Dimension Of Attractor Research Articles

Exploring the Tropical Pacific Manifold in Models and Observations

The threat of global warming and the demand for reliable climate predictions pose a formidable challenge because the climate system is multiscale, high-dimensional and nonlinear. Spatiotemporal recurrences of the system hint to the presence of a low-dimensional manifold containing the high-dimensional climate trajectory that could make the problem more tractable. Here we argue that reproducing the geometrical and topological properties of the low-dimensional attractor should be a key target for models used in climate projections. In doing so, we propose a general data-driven framework to characterize the climate attractor and showcase it in the tropical Pacific Ocean using a reanalysis as observational proxy and two state-of-the-art models. The analysis spans four variables simultaneously over the periods 1979–2019 and 2060–2100. At each time t, the system can be uniquely described by a state space vector parametrized by N variables and their spatial variability. The dynamics is confined on a manifold with dimension lower than the full state space that we characterize through manifold learning algorithms, both linear and nonlinear. Nonlinear algorithms describe the attractor through fewer components than linear ones by considering its curved geometry, allowing for visualizing the high-dimensional dynamics through low-dimensional projections. The local geometry and local stability of the high-dimensional, multivariable climate attractor are quantified through the local dimension and persistence metrics. Model biases that hamper climate predictability are identified and found to be similar in the multivariate attractor of the two models during the historical period while diverging under the warming scenario considered. Finally, the relationships between different subspaces (univariate fields), and therefore among climate variables, are evaluated. The proposed framework provides a comprehensive, physically based, test for assessing climate feedbacks and opens new avenues for improving their model representation.13 MoreReceived 8 October 2021Revised 14 March 2022Accepted 20 April 2022Corrected 15 July 2022DOI:https://doi.org/10.1103/PhysRevX.12.021054Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAtmospheric fluid dynamicsPatterns in complex systemsSpatiotemporal chaosInterdisciplinary PhysicsNonlinear DynamicsFluid Dynamics

Языковые средства реализации аттрактивности в предтекстовых образованиях (на материале французского и русского языков)

The aim of the study is to identify and describe the linguistic means of attractiveness used in pretext formations of electronic versions of the Russian and French press. The article reveals the main features and functions of the pretexts of news articles. The main attention is focused on the notion of attractiveness and the properties of attractors that ensure the attractiveness of information presentation. The scientific novelty lies in the definition of the linguistic means that are most actively used as attractors in secondary news texts in the French and Russian languages. As a result, it has been found that the selected attractors are represented by a wide range of means reflecting general trends in use and language specifics.

Hausdorff dimension of thin higher-dimensional solenoidal attractors

In this work we study the Hausdorff dimension of hyperbolic attractors that are given by skew-products. We prove that if the contraction is sufficiently strong, the weak-contraction is conformal and the attractor satisfies a geometric condition of transversality between its components, then the Hausdorff dimension and the box-counting dimension of the attractor satisfy Bowen’s formula. We also prove that for any attractor whose contraction is sufficiently strong, there exist dynamics C r -close such that the Hausdorff dimension and the box-counting dimension of the attractor have the same value given by Bowen’s formula.

Higher-order robust attractors for stochastic retarded degenerate parabolic equations

We study the existence, measurability and time-dependent property of pullback random attractors in the higher-order space for stochastic degenerate parabolic equations with variable delay defined on R n . Let X, Z be the square integrable space and the p-times (p > 2) integrable space, respectively. We first prove the existence of a unique pullback ( X ϱ , Z ϱ ) -random attractor A with X ϱ = C ( [ − ϱ , 0 ] ; X ) and Z ϱ = C ( [ − ϱ , 0 ] ; Z ) , and establish the forward compactness, measurability and long-time stability of the bi-spatial attractor A in the higher-order space Z ϱ . We then investigate the higher-order delay-free stability of A in Z ϱ , more precisely, the upper semicontinuity of A under the topology of Z ϱ as the memory time tends to zero is established.

Sublinear measures, Menger curvature, and Hausdorff dimension

We study metric properties of sets which are characterized by their approximation properties by smooth curves at scales of positive density and their relationships with known problems in complex dynamics and analysis. Three direct applications towards removability for bounded analytic functions, optimal bounds for Traveling Salesman problem, and Hausdorff dimension of unfolding attractors are discussed in the paper.The main technical ingredient is a construction, for every continuum K, of a finite Radon measure of bounded Menger curvature such that on every ball B(x,r), x∈K, the measure is bounded by a universal constant multiple of rexp⁡(−g(x,r)), where g(x,r)≥0 is an explicit function. Theorem A is the main analytical result of the paper and provides a precise estimate on the growth of the function g(x,r) in terms of Jones' β numbers. Its dimension theory counterpart is stated as Theorems 1 and 2 and yields sharp estimates of the Hausdorff dimension of compacta in terms of asymptotic properties of log⁡g(x,r)/log⁡r for all x∈K except for a set of finite linear measure.

Learning to represent continuous variables in heterogeneous neural networks.

Animals must monitor continuous variables such as position or head direction. Manifold attractor networks-which enable a continuum of persistent neuronal states-provide a key framework to explain this monitoring ability. Neural networks with symmetric synaptic connectivity dominate this framework but are inconsistent with the diverse synaptic connectivity and neuronal representations observed in experiments. Here, we developed a theory for manifold attractors in trained neural networks, which approximates a continuum of persistent states, without assuming unrealistic symmetry. We exploit the theory to predict how asymmetries in the representation and heterogeneity in the connectivity affect the formation of the manifold via training, shape network response to stimulus, and govern mechanisms that possibly lead to destabilization of the manifold. Our work suggests that the functional properties of manifold attractors in the brain can be inferred from the overlooked asymmetries in connectivity and in the low-dimensional representation of the encoded variable.

Global attractors of the degenerate fractional Kirchhoff wave equation with structural damping or strong damping

Abstract This article deals with the degenerate fractional Kirchhoff wave equation with structural damping or strong damping. The well-posedness and the existence of global attractor in the natural energy space by virtue of the Faedo-Galerkin method and energy estimates are proved. It is worth mentioning that the results of this article cover the case of possible degeneration (or even negativity) of the stiffness coefficient. Moreover, under further suitable assumptions, the fractal dimension of the global attractor is shown to be infinite by using Z 2 {{\mathbb{Z}}}_{2} index theory.

Well-posedness and attractors for a 2D Boussinesq system with partial dissipation

This article is devoted to the global well-posedness and the long-time behavior of solutions of a 2D Boussinesq equations with partial dissipation. We prove that this system is global well-posed under some weaker assumptions on the initial data and has a weak sigma-attractor which retains some of the common properties of global attractors for the dissipative dynamical system, moreover, the local attractor which is the composition of the weak sigma-attractor is upper semicontinuous under small viscosity perturbations.

Chaos along the rc-branch of [formula omitted]-shunted intrinsic Josephson junctions

The appearance of chaos along the “charged” resonance circuit branch of the current-voltage characteristics is examined in a model of intrinsic Josephson junctions shunted by resistive, inductive and capacitive circuit elements. Detailed numerical simulations of the electric charge density, current-voltage characteristics and various chaos indicators, such as Lyapunov exponents and Poincaré sections are performed over a wide range of relevant system parameters. In mapping the parameter space, several distinctly different regions are found: some completely regular, while others are dominated by chaos, where the rc-frequency determines the appearance and properties of those regions. At higher values of the rc-frequency chaos appears at first over relatively narrow regions at the lower-current end of the rc-branch. As the rc-frequency is lowered, the chaotic region at the lowered-current end of the rc-branch may becomes wider, depending sensitively on the combination of system parameters. Examination of the metric entropy and maximal Kaplan–Yorke dimension shows that the dimensions of the chaotic attractors associated with the chaos do not plateau, as in the overdamped case, but surprisingly increase indefinitely with the number of junctions, indicating that the system is capable of truly high-dimensional chaos. The onset of the chaos along the rc-branch is found to occur via a two-frequency quasi-periodic route. Our results provided a guide for specific parameter combinations that could minimize the chaos, thereby making certain applications potentially more viable.

Structure and behavior in Boolean monotonic model pools

One of the main theoretical questions in the field of discrete regulatory networks is the question what aspects of the dynamics – the structure of the state transition graph – are already imposed by structural descriptions of the network such as the interaction graph. For Boolean networks, prior work has concentrated on different versions of the Thomas conjectures that link feedback cycles in the network structure to attractor properties. Other approaches check algorithmically whether certain properties hold true for all models sharing specific structural constraints, e.g. by using model checking techniques. In this work we investigate the behavior of the pool of Boolean networks in agreement with a given interaction graph using a different approach. Grouping together states that are updated consistently across the pool we derive an equivalence relation and analyze a corresponding quotient graph on the state space. By construction this graph yields information about the dynamics of all functions in the pool. Our main result is that this graph can be computed efficiently without enumerating and analyzing all individual functions. This opens up new possibilities for applications, where such model pools arise when modeling under uncertainty.

Estimates for the Dimension of Attractors of a Regularized Euler System with Dissipation on the Sphere

Estimates for the Dimension of Attractors of a Regularized Euler System with Dissipation on the Sphere

Sharp upper and lower bounds of the attractor dimension for 3D damped Euler–Bardina equations

The dependence of the fractal dimension of global attractors for the damped 3D Euler–Bardina equations on the regularization parameter α>0 and Ekman damping coefficient γ>0 is studied. We present explicit upper bounds for this dimension for the case of the whole space, periodic boundary conditions, and the case of bounded domain with Dirichlet boundary conditions. The sharpness of these estimates when α→0 and γ→0 (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.

A Damping-Tunable Snap System: From Dissipative Hyperchaos to Conservative Chaos.

A hyperjerk system described by a single fourth-order ordinary differential equation of the form has been referred to as a snap system. A damping-tunable snap system, capable of an adjustable attractor dimension () ranging from dissipative hyperchaos () to conservative chaos (), is presented for the first time, in particular not only in a snap system, but also in a four-dimensional (4D) system. Such an attractor dimension is adjustable by nonlinear damping of a relatively simple quadratic function of the form , easily tunable by a single parameter A. The proposed snap system is practically implemented and verified by the reconfigurable circuits of field programmable analog arrays (FPAAs).

Энтропия и размерность хаотического аттрактора в зависимости от управляющих параметров

The dependence of the entropy and dimension of the chaotic attractor on the control parameter is investigated by the numerical experiment. Calculations are carried out for one of the simplest systems described by nonlinear equations of dynamics — a rotator driven by an external periodic field. Here, regular and chaotic solutions alternate when the control parameter changes. The numerical experiment shows that the dimension of the chaotic attractor and, as a consequence, its entropy change significantly when the control parameter is in the ranges where, due to intermittency, the transition from chaotic motion to regular motion occurs.

Bistability, bifurcations and chaos in the Mackey-Glass equation

<p style='text-indent:20px;'>Numerical bifurcation analysis, and in particular two-parameter continuation, is used in consort with numerical simulation to reveal complicated dynamics in the Mackey-Glass equation for moderate values of the delay close to the onset of chaos. In particular a cusp bifurcation of periodic orbits and resulting branches of folds of periodic orbits effectively partition the parameter space into regions where different behaviours are seen. The cusp bifurcation leads directly to bistability between periodic orbits, and subsequently to bistability between a periodic orbit and a chaotic attractor. This leads to two different mechanisms by which the chaotic attractor is destroyed in a global bifurcation with a periodic orbit in either an interior crisis or a boundary crisis. In another part of parameter space a sequence of subcritical period-doublings is found to give rise to bistability between a periodic orbit and a chaotic attractor. Torus bifurcations, and a codimension-two fold-flip bifurcation are also identified, and Lyapunov exponent computations are used to determine chaotic regions and attractor dimension.</p>

Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise

<p style='text-indent:20px;'>In this paper, we mainly consider the existence of random attractor and random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. First step, the well-posedness and the existence of a random attractor for the cocycle associated with the considered system is established. Second step, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero. Third step, we prove the regularity of random attractor in a higher regular space by the "iteration" method. Finally, we give the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor.</p>

Long-time dynamical behavior for a piezoelectric system with magnetic effect and nonlinear dampings

<abstract><p>This paper is concerned with the long-time dynamical behavior of a piezoelectric system with magnetic effect, which has nonlinear damping terms and external forces with a parameter. At first, we use the nonlinear semigroup theory to prove the well-posedness of solutions. Then, we investigate the properties of global attractors and the existence of exponential attractors. Finally, the upper semicontinuity of global attractors has been investigated.</p></abstract>

Attractors for a class of extensible beams with strong nonlinear damping

<p style='text-indent:20px;'>We concern to stablish the existence and qualitative properties of the compact global attractor associate to solutions of a class of extensible beam equations with strong nonlinear damping arising from the wave model proposed by Prestel [<xref ref-type="bibr" rid="b18">18</xref>].</p>

Resonance three-wave interactions and strange attractor

It is shown that the incorporation of linear sink/source terms in the three-wave resonance interaction model results in the time dependence of the wave amplitudes, which could exhibit the properties of a strange attractor. This finding demonstrates that the transition to turbulent dynamics of the waves could be related not only to the coupling of wave triads but also to the establishing of the strange attractor-like dynamics within individual wave triads.

Estimates for the Dimension of Attractors of a Regularized Euler System with Dissipation on the Sphere

Доказывается существование глобального аттрактора регуляризированной системы Эйлера-Бардины с диссипацией на двумерной сфере и в произвольных областях на сфере. Получены явные оценки фрактальной размерности аттрактора в терминах физических параметров. Библиография: 20 названий.