Dimension Of Attractor Research Articles

Multi-vortices and Lower Bounds for the Attractor Dimension of 2D Navier–Stokes Equations

Multi-vortices and Lower Bounds for the Attractor Dimension of 2D Navier–Stokes Equations

Strong solutions and attractor dimension for 2D NSE with dynamic boundary conditions

We consider incompressible Navier–Stokes equations in a bounded 2D domain, complete with the so-called dynamic slip boundary conditions. Assuming that the data are regular, we show that weak solutions are strong. As an application, we provide an explicit upper bound of the fractal dimension of the global attractor in terms of the physical parameters. These estimates comply with analogous results in the case of Dirichlet boundary condition.

Fractal dimension of random attractors for nonautonomous stochastic strongly damped wave equations on ℝN

In this article, we investigate the dynamics of a nonautonomous stochastic strongly damped wave equation defined on . We first use the energy equation and tail‐estimates to prove the asymptotic compactness of the solutions and obtain the existence of a unique pullback random attractor for the equation with critical nonlinearity. Then, we give an upper bound of fractal dimension of the random attractor when the nonlinearity is of subcritical growth. The unboundedness of the physical space will impose difficulties on the estimation for the upper bound of fractal dimension of the random attractor.

Fractal dimension of global attractors for a Kirchhoff wave equation with a strong damping and a memory term

This paper is concerned with the dimension of the global attractors for a time-dependent strongly damped subcritical Kirchhoff wave equation with a memory term. A careful analysis is required in the proof of a stabilizability inequality. The main result establishes the finite dimensionality of the global attractor.

Remarks on the Mathematical Modeling of Gene and Neuronal Networks by Ordinary Differential Equations

In the theory of gene networks, the mathematical apparatus that uses dynamical systems is fruitfully used. The same is true for the theory of neural networks. In both cases, the purpose of the simulation is to study the properties of phase space, as well as the types and the properties of attractors. The paper compares both models, notes their similarities and considers a number of illustrative examples. A local analysis is carried out in the vicinity of critical points and the necessary formulas are derived.

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

Topological dimensions of random attractors for a stochastic reaction-diffusion equation with delay

The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a stochastic reaction-diffusion equation with delay. The stochastic equation is firstly transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the almost surely differentiable properties of the RDS, upper estimates of both Hausdorff and fractal dimensions of the random attractors are obtained.

NUMERICAL SIMULATION OF DYNAMICS OF BLOCK MEDIA BY MOVABLE LATTICE AND MOVABLE AUTOMATA METHODS

Two versions of modified Burridge-Knopoff model including state dependent friction, elastic force and thermal conductivity are derived. The friction model describes a velocity weakening of friction and elasticity between moving blocks and an increase of both static friction and rigidity during stick periods as well their weakening during motion. It provides a simplified but qualitatively correct behavior including the transition from smooth sliding to stick-slip behavior, which is often observed in various tribological and tectonic systems. Attractor properties of the model dynamics is studied also. The alternative versions of the model are proposed which apply a simulation of the motion of interacting elastically connected mesh elements and motion of relatively large solid blocks, utilizing technique of the movable cellular automata. First version of the model was already basically studied before. Its advanced version here involves all components of the real system: state-depending friction and changeable rigidity, as well as heat production and thermal conductivity. Model based on the movable automata also involves the components included into traditional lattice model. It has its own ad-vantages and disadvantages which are also discussed in the paper.

Finite Fractal Dimensional Pullback Attractors for a Class of 2D Magneto-Viscoelastic Flows

Finite Fractal Dimensional Pullback Attractors for a Class of 2D Magneto-Viscoelastic Flows

Quantifying strange property of attractors in quasiperiodically forced systems

Quasiperiodically forced systems are an important class of dynamical systems exhibiting quasiperiodic, strange nonchaotic, and chaotic attractors. A major concern is the identification of the parameter range in which each one of the attractors is present. In this work, based on the phase sensitivity proposed by Pikovsky and Feudel, we define a measure to quantitatively distinguish quasiperiodic attractors, strange nonchaotic attractors, and chaotic attractors. Particularly, we can determine the boundary points of these three attractors in parameter space. The reliability of this measure is verified in smooth and non-smooth systems.

On discrete Lorenz-like attractors in three-dimensional maps with axial symmetry.

We describe a class of three-dimensional maps with axial symmetry {x→-x,y→-y,z→z} and the constant Jacobian. We study bifurcations and chaotic dynamics in quadratic maps from this class and show that these maps can possess discrete Lorenz-like attractors of various types. We give a description of bifurcation scenarios leading to such attractors and show examples of their implementation in our maps. We also describe the main geometric properties of the discrete Lorenz-like attractors including their homoclinic structures.

Linear Program-Based Policies for Restless Bandits: Necessary and Sufficient Conditions for (Exponentially Fast) Asymptotic Optimality

We provide a framework to analyze control policies for the restless Markovian bandit model under both finite and infinite time horizons. We show that when the population of arms goes to infinity, the value of the optimal control policy converges to the solution of a linear program (LP). We provide necessary and sufficient conditions for a generic control policy to be (i) asymptotically optimal, (ii) asymptotically optimal with square root convergence rate, and (iii) asymptotically optimal with exponential rate. We then construct the LP-index policy that is asymptotically optimal with square root convergence rate on all models and with exponential rate if the model is nondegenerate in finite horizon and satisfies a uniform global attractor property in infinite horizon. We next define the LP-update policy, which is essentially a repeated LP-index policy that solves a new LP at each decision epoch. We conclude by providing numerical experiments to compare the efficiency of different LP-based policies.Funding: This work was supported by Agence Nationale de la Recherche [Grant ANR-19-CE23-0015].

Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties

Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties

Continuity Properties of Pullback and Pullback Exponential Attractors for Non-autonomous Plate with $$p-$$Laplacian

Continuity Properties of Pullback and Pullback Exponential Attractors for Non-autonomous Plate with $$p-$$Laplacian

DYNAMICS OF EVOLUTION OF CHIALVO SYSTEM: STUDY OF CHAOS AND COMPLEXITY

Chialvo map modelling a neuron has been investigated numerically as it exhbit generically some of the intriguing/ complex features of several excitable biological systems. Bifurcation diagram obtained for this map shows pattern of repeated period-doubling route to chaos. Appearance of complicated periodic windows within bifurcation suggests presence of complexity within the map. Regular and chaotic attractors are drawn here in different parameter space. Lyapunov exponents describing the stability of the systems have been computed for different cases, clearly revealing complexity involved dynamics. Numerical simulation have been extended to calculate topological entropy as a measure of complexity and presented through graphics. Topological entropy graphs showing significant increase within certain parameter range. Correlation dimension of certain chaotic attractors also obtained.

Cumulative Processes in the Modern World-System in the Prism of Dialectics and Laws of Synergetics: to the Problem of Transition to Egalitar Society under Conditions of Multipolarity

Introduction. The paper examines the state of the modern world and the prospects for its multipolarity from the standpoint of synergetics, dialectics, the world-systems analysis and geopolitics. The world-systems approach substantiates the crisis and finiteness of the modern world-economy, the impossibility of its modernization and the emergence of a demand for the formation of an egalitarian world order. The idea of multipolarity is one of the answers to this request.Methodology and sources. Ideas of social synergetics in the works of E.N. Knyazeva, S.P. Kurdyumov and V.V. Tuzov made it possible to describe the state of the modern world system as a state of a non-equilibrium system at the stage of self-organization. Hegelian dialectical laws were used to describe the most general characteristics in the logic of the historical stages in the formation of a social system. The ideas of I. Wallerstein and the geopolitical theory of R. Collins made it possible to designate the properties of attractors that lead to the emergence of poles in the world-economy.Results and discussion. From the point of view of synergetics, multipolarity in the world-system is a state of chaos; from the point of view of dialectics, this is a state of a leap, a transition to a new quality. A geopolitical-world-system typology of attractors that can lead to the formation of poles is proposed. It is shown that the multipolar world contains different options for development, depending on what kind of economic relations will develop between the participants. Within the framework of the existing economic system (capitalist relations, relations of self-organization), multipolarity acts as an intermediate state from which a unipolar world is born. The authors came to the conclusion that for a new result, it is necessary to change the economic paradigm from a self-organizing world-system to one organized by the will of people and based on science. The requirements of a multipolar world on the part of the BRICS, SCO, and EAEU states are within the framework of the existing development paradigm, so the formation of egalitarian relations on this basis is problematic.Conclusion. Humanity gets a chance to overcome the spontaneity of self-organization processes and form a more egalitarian society in the course of the bifurcation of the existing system. Multipolarity can act as a stage of such a transition. However, if the issue of a qualitative change in social relations is not resolved, then multipolarity will turn out to be only a stage in the return to a unipolar world.

A new result on the fractal dimension estimates of random attractor for non-autonomous random 2D stochastic dynamical type systems

We prove some conditions for bounding the fractal dimension of random invariant sets of non-autonomous random dynamical systems on separable Banach spaces. Then we apply these conditions to prove the finiteness of fractal dimension of random attractor for stochastic 2D hydrodynamical type equations with linear additive white noise in bounded domains or unbounded domains satisfying the Poincaré inequality.

Optimal Conditions for a Multimode Laser Diode with Delayed Optical Feedback in Terahertz Time-Domain Spectroscopy

Recent studies have indicated that terahertz time-domain spectroscopy (THz-TDS) can stably and efficiently acquire output spectra using an affordable and compact multimode laser diode (MMLD) with delayed optical feedback as the light source. This research focused on a numerical analysis of the optimal conditions for employing an MMLD with delayed optical feedback (a chaotic oscillating laser diode) in THz-TDS utilizing multimode rate equations. The findings revealed that the intermittent chaotic output generated by the MMLD, characterized by concurrent picosecond pulse oscillations lasting several tens of picoseconds, proved to be highly effective for THz-TDS. By appropriately setting the amounts for the injection current and optical feedback and the delay time for the optical feedback, intermittent chaotic oscillation could be attained within a considerably broad parameter range. The generation of intermittent chaotic oscillations was confirmed by observing their characteristic asymmetric spectral shapes. Moreover, both the MMLD output spectrum and the THz-TDS output spectrum exhibited consistently stable shapes at the microsecond scale, demonstrating the attractor properties inherent in an MMLD with delayed optical feedback.

Nonlinear dynamical analysis for globally modified incompressible non-Newtonian fluids

We present the global modification of the Ladyzhenskaya equations, for incompressible non-Newtonian fluids. This modification is through a cut-off function that multiplies the convective term of the equation and an additional artificial smoothing dissipation term as part of the viscous term of the equation. The goal of this work is the comparative analysis between the modified system and the non-modified system. Therefore, we show the existence and regularity of weak solutions, the existence of global attractors, the estimation of the fractal dimension of the global attractors, and finally, the relationship of the autonomous dynamics between the modified system and the non-modified system.

Systematic Bias Correction in Ocean Mesoscale Forecasting Using Machine Learning

AbstractThe ocean circulation is modulated by meandering currents and eddies. Forecasting their evolution is a key target of operational models, but their forecast skill remains limited. We propose a machine learning approach that improves the output of an ocean circulation model by learning and predicting its systematic biases. This method can be applied a priori to any region, and is tested in the Gulf of Mexico, where the Loop Current (LC) and the large anticyclonic eddies that detach from it are major forecasting targets. The LC dynamics are recurrent and lie on a low‐dimensional dynamical attractor. Building upon the information gained analyzing this low dimensional attractor, we improve the representation of sea surface anomalies in model outputs through information from satellite altimeter data using a Sequence‐to‐Sequence model, which is a special class of Recurrent Neural Network. Building upon the HYCOM‐NCODA analysis system, we deliver a correction to the forecast at the observation resolution. For at least 15 days the proposed method learns to forecast the systematic bias in the HYCOM‐NCODA, outperforming persistence, and improving the forecast. This data‐driven approach is fast and can be implemented as an added step to any dynamical hindcasting or forecasting model. It offers an interesting avenue for further developing hybrid modeling tools. In these tools, fundamental physical conservations are preserved through the integration of partial differential equations which obey them. In addition, the method highlights specific deficiencies of the hindcast system that deserve further investigation in the future.