Trajectory Attractors Research Articles

Uniform attractors for the Bingham model

In this paper, on the base of the theory of attractors for non-invariant trajectory spaces, the limiting behavior of solutions to a periodic in spatial variables problem for the Bingham model is investigated. For the considered problem, necessary exponential estimates are established, the trajectory space is determined, and the existence of a minimal uniform trajectory and uniform global attractors is proved.

Emergence of Self-Replicating Hierarchical Structures in a Binary Cellular Automaton.

We have discovered a novel transition rule for binary cellular automata (CAs) that yields self-replicating structures across two spatial and temporal scales from sparse random initial conditions. Lower-level, shape-shifting clusters frequently follow a transient attractor trajectory, generating new clusters, some of which periodically self-duplicate. When the initial distribution of live cells is sufficiently sparse, these clusters coalesce into larger formations that also self-replicate. These formations may further form the boundaries of an expanding complex on an even larger scale. This rule, dubbed "Outlier," is rotationally symmetric and applies to 2-D Moore neighborhoods. It was evolved through genetic programming during an extensive search for rules that foster open-ended evolution in CAs. While self-replicating structures, both crafted and emergent, have been created in CAs with state sets intentionally designed for this purpose, the Outlier may be the first known rule to facilitate nontrivial emergent self-replication across two spatial scales in binary CAs.

Nonlinear analysis and recognition of epileptic EEG signals in different stages.

It is a hot problem in epilepsy research to detect and predict seizures by EEG signals. Clinically, it is generally observed that there are only sudden abnormal signals during the ictal stage, but there is no significant difference in the EEG signal between the interictal and preictal stages. To solve the problem that preictal signals are difficult to recognize clinically, and then effectively improve the recognition efficiency of epileptic seizures, so, in this paper, some nonlinear methods are comprehensively used to extract the hidden information in the EEG signals in different stages, namely, phase space reconstruction (PSR), Poincaré section (PS), synchroextracting transform (SET), and machine learning for EEG signal analysis. First, PSR based on C-C method is used, and the results show that there are different diffuse attractor trajectories of the signals in different stages. Second, the confidence ellipse (CE) is constructed by using the scatter diagram of the corresponding trajectory on PS, and the aspect ratio and area of the ellipse are calculated. The results show that there is an interesting transitional phenomenon in preictal stage. To recognize ictal and preictal signals, time-frequency (TF) spectrums, which are processed by SET, are fed into the convolutional neural network (CNN) classifier. The accuracy of recognizing ictal and preictal signals reaches 99.7% and 93.7%, respectively. To summarize, our results based on nonlinear method provide new research ideas for seizure detection and prediction.NEW & NOTEWORTHY Our results based on nonlinear method have better practical significance and clinical application value and improved the prediction efficiency of epileptic EEG signals effectively. This work provides direct insight into the application of these biomarkers for seizure detection and prediction.

Characterization of the friction-induced attractor trajectories during the running-in process in ball-on-disk tribosystem

PurposeThe purpose of this study is to solve the issues of time-consuming and complicated computation of traditional measures, as well as the underutilization of two-dimensional (2D) phase-trajectory projection matrix, so a new set of features were proposed based on the projection of attractors trajectory to characterize the friction-induced attractors and to reveal the tribological behavior during the running-in process.Design/methodology/approachThe frictional running-in experiments were conducted by sliding a ball against a static disk, and the friction coefficient was collected to reconstruct the friction-induced attractors. The projection of the attractors in 2D subspace was then mapped and the distribution of phase points was adapted to conduct the feature extraction.FindingsThe evolution of the proposed moment measures could be described as “initial rapid decrease/increase- midterm gradual decrease/increase- finally stable,” which could effectively reveal the convergence degree of the friction-induced attractors. Moreover, the measures could also describe the relative position of the attractors in phase–space domain, which reveal the amplitude evolution of signals to some extent.Originality/valueThe proposed measures could reveal the evolution of tribological behaviors during the running-in process and meet the more precise real-time running-in status identification.

An exact model for enhancing/suppressing primordial fluctuations

Abstract Enhancements of primordial curvature fluctuations in single field inflation often involve departures from attractor trajectories in the phase space. We study enhancement/suppression of primordial fluctuations in one of the simplest models with exact background solutions for arbitrary initial conditions: a single field inflationary model with a piecewise exponential potential. We then present close to exact analytical solutions for primordial fluctuations in a general transition between two slow-roll attractors, valid whether the first slow parameter increases or decreases. The main features in the primordial spectrum are determined by the ratio of exponents of the potential. We also discuss the imprint of such features in the induced GW spectrum. Lastly, we apply the δN formalism to discuss non-Gaussianities and the tail of the probability distribution. We find that while non-Gaussianities are at most 𝒪(1) in the case of enhancement, they can be very large in the case of suppression. Our work can be easily generalized to multiple piecewise exponential potentials.

Photoplethysmogram beat detection using Symmetric Projection Attractor Reconstruction.

Many methods have been proposed to detect beats in photoplethysmogram (PPG) signals. We present a novel method which uses the Symmetric Projection Attractor Reconstruction (SPAR) method to generate an attractor in a two dimensional phase space from the PPG signal. We can then define a line through the origin of this phase space to be a Poincaré section, as is commonly used in dynamical systems. Beats are detected when the attractor trajectory crosses the Poincaré section. By considering baseline drift, we define an optimal Poincaré section to use. The performance of this method was assessed using the WESAD dataset, achieving median F 1 scores of 74.3% in the Baseline phase, 63.0% during Stress, 93.6% during Amusement, and 97.7% during Meditation. Performance was better than an earlier version of the method, and comparable to one of the best algorithms identified in a recent benchmarking study of 15 beat detection algorithms. In addition, our method performed better than two others in the accuracy of the inter-beat intervals for two resting subjects.

Trajectory and Global Attractors for the Kelvin–Voigt Model Taking into Account Memory along Fluid Trajectories

This article is devoted to the study of the existence of trajectory and global attractors in the Kelvin–Voigt fluid model, taking into account memory along the trajectories of fluid motion. For the model under study, the concept of a weak solution on a finite segment and semi-axis is introduced and the existence of their solutions is proved. The necessary exponential estimates for the solutions are established. Then, based on these estimates, the existence of trajectory and global attractors in the problem under study is proved.

Self-Localization Using Trajectory Attractors in Outdoor Environments

Self-localization in probabilistic robotics requires detailed, geographically consistent environmental maps, which increases the computational cost. In this study, we propose a simple self-localization method that does not require such maps. In the proposed method, the order structure, such as the mobile robot’s navigation route, is embedded as trajectory attractors in the state space of a nonmonotone neural network, and self-position estimation is performed by processing based on the autonomous dynamics of the network. From experiments, we demonstrated the basic performance of the proposed method, including robust self-localization in complex outdoor environments. Furthermore, self-localization is possible on multiple courses with overlapping paths by suitably varying the network dynamics based on environmental information. While issues remain, this study points to the great potential of neurodynamics-based robotic self-localization.

Homogenization of Attractors to Ginzburg-Landau Equations in Media with Locally Periodic Obstacles: Critical Case

In this paper the Ginzburg-Landau equation is considered in locally periodic porous medium, with rapidly oscillating terms in the equation and boundary conditions. It is proved that the trajectory attractors of this equation converge in a weak sense to the trajectory attractors of the limit Ginzburg-Landau equation with an additional potential term. For this aim we use an approach from the papers and monographs of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of evolution equations. Also we apply homogenization methods appeared at the end of the XX-th century. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) equation and prove the existence of trajectory attractors for this equation. Then we formulate the main theorem and prove it with the help of axillary lemmas.

ON ATTRACTORS OF GINZBURG–LANDAU EQUATIONS IN DOMAIN WITH LOCALLY PERIODIC MICROSTRUCTURE. SUBCRITICAL, CRITICAL AND SUPERCRITICAL CASES

In the paper we consider a problem for complex Ginzburg–Landau equations in a medium with locally periodic small obstacles. It is assumed that on the obstacle surface one can have different conductivity coefficients. We prove that the trajectory attractors of this system converge in a certain weak topology to the trajectory attractors of the homogenized Ginzburg–Landau equations with an additional potential (in the critical case), without the additional potential (in the subcritical case) in a medium without obstacles, or simply disappear (in the supercritical case).

Cosmology in holographic non-minimal derivative coupling theory: Constraints from inflation and variation of gravitational constant

We consider a cosmological model of non-minimal derivative coupling (NMDC) to gravity with holographic effect from Bekenstein–Hawking entropy using Hubble horizon IR cutoff. Holographic parameter c is considered constant with value in a range, 0≤c<1. The NMDC effect is considered either as modification in kinetic scalar field or in gravitational constant. The NMDC effect allows gravitational constant to be time-varying. Since the NMDC effect is cosmological, definition of holographic density should include time-varying part of the gravitational constant. The NMDC part reduces strength of gravitational constant for κ>0 and opposite for κ<0. The holographic part enhances gravitational strength. Slow-roll parameters are derived. We use spectral index and tensor-to-scalar ratio to test the model against CMB constraint. Number of e-folding is chosen to be N≥60. Power-law scalar potentials, V=V0ϕn with n=2,4, and exponential potential, V=V0exp(−βϕ) are considered. Combined parametric plots of κ and ϕ show that the allowed regions of the power spectrum index and of the tensor-to-scalar ratio are not overlapping. NMDC inflation is ruled out and the holographic NMDC inflation is also ruled out for 0<c<1. NMDC significantly changes major anatomy of the dynamics, i.e. it gives new late-time attractor trajectories in acceleration regions. The holographic part clearly affects pattern of trajectories. However, for the holographic part to affect shape of the acceleration region, the NMDC field must be in presence. To constrain the model at late time, variation of gravitational constant is considered. Gravitational-wave standard sirens and supernovae data give a constraint, Ġ/G|t0≲3×10−12year−1 (Zhao et al., 2018) which, for this model, results in 10−12year−1≳−κϕ̇ϕ̈/MP2. Positive κ is favored and greater c2 results in lifting up lower bound of κ.

ON ASYMPTOTICS OF ATTRACTORS TO NAVIER-STOCKES SYSTEM IN ANISOTROPIC MEDIUM WITH SMALL PERIODIC OBSTACLES

The paper considers a two-dimensional system of Navier–Stokes equations in medium with anisotropic variable viscosity and periodic small obstacles. It is proved that the trajectory attractors of this system tend in a certain weak topology to the trajectory attractors of the averaged system of Navier–Stokes equations with an additional potential in a medium without obstacles.

FUNDAMENTALS OF MANAGEMENT OF INCLUSIVE DEVELOPMENT OF BUSINESS STRUCTURES OF AGRIBUSINESS

Entrepreneurship plays a vital role in the development of agriculture. Its development in agricultural production requires a balanced approach to implementing agricultural and food policy. It is emphasized that an essential component of managing the inclusive development of agribusiness entrepreneurial structures is the control over implementing an investment business project in the inclusive sphere of activity. The need to establish an inefficient management method in months of a significant decrease in profitability was emphasized. The structure of socioeconomic standards for checking the methods of managing the inclusive development of enterprises on the example of an agricultural firm is considered. Management of the inclusive development of entrepreneurial structures of agribusiness involves the implementation of evaluation, restoration, and increase of its pace. The evaluation takes place with the help of the indicator of inclusive development and the restoration and increase of development rates – with the help of management technology by goals and a bank of methods and techniques for adjusting methods of managing agribusiness business structures. The scheme of using business methods and techniques for adjusting the methods of managing the inclusive development of enterprises is detailed. Examples of the restoration of inclusive development by the rings of the chain of adding consumer properties to the product and the components of the complex activity of the entrepreneurial structure of agribusiness are considered. The practical difficulties that today accompany the development of an inclusive model of economic growth in agribusiness are analyzed. It is proven that the management of an enterprise or business is more concrete and is explained as a purposeful management of all divisions and links of the enterprise for organization and coordination. The theoretical basis of the above provisions is the mechanism of transferring entrepreneurial structures of agribusiness to the state of inclusive development, which, unlike the existing one, is based on the coordination of the market signal (about the vector of the leading consumer property of the product) with a particular trajectory (attractor) of the inclusive development of the business entity.

On the constant roll complex scalar field inflationary models

In this paper we wish to point out the possibility of using a complex scalar field in a constant roll inflationary model, as needed for observational viability. We extend the idea of real field inflaton with constant rate of roll to a complex field, showing the feasibility of solving Einstein Klein-Gordon equations constrained by an appropriate form of constant roll definition. As compared to the well known (two-parametric class of) real field models, there is one more degree of flexibility in constant roll inflationary solutions which is represented by an arbitrary function of time, γ(t). We work with an arbitrary but constant function γ (where γ = 0 refers to the corresponding real field model) and find new inflationary class of potentials. In this class of models, the behavior of real and complex field models are similar in some aspects, for example the solutions with large constant roll parameter are not stable and should be considered as early time transients. These field solutions relax at late time on a dual attractor trajectory. However, complex fields phase space trajectories reach this stable regime after real fields. We performed the stability analysis on γ function space solutions and found that dynamically stable trajectories in phase space are stable under γ variations. We extended this study by considering multifield models of constant roll inflation with non-canonical kinetic terms. By enlarging the size of field space, we showed that a multifield constant roll model is dynamically a single field effective theory. If field space is parametrized by N non-canonical fields, there will be N free parameters in the potential that can be attributed to the interaction between the fields.

Ramified continua as global attractors of C 1-smooth self-maps of a cylinder close to skew products

In the paper we consider -smooth self-maps of a cylinder close to -smooth skew products (and satisfying some additional conditions). We study such geometric property of the maps, as existence of -smooth invariant local lamination, and apply this geometric property to the proof of the geometric integrability of maps under consideration. Using obtained results we construct the example of the family of -smooth maps close to skew products so that each map from this family admits the global attractor, which is a one-dimensional ramified continuum with a complicated topological structure. The global attractor of every map from the family under consideration consists of arcs of two types. On the unique circle (which is the arc of first type) the map is mixing; on arcs of second type of different lengths homeomorphic to a closed interval (the family of such arcs has continuum cardinality) the map is not mixing. The topological structure of the global attractor and dynamical properties of trajectories on the attractor lead to the property of dense intermittency (in the complement to the attractor) of attraction sets of different ω-limit sets, the union of which coincides with the global attractor.

Strongly compact strong trajectory attractors for evolutionary systems and their applications

We show that for any fixed accuracy and time length T, a finite number of T-time length pieces of the complete trajectories on the global attractor are capable of uniformly approximating all trajectories within the accuracy in the natural strong metric after sufficiently large time when the observed dissipative system is asymptotically compact. Moreover, we obtain the strong equicontinuity of all the complete trajectories on the global attractor. These results follow by proving the existence of a strongly compact strong trajectory attractor. The notion of a trajectory attractor was previously constructed for a family of auxiliary systems including the originally considered one without uniqueness. Recently, Cheskidov and the author developed a new framework called evolutionary system, with which a (weak) trajectory attractor can be actually defined for the original system. In this paper, the theory of trajectory attractors is further developed in the natural strong metric for our purpose. We then apply it to both the 2D and the 3D Navier–Stokes equations and a general nonautonomous reaction–diffusion system.

Application of Fatou’s Lemma for Strong Homogenization of Attractors to Reaction–Diffusion Systems with Rapidly Oscillating Coefficients in Orthotropic Media with Periodic Obstacles

We study reaction–diffusion systems with rapidly oscillating terms in the coefficients of equations and in the boundary conditions, in media with periodic obstacles. The non-linear terms of the equations only satisfy general dissipation conditions. We construct trajectory attractors for such systems in the strong topology of the corresponding trajectory dynamical systems. By means of generalized Fatou’s lemma we prove the strong convergence of the trajectory attractors of considered systems to the trajectory attractors of the corresponding homogenized reaction–diffusion systems which contain an additional potential.

Attractors of Ginzburg–Landau equations with oscillating terms in porous media: homogenization procedure

We consider the Ginzburg–Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg–Landau equation with a ‘strange term’ (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor.

Pullback trajectory attractor for nonautonomous wave equations

We consider the dynamical behavior for nonautonomous wave equations without uniqueness. The existence of both pullback weak and strong trajectories is proved. In order to obtain pullback strong trajectory attractor, we use the measure of noncompactness to verify evolutionary process is pullback strong asymptotically compact. The external force is not needed to be translation bounded.

The Rayleigh–Bénard Problem for Compressible Fluid Flows

We consider the physically relevant fully compressible setting of the Rayleigh–Bénard problem of a fluid confined between two parallel plates, heated from the bottom, and subjected to gravitational force. Under suitable restrictions imposed on the constitutive relations we show that this open system is dissipative in the sense of Levinson, meaning there exists a bounded absorbing set for any global-in-time weak solution. In addition, global-in-time trajectories are asymptotically compact in suitable topologies and the system possesses a global compact trajectory attractor mathcal {A}. The standard technique of Krylov and Bogolyubov then yields the existence of an invariant measure—a stationary statistical solution sitting on mathcal {A}. In addition, the Birkhoff–Khinchin ergodic theorem provides convergence of ergodic averages of solutions belonging to mathcal {A} a.s. with respect to the invariant measure.