Attractor Research Articles

Controlling Chaotic Fluctuations through Monetary Policy

Abstract This paper applies the chaos control method (the OGY method) proposed by Ott, E., C. Grebogi, and J. A. Yorke. (1990. “Controlling Chaos.” Physical Review Letters 64: 1196–9) to policy-making in macroeconomics. This paper demonstrates that the monetary equilibrium paths in a discrete-time, two-dimensional overlapping generations model exhibit chaotic fluctuations depending on the money supply rate and the elasticity of substitution between capital and labor under the assumption of the constant elasticity of substitution (CES) production function. We also show that the chaotic fluctuations can be stabilized by controlling the money supply rate by using the OGY method and that even when the OGY method does not work due to periodic attractors, adding moderate stochastic shocks to the model can successfully stabilize the economy.

Heterogeneous multistability and antimonotonicity for a new 3D system with a triple well nonlinearity: theoretical study, control and microcontroller implementation

We propose a new 3D autonomous multistable jerk-like system with a nonlinear term consisting of a six-order triple well function. The presence of six equilibrium points with symmetrical locations along the x-axis represents one of the main distinguishing properties of the new system. Strikingly, the stability analysis of equilibria reveals a cascade of Hopf bifurcations at three specific values of a single control parameter, which results in several forms of complexity. Accordingly, various forms of coexisting attractors such as stable fixed points, limit cycles of diverse periodicities, and chaotic attractors are depicted for some special parameter values. Moreover, It is found that the new jerk-like system with six order triple well polynomial function exhibit extremely complex nonlinear behaviors such as anti-monotone bifurcations, hysteresis and parallel bifurcation branches. These latter aspects explain the presence of multiple (i.e. up to four) coexisting asymmetric attractors for some special rank of parameters. In the presence of multiple competing dynamics, we resort to basins of attraction in order to highlight the how the state space is magnetized. The combination of dynamic features discussed in the new jerk-like system with triple well polynomials nonlinearity introduced in this article is unique and rarely reported. An electronic version of the new system with triple well polynomial nonlinearity is implemented in PSpice. Moreover, a hardware digital implementation of the system is also carried out using an Arduino microcontroller. A very good agreement is captured between PSpice simulation results, the laboratory measurements and the theoretical predictions.

A study on offline handwritten Chinese character recognition based on chaotic iteration

Offline handwritten Chinese character recognition technology has important practical value in the field of automatic machine entryomit, and is one of the research hotspots of intelligent technology. In this paper, a handwritten Chinese character feature extraction method based on dynamic system iteration is constructed by using chaos theory. Firstly, from the perspective of system iteration, the Chinese character image is preprocessed to optimize the image information. Secondly, the iterative matrix constructed by the auxiliary function is combined with the handwritten Chinese character image dot matrix to avoid repetiton the discrete dynamic system, and the approximate chaotic attractor is iteratively generated as the image feature. Then, the similarity of the image is determined according to the size of the Pearson correlation coefficient. Finally, multiple auxiliary functions with large differences are selected to generate multiple discrete chaotic dynamic systems. The above operations are repeated, and the unrecognized image set is used as the next data set to be identified. The experimental results show that the recognition algorithm has achieved good recognition results on the HCL2000 Chinese character database.

Mathematical Modeling for a CHIKV Transmission Under the Influence of Periodic Environment

We studied a simple mathematical model for the chikungunya virus (CHIKV) spread under the influence of a seasonal environment with two routes of infection. We investigated the existence and the uniqueness of a bounded positive solution, and we showed that the system admits a global attractor set. We calculated the basic reproduction number R0 for the both cases, the fixed and seasonal environment which permits us to characterise both, the extinction and the persistence of the disease with regard to the values of R0. We proved that the virus-free equilibrium point is globally asymptotically stable if R0≤1, while the disease will persist if R0>1. Finally, we gave some numerical examples confirming the theoretical findings.

Stable non-BPS black holes and black strings in five dimensions

In this paper we study black hole and black string solutions in five dimensional N=2 supergravity theories arising from the compactification of M-theory on Calabi-Yau manifolds. In particular, we consider explicit examples of three parameter Calabi-Yau manifolds which are obtained as hypersurfaces in toric varieties. Using the attractor mechanism, we obtain Bogomol’nyi-Prasad-Sommerfield (BPS) as well as non-BPS black holes in these compactified supergravity theories. We also consider the black string solutions in these models. We analyze the stability of these extremal black brane configurations by computing the recombination factor. We find multiple stable non-BPS attractor solutions in some of these models. Published by the American Physical Society 2024

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.

On bifurcations of chaotic attractors in a pulse width modulated control system

This paper discusses bifurcational phenomena in a control system with pulse-width modulation of the first kind. We show that the transition from a regular dynamics to chaos occurs in a sequence of classical supercritical period doubling and border collision bifurcations. As a parameter is varied, one can observe a cascade of doubling of the cyclic chaotic intervals, which are associated with homoclinic bifurcations of unstable periodic orbits. Such transition are also refereed as merging bifurcation (known also as merging crisis). At the bifurcation point, the unstable periodic orbit collides with some of the boundaries of a chaotic attractor and as a result, the periodic orbit becomes a homoclinic. This condition we use for obtain equations for bifurcation boundaries in the form of an explicit dependence on the parameters. This allow us to determine the regions of stability for periodic orbits and domains of the existence of four-, two- and one-band chaotic attractors in the parameter plane.

Removing Chaos while preserving equilibria by means of 2-contraction

Conditions for designing a feedback control law to remove the dense set of Unstable Periodic Orbits and chaotic attractors while preserving equilibria are formulated in terms of 2-contractivity of the closed loop system, by means of the second additive compound of the system's Jacobian. Matrix inequalities to allow computing the control gain matrix are discussed. An example of application of the method to the Thomas’ system is presented.

Global Dynamics of a Fisheries Economic Model with Gradient Adjustment

Taking into account the nonlinear demand function, we have developed a multi-agent fishery economic model, where a multitude of agents are bounded by rationality. The fishing decisions of these agents are driven by a profit gradient mechanism. To assess the local stability of the system, stability analysis is performed with the Jury criterion. The investigation has revealed the presence of two conventional paths to chaos, namely, the flip bifurcation and the Neimark–Sacker bifurcation. This was achieved by mapping the stability regions and stability curves of the Nash equilibrium. The multistability of the system is further explored on two-dimensional planes on which the influence of joint parameters on the system’s stability is demonstrated. The existence of Arnold’s tongue has demonstrated unparalleled complexity and intricate interactions across different scales of the system. Both critical curves and basins of attraction are illustrated to gain insight into global bifurcations. The chaotic attractor is found to be confined within specific boundaries. The findings clearly show higher maximum instantaneous demand, relatively slower adjustment speed, and lower price sensitivity. Arguably, a controlled cost would lead to sustainable fishing resources. Moreover, the results also suggest that the agents would benefit more from confined conditions.

Bifurcation Analysis of the Dynamics in COVID‐19 Transmission through Living and Nonliving Media

Transmission of COVID‐19 occurs either through living media, such as interaction with a sufferer, or nonliving objects contaminated with the virus. Recovering sufferers and disinfectant spraying prevent interaction between people and virus become the treatment to overcome it. In this research, we formulate a new mathematical model as a three‐dimensional ordinary differential equation system representing an interaction between viruses attached in nonliving media, susceptible, and infected subpopulations, including the treatment to investigate its effect. Disease‐free, sterile‐media endemic, and two nonsterile media endemic equilibriums exist in the model. The nonexistence of sterile‐media equilibria interpreting the nonendemic condition is achieved by crossing the branch point bifurcation of the equilibria point as the infected subpopulation recovery rate increases. Continuation of the limit cycle generated at a Hopf bifurcation point as susceptible‐coronavirus interaction prevention rate and period increase trigger two saddle‐node bifurcations and a branch point bifurcation of cycle. Stable symmetric cycles with decreasing amplitude that make the dynamic of subpopulation easier to control start to be gained at the branch point bifurcation of cycle between the two saddle‐node bifurcation points as the prevention rate increases. Some chaotic attractors which describe a complex and unpredictable pattern of the dynamic in the population are also found at inclination flip bifurcation by a continuation of a homoclinic orbit generated near the Bogdanov‐Takens bifurcation point as the prevention rate increases while the recovery rate decreases. Increasing the recovery and prevention rate along with avoiding an increase of the prevention rate while the recovery rate decreases becomes the treatment to optimize the effort in overcoming COVID‐19 transmission.

A multivariate method of forecasting the nonlinear dynamics of production network based on multilayer neural models

Design of production network based on multilayer neural models is considered in this paper. Design of production network is crucial because it determines the optimal location of production and logistics facilities, affects cost efficiency, customer service level and overall competitiveness in the global market. Multi-layer neural networks play an important role in this process, using advanced algorithms, machine learning models and optimization techniques to analyze huge amounts of data. Special attention is focused on qualitative analysis of dynamic behavior, dynamic lattice model. The model includes rate constants and initial conditions affecting the model trajectories, which can be classified as a stable site, limit cycle, or chaotic attractor. We aim to solve the problem of qualitative behavior of the model as a problem of multilayer neural models. A multivariate method of predicting nonlinear dynamics was used to construct the training data set. Neural networks defined by regenerative architectures with linear and non-linear outputs were analyzed and compared. As a result of the analysis, it was found that architectures with linear outputs show better correspondence between expected and predicted values. Architectures with non-linear outputs, despite their complexity, exhibit less accuracy and more deviation compared to linear ones. The single-layer architecture with linear outputs shows the best accuracy, although the two-layer architecture with linear outputs has the lowest rms error. Architectures with non-linear outputs have faster training times but poor accuracy, while architectures with linear outputs require more training time but have lower errors. The results obtained in the work indicate the importance of choosing the right architecture of the neural network depending on the tasks and requirements for accuracy and training time of the model.

Regular and chaotic dynamics in a 2D discontinuous financial market model with heterogeneous traders

We develop a financial market model where three types of traders operate simultaneously: fundamentalists and chartists of two types, namely, trend followers and contrarians. The dynamics of this model is described by a two-dimensional discontinuous map defined by two linear functions, where one acts in the partition between two (parallel) discontinuity lines and the other one acts outside this partition. Our analysis shows that despite the linearity of the map components, its dynamics can be quite complex, with various, possibly coexisting attracting cycles and chaotic attractors. As a first step towards the understanding how the overall bifurcation structure observed in the parameter space of the map is organized, we obtain analytically the boundaries of periodicity regions related to the simplest attracting period-n cycles, n≥3, with one point in the middle partition and n−1 points outside it. These boundaries can be related to border-collision bifurcations (when a point of the cycle collides with a discontinuity line) as well as to degenerate bifurcations (associated with eigenvalues on the unit circle). We show also some elements of period-adding and period-incrementing bifurcation structures for which the cycles mentioned above are basic. From an economic point of view, our study confirms that a fairly simple financial market model with heterogeneous agents is able to produce complicated boom-bust dynamics typical of real financial markets.

A study on the nonlinear relationship between market, subsidy, and income of photovoltaic enterprises based on chaos theory

With the annual promotion of the international “dual carbon” goals, countries attach great importance to the development and innovation of clean energy. The United States, Japan, and China have all created many policies for the research and market development of photovoltaic energy. This article incorporates market dynamic regulation capability into a two-dimensional system of government subsidy policies and photovoltaic revenue, constructs a three-dimensional dynamic nonlinear model based on market dynamic regulation capability, government subsidies, and enterprise revenue, and numerically simulates and analyzes the impact of parameter and initial value changes in the equation on enterprise revenue. The market dynamic regulation capability is obtained from Chaotic attractors and dynamic evolution graphs of the nonlinear evolution between government subsidies and corporate profits in different scenarios. Research has shown that: (1) Rapidly improving the dynamic regulation ability of the market cannot continuously increase the revenue of the photovoltaic industry; (2) The changes in market dynamics affect the dependence of enterprises on government subsidies; (3) The demand for government subsidies by enterprises gradually decreases with the increase of their own profits.

Controllable multistability of fractional-order memristive Henon map and its application in video encryption

In recent years, the use of discrete memristors to enhance chaotic maps has received increasing attention. The introduction of memristors increases the complexity of chaotic maps, making them suitable for engineering applications based on chaotic systems. In this work, a fractional-order discrete memristor exhibiting local activity and controllable asymptotic stability points is constructed by using multiband nonlinear functions. The locally active property of this memristor is demonstrated by using the power-off plot and DC <i> <b>v</b> </i> -<i> <b>i</b> </i> plot. It is then introduced into the Henon map to construct a fractional-order memristive Henon map that can generate any number of coexisting attractors. Simulation results show that the number of fixed points in the system is controlled by the memristor parameters and related to the number of coexisting attractors, thus achieving controllable homogeneous multistability. The complex dynamical behaviors of this map are analyzed by using phase portraits, bifurcation diagrams, maximum Lyapunov exponent (MLE), and attractor basins. Numerical simulations show that the fractional-order map can generate various periodic orbits, chaotic attractors, and period-doubling bifurcations. The system is then implemented on an ARM digital platform. The experimental results are consistent with the simulation results, confirming the accuracy of the theoretical analysis and its physical feasibility. Finally, a parallel video encryption algorithm is designed by using the chaotic sequence iteratively generated by fraction-order memory Henon mapping, which mainly includes frame pixel scrambling and diffusion. Comprehensive security analyses are conducted, proving the robustness and reliability of the proposed encryption scheme. The results show that the encryption algorithm can effectively protect video information. In the future, we will explore other methods of constructing chaotic or hyperchaotic systems with controllable multistability and study their circuit implementation, synchronization control, and chaos-based engineering applications.

An FHN-HR Neuron Network Coupled With a Novel Locally Active Memristor and Its DSP Implementation.

In this article, a novel locally active memristor (LAM) model is designed and its characteristics are studied in detail. Then, the LAM model is applied to couple FitzHugh-Nagumo (FHN) and Hindmarsh-Rose (HR) neuron. The simple neuron network is built to emulate connection of separate neurons and transmission of information from FHN neuron to HR neuron. The equilibrium point about this FHN-HR model is analyzed. Under the influence of varied parameters, dynamical characteristics for the model are explored with various analysis methods, including phase diagram, time series, bifurcation diagram, and Lyapunov exponent spectrum (LEs). The spectral entropy (SE) complexity and sequence randomness of the model are studied. In addition to observing chaotic and periodic attractors, multiple types of attractor coexistence and particular state transition phenomena are also found in the coupled FHN-HR model. Furthermore, geometric control is used for modulating the amplitude and offset of attractor and neuron firing signals, involving amplitude control and offset control. Finally, DSP implementation is finished, proving digital circuit feasibility of the FHN-HR model. The research imitates the coupling and information transmission between different neurons and has potential applications to secrecy or encryption.

Design and analysis of a novel self-excited 5-D hyperchaotic system with maximum (n-2) positive lyapunov exponents

In this work, analytic and designed a new self -existed 5-D with thirteen terms nine of the term exhibit linearity, while the remaining one demonstrates nonlinearity. It has the highest Lyapunov Exponent with three positive ones to leverage the sequence in data security, encryption, and other related aspects, it’s advantageous to utilize a complex system where the key remains unbreakable It contains fifteen keys sensitive ICs and parameters. The system is a highly intricate dynamical system characterized by two equilibrium points, as demonstrated through examinations such as observation of phase plane trajectories, the investigation into eigenvalue structure, and assessment of dissipative stability. These analyses provide evidence supporting the existence of chaotic and hyperchaotic attractors within the system. The results of the simulations are derived from MATLAB 2021b, and Mathematica 11. A detailed and comprehensive evaluation of the recommended techniques has been carried out, offering insights into their effectiveness and suitability for theoretical computation and numerical simulation.

A New Memristive System with Extreme Multistability and Hidden Chaotic Attractors and with Application to Image Encryption

Chaotic systems have proven highly beneficial in engineering applications. Pseudo-random numbers produced by chaotic systems have been used for secure communication, notably image encryption. Specific characteristics can increase the chaotic behavior of the system by adding complexity and nonlinearity. The three most well-known characteristics are memristive properties, multistability (coexisting attractors), and hidden attractors. These characteristics strengthen the produced time series’ unpredictability and randomness, strengthening an encryption algorithm’s resistance to many attacks. This study introduces a unique four-dimensional chaotic system with extreme multistability with respect to three initial conditions (including the memristor initial condition) and all previously known properties. It is rare to find an extreme multistable system like this. This system is coupled with a quadratic flux-controlled memristor based on the well-known Sprott J system. This system has a line of unstable equilibrium points with hidden attractors. The memristor displays the characteristic pinched hysteresis loops, where the area inside a loop and the voltage frequency are inversely related. A comprehensive dynamical analysis thoroughly examines all system characteristics and initial conditions. The numerical findings are carefully verified, and an analog circuit is successfully built and simulated. The chaotic sequences generated by this system are combined with deoxyribonucleic acid (DNA) operations and the global bit scrambling (GBS) technique to create an image encryption algorithm that has strong resistance to a variety of potential attacks, including noise, statistical, exhaustive, differential, and cropping attacks.

Bifurcation and chaos in simple discontinuous systems separated by a hypersurface

<abstract><p>This research focuses on a mathematical examination of a path to sliding period doubling and chaotic behaviour for a novel limited discontinuous systems of dimension three separated by a nonlinear hypersurface. The switching system is composed of dissipative subsystems, one of which is a linear systems, and the other is not linked with equilibria. The non-linear sliding surface is designed to improve transient response for these subsystems. A Poincaré return map is created that accounts for the existence of the hypersurface, completely describing each individual sliding period-doubling orbits that route to the sliding chaotic attractor. Through a rigorous analysis, we show that the presence of a nonlinear sliding surface and a set of such hidden trajectories leads to novel bifurcation scenarios. The proposed system exhibits period-$ m $ orbits as well as chaos, including partially hidden and sliding trajectories. The results are numerically verified through path-following techniques for discontinuous dynamical systems.</p></abstract>

Bifurcation analysis of double cavity flows

The first few bifurcations in a two-dimensional incompressible double cavity flow are investigated using the linear stability analysis, the Floquet analysis, and the nonlinear direct numerical simulations (DNS). The prediction of the critical Reynolds number and the type of bifurcation (Hopf, pitchfork, inverse pitchfork, and Neimark–Sacker), which depend on cavity configuration, by the linear stability analysis and the Floquet analysis is consistent with nonlinear DNS. The nonlinear DNS results show that the state of the system passes through multiple intermediate (unstable) states before it reaches the stable attractor (heteroclinic chain), and the type of intermediate states depends on initial conditions. The intermediate states are reported as the asymptotic state in the literature for some flow conditions because it is not known a priori how long it will take to reach the asymptotic state in nonlinear simulations. The present study reports the actual asymptotic state for those flow conditions.

Influence of seasonality on <i>Zika virus</i> transmission

<abstract><p>In order to study the impact of seasonality on <italic>Zika virus</italic> dynamics, we analyzed a non-autonomous mathematical model for the <italic>Zika virus</italic> (<italic>ZIKV</italic>) transmission where we considered time-dependent parameters. We proved that the system admitted a unique bounded positive solution and a global attractor set. The basic reproduction number, $ \mathcal{R}_0 $, was defined using the next generation matrix method for the case of fixed environment and as the spectral radius of a linear integral operator for the case of seasonal environment. We proved that if $ \mathcal{R}_0 $ was smaller than the unity, then a disease-free periodic solution was globally asymptotically stable, while if $ \mathcal{R}_0 $ was greater than the unity, then the disease persisted. We validated the theoretical findings using several numerical examples.</p></abstract>