Forward Attractors Research Articles

Far-from-equilibrium attractors with full relativistic Boltzmann approach in boost-invariant and non-boost-invariant systems

We study the universal behavior associated with a Relativistic Boltzmann Transport (RBT) approach with the full collision integral in 0+1D conformal systems. We show that all momentum moments of the distribution function exhibit universal behavior. Furthermore, the RBT approach allows to calculate the full distribution function, showing that an attractor behavior is present in both the longitudinal and transverse momentum dependence. We compare our results to the far-from-equilibrium attractors determined with other approaches, such as kinetic theory in Relaxation Time Approximation (RTA) and relativistic hydrodynamic theories, both in their viscous (DNMR) an anisotropic (aHydro) formulations, finding a very similar evolution, but an even faster thermalization in RBT for higher order moments. For the first time, we extended this analysis also to study the attractor behavior under a temperature-dependent viscosity η/s(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta /s(T)$$\\end{document}, accounting also for the rapid increase toward the hadronic phase. We find that a partial breaking of the scaling behavior with respect to τ/τeq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au /\ au _{eq}$$\\end{document} emerges only at T≈Tc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T \\approx T_c$$\\end{document} generating a transient deviation from attractors; interestingly this in realistic finite systems may occur around the freeze-out dynamics. Finally, we investigate for the first time results beyond the boost-invariant picture, finding that also in such a case the system evolves toward the universal attractor. In particular, we present the forward and pull-back attractors at different space-time rapidities including rapidity regions where initially the distribution function is even vanishing.

Forwards Attractor Structures in a Planar Cooperative Non-autonomous Lotka–Volterra System

The global attractor of a dissipative dynamical system provides the necessary information to understand the asymptotic dynamics of all the system’s solutions. A crucial question consists in finding the structure of this set. In this paper we provide a full characterization of the structure of attractors for a planar non-autonomous Lotka–Volterra cooperative system. We show sufficient conditions for the existence of forward attractors and give a full description of them by proving the existence of such bounded global solutions that all bounded global solutions join them, i.e. converge towards them when time tends to plus and minus infinity. These results generalize those known in an autonomous framework. The case of particular interest in our work is the situation where globally forward-stable non-autonomous solutions have both coordinates strictly positive. We study this case in detail and obtain sufficient conditions that the problem parameters must satisfy in order to obtain various structures of non-autonomous attractors. This allows us to understand different paths of the solutions towards the unique globally stable one.

Attractor-repeller construction of Shintani domains for totally complex quartic fields

The units of a number field k act naturally on the real vector space k⊗QR, and so on open subsets of (k⊗QR)⁎ that are stable under the units. A Shintani domain for this action consists of a finite number of polyhedral cones, all having generators in k, whose union is a fundamental domain. Aside from the trivial case of imaginary quadratic fields, no practical method for computing Shintani domains for totally complex number fields has been published. Here we give a quick way to compute Shintani domains for totally complex quartic number fields.To construct these domains we exploit the existence of a forward attractor and backward repeller set for this action. We prove that the number of polyhedral cones needed for the Shintani domain is bounded by an absolute constant, a fact previously known only for cubic or quadratic fields. We also show that our algorithm for finding a Shintani domain runs in polynomial time and we give a table describing the results of running the algorithm on more than 168,000 fields.

Novel features of attractors and transseries in nonconformal Bjorken flows

In this work we investigate the impact of conformal symmetry breaking on hydrodynamization of a far-from-equilibrium fluid. We find a new kind of transseries solutions for the nonconformal hydrodynamic equations of a longitudinal boost invariant expanding plasma. The new transseries solutions unveil a rich physical structure which arises due to the interplay of different physical scales. In the perfect fluid case the nonconformal speed of sound slows down the cooling of the temperature due to the emergence of logarithmic corrections that depends on the mass of the particle. These terms propagate into the perturbative and nonperturbative sectors of the transseries once viscous corrections are included. The logarithmic mass contributions increase the asymptotic value of the Knudsen number while decreasing the damping rate of the transient nonhydrodynamic modes and thus, yielding to an extremely slow hydrodynamization process where flow lines merge to their forward attractor at extremely late times. The early time free streaming expansion is modified and receives logarithmic mass corrections induced by the shear-bulk couplings. The global flow structure and numerical analyses carried out in our work demonstrate the existence of local attraction at early and late times for the shear viscous tensor and bulk viscous pressure.

Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors

In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior need to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pullback attractors, as well as forward limit sets persist and that the latter two notions even converge under perturbation.As concrete application, we study integrodifference equations over the continuous functions under spatial discretization of collocation type. Integrodifference equation and Pullback attractor and Forward attractor and Urysohn operator

Asymptotically autonomous dynamics for non-autonomous stochastic $ g $-Navier-Stokes equation with additive noise

<p style='text-indent:20px;'>Both sufficient and necessary criteria for the existence of a bi-parametric attractor which attaches with forward compactness is established. Meanwhile, we prove that, under certain conditions, the components of the random attractor of a non-autonomous dynamical system can converge in time to those of the random attractor of the limiting autonomous dynamical system. As an application of the abstract theory, we show that the non-autonomous stochastic <inline-formula><tex-math id="M2">\begin{document}$ g $\end{document}</tex-math></inline-formula>-Navier-Stokes (g-NS) equation possesses a forward compact random attractor such that it is forward asymptotically autonomous to a random attractor of the autonomous g-NS equation.</p>

Forward attractor for stochastic chemostat model with multiplicative noise

In this paper, we consider a stochastic chemostat model with multiplicative noise. By appropriate variable substitution, we get a random chemostat system driven by Ornstein-Uhlenbeck process, which will no longer contain white noise. Firstly, we prove the existence and uniqueness of the global positive solution for any positive initial value for random chemostat system. Secondly, we state some results regarding the existence of a compact forward absorbing set as well as a forward attracting one, its internal structure will provide us some useful information about the long-time behavior of microorganism in random chemostat model. Finally, we make some comparison and analysis between both ways of modeling randomness and stochasticity in the chemostat model and show some numerical simulations to support our theoretical results.

Transasymptotics and hydrodynamization of the Fokker-Planck equation for gluons

We investigate the non-linear transport processes and hydrodynamization of a system of gluons undergoing longitudinal boost-invariant expansion. The dynamics is described within the framework of the Boltzmann equation in the small-angle approximation. The kinetic equations for a suitable set of moments of the one-particle distribution function are derived. By investigating the stability and asymptotic resurgent properties of this dynamical system, we demonstrate, that its solutions exhibit a rather different behavior for large (UV) and small (IR) effective Knudsen numbers. Close to the forward attractor in the IR regime the constitutive relations of each moment can be written as a multiparameter transseries. This resummation scheme allows us to extend the definition of a transport coefficient to the non-equilibrium regime naturally. Each transport coefficient is renormalized by the non-perturbative contributions of the non-hydrodynamic modes. The Knudsen number dependence of the transport coefficient is governed by the corresponding renormalization group flow equation. An interesting feature of the Yang-Mills plasma in this regime is that it exhibits transient non-Newtonian behavior while hydrodynamizing. In the UV regime the solution for the moments can be written as a power-law asymptotic series with a finite radius of convergence. We show that radius of convergence of the UV perturbative expansion grows linearly as a function of the shear viscosity to entropy density ratio. Finally, we compare the universal properties in the pullback and forward attracting regions to other kinetic models including the relaxation time approximation and the effective kinetic Arnold-Moore-Yaffe (AMY) theory.

Monotone skew-Product Semiflows for Carathéodory Differential Equations and Applications

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carath\'eodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained. These results are important for the study of the long-term behavior of the trajectories. In particular, the construction of semicontinuous semiequilibria and equilibria is extended to the context of ordinary and delay Carath\'eodory differential equations. Under appropriate assumptions of sublinearity, the existence of a unique continuous equilibrium, whose graph coincides with the pullback attractor for the evolution processes, is shown. The conditions under which such a solution is the forward attractor of the considered problem are outlined. Two examples of application of the developed tools are also provided.

Pullback and forward attractors of contractive difference equations

The construction of attractors of a dissipative difference equation is usually based on compactness assumptions. In this paper, we replace them with contractivity assumptions under which the pullba...

Pullback and forward attractors of contractive difference equations

Pullback and forward attractors of contractive difference equations

Forward and Pullback Dynamics of Nonautonomous Integrodifference Equations: Basic Constructions

In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture their long-term behaviour comprehensively, both pullback and forward attractors, as well as forward limit sets are constructed for general infinite-dimensional nonautonomous dynamical systems in discrete time. While the theory of pullback attractors, but not their application to integrodifference equations, is meanwhile well-established, the present novel approach is needed in order to understand their future behaviour.

Local Approximate Forward Attractors of Nonautonomous Dynamical Systems

Pullback dynamics of nonautonomous dynamical systems has been considerably developed. However, it is still a tough job to study forward dynamics of nonautonomous dynamical systems, since forward attractors were only obtained in some particular cases. In the paper, under some reasonable conditions, it is shown that closing to a local pullback attractor, there is an approximate forward attractor. Specifically, let ϕ be a cocycle semiflow on a Banach space X with driving system θ on a base space P. Suppose that the base space P is compact and ϕ is uniformly asymptotically compact. Let A(∙) be a local pullback attractor with being compact. We prove that every e-extended neighborhood Ae(∙) of A(∙) will forward attract every bounded set B(∙) that is pullback attracted by A(∙). We then call Ae(∙) an approximate forward attractor of ϕ.

Stability and forward attractors for non-autonomous impulsive semidynamical systems

In this paper, we study the theory of forward attractors for non-autonomous impulsive semidynamical systems. Moreover, we investigate some types of stability of the global attractor as orbital stability, asymptotic stability and stability in the sense of Lyapunov-Barbashin. We present an example to illustrate the theory.

Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain

A pullback random attractor is called forward controllable if its time-component is semi-continuous to a compact set in the future, and the minimum among all such compact limit-sets is called a forward controller. The existence of a forward controller closely relates to the forward compactness of the attractor, which is further argued by the forward-pullback asymptotic compactness of the system. The abstract results are applied to the non-autonomous stochastic sine-Gordon equation on an unbounded domain. The existence of a forward compact attractor is proved, which leads to the existence of a forward controller. The measurability of the attractor is proved by considering two different universes.

The Structure of Global Attractors for Non-autonomous Perturbations of Gradient-Like Dynamical Systems

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of autonomous gradient-like dynamical systems under the assumption that the original autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular $$\tau $$ -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of $$\tau $$ -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.

Approximate Forward Attractors of Non-Autonomous Dynamical Systems

In this paper the forward asymptotical behavior of non-autonomous dynamical systems and their attractors are investigated. Under general conditions, the authors show that every neighborhood of pullback attractor has forward attracting property.

Forward omega limit sets of nonautonomous dynamical systems

The forward $ \omega $-limit set $ \omega_{\mathcal{B}} $ of a nonautonomous dynamical system $ \varphi $ with a positively invariant absorbing family $ \mathcal{B} $ $ = $ $ \{ B(t), t \in \mathbb{R}\} $ of closed and bounded subsets of a Banach space $ X $ which is asymptotically compact is shown to be asymptotically positive invariant in general and asymptotic negative invariant if $ \varphi $ is also strongly asymptotically compact and eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. In addition, a necessary and sufficient condition for a $ \varphi $-invariant family $ \mathcal{A} $ $ = $ $ \left\{A(t), t \in \mathbb{R}\right\} $ in $ \mathcal{B} $ of nonempty compact subsets of $ X $ to be a forward attractor is generalised to this context.

Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations

We consider the nonautonomous perturbation $ x_t+Ax = f(x)+\varepsilon h(t) $ of a gradient-like system $ x_t+Ax = f(x) $ in a Banach space $ X $, where $ A $ is a sectorial operator with compact resolvent. Assume the non-perturbed system $ x_t+Ax = f(x) $ has an attractor $ {\mathscr A} $. Then it can be shown that the perturbed one has a pullback attractor $ {\mathscr A} _\varepsilon $ near $ {\mathscr A} $. If all the equilibria of the non-perturbed system in $ {\mathscr A} $ are hyperbolic, we also infer from [4,6] that $ {\mathscr A} _\varepsilon $ inherits the natural Morse structure of $ {\mathscr A} $. In this present work, we introduce the notion of nonautonomous equilibria and give a more precise description on the Morse structure of $ {\mathscr A} _\varepsilon $ and the asymptotically synchronizing behavior of the perturbed system. Based on the above result we further prove that the sections of $ {\mathscr A} _\varepsilon $ depend on time symbol continuously in the sense of Hausdorff distance. Consequently, one concludes that $ {\mathscr A} _\varepsilon $ is a forward attractor of the perturbed nonautonomous system. It will also be shown that the perturbed system exhibits completely a global forward synchronizing behavior with the external force.

Invariant forward attractors of non-autonomous random dynamical systems

Forward attractors, especially invariant ones, of non-autonomous and random dynamical systems have not been as well studied as pullback attractors. This is mainly due to the different role that time plays in each case. Pullback attractors involve convergence at each instant of current time as the initial data is pulled back to distant past, whereas forward attractors are concerned with what happens in the distant future. Moreover, if they exist, they need not be unique. In this paper, ideas introduced in Kloeden [20] for asymptotically invariant forward attracting sets of non-autonomous ODEs are generalized and extended to study strictly invariant forward attractors of non-autonomous random dynamical systems on metric spaces. These are formulated as processes that include the effects of both noise and deterministic time variation. It is shown that the random forward attractors, when they exist, can be constructed by a pullback argument inside a positively invariant set. However, this is only a necessary condition, so additional sufficiency conditions are also given. Since forward attractors need not exist in dissipative systems, an alternative object, specifically the closed union of the forward omega limit sets for each initial time of a forward absorbing set is investigated. It is shown to be asymptotically invariant under certain conditions, so looks more and more like an invariant attractor as conventionally understood. It resembles the Vishik uniform attractor, but need not attract uniformly in the initial times and does not even require the system to be defined in the distant past.