Attractors Of Dynamical Systems Research Articles

On maximal attractors for dynamical systems. Application to a semilinear parabolic problem with controlled growth

On maximal attractors for dynamical systems. Application to a semilinear parabolic problem with controlled growth

Hidden Attractors in Chaotic Systems with Nonlinear Functions

In the present work, an interesting mini-review of hidden attractors in dynamical systems with associated nonlinear functions is carried out. Chaotic systems with nonlinear functions often possess hidden attractors due to their inherent complexity. These attractors can arise in various mathematical models, such as the Lorenz system, Rössler system, or Chua's circuit. The identification and comprehension of hidden attractors broaden our understanding of complex systems and provide new directions for future study and technological development. The discovery and characterization of hidden attractors in chaotic systems have profound implications for various scientific disciplines, including physics, biology, and engineering.

Topological stability for homeomorphisms with global attractor

Abstract We prove that every topologically stable homeomorphism with global attractor of $\mathbb {R}^n$ is topologically stable on its global attractor. The converse is not true. On the other hand, if a homeomorphism with global attractor of a locally compact metric space is expansive and has the shadowing property, then it is topologically stable. This extends the Walters stability theorem (Walters, On the pseudo-orbit tracing property and its relationship to stability. The structure of attractors in dynamical systems, 1978, pp. 231–244).

On attractors in dynamical systems modeling genetic networks

The system of ordinary differential equations that arises in the theory of genetic networks is studied. Attracting sets of a special kind is the focus of the study. These attractors appear as combinations of attractors of lower dimensions,
 which are stable limit cycles. The properties of attractors are studied. Visualizations and examples are provided.

Wild Cantor actions

The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations.

On a High-Precision Method for Studying Attractors of Dynamical Systems and Systems of Explosive Type

The author of this article considers a numerical method that uses high-precision calculations to construct approximations to attractors of dynamical systems of chaotic type with a quadratic right-hand side, as well as to find the vertical asymptotes of solutions of systems of explosive type. A special case of such systems is the population explosion model. A theorem on the existence of asymptotes is proved. The extension of the numerical method for piecewise smooth systems is described using the Chua system as an example, as well as systems with hysteresis.

Properties of Neighborhoods of Attractors of Dynamical Systems

Properties of Neighborhoods of Attractors of Dynamical Systems

Climate Patterns: Origin and Forcing

This brief review described spatial-time climate patterns generated by the dynamics and thermodynamics of the Earth’s climate system and methods of identifying these patterns. Specifically, it does discuss the following major climate patterns: El Ni?o-Southern Oscillation (ENSO), Cold Ocean-Warm Land (COWL) pattern, Northern and Southern Annular Patterns (NAM and SAM), Atlantic Multidecadal Oscillation (AMO) and Atlantic Meridional Overturning Circulation (AMOC), Pacific North-American Pattern (PNA) and Pacific Decadal Oscillation Pattern (PDO). In view of an extensive number of publications on some climate patterns, such as the ENSO, which discussed in many hundred of publications, this review is not intended to cover all the details of individual climate patterns but intends only to give a general overview of their structure, mechanisms of their formation and response to external forcing. It is assumed that the climate patterns can be treated as attractors of dynamical systems allowing us to extract and predict some specific features of the patterns such as the origin and evolution of the climate patterns and their role in climate change. Thus the knowledge of patterns allows the climate prediction on long time scales and understanding of how an external forcing affects the frequency of occurrence of climate patterns and their magnitude but not the spatial structure.

Properties of Neighborhoods of Attractors of Dynamical Systems

Статья посвящена изучению динамических систем, допускающих компактные аттракторы, которые описывают поведение траекторий системы на бесконечности и играют важную роль в качественной теории устойчивости движения. Устанавливаются свойства траекторий динамической системы в области притяжения аттракторов и на границе с привлечением таких понятий, как эллиптические и слабо эллиптические множества, а также свойств псевдоустойчивости и псевдопролонгаций. Библиография: 41 название.

Investigation of denoising effects on forecasting models by statistical and nonlinear dynamic analysis

Abstract In this study, the denoising effect on the performance of prediction models is evaluated. The 13-year daily data (2002–2015) of hydrological time series for the sub-basin of Parishan Lake of the Helle Basin in Iran were used to predict time series. At first, based on observational precipitation and temperature data, the prediction was performed, using the ARIMA, ANN-MLP, RBF, QES, and GP prediction models (the first scenario). Next, time series were denoised using the wavelet transform method, and then the prediction was made based on the denoised time series (the second scenario). To investigate the performance of the models in the first and second scenarios, nonlinear dynamic and statistical analysis, as well as chaos theory, was used. Finally, the analysis results of the second scenario were compared with those of the first scenario. The comparison revealed that denoising had a positive impact on the performance of all the models. However, it had the least influence on the GP model. In the time series produced by all the models, the error rate, embedding dimension needed to describe the attractors in dynamical systems and entropy decreased, and the correlation and autocorrelation increased.

On boundaries of attractors in dynamical systems

Fractal geometry is one of the beautiful and challenging branches of mathematics. Self similarity is an important property, exhibited by most of the fractals. Several forms of self similarity have been discussed in the literature. Iterated Function System (IFS) is a mathematical scheme to generate fractals. There are several variants of IFSs such as condensation IFS, countable IFS, etc. In this paper, certain properties of self similar sets, using the concept of boundary are discussed. The notion of boundaries like similarity boundary and dynamical boundary are extended to condensation IFSs. The relationships and measure theoretic properties of boundaries in dynamical systems are analyzed. Self similar sets are characterized using the Hausdorff measure of their boundaries towards the end.

Knots and Solenoids that Cannot Be Attractors of Self-homeomorphisms of ℝ3

AbstractAs a 1st step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a homeomorphism of $\mathbb{R}^3$. In this paper we introduce toroidal sets as those continua $K \subseteq \mathbb{R}^3$ that have a neighbourhood basis comprised of solid tori and, generalizing the classical notion of genus of a knot, give a natural definition of the genus of toroidal sets and study some of its properties. Using these tools we exhibit knots and solenoids for which the answer to the realizability problem stated above is negative.

What can Koopmanism do for attractors in dynamical systems?

We characterize the longterm behavior of a semiflow on a compact space K by asymptotic properties of the corresponding Koopman semigroup. In particular, we compare different concepts of attractors, such as asymptotically stable attractors, Milnor attractors and centers of attraction. Furthermore, we give a characterization for the minimal attractor for each mentioned property. The main aspect is that we only need techniques and results for linear operator semigroups, since the Koopman semigroup permits a global linearization for a possibly non-linear semiflow.

Attractors of dynamical systems in locally compact spaces

Abstract In this article the properties of attractors of dynamical systems in locally compact metric space are discussed. Existing conditions of attractors and related results are obtained by the near isolating block which we present.

Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems

In this paper we study pullback attractors of multi-valued dynamical systems that are asymptotically convergent. It is shown that, under certain conditions, the components of the pullback attractor of a dynamical system can converge in time to those of the pullback attractor of the limiting dynamic al system. Particular examples are asymptotically autonomous and asymptotically periodic pullback attractors. Different criteria theorems requiring different conditions are established and their applicability and advantages are highlighted.

Description of the information process as a discrete stream.

The work considers the forecasting of the process of functioning of a dynamic complex system, which is a big problem because of the considerable volume of information circulating in it and many random factors affecting its operation. In this paper, the structure of the information process described in the form of a discrete information flow is considered. To determine portions of the information flow information for the formation of a control action, these flows are identified and structured based on determining the dimension of the attractors of dynamical systems in finite-dimensional spaces. To manage a discrete information process, the number of control checks that make it possible to determine the duration of this process predicts it. In this case, the pieces of information and time intervals for each portion of the discrete information stream are considered the same. As a result of the conducted researches the program complex on formation of predicted parameters of training with definition of portions of training information is developed. The paper gives comparative results of the duration of training - experimental and predicted.

Estimation of Lyapunov exponents for quasi-stable attractors of dynamical systems with time delay

Lyapunov exponents play an important role among the invariant characteristics of dynamical systems. Analysis of the spectrum of Lyapunov exponents is widely used to study the complex dynamics in systems of ordinary differential equations and models that can be reduced to maps. In this paper we consider the question of numerical evaluation of Lyapunov exponents for delay differential equations. We used a new algorithm with FFT, tested it on Hutchinson equation with known Lyapunov exponents spectrum and compared with results of the old algorithm. The so-called quasi-stable behavior is typical for a number of gene networks models and neuronal associations that have been studied recently. The phenomenon of cycle (k-dimensional torus) quasistability in dynamical system is characterized by the fact that some of its multipliers are asymptotically close to the unit circle, and the remaining multipliers are modulo less than one (with the exception of a simple root 1 (k such roots)). In some cases, it is possible to prove the existence and give an asymptotic estimate of the multipliers of considered system by using the large parameter methods. However, if these methods are not applicable, it is necessary to obtain a tool for the numerical estimation of multipliers. Such a tool is provided by algorithms for Lyapunov exponents estimation. Given that the equations with delay are often applied in the models of neural and gene networks, the algorithm for Lyapunov exponents estimation for such systems will be in demand.

The Relationship between Deterministic and Ensemble Mean Forecast Errors Revealed by Global and Local Attractor Radii

It has been demonstrated that ensemble mean forecasts, in the context of the sample mean, have higher forecasting skill than deterministic (or single) forecasts. However, few studies have focused on quantifying the relationship between their forecast errors, especially in individual prediction cases. Clarification of the characteristics of deterministic and ensemble mean forecasts from the perspective of attractors of dynamical systems has also rarely been involved. In this paper, two attractor statistics—namely, the global and local attractor radii (GAR and LAR, respectively)—are applied to reveal the relationship between deterministic and ensemble mean forecast errors. The practical forecast experiments are implemented in a perfect model scenario with the Lorenz96 model as the numerical results for verification. The sample mean errors of deterministic and ensemble mean forecasts can be expressed by GAR and LAR, respectively, and their ratio is found to approach $$\sqrt 2 $$ with lead time. Meanwhile, the LAR can provide the expected ratio of the ensemble mean and deterministic forecast errors in individual cases.

Describing chaotic attractors: Regular and perpetual points.

We study the concepts of regular and perpetual points for describing the behavior of chaotic attractors in dynamical systems. The idea of these points, which have been recently introduced to theoretical investigations, is thoroughly discussed and extended into new types of models. We analyze the correlation between regular and perpetual points, as well as their relation with phase space, showing the potential usefulness of both types of points in the qualitative description of co-existing states. The ability of perpetual points in finding attractors is indicated, along with its potential cause. The location of chaotic trajectories and sets of considered points is investigated and the study on the stability of systems is shown. The statistical analysis of the observing desired states is performed. We focus on various types of dynamical systems, i.e., chaotic flows with self-excited and hidden attractors, forced mechanical models, and semiconductor superlattices, exhibiting the universality of appearance of the observed patterns and relations.

СКРЫТЫЕ АТТРАКТОРЫ НЕКОТОРЫХ МУЛЬТИСТАБИЛЬНЫХ СИСТЕМ С БЕСКОНЕЧНЫМ ЧИСЛОМ СОСТОЯНИЙ РАВНОВЕСИЯ

It is well known that mathematically simple systems of nonlinear differential equations can exhibit chaotic behavior. Detection of attractors of chaotic systems is an important problem of nonlinear dynamics. Results of recent researches have made it possible to introduce the following classification of periodic and chaotic attractors depending on the presence of neighborhood of equilibrium into their basin of attraction – self-excited and hidden attractors. The presence of hidden attractors in dynamical systems has received considerable attention to both theoretical and applied research of this phenomenon. Revealing of hidden attractors in real engineering systems is extremely important, because it allows predicting the unexpected and potentially dangerous system response to perturbations in its structure. In the past three years after discovering by S. Jafari and J. C. Sprott chaotic system with a line and a plane of equilibrium with hidden attractors there has been much attention to systems with uncountable or infinite equilibria. In this paper it is offered new models of control systems with an infinite number of equilibrium possessing hidden chaotic attractors: a piecewise-linear system with a locally stable segment of equilibrium and a system with periodic nonlinearity and infinite number of equilibrium points. The original analytical-numerical method developed by the author is applied to search hidden attractors in investigated systems.