Attractor Solution Research Articles

Dynamic Time Warping as Elementary Effects Metric for Morris-Based Global Sensitivity Analysis of High-Dimension Dynamical Models

This work focused on demonstrating the use of dynamic time warping (DTW) as a metric for the elementary effects computation in Morris-based global sensitivity analysis (GSA) of model parameters in multivariate dynamical systems. One of the challenges of GSA on multivariate time-dependent dynamics is the modeling of parameter perturbation effects propagated to all model outputs while capturing time-dependent patterns. The study establishes and demonstrates the use of DTW as a metric of elementary effects across the time domain and the multivariate output domain, which are all aggregated together via the DTW cost function into a single metric value. Unlike the commonly studied coefficient-based functional approximation and covariance decomposition methods, this new DTW-based Morris GSA algorithm implements curve alignment via dynamic programing for cost computation in every parameter perturbation trajectory, which captures the essence of “elementary effect” in the original Morris formulation. This new algorithm eliminates approximations and assumptions about the model outputs while achieving the objective of capturing perturbations across time and the array of model outputs. The technique was demonstrated using an ordinary differential equation (ODE) system of mixed-order adsorption kinetics, Monod-type microbial kinetics, and the Lorenz attractor for chaotic solutions. DTW as a Morris-based GSA metric enables the modeling of parameter sensitivity effects on the entire array of model output variables evolving in the time domain, resulting in parameter rankings attributed to the entire model dynamics.

Swampland conjectures constraints on dark energy from a highly curved field space

We study the interplay of the trans-Planckian censorship conjecture (TCC) and the swampland distance conjecture (SDC) in the context of multifield dark energy in a curved field space. In this scenario, the phase of accelerated expansion is realized as nongeodesic motion in a highly curved field space, reminiscent of models developed in the context of inflation. The model features a stable attractor solution with near constant equation of state w≃−1, and predicts that the current era of accelerated expansion is eternal. The latter implies an eventual conflict with the TCC, which holds that the duration of any epoch of cosmic acceleration is bounded by the requirement that the large-scale observable universe is blind to Planck-scale early universe physics. This tension can be resolved by an interplay with the distance conjecture: for suitable parameter values, the apparent violation of the TCC occurs well after the fields have traversed a Planckian distance. The SDC then predicts a breakdown of the effective field theory before the TCC can be violated. We derive the constraints on the model arising from the SDC+TCC and the de Sitter conjecture. We demonstrate that the model can be consistent with both swampland conjectures and observational data from Planck 2018 and the Dark Energy Spectroscopic Instrument. Published by the American Physical Society 2024

Hydrodynamic Attractor in Ultracold Atoms.

The hydrodynamic attractor is a concept that describes universal equilibration behavior in which systems lose microscopic details before hydrodynamics becomes applicable. We propose a setup to observe hydrodynamic attractors in ultracold atomic gases, taking advantage of the fact that driving the two-body s-wave scattering length causes phenomena equivalent to isotropic fluid expansions. We specifically consider two-component fermions with contact interactions in three dimensions and discuss their dynamics under a power-law drive of the scattering length in a uniform system. By explicit computation, we derive a hydrodynamic relaxation model. We analytically solve their dynamics and find the hydrodynamic attractor solution. Our proposed method using the scattering length drive is applicable to a wide range of ultracold atomic systems, and our results establish these as a new platform for exploring hydrodynamic attractors.

Weyl-Lewis-Papapetrou coordinates, self-dual Yang-Mills equations and the single copy

We consider the dimensional reduction to two dimensions of certain gravitational theories in D ≥ 4 dimensions at the two-derivative level. It is known that the resulting field equations describe an integrable system in two dimensions which can also be obtained by a dimensional reduction of the self-dual Yang-Mills equations in four dimensions. We use this relation to construct a single copy prescription for classes of gravitational solutions in Weyl-Lewis-Papapetrou coordinates. In contrast with previous proposals, we find that the gauge group of the Yang-Mills single copy carries non-trivial information about the gravitational solution. We illustrate our single copy prescription with various examples that include the extremal Reissner-Nordstrom solution, the Kaluza-Klein rotating attractor solution, the Einstein-Rosen wave solution and the self-dual Kleinian Taub-NUT solution.

Attractive (s)axions: cosmological trackers at the boundary of moduli space

We study the cosmological evolution of a FLRW universe dominated by the energy density of moduli close to asymptotic regions of moduli space. Due to the structure of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 1 SUGRA kinetic term, a saxion and an axion residing in the same chiral multiplet are (universally) coupled even if the latter is a flat direction of the potential, resulting in non-trivial dynamics. We generalise known results in the literature to the case of multiple moduli, showing the existence of various “tracker” attractor solutions where the relative energy densities of many components (axions included) stay in a fixed ratio throughout the evolution. We conclude with some phenomenological applications, relevant for both the early and late universe.

Scaling solutions as Early Dark Energy resolutions to the Hubble tension

A wide class of scalar field models including Quintessence and K-essence have the attractive property of tracker regimes, where the energy density stored in the field evolves so as to mimic that of the dominant background component. During this evolution, for a brief period of time, there is an increase in the energy density of the field as it spirals in towards its attractor solution. We show that when the peak of this energy density occurs around the epoch of equality, we can address a key requirement of early dark energy (EDE), postulated as a solution to the Hubble tension. In particular we demonstrate how this can occur in a wide class of Quintessence, axion and K-essence models, before showing that the Quintessence models suffer in that they generally lead to sound speeds incompatible with the requirements of EDE, whereas the K-essence and axion models can do a better job of fitting the data.

Anisotropic dark energy from string compactifications

We explore the cosmological dynamics of a minimalistic yet generic string-inspired model for multifield dark energy. Adopting a supergravity four-dimensional viewpoint, we motivate the model’s structure arising from superstring compactifications involving a chiral superfield and a pure U(1) gauge sector. The chiral sector gives rise to a pair of scalar fields, such as the axio-dilaton, which are kinetically coupled. However, the scalar potential depends on only one of them, further entwined with the vector field through the gauge kinetic function. The model has two anisotropic attractor solutions that, despite a steep potential and thanks to multifield dynamics, could explain the current accelerated expansion of the Universe while satisfying observational constraints on the late-times cosmological anisotropy. Nevertheless, justifying the parameter space allowing for slow roll dynamics together with the correct cosmological parameters, would be challenging within the landscape of string theory. Intriguingly, we find that the vector field, particularly at one of the studied fixed points, plays a crucial role in enabling geodesic trajectories in the scalar field space while realizing slow-roll dynamics with a steep potential. This observation opens a new avenue for exploring multifield dark energy models within the superstring landscape.

Backreaction of axion-SU(2) dynamics during inflation

We consider the effects of backreaction on axion-SU(2) dynamics during inflation. We use the linear evolution equations for the gauge field modes and compute their backreaction on the background quantities numerically using the Hartree approximation. We show that the spectator chromo-natural inflation attractor is unstable when back-reaction becomes important. Working within the constraints of the linear mode equations, we find a new dynamical attractor solution for the axion field and the vacuum expectation value of the gauge field, where the latter has an opposite sign with respect to the chromo-natural inflation solution. Our findings are of particular interest to the phenomenology of axion-SU(2) inflation, as they demonstrate the instability of the usual trajectory due to large backreaction effects. The viable parameter space of the model becomes significantly altered, provided future non-Abelian lattice simulations confirm the existence of the new dynamical attractor. In addition, the backreaction effects lead to characteristic oscillatory features in the primordial gravitational wave background that are potentially detectable with upcoming gravitational wave detectors.

Global Attractor and Non-trivial Equilibrium Solution for Neural Fields with External Stimuli

Global Attractor and Non-trivial Equilibrium Solution for Neural Fields with External Stimuli

Multiple power-law tails in the density and column-density distribution in contracting star-forming clumps

ABSTRACT We present a numerical study of the evolution of power-law tails (PLTs) in the (column-)density distributions (N-PDF, ρ-PDF) in contracting star-forming clumps in primordial gas, without and with some initial rotational and/or turbulent support. In all considered runs multiple PLTs emerge shortly after the formation of the first protostar. The first PLT (PLT 1) in the ρ-PDF is a stable feature with slope q1 ≃ −1.3 which corresponds – under the condition of preserved spherical symmetry – to the outer envelope of the protostellar object with density profile ρ ∝ l−2 in the classical Larson–Penston collapse model, where l is the radius. The second PLT (PLT 2) in the ρ-PDF is stable in the pure-infall runs but fluctuates significantly in the runs with initial support against gravity as dozens of protostars form and their mutual tidal forces change the density structure. Its mean slope, 〈q2〉 ≃ −2, corresponds to a density profile of ρ ∝ l−3/2 which describes a core in free fall in the classical Larson–Penston collapse model or an attractor solution at scales with dominating protostellar gravity. PLT 1 and PLT 2 in the N-PDFs are generally consistent with the observational data of Galactic low-mass star-forming regions from Herschel data. In the runs with initial support against gravity a third PLT (PLT 3) in the ρ-PDFs appears simultaneously with or after the emergence of PLT 2. It is very shallow, with mean slope of 〈q3〉 ≃ −1, and is associated with the formation of thin protostellar accretion discs.

Stable non-BPS black holes and black strings in five dimensions

In this paper we study black hole and black string solutions in five dimensional N=2 supergravity theories arising from the compactification of M-theory on Calabi-Yau manifolds. In particular, we consider explicit examples of three parameter Calabi-Yau manifolds which are obtained as hypersurfaces in toric varieties. Using the attractor mechanism, we obtain Bogomol’nyi-Prasad-Sommerfield (BPS) as well as non-BPS black holes in these compactified supergravity theories. We also consider the black string solutions in these models. We analyze the stability of these extremal black brane configurations by computing the recombination factor. We find multiple stable non-BPS attractor solutions in some of these models. Published by the American Physical Society 2024

On $ L^{\infty } $-estimates and the structure of the global attractor for weak solutions of reaction-diffusion equations

On $ L^{\infty } $-estimates and the structure of the global attractor for weak solutions of reaction-diffusion equations

Universal dynamics and non-thermal fixed points in quantum fluids far from equilibrium

AbstractClosed quantum systems far from thermal equilibrium can show universal dynamics near attractor solutions, known as non-thermal fixed points, generically in the form of scaling behaviour in space and time. A systematic classification and comprehensive understanding of such scaling solutions are tasks of future developments in non-equilibrium quantum many-body theory. In this tutorial review, we outline several analytical approaches to non-thermal fixed points and summarise corresponding numerical and experimental results. The analytic methods include a non-perturbative kinetic theory derived within the two-particle irreducible effective action formalism, as well as a low-energy effective field theory framework. As one of the driving forces of this research field are numerical simulations, we summarise the main results of exemplary cases of universal dynamics in ultracold Bose gases. This encompasses quantum vortex ensembles in turbulent superfluids as well as recently observed real-time instanton solutions in one-dimensional spinor condensates.

(Slow-)twisting inflationary attractors

We explore in detail the dynamics of multi-field inflationary models. We first revisit the two-field case and rederive the coordinate independent expression for the attractor solution with either small or large turn rate, emphasizing the role of isometries for the existence of rapid-turn solutions.Then, for three fields in the slow-twist regime we provide elegant expressions for the attractor solution for generic field-space geometries and potentials and study the behaviour of first order perturbations. For generic 𝒩-field models, our method quickly grows in algebraic complexity. We observe that field-space isometries are common in the literature and are able to obtain the attractor solutions and deduce stability for some isometry classes of 𝒩-field models. Finally, we apply our discussion to concrete supergravity models. These analyses conclusively demonstrate the existence of 𝒩 > 2 dynamical attractors distinct from the two-field case, and provide tools useful for future studies of their phenomenology in the cosmic microwave background and stochastic gravitational wave spectrum.

Attractor solutions in interacting dark energy models

We investigate a cosmological model in which dark energy, represented by a quintessential scalar field, is coupled to a dark-matter perfect fluid in the spatially flat Friedmann–Robertson–Walker Universe. This allows an energy exchange in the dark sector which could happen both at early times before recombination era or at late times. We use the coupling function Q=γρdmφ̇ which is induced by conformal transforming scalar-tensor and f(R) gravity theories to Einstein frame. It is argued that there is a connection between this coupling function and Q∝ρdmH. A dynamical analysis is used to show that there are early- and late-time attracting solutions for which the system evolves for a wide range of initial conditions. These attractors generalize the scaling solutions which have been already found in the non-interacting case.

Pole inflation and primordial black holes formation in Starobinsky-like supergravity

We extend the Cecotti–Kallosh model of Starobinsky inflation in supergravity by adding a holomorphic function to the superpotential in order to generate a large peak in the power spectrum of scalar (curvature) perturbations. In our approach, the singular non-canonical kinetic terms are largely responsible for inflation (as an attractor solution), whereas the superpotential is engineered to generate a production of PBH. We study the cases with (a) a linear holomorphic function, (b) a quadratic holomorphic function, and (c) an exponential holomorphic function, as regards the dependence of inflation and PBH production upon parameters of those functions and initial conditions, as well as verify viability of inflation with our superpotentials. We find that an efficient production of PBH consistent with cosmic microwave background measurements is only possible in the second (b) case. We calculate the masses of the produced PBH and find that they are below the Hawking (black hole) evaporation limit, so that they cannot be part of the current dark matter in our Universe.

Dimension Estimate of the Global Attractor for a 3D Brinkman- Forchheimer Equation

In this paper, we study the dimension estimate of global attractor for a 3D Brinkman-Forchheimer equation. Based on the differentiability of the semigroup with respect to the initial data, we show that the global attractor of strong solution of the 3D Brinkman-Forchheimer equation has finite Hausdorff and fractal dimensions.

An overview of dynamical methods for studying transitions between states in sheared plasma flows.

The self-organization of structures in a tokamak plasma as it undergoes an [Formula: see text]-mode transition shows properties similar to simpler shear flow configurations. We will describe recent dynamical studies of plasma shear flows, including the idea of tracking the edge of chaos that separates two bistable states, computing the nonlinear minimal seed that can lead to turbulence, finding the attractor solution on the edge and seeing how starting from this solution we can understand the stability of relative period orbits that permeate the turbulent basin of attraction. We present a modus operandi developed for these simple configurations that can be adapted to understand the [Formula: see text]-mode transition. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

A time-periodic reaction-diffusion population model of a species subjected to harvesting on the boundary of a protected habitat

The objective in this paper is to determine analytically a maximally sustainable pro-rata density-dependent harvest rate of a hypothetical biological species on the spatial boundary of a protected habitat (i.e. no harvesting of the species is allowed in the interior of the spatial domain). This is achieved by analysing an initial-boundary value model involving a reaction-diffusion equation in which the reaction term is a concave function of the population density, depends periodically on time, and varies spatially. The model is equipped with Robin boundary conditions modelling a continuum of potential pro-rata density-dependent harvest rates on the spatial boundary. A set of necessary and sufficient criteria for the existence of a unique positive periodic attractor of solutions to the model is established by employing the theory of comparison principles and monotone iteration schemes. Thereafter, a long-time asymptotic analysis of the population density is undertaken by invoking classical results from the theory of eigenproblems. In this way, the aforementioned necessary and sufficient conditions are rendered more practical in terms of an upper bound on the maximal allowable pro-rata density-dependent harvest rate. Moreover, important properties of the time-periodic attractor of solutions are established to guarantee the existence of a density pro-rata harvest rate which maximises the total harvest per unit time at equilibrium.

A minimal self-tuning model to solve the cosmological constant problem

The expansion of the Universe is observed to be accelerating, with the simplest solution being a classical cosmological constant. However, this receives contributions from the quantum vacuum, which are predicted to be many orders of magnitude larger than observations, and suffers from radiative instabilities requiring repeated fine-tuning. In this paper we present a minimal, self-tuning scalar field model that can dynamically cancel a large quantum vacuum energy, avoiding Weinberg's No-Go Theorem, and produce accelerated de Sitter expansion at a lower energy scale as a solution to the problem. Our minimal model, which contains a non-canonical kinetic energy and a linear potential, belongs to the Kinetic Gravity Braiding sub-class of Horndeski theory which is not observationally excluded, and lies outside of the known Fab-Four or Well-Tempered models. We find analytic solutions in the limits of slow-roll and fast-roll, and numerically solve the equations of motion to illustrate our model. We show that the model allows for a matter dominated era, and that the attractor solution is stable under a phase transition in the vacuum energy density. We also consider the energy-scales required to match observations. Our model shows the existence of a wider class of successful self-tuning models than previously assumed.