Attractor Research Articles

Artificial Intelligence, Smart Topological Data Analysis and Chaos in Business Continuity Management: The Case of COVID-19 in Birmingham Airport

The latest state-of-the-art empirical methods from chaos theory have incorporated smart topological data analysis (STDA) combining chaos theory methods, topological machine learning, adaptive artificial intelligence systems, topological data analysis and fractal analysis methods for attractor reconstruction and the topological study of the dynamics, with impact on risk science and complexity research. In the current work, we apply a topological adaptive AI system to the study of Birmingham airport’s air traffic dynamics, and employ topological data analysis, chaos theory methods and multifractal analysis to research the resulting dynamics. Our results show the presence of a form of stochastic chaos with a low-dimensional attractor associated with a long wave dynamics in the pre-COVID-19 period, which continues in the COVID-19 crisis and subsequent recovering around a rising trend, with the topological AI system able to adapt to the COVID-19 crisis and predict with high performance the dynamics during this period. Multifractal analysis methods, applied to the adaptive topological AI system’s residuals, show that the dynamical noise affecting the chaotic attractor is multifractal, with a multifractal phase transition occurring during the COVID-19 recovery period. Implications of the methods and results for business continuity management are drawn.

Periodic solutions and chaotic attractors of a modified epidemiological SEIS model.

We consider a generalized SEIS (susceptible, exposed, infectious, and susceptible) model where individuals are divided into three compartments: S (healthy and susceptible), E (infected but not just infectious, or exposed), and I (infectious). Finite waiting times in the compartments yield a system of delay-differential or memory equations and may exhibit oscillatory (Hopf) instabilities of the otherwise stationary endemic state, leading normally to regular oscillations in the form of an attractive limit cycle in the phase space spanned by the compartment rates. In the present paper, our aim is to demonstrate that in the dynamics of delayed SEIS models, persistent chaotic attractors can bifurcate from these limit cycles and become accessible if the nonlinear interaction terms fulfill certain basic requirements, which to our knowledge were not addressed in the literature so far. Computing the largest Lyapunov exponent, we show that chaotic behavior exists in a wide parameter range. Finally, we discuss a more general system and show that a sudden falloff of the infection rate with respect to increasing infection number may be responsible for the emergence of chaotic time evolution. Such a falloff can describe mitigation measures, such as wearing masques, individual isolation, or vaccination. The model may have a wide range of interdisciplinary applications beyond epidemic spreading, for instance, in the kinetics of certain chemical reactions.

Dynamical analysis and preassigned-time intermittent control of memristive chaotic system via T-S fuzzy method.

In this paper, we propose a novel fourth-order memristive chaotic system (MCS), in which both its dynamical behaviors and the preassigned-time stabilization problem are analyzed. First, the dynamical behaviors of the proposed MCS are studied in detail, such as the infinite unstable equilibrium points, the chaotic attractor, the Lyapunov exponents, the Kaplan-Yorke dimension, and the bifurcation. Then, the T-S fuzzy method is employed to characterize the MCS, and a simpler model is built to deal with the nonlinearity caused by the memristor in the MCS. In addition, two intermittent controllers are proposed to guarantee the preassigned-time stability and the settling time, which can be set freely, independent of system parameters and initial state. Finally, numerical simulations provide solid confirmation for the validity of these theoretical results.

Analysis of Hopf bifurcation and time-delayed control in multi-scroll chaotic attractors produced by parallel transformation

Analysis of Hopf bifurcation and time-delayed control in multi-scroll chaotic attractors produced by parallel transformation

Construction and Attractor Analysis of Discrete Chaotic Systems Based on Julia Fractal Transformation

Fractals and chaotic systems play an important role in science and engineering. However, few scholars have focused on further deep integration of multiple fractals and multi-level cascaded chaotic systems to create new discrete chaotic maps. In this paper, we propose an innovative method to construct a series of new fractal chaotic maps by borrowing the concept of Julia fractal. Unlike previous fractal discrete map designs, this method enriches the dynamical behavior of the new maps by introducing the topological complexity of multiple fractal transformations into chaotic systems in terms of system structure. More importantly, it ensures that the new system dynamical structure satisfies the necessary condition for generating chaos through an internal feedback mechanism, i.e. the initial error is continuously transmitted in each state of the system iteration. The paper applies the method to eight types of discrete chaotic maps, generating various fractal chaotic maps, revealing new chaotic attractors as a result, and discussing the dynamical characteristics of discrete fractal chaotic maps through Lyapunov exponent, information entropy, spectral entropy, C[Formula: see text] complexity and bifurcation diagram analysis. The dynamical analysis indicates that the proposed fractal chaotic maps have higher complexity and stronger randomness. Consequently, the method proposed in this paper for generating new discrete chaotic maps through Julia fractal offers a novel approach to the design of chaotic systems, with promising potential for engineering and scientific applications.

On Product Neutrosophic Fractal Spaces and α-Density Theory with Arbitrarily Small and Controlled Error

In this manuscript, we present the classical Hutchinson–Barnsley theory on the product neutrosophic fractal spaces by utilizing an iterated function system, which is enclosed by neutrosophic Edelstein contractions and a finite number of neutrosophic b-contractions. Further, we provide a sequence of sets that, under appropriate conditions and in terms of the Hausdorff neutrosophic metric, converge to the attractor set of specific neutrosophic iterated function systems. Furthermore, we present a fuzzy variant of α-dense curves that can accurately approximate the attractor set of certain iterated function systems with barely noticeable and controlled errors. In the end, we make a connection between the above-discussed concepts of neutrosophic theory and α-density theory.

Dynamics of predator-prey system with the consequences of double Allee effect in prey population.

A underlying complex dynamical behavior of double Allee effects in predator-prey system is studied in this article to understand the predator-prey relation more intensely from different aspects. We first propose a system with the Caputo sense fractional-order predator-prey system incorporating the Allee effect in prey populations to explain how the memory effect can change the different emergent states. Local stability analysis is analyzed by applying Matignon's condition for the FDE system. Further, we consider a discrete-time system to show the influence of double Allee effects in non-overlapping generations. For discrete-time system, different bifurcations like Neimark-Sacker, flip bifurcations, irregularity in periodic oscillations, are observed. Irregularity occurs through a period-doubling cascade which is a common route to chaos in a dynamical sense. Maximum Lyapunov exponent (MLE) is shown to illustrate the irregular behaviors of discrete-time systems. The Allee effect influences system stability where the strong Allee effect enhances system stability whereas the stability is lost for the weak Allee effect. The extinction risk of populations in the presence of the Allee effect is a concerning issue. We have insight into how all populations survive along with stable extinction equilibrium. Our proposed systems exhibit different alternative states. Multiple stable attractor basins are plotted to depict the different alternative states of the FDE system as well as the discrete-time system. Initial population densities play a key role in the coexistence of all the populations otherwise there is a risk of species extinction. Besides analytical results, numerical simulation is performed to valid our analytical findings of different dynamical states like bifurcation, stability, irregularity as well as multi-stability.

Hyperbolic Sine Function Control-Based Finite-Time Bipartite Synchronization of Fractional-Order Spatiotemporal Networks and Its Application in Image Encryption

This work is devoted to the hyperbolic sine function (HSF) control-based finite-time bipartite synchronization of fractional-order spatiotemporal networks and its application in image encryption. Initially, the addressed networks adequately take into account the nature of anisotropic diffusion, i.e., the diffusion matrix can be not only non-diagonal but also non-square, without the conservative requirements in plenty of the existing literature. Next, an equation transformation and an inequality estimate for the anisotropic diffusion term are established, which are fundamental for analyzing the diffusion phenomenon in network dynamics. Subsequently, three control laws are devised to offer a detailed discussion for HSF control law’s outstanding performances, including the swifter convergence rate, the tighter bound of the settling time and the suppression of chattering. Following this, by a designed chaotic system with multi-scroll chaotic attractors tested with bifurcation diagrams, Poincaré map, and Turing pattern, several simulations are pvorided to attest the correctness of our developed findings. Finally, a formulated image encryption algorithm, which is evaluated through imperative security tests, reveals the effectiveness and superiority of the obtained results.

False Strange Attractors as Sources of Pseudo Randomness

Abstract In this work, the use of false chaotic attractors in enhancing the security of chaos-based encryption applications will be explored. False attractors result by rearranging the solution trajectories of a nonlinear dynamical system, leading to novel shapes in the 3D space. This way, a single solution trajectory of a chaotic system can be used to generate numerous new false trajectories. Examples of such attractors are provided for the Lorenz system, and a system with line equilibrium from Moysis et al. (2019). By formalizing the method of constructing false attractors, and generalizing it by adding a series of elementary operations, the key parameters of this procedure can be used as part of the secret key in symmetric encryption applications, effectively increasing the key space. Specifically, the key space increases significantly with respect to the simulation time, and the system’s dimension. Moreover, due to its fast implementation, this technique can be used as a less complex, but much lighter alternative to full scale permutation. The usability of the false attractors as sources of randomness is showcased through the design of a chaotic pseudo-random bit generator. The generator can output 64 bits per iteration, and is equally effective under any false attractor of the chaotic system.

The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization

The Ueda oscillator is one of the most popular and studied nonlinear oscillators. Whenever subjected to external periodic excitation, it exhibits a fascinating array of nonlinear behaviors, including chaos. This research introduces a novel fractional discrete Ueda system based on Y-th Caputo fractional difference and thoroughly investigates its chaotic dynamics via commensurate and incommensurate orders. Applying numerical methods like maximum Lyapunov exponent spectra, bifurcation plots, and phase plane. We demonstrate the emergence of chaotic attractors influenced by fractional orders and system parameters. Advanced complexity measures, including approximation entropy (ApEn) and C0 complexity, are applied to validate and measure the nonlinear and chaotic nature of the system; the results indicate a high level of complexity. Furthermore, we propose a control scheme to stabilize and synchronize the introduced Ueda map, ensuring the convergence of trajectories to desired states. MATLAB R2024a simulations are employed to confirm the theoretical findings, highlighting the robustness of our results and paving the way for future works.

Chaotic and Coexistence Attractors of Classical Complex Exotic Oscillator with Position-Dependent Mass

Chaotic and Coexistence Attractors of Classical Complex Exotic Oscillator with Position-Dependent Mass

The topology of a chaotic attractor in the Kuramoto-Sivashinsky equation.

The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within a chaotic attractor and the topology of that attractor, for three-dimensional systems. In certain cases, the fractal dimension of a chaotic attractor in a partial differential equation (PDE) is less than three, even though that attractor is embedded within an infinite-dimensional space. Here, we study the Kuramoto-Sivashinsky PDE at the onset of chaos. We use two different dimensionality-reduction techniques-proper orthogonal decomposition and an autoencoder neural network-to find two different mappings of the chaotic attractor into three dimensions. By finding the image of the attractor's UPOs in these reduced spaces and examining their linking numbers, we construct templates for the branched manifold, which encodes the topological properties of the attractor. The templates obtained using two different dimensionality reduction methods are equivalent. The organization of the periodic orbits is identical and consistent symbolic sequences for low-period UPOs are derived. While this is not a formal mathematical proof, this agreement is strong evidence that the dimensional reduction is robust, in this case, and that an accurate topological characterization of the chaotic attractor of the chaotic PDE has been achieved.

Comparison of integer-order chaotic attractors as randomness source in collision-free robotic exploration methods

Comparison of integer-order chaotic attractors as randomness source in collision-free robotic exploration methods

Multistability and organization of chaos and quasiperiodicity in a memristor-based Shimizu-Morioka oscillator under two-frequency excitation

In this paper we investigate the organization of chaos and quasiperiodicity in a parameter plane of a continuous-time three-dimensional nonautonomous dynamical system. More specifically, we investigate a memristor-based Shimizu-Morioka oscillator, where the external excitation is represented by the sum of two different sinusoidal functions with angular frequencies ω1 and ω2. Through a scan carried out in the (ω1, ω2) parameter plane, with the dynamical behavior of each point in the phase-space being characterized by the Lyapunov exponents spectrum, we show that this system presents chaos and quasiperiodicity regions, without presenting, however, periodicity regions. Parameter regions for which the multistability phenomenon was detected, also are observed. Basins of attraction of coexisting chaotic and quasiperiodic attractors, as well as the attractors themselves, are reported.

Vibrational resonance and chaos control in the canonical Chua’s circuit with a smooth nonlinear resistor

Vibrational resonance and chaos control in the canonical Chua’s circuit with a smooth cubic nonlinear resistor is investigated by an analog circuit experiment and a dynamical model. By adjusting the amplitude and frequency of the high-frequency signal while keeping other parameters constant, the system exhibits a resonant peak in its response to the weak low-frequency signal. Notably, when the amplitude of the high-frequency signal exceeds the critical threshold, the system undergoes a transition from a single-scroll chaotic attractor to a double-scroll chaotic attractor, marking the emergence of vibrational resonance. In particular, the maximum of the system’s response amplitude is insusceptible when the frequency of the high-frequency signal varies over a broad range, which indicates the strong robustness of the vibrational resonance in the present system. The experimental results are coincident with the numerical simulations. This research has potential applications in chaos control and weak signal detection.

Global Analysis of a Duopoly Game with Privatization of State-Owned Enterprises

This paper constructs a nonlinear dynamical model of a mixed ownership market duopoly game for environmental tax collection within the framework of bounded rationality expectations. A stability analysis of the proposed model is conducted, and a Nash equilibrium stability region is estimated using the Jury criterion. The effects of the degree of privatization and environmental tax rate on the stability are investigated through numerical simulations. Subsequently, the comprehensive global analysis is carried out using the composite cell coordinate system method. Two types of crisis phenomena, namely, interior and boundary crises, have been identified. The former is caused by the collision between chaotic or periodic attractors and interior chaotic saddle within the basin of attraction, and the latter is caused by the collision between chaotic attractor and periodic saddle on the basin boundary. Meanwhile, the global dynamic behavior of two private enterprises engaged in synchronized output adjustment is investigated. When the output adjustment speed varies, the structure of the basin of attraction undergoes a transformation as changes are made to the number of coexisting attractors. The sudden disappearance of one attractor is due to the collision of the chaotic attractor with the saddle, on the basin boundary, merging into a larger saddle and resulting in a boundary crisis. The emergence of numerous “holes” can also be observed. It is found that the presence of periodic or chaotic saddles enriches the dynamical phenomena. Finally, the chaotic phenomenon occurring in the mixed ownership market is controlled using the time-delay feedback control method, stabilizing the system in a stable state, which implies a steady-state of total market output. The study of this model can provide theoretical guidance for the decision-making process of enterprises, helping minimize profit losses while stabilizing the market.

Analysis and circuit implementation of fractional-order memristive hyperchaotic system with enhanced memory

Abstract In this paper, based on the memory characteristics of fractional calculus, a new fractional-order memristor is proposed. Fractional-order memristor is a more accurate description of memristor, which has richer dynamic behavior and better memory performance. Which has a stronger memorizability compared to other fractional-order memristor by analyzing the pinched hysteresis loop area. Based on the above fractional-order memristor, a fractional-order memristive hyperchaotic circuit is designed, such system is analyzed by using the Lyapunov Exponents and the bifurcation diagrams. With the change of system parameters, the phase trajectory of the system expands and narrows, and the amplitude of the chaotic attractor also changes. In addition, double chaotic attractors and coexisting attractors are found under different parameters and initial values. Finally, the fractional order memristor and the fractional order memristor hyperchaos circuit are realized by analog circuit in Multisim.

Phase-amplitude reduction-based imitation learning

In this study, we propose the use of the phase-amplitude reduction method to construct an imitation learning framework. Imitating human movement trajectories is recognized as a promising strategy for generating a range of human-like robot movements. Unlike previous dynamical system-based imitation learning approaches, our proposed method allows the robot not only to imitate a limit cycle trajectory but also to replicate the transient movement from the initial or disturbed state to the limit cycle. Consequently, our method offers a safer imitation learning approach that avoids generating unpredictable motions immediately after disturbances or from a specified initial state. We first validated our proposed method by reconstructing a simple limit-cycle attractor. We then compared the proposed approach with a conventional method on a lemniscate trajectory tracking task with a simulated robot arm. Our findings confirm that our proposed method can more accurately generate transient movements to converge on a target periodic attractor compared to the previous standard approach. Subsequently, we applied our method to a real robot arm to imitate periodic human movements.

A Novel Megastable Chaotic System with Hidden Attractors and Its Parameter Estimation Using the Sparrow Search Algorithm

This work proposes a new two-dimensional dynamical system with complete nonlinearity. This system inherits its nonlinearity from trigonometric and hyperbolic functions like sine, cosine, and hyperbolic sine functions. This system gives birth to infinite but countable coexisting attractors before and after being forced. These two megastable systems differ in the coexisting attractors’ type. Only limit cycles are possible in the autonomous version, but torus and chaotic attractors can emerge after transforming to the nonautonomous version. Because of the position of equilibrium points in different attractors’ attraction basins, this system can simultaneously exhibit self-excited and hidden coexisting attractors. This system’s dynamic behaviors are studied using state space, bifurcation diagram, Lyapunov exponents (LEs) spectrum, and attraction basins. Finally, the forcing term’s amplitude and frequency are unknown parameters that need to be found. The sparrow search algorithm (SSA) is used to estimate these parameters, and the cost function is designed based on the proposed system’s return map. The simulation results show this algorithm’s effectiveness in identifying and estimating parameters of the novel megastable chaotic system.

Periodic and chaotic behaviors of a compound pendulum driven by a horizontal periodic external force

Abstract Based on the oscillation experiments of a compound pendulum driven by a horizontal periodic external force, a nonlinear dynamical model is established and studied numerically. The periodic, quasiperiodic and chaotic orbits are found in the motions of the compound pendulum in response to several driving forces. The numerical results are in qualitative agreement with those in the experiments. It is found that the chaotic attractor in the three-dimensional phase space displays a two-dimensional torus structure. On a Poincare section, the chaotic attractor consists of the relative rotating core and its trailing tails. The fractal dimension of the chaotic attractor on the Poincare section is determined by using the box-counting method. The bifurcation diagram shows that the transition of the system from periodic motion to chaos is realized by the period-doubling bifurcation. The first Feigenbaum universal constant is approximately determined in the period-doubling bifurcation process. The competition between the inherent vibration and the external driving vibration of the system is thus thought as the physical mechanism for leading to the complex phenomena such as the period-doubling bifurcation and chaos. The numerical results combining with the experimental ones can provide an intuitive and in-depth understanding of chaotic phenomena, which is of great significance for the optimization design and the stability control in the engineering technology.