixing the fractal domain, and variation of the fractal domain as well as the fractional domain. Finally, a comparative analysis of chaotic chameleon systems based on singularity versus non-singularity and locality versus non-locality is depicted in terms of chaotic illustrations.

Complexity analysis and discrete fractional difference implementation of the Hindmarsh–Rose neuron system

Nowadays, multistability assessment in discrete nonlinear fractional difference systems is an emotive topic. This paper introduces a novel discrete, nonequilibrium, memristor-based Hindmarsh–Rose neuron (HRN) with the Caputo fractional difference scheme. Furthermore, in a setting involving commensurate and incommensurate scenarios, the complex structure of the proposed discrete fractional model, which includes its multistability, concealed chaos and hyperchaotic attractor, is examined using a wide range of computational approaches, such as Lyapunov exponents, phase portraits, bifurcation illustrations, and the 0–1 evaluation emergence of synchronization. These evolving properties imply that the fractional discrete HRN has an undetected multistability. Ultimately, an elaborate investigation is performed to confirm the existence of unpredictability employing approximation entropy (ApEn) and the ℂ0 measurements. It is demonstrated that whenever the immediate synchronization indicates that it happens, the neurons’ interactions modify, recurring in decreasing fractional orders. Furthermore, minimizing the derivative order increases the incidence of explosions in the synchronization manifold, which is opposite to the behaviour of one nerve cell.

Dynamic Behavior Analysis and Synchronization of Memristor-Coupled Heterogeneous Discrete Neural Networks

Continuous memristors have been widely studied in recent years; however, there are few studies on discrete memristors in the field of neural networks. In this paper, a four-stable locally active discrete memristor (LADM) is proposed as a synapse, which is used to connect a two-dimensional Chialvo neuron and a three-dimensional KTZ neuron, and construct a simple heterogeneous discrete neural network (HDNN). Through a bifurcation diagram and Lyapunov exponents diagram, the period and chaotic regions of the discrete neural network model are shown. Through numerical analysis, it was found that the chaotic region and periodic region of the neural network based on DLAM are significantly improved. In addition, coexisting chaos and chaos attractors, coexisting periodic and chaotic attractors, and coexisting periodic and periodic attractors will appear when the initial value of the LADM is changed. Coupled by a LADM synapse, two heterogeneous discrete neurons are gradually synchronized by changing the coupling strength. This paper lays a good foundation for the future analysis of LADMs and the related research of discrete neural networks coupled by LADMs.

Bifurcations in a forced Wilson-Cowan neuron pair

We investigate bifurcations of periodic solutions observed in the forced Wilson-Cowan neuron pair by both the brute-force computation and the shooting method. By superimposing the results given by both methods, a detailed topological classification of periodic solutions is achieved that includes tori and chaos attractors in the parameter space is achieved. We thoroughly explore the parameter space composed of threshold values, amplitude, and angular velocity of an external forcing term. Many bifurcation curves that are invisible when using brute-force method are solved by the shooting method. We find out a typical bifurcation structure including Arnold tongue in the angular velocity and the amplitude of the external force parameter plane, and confirm its fractal structure. In addition, the emergence of periodic bursting responses depending on these patterns is explained.

Dynamics of plant disease models with continuous and pulse farming awareness control

&lt;p style='text-indent:20px;'&gt;How to develop the mathematical model of plant disease transmission and consider the effect of saturated farming awareness strategies on plant disease control is of great practical significance. To do this, the plant disease transmission models with continuous and pulse farming awareness control strategy have been proposed in the present paper. For the model with continuous control strategies, the threshold values for the existence and stability of multiple equilibria have been given, and the effect of farming awareness on disease persistence is discussed; For the model with nonlinear impulsive control tactics, the existence and stability of the plant disease-free periodic solutions are investigated and the threshold condition is given, further we get the conditions for the permanence of the system and obtain the sufficient conditions under which the positive periodic solution exists by bifurcation theory. Numerical simulations reveal that people's farming awareness, especially global awareness, plays an important role in controlling and eradicating the plant disease. Moreover, the nonlinear impulsive control could result in the complex dynamic behavior, including period doubling, chaos and multiple attractors, which makes it difficult to design a successful plant disease control strategy.&lt;/p&gt;

Pest control switching models with instantaneous and non-instantaneous impulsive effects

In this paper, it is considered that the predation rate and transformation rate of natural enemies on pests in different time periods of pesticide action and non-action are different, and after spraying pesticides, the pesticides kill the pests and natural enemies instantaneously, and at the same time bring non-instantaneous residual effects on them. Firstly, an integrated pest control switching model with instantaneous and non-instantaneous impulsive effects at the fixed time is proposed. It assumes that spraying pesticides is more frequently used than releasing natural enemies. The sufficient condition for the global asymptotic stability of the pest-eradication periodic solution is obtained. By numerical simulations, we find that the system we study has many complex dynamics including period-doubling bifurcation, period-halving bifurcation, chaos and multiple attractors. Further, the effects of some key parameters on the threshold for pest eradication are analyzed. The results imply that it is not that the more frequent, the more conductive to pest control, and within each releasing period, applying pesticides too early or too late is not good for pest control. Finally, with the purpose of preventing pests below the economic injury level, a state-dependence pest control switching model with instantaneous and non-instantaneous impulsive effects is investigated numerically. Our results show that the frequency of applying pesticides depends on the key parameters of the established system, such as initial population density, the releasing amount of natural enemies and the period of releasing natural enemies, etc.

Bifurcation, chaos, and circuit realisation of a new four-dimensional memristor system

AbstractThis paper discusses the complex dynamic behavior of a novel chaotic system, which was firstly established by introducing a memristor into a similar Chen’s system. Then by choosingaas the key parameter, we analyze the stability of memristor system based on eigenvalue theory. It is also found that whenacross some critical values, the system can exhibit Neimark–Sacker bifurcation and chaos behaviors. Some numerical simulations including phase diagrams and maximum Lyapunov exponent graph of the memristor-based systems are presented to verify the existence of chaos attractors. Finally, to make the results of this paper useful in the actual situation, such as the design of chaos security algorithm, analog electronic circuit of memristor chaotic system is designed.

Bifurcation analysis and chaos control in Zhou's dynamical system

PurposeThe purpose of the study is to obtain explicit formulas to determine the stability of periodic solutions to the new system and study the extent of the stability of those periodic solutions and the direction of bifurcated periodic solutions. More than that, the authors did a numerical simulation to confirm the results that the authors obtained and presented through numerical analysis are the periodic and stable solutions and when the system returns again to the state of out of control.Design/methodology/approachThe authors studied local bifurcation and verified its occurrence after choosing the delay as a parameter of control in Zhou 2019’s dynamical system with delayed feedback control. The authors investigated the normal form theory and the center manifold theorem.FindingsThe occurrence of local Hopf bifurcations at the Zhou's system is verified. By using the normal form theory and the center manifold theorem, the authors obtain the explicit formulas for determining the stability and direction of bifurcated periodic solutions. The theoretical results obtained and the corresponding numerical simulations showed that the chaos phenomenon in the Zhou's system can be controlled using a method of time-delay auto-synchronization.Originality/valueAs the delay increases further, the numerical simulations show that the periodic solution disappears, and the chaos attractor appears again. The obtained results can also be applied to the control and anti-control of chaos phenomena of system (1). There are still abundant and complex dynamical behaviors, and the topological structure of the new system should be completely and thoroughly investigated and exploited.

Numerical analysis on chaos attractors using a backward difference formulation

The chaos attractor is a system of ordinary differential equations which is known for having chaotic solutions for certain parameter values and an initial condition. Research conducted in the current work establishes a backward difference algorithm to study these chaos attractors. Different types of chaos mapping, namely the Lorenz chaos, 'sandwich' chaos and 'horseshoe' chaos will be analyzed. Compared to other numerical methods, the proposed backward difference algorithm will show to be an efficient tool for analyzing solutions for the chaos attractors.

Analysis and FPGA implementation of memristor chaotic system with extreme multistability

&lt;sec&gt;The memristor is a kind of nonlinear element with nanometer size, which can enhance the complexity of a chaotic system. With the further research of chaos, several novel nonlinear phenomena have been found by scholars, such as hidden attractors, coexisting attractors and multi-stability. Meanwhile, the extremely multi-stability representation system coexists with the infinite attractors, which has become a hot spot in the field of memristor chaos research in recent years. A general method to construct a chaotic systems of multiple coexistence is to increase the number of equilibrium points of chaotic system by means of control. The introduction of memristor results in the linear distribution of the equilibrium points of chaotic system in space, which are the linear equilibrium points. The existing researches show that chaotic system with extremely multi-stability can produce better chaotic sequence, which can be used in engineering fields such as secure communication. Therefore, it is of great significance to construct chaotic systems with rich dynamic behaviors by using memristors.&lt;/sec&gt;&lt;sec&gt;In order to further improve the complexity of the chaotic system, a five-dimensional memristor chaotic system is constructed by replacing the coupling parameters in the four-dimensional chaotic system based on Sprott-B with a magnetically controlled memristor. The dynamic behavior of the system is analyzed by bifurcation diagram, Lyapunov exponent spectrum, phase portrait, Poincaré map, dynamic map and other conventional means. The analysis shows that the new system has rich dynamic behaviors: when the system parameters change, the system can produce a large number of chaotic attractors with different topological structures and periodic limit cycles with different periods. When different parameters change, the dynamic characteristics of the system also change; when the system parameters are fixed, the system not only has an offset enhancement phenomenon that depends on the change of the initial conditions, but also shows a very strong sensitivity to the initial values and a great adjustment range of the initial values, which leads the infinite chaos and periodic attractors to coexist, namely extremely multi-stability appears. Finally, the digital circuit of the memristor chaotic system is implemented based on the field programmable gate array (FPGA) technology. The phase portrait captured on the oscilloscope is consistent with that from the numerical simulation, which verifies the correctness and realizability of the memristor system.&lt;/sec&gt;

Analysis and FPGA implementation of a Memristor chaotic system with extreme multistability

The memristor is a kind of nonlinear element with nanometer size, which can enhance the complexity of a chaotic system. With the further research of chaos, several novel nonlinear phenomena have been found by scholars, such as hidden attractors, coexisting attractors and multi-stability. Meanwhile, the extremely multi-stability representation system has the coexistence of infinite attractors, which has become a hot spot in the field of memristor chaos research in recent years. The common method to construct a chaotic systems of multiple coexistence is to increase the number of equilibrium points of chaotic systems by means of control. The introduction of memristor results in the linear distribution of the equilibrium points of chaotic systems in space, which is the linear equilibrium points. The extant research shows that chaotic systems with extremely multi-stability can produce better chaotic sequences, which can be used in engineering fields such as secure communication. Therefore, it is of great significance to construct chaotic systems with rich dynamic behaviors by using memristors.&lt;br&gt;In order to further improve the complexity of the chaotic system, a five-dimensional memristor chaotic system is constructed by replacing the coupling parameters in the four-dimensional chaotic system based on Sprott-B with a magnetically controlled memristor. The dynamic behavior of the system is analyzed by bifurcation diagram, Lyapunov exponent spectrum, phase portrait, Poincaré map, dynamic map and other conventional means. The analysis shows that the new system has rich dynamic behaviors:when the system parameters change, the system can produce a large number of chaotic attractors with different topological structures and periodic limit cycles with different periods. When different parameters change, the dynamic characteristics of the system also change; When the system parameters are fixed, the system not only has an offset enhancement phenomenon that depends on the change of the initial conditions, but also shows a very strong sensitivity to the initial values and a great adjustment range of the initial values, which leads to the coexistence of infinite chaos and periodic attractors namely extremely multi-stability. Finally, the digital circuit of the memristor chaotic system is implemented based on PFGA technology. The phase portrait captured on the oscilloscope is consistent with the numerical simulation, which verifies the correctness and realizability of the memristor system.