Types Of Attractors Research Articles

Sub-Synchronous Oscillations and Transient Stability of Islanded Microgrid

Sustained sub-synchronous oscillations are observed in an islanded microgrid consisting of grid-forming converters under a large transient disturbance. Such oscillations are studied in terms of various types of hidden attractors, namely, hidden periodic orbit, hidden quasi-periodic orbit, and hidden chaotic attractor, which are shown to coexist with the stable equilibrium point, share different basins of attraction, and be transformed mutually for different parameter values. Moreover, the basins of attraction have fractal boundaries, making the nearby trajectories unpredictable. The transient stability of the islanded microgrid can be improved by eliminating the hidden attractors through adjusting control parameters and triggering a homoclinic bifurcation. Furthermore, transient disturbances can be mitigated by protecting critical converters and accelerating the restarting process. Besides, the emergence of sub-synchronous oscillations can be eliminated through tripping off one or more specific converters. Full-circuit cycle-by-cycle simulations are performed for islanded microgrids comprising i) 5 grid-forming converters, ii) 5 grid-forming converters and a synchronous generator, and iii) 30 grid-forming converters to verify the analytical findings.

Asymptotics of singularly perturbed damped wave equations with super-cubic exponent

<p style='text-indent:20px;'>This work is devoted to studying the relations between the asymptotic behavior for a class of hyperbolic equations with super-cubic nonlinearity and a class of heat equations, where the problem is considered in a smooth bounded three dimensional domain. Based on the extension of the Strichartz estimates to the case of bounded domain, we show the regularity of the pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation. Then we prove that all types of non-autonomous attractors converge, upper semicontiously, to the natural extension global attractor of the limit parabolic equations.</p>

Complexity Analysis of a 2D‐Piecewise Smooth Duopoly Model: New Products versus Remanufactured Products

Recent studies on remanufacturing duopoly games have handled them as smooth maps and have observed that the bifurcation types that occurred in such maps belong to generic classes like period‐doubling or Neimark‐Sacker bifurcations. Since those games yield piecewise smooth maps, their bifurcations belong to the so‐called border‐collision bifurcations, which occur when the map’s fixed points cross the borderline between the smooth regions in the phase space. In the current paper, we present a proper systematic analysis of the local stability of the map’s fixed points both analytically and numerically. This includes studying the border‐collision bifurcation depending on the map’s parameters. We present different multistability scenarios of the dynamics of the game’s map and show different types of periodic cycles and chaotic attractors that jump from one region to another or just cross the borderline in the phase space.

The influence of fear on the dynamic of an eco-epidemiological system with predator subject to the weak Allee effect and harvesting

In this paper, an eco-epidemiological prey-predator system when the predator is subjected to the weak Allee effect, and harvesting was proposed and studied. The set of ordinary differential equations that simulate the system’s dynamic is constructed. The impact of fear and Allee’s effect on the system's dynamic behavior is one of our main objectives . The properties of the solution of the system were studied. All possible equilibrium points were determined, and their local, as well as global stabilities, were investigated. The possibility of the occurrence of local bifurcation was studied. Numerical simulation was used to further evaluate the global dynamics and understood the effects of varying parameters on the asymptotic behavior of the system. It is observed that the system has different types of attractors including stable point, periodic, and even bi-stable behavior.

Коммуникативное текстовое событие и способы его аттрактивного конструирования в медийном пространстве (на материале русского и корейского языков)

The study aims to reveal the role that the attractivity category plays in the construction of communicative event discourse. Linguocognitive attractivisation mechanisms are modelled, the characteristics of the main types of attractors are described using the example of the Russian and Korean languages. Scientific novelty of the study is determined by a new approach to developing ways to achieve attractivity and effectiveness in a discourse of the promoting type. It is the first time that these ways are considered as linguocognitive mechanisms that activate the addressee’s thinking on the basis of breaking the coherence of media message interpretation processes. As a result of the study, the researchers have identified the main types of cognitive mechanisms of attraction, have characterised the said types.

Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System

In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem.

On rigorous integration of continuous piecewise linear systems

In this work, rigorous methods to compute solutions of continuous piecewise linear systems are presented. A general technique, based on integrating a perturbed linear system, is described to compute enclosures of trajectories crossing planes where the vector field in not smooth. Algorithms to compute enclosures of trajectories passing arbitrarily close to unstable equilibria are proposed. The usefulness of the methods is shown by proving the existence of trapping regions enclosing three different types of chaotic attractors existing in the Chua’s circuit.

Functional diversity buffers the effects of a pulse perturbation on the dynamics of tritrophic food webs.

Biodiversity decline causes a loss of functional diversity, which threatens ecosystems through a dangerous feedback loop: This loss may hamper ecosystems’ ability to buffer environmental changes, leading to further biodiversity losses. In this context, the increasing frequency of human‐induced excessive loading of nutrients causes major problems in aquatic systems. Previous studies investigating how functional diversity influences the response of food webs to disturbances have mainly considered systems with at most two functionally diverse trophic levels. We investigated the effects of functional diversity on the robustness, that is, resistance, resilience, and elasticity, using a tritrophic—and thus more realistic—plankton food web model. We compared a non‐adaptive food chain with no diversity within the individual trophic levels to a more diverse food web with three adaptive trophic levels. The species fitness differences were balanced through trade‐offs between defense/growth rate for prey and selectivity/half‐saturation constant for predators. We showed that the resistance, resilience, and elasticity of tritrophic food webs decreased with larger perturbation sizes and depended on the state of the system when the perturbation occurred. Importantly, we found that a more diverse food web was generally more resistant and resilient but its elasticity was context‐dependent. Particularly, functional diversity reduced the probability of a regime shift toward a non‐desirable alternative state. The basal‐intermediate interaction consistently determined the robustness against a nutrient pulse despite the complex influence of the shape and type of the dynamical attractors. This relationship was strongly influenced by the diversity present and the third trophic level. Overall, using a food web model of realistic complexity, this study confirms the destructive potential of the positive feedback loop between biodiversity loss and robustness, by uncovering mechanisms leading to a decrease in resistance, resilience, and potentially elasticity as functional diversity declines.

A Hidden Chaotic System with Multiple Attractors.

This paper reports a hidden chaotic system without equilibrium point. The proposed system is studied by the software of MATLAB R2018 through several numerical methods, including Largest Lyapunov exponent, bifurcation diagram, phase diagram, Poincaré map, time-domain waveform, attractive basin and Spectral Entropy. Seven types of attractors are found through altering the system parameters and some interesting characteristics such as coexistence attractors, controllability of chaotic attractor, hyperchaotic behavior and transition behavior are observed. Particularly, the Spectral Entropy algorithm is used to analyze the system and based on the normalized values of Spectral Entropy, the state of the studied system can be identified. Furthermore, the system has been implemented physically to verify the realizability.

A New Criterion Beyond Divergence for Determining the Dissipation of a System: Dissipative Power

Divergence is usually used to determine the dissipation of a dynamical system, but some researchers have noticed that it can lead to elusive contradictions. In this article, a criterion, dissipative power, beyond divergence for judging the dissipation of a system is presented, which is based on the knowledge of classical mechanics and a novel dynamic structure by Ao. Moreover, the relationship between the dissipative power and potential function (or called Lyapunov function) is derived, which reveals a very interesting, important, and apparently new feature in dynamical systems: to classify dynamics into dissipative or conservative according to the change of “energy function” or “Hamiltonian,” not according to the change of phase space volume. We start with two simple examples corresponding to two types of attractors in planar dynamical systems: fixed points and limit cycles. In judging the dissipation by divergence, these two systems have both the elusive contradictions pointed by researchers and new ones noticed by us. Then, we analyze and compare these two criteria in these two examples, further consider the planar linear systems with the coefficient matrices being the four types of Jordan’s normal form, and find that the dissipative power works when divergence exhibits contradiction. Moreover, we also consider another nonlinear system to analyze and compare these two criteria. Finally, the obtained relationship between the dissipative power and the Lyapunov function provides a reasonable way to explain why some researchers think that the Lyapunov function does not coexist with the limit cycle. Those results may provide a deeper understanding of the dissipation of dynamical systems.

On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities

This article discusses the search procedure for Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system, using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated, using the modified Benettin’s algorithm to study the stability of the found regime and confirm the type of attractor.

Multistability analysis and nonlinear vibration for generator set in series hybrid electric vehicle through electromechanical coupling.

The non-linear analysis of undesired vibrations observed on hybrid electric vehicle (HEV) powertrains is hardly developed in the literature. In this paper, a mathematical modeling of the vibrations observed at the level of the electromechanical coupling between the internal combustion engine and the generator in the series architecture of HEVs, named (SHEVs), is established using the Lagrangian theory. The stability and instability motions of this SHEV are perfectly detailed using amplitude-frequency response curves. An analysis of the electromagnetic torque amplitude of the new SHEV demonstrates the presence of multistability with the coexistence of two or three different types of attractors. In addition, this new SHEV model has other dynamic regimes of chaotic and periodic oscillations. Coexisting bifurcations with parallel branches, hysteresis, and period-doubling are also discovered. A unique contribution of this work is the abundance and complicated dynamical behaviors found in such types of systems compared with some rare cases previously reported on HEV powertrain models. The simulation results obtained using non-linear analysis tools sufficiently demonstrate that the objectives of this paper are achieved.

Agglomerative Effects of Crime Attractors and Generators on Street Robbery? An Assessment by Luojia 1-01 Satellite Nightlight

Scholars have long confirmed the agglomerative effect in retail geography. The colocation of multiple shops would encourage customers to make multiple stops in a single shopping trip. Does the agglomerative effect exist in illegal activities such as crime? What is a viable measurement of the colocation of facilities that attract or generate crime? These questions have never been explicitly addressed in crime geography. Numerous studies explain crime by the count of facilities, but the count variable ignores the size variation among the facilities. Because facilities are typically associated with lights at night, this study uses Luojia 1-01 satellite nightlight and a Gini coefficient–based adjuster to infer the agglomerative impact of crime attractors and generators in Cincinnati, Ohio. Results show that nightlights have a strong spatial correlation with the facilities and they can effectively capture large-sized and moderate-sized clusters of diverse types of crime attractors and generators. Additionally, negative binomial regression models compare the impacts of these different measures on crime by controlling potential confounding variables representing social disorganization. Results show that the nightlight models outperform the count models. Such advantage is more pronounced in areas where the crime rates are high. This is certainly an encouraging outcome, because both crime research and crime prevention tend to focus on high-crime areas. In sum, this preliminary study on the possible agglomerative effect in illegal activities reveals that nightlight data can effectively measure the agglomeration of crime attractors and generators and that the agglomerative nightlight explains crime better than the popular measure of facility counts.

A new 3D multi-scroll chaotic system generated with three types of hidden attractors

A new 3D multi-scroll chaotic system generated with three types of hidden attractors

Asymmetry Evolvement and Controllability of a Symmetric Hyperchaotic Map

Trigonometric functions were used to construct a 2-D symmetrical hyperchaotic map with infinitely many attractors. The regime of multistability depends on the periodicity of the trigonometric function, which is closely related to the initial condition. For this trigonometric nonlinearity and the introduction of an offset controller, the initial condition triggers a specific multistability evolvement, in which infinitely countless symmetric and asymmetric attractors are produced. Initial condition-triggered offset boosting is explored, combined with constant controlled offset regulation. Furthermore, this symmetric map gives the sequences in various types of asymmetric attractors, in which the polarity balance is maintained by the initial condition and a negative coefficient due to the trigonometric function. Finally, as determined through the hardware implementation of STM32, the corresponding results agree with the numerical simulation.

Urban attractors: Discovering patterns in regions of attraction in cities.

Understanding the dynamics by which urban areas attract visitors is important in today’s cities that are continuously increasing in population towards higher densities. Identifying services that relate to highly attractive districts is useful to make policies regarding the placement of such places. Thus, we present a framework for classifying districts in cities by their attractiveness to daily commuters and relating Points of Interests (POIs) types to districts’ attraction patterns. We used Origin-Destination matrices (ODs) mined from cell phone data that capture the flow of trips between each pair of places in Riyadh, Saudi Arabia. We define the attraction profile for a place based on three main statistical features: The number of visitors a place received, the distribution of distance traveled by visitors on the road network, and the spatial spread of locations from where trips started. We used a hierarchical clustering algorithm to classify all places in the city by their features of attraction. We discovered three main types of Urban Attractors in Riyadh during the morning period: Global, which are significant places in the city, Downtown, which contains the central business district, and Residential attractors. In addition, we uncovered what makes districts possess certain attraction patterns. We used a statistical significance testing approach to quantify the relationship between Points of Interests (POIs) types (services) and the patterns of Urban Attractors detected.

A new chaotic multi-stable hyperjerk system with various types of attractors

In this paper, a multi-stable chaotic hyperjerk system with both self-excited and hidden attractors is proposed. Such a system is infrequent between dynamical systems. State-space, bifurcation, and Lyapunov exponent plots are presented to show the existence of chaotic dynamics. The fractional-order model of the system and its dynamical properties are studied in this paper.

On scenarios of the onset of homoclinic attractors in three-dimensional non-orientable maps

We study scenarios of the appearance of strange homoclinic attractors (which contain only one fixed point of saddle type) for one-parameter families of three-dimensional non-orientable maps. We describe several types of such scenarios that lead to the appearance of discrete homoclinic attractors including Lorenz-like and figure-8 attractors (which contain a saddle fixed point) as well as two types of attractors of spiral chaos (which contain saddle-focus fixed points with the one-dimensional and two-dimensional unstable manifolds, respectively). We also emphasize peculiarities of the scenarios and compare them with the known scenarios in the orientable case. Examples of the implementation of the non-orientable scenarios are given in the case of three-dimensional non-orientable generalized Hénon maps.

Hysteresis bifurcation and application to delayed FitzHugh-Nagumo neural systems

Hysteresis dynamics has been described in a vast number of biological experimental studies. Many such studies are phenomenological and a mathematical appreciation has not attracted enough attention. In the paper, we explore the nature of hysteresis and study it from the dynamical system point of view by using the bifurcation and perturbation theories. We firstly make a classification of hysteresis according to the system behaviours transiting between different types of attractors. Then, we focus on a mathematically amenable situation where hysteretic movements between the equilibrium point and the limit cycle are initiated by a subcritical Hopf bifurcation and a saddle-node bifurcation of limit cycles. We present a analytical framework by using the method of multiple scales to obtain the normal form up to the fifth order. Theoretical results are compared with time domain simulations and numerical continuation, showing good agreement. Although we consider the time-delayed FitzHugh-Nagumo neural system in the paper, the generalization should be clear to other systems or parameters. The general framework we present in the paper can be naturally extended to the notion of bursting activity in neuroscience where hysteresis is a dominant mechanism to generate bursting oscillations.

BIFURCATIONS AND AMPLITUDE DEATH FROM DISTRIBUTED DELAYS IN COUPLED LANDAU STUART OSCILLATORS AND A CHAOTIC PARAMETRICALLY FORCED VAN DER POL-RAYLEIGH SYSTEM

Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are close via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all three systems, with oscillations at larger values (corresponding to weaker delay). In the Landau-Stuart system, the Hopf-generated limit cycles for $a>a_{Hopf}$ turn out to be remarkably stable under very large variations of all other system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. This is to be expected as the corresponding undelayed systems are robust oscillators over wide ranges of their respective parameters. Numerical simulations reveal strong distortion and rotation of the limit cycles in phase space as the parameters are pushed far into the post-Hopf regime, and reveal other features, such as how the oscillation amplitudes and time periods of the physical variables on the limit cycle attractor change as the delay and other parameters are varied. For the chaotic system, very strong delays may still lead to the cessation of oscillations and the onset of AD (even for relatively large values of the system forcing which tends to oppose this phenomenon). Varying of the other important system parameter, the parametric excitation, leads to a rich sequence of dynamical behaviors, with the bifurcations leading from one regime (or type of attractor) into the next being carefully tracked.