Future Attractors Research Articles

On the predictability of possible storylines for forced complex systems

Abstract It is well-known that even for fairly simple deterministic nonlinear systems, exact prediction of future state is, on average, impossible beyond some small multiple of the Lyapunov time that quantifies the rate of separation of trajectories within an attractor. Nonetheless, it may be possible to find a physical measure that is the distribution of a trajectory within the attractor. In that sense, there can be a still weaker form of predictability. In this paper, we show that this can also fail but an even weaker form of predictability can appear for non-autonomous (i.e. forced) systems in the presence of tipping points. The {\em predictability of possible storylines} appears when one can interpret the frequencies of runs within an ensemble arriving at one of several possible future attractors (storylines) in a probabilistic manner.
As predictability is a major concern and a challenge in climate science, we illustrate this notion of predictability with two climate-related examples: a chaotic energy balance model and a global ocean model featuring a tipping point of the Atlantic Meridional Overturning Circulation.

Dynamical system method of viscous fluid in f(T) gravity theory

In this paper, using the recently introduced f(T) gravity framework, we have analyzed the viscous fluid cosmological model in FLRW cosmological model by assuming a specific form of the bulk viscosity coefficient as, ζ=ζ0+ζ1H+ζ2ḢH+H and a non-linear f(T) model particularly, f(T)=T−α−T where α is the model parameter. The bulk viscosity coefficients are estimated by using Hubble 31 data set. Using the phase space technique, we examine the asymptotic behaviour of our cosmological bulk viscous model. We found one stable critical point. Phase space analysis and the geometrical interpretations are given. We discover that according to our model, the Universe evolved from a matter-dominated decelerated phase (a past attractor) to a stable de-Sitter accelerated epoch (a future attractor). The evolution of EoS parameter shows the acceleration phase of the cosmic expansion whereas the negative behavior of viscosity-induced pressure indicates the accelerated expansion of the Universe. Additionally, the cosmic matter-energy density exhibits positive behavior. We look into how statefinder parameters behave for our model . We discover that the trajectory of our model starts at the quintessence and ends at the ΛCDM region. We use the Om diagnostic test, which reveals that our model displays quintessence behaviour. The energy condition criteria are analyzed, and we discover that SEC has been violated, whereas WEC, NEC and DEC satisfy the positivity criteria. Our f(T) cosmological model, with the influence of bulk viscosity successfully describes the expansion history of the Universe and provides a good fit to recent observational data.

Global phase space analysis for a class of single scalar field bouncing solutions in general relativity

We carry out a compact phase space analysis of a non-canonical scalar field theory whose Lagrangian is of the form F(X)-V(ϕ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X)-V(\\phi )$$\\end{document} within general relativity. In particular, we focus on a kinetic term of the form F(X)=βXm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X)=\\beta X^m$$\\end{document} (m≠1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m\ e 1/2$$\\end{document}) with power-law potential V0ϕn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_0 \\phi ^n$$\\end{document} and exponential potential V0e-λϕ/MPl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_0 e^{-\\lambda \\phi /M_{Pl}}$$\\end{document} of the scalar field. The Cuscuton case m=1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m=1/2$$\\end{document} where the scalar field is non-dynamical is left out of consideration. The main aim of this work is to investigate the genericity of nonsingular bounce in these models and to investigate the cosmic future of the bouncing cosmologies when they are generic. A global dynamical system formulation that is particularly suitable for investigating nonsingular bouncing cosmologies is used to carry out the analysis. We show that when F(X)=βXm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(X)=\\beta X^m$$\\end{document} (β<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta <0$$\\end{document}), nonsingular bounce is generic for a power law potential V(ϕ)=V0ϕn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\phi ) = V_0 \\phi ^n$$\\end{document} only within the parameter range 12<m<1,n<2mm-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\{ \\frac{1}{2}<m<1,\\,n<\\frac{2\\,m}{m-1}\\right\\} $$\\end{document} and for an exponential potential V(ϕ)=V0e-λϕ/MPl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\phi ) = V_0 e^{-\\lambda \\phi /M_{Pl}}$$\\end{document} only within the parameter range 12<m≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\{ \\frac{1}{2}<m\\le 1\\right\\} $$\\end{document}. Except in these cases, nonsingular bounce in these models is not generic due to the non-existence of global past or future attractors. Our analysis serves to show the importance of a global phase space analysis to address important questions about nonsingular bouncing solutions, an idea that may and must be adopted for such solutions even in other theories.

Phase‐space analysis of the anisotropies in Weyl integrable spacetime: Kantowski–Sachs cosmology

We consider anisotropic Kantowski–Sachs cosmology in Weyl integrable spacetime with an ideal gas. We perform a detailed analysis of the dynamics by investigating the stationary points and reconstruct the cosmological history. This analysis gives important information for the initial value problem in Weyl integrable spacetime. We study three cases, in which the scalar field mass depends only on the density of the ideal gas, with or without cosmological constant term, and when there exists an exponential scalar field potential. We conclude that the existence of the nonzero potential function is essential in order that a spatially flat FLRW universe which can describe inflation can be a future attractor.

Dynamical Analysis in Chameleon Dark Energy

AbstractA detailed analysis of the phase‐space for the field equations in scalar field cosmology with a chameleon cosmology in a spatially flat Friedmann–Lemaître–Robertson–Walker spacetime is presented. For the matter source it is assumed that it is an ideal gas with a constant equation of state parameter, while for the scalar field potential and the coupling function of the chameleon mechanism it is considered that four different sets which provide four different models. The H‐normalization approach and the field equations with the help of dimensionless variables written is considered. The asymptotic solutions are determined from where the theory can describe the main eras of cosmological history and evolution that are found. Future attractors which describe acceleration exist; however, acceleration solutions related to the inflationary era is found in past, as also the radiation epoch and the matter dominated eras are provided by the dynamics. The Chameleon dark energy model can be used as a unified model for the elements which contribute to the dark sector of the universe that are concluded.

Phantom attractors in kinetic gravity braiding theories: a dynamical system approach

We revise the expansion history of the scalar field theories known as Kinetic Gravity Braiding. These theories are well-known for the possibility of driving the expansion of the cosmos towards a future self-tuning de Sitter state when the corresponding Lagrangian is invariant under constant shifts in the scalar field. Nevertheless, this is not the only possible future fate of these shift-symmetric models. Using a dynamical system formulation we show that future cosmological singularities can also appear in this framework. Moreover, we present explicit examples where the future attractor in the configuration space of the theory corresponds to a big rip singularity.

Viscous Fluid Cosmology in Symmetric Teleparallel Gravity

AbstractIn this article, the viscous fluid cosmological model is analyzed in the framework of recently proposed gravity by assuming three different forms of bulk viscosity coefficients, specifically, , , and and a linear model, particularly, where is free model parameter. The bulk viscosity coefficients and the model parameter values using the combined H(z)+Pantheon+BAO data set are estimated. The asymptotic behavior of our cosmological bulk viscous model is studied by utilizing the phase space method. It is found that corresponding to all three cases, our model depicts the evolution of the universe from matter dominated decelerated epoch (a past attractor) to a stable de‐sitter accelerated epoch (a future attractor). Furthermore, the physical behavior of effective pressure, effective equation of state (EoS), and the statefinder parameters are studied. It is also found that the pressure component in the presence of bulk viscosity shows negative behavior and the effective EoS parameter predicts the accelerated expansion phase of the universe for all three cases. Moreover, the trajectories of the model lie in the quintessence region and it converges to the ΛCDM fixed point in the far future. It is found that the accelerated de‐Sitter like phase comes purely from the case without any geometrical modification to GR. Moreover, the late‐time behavior of all three cases of viscosity coefficients are identical. Furthermore, a non‐linear model is considered, specifically, , and then analyzed the behavior of model using dynamical approach. It is also found that the late‐time behavior of the considered non‐linear model with is similar to the linear case, whereas for the case results are quite different.

Scalar Field Cosmology from a Modified Poisson Algebra

We investigate the phase space of a scalar field theory obtained by minisuperspace deformation. We consider quintessence or phantom scalar fields in the action that arises from minisuperspace deformation on the Einstein–Hilbert action. We use a modified Poisson algebra where Poisson brackets are the α-deformed ones and are related to the Moyal–Weyl star product. We discuss early- and late-time attractors and reconstruct the cosmological evolution. We show that the model can have the ΛCDM model as a future attractor if we initially consider a massless scalar field without a cosmological constant term.

Dynamical analysis of logarithmic energy–momentum squared gravity

We perform the dynamical system analysis of a cosmological model in the energy–momentum squared gravity (EMSG) of the form f ( T μ ν T μ ν ) = α ln ( λ T μ ν T μ ν ) , which is known as energy–momentum log gravity (EMLG). In particular, we show that the analytical cosmological solution of EMLG presented by Akarsu et al. (Eur. Phys. J. C 79:846, 2019) is a future attractor. It includes new terms in the right-hand side of the Einstein field equations, which yield constant inertial mass density and provide a dynamical dark energy with a density passing below zero at large redshifts, accommodating a mechanism for screening Λ in the past for α < 0 , suggested for alleviating some cosmological tensions. We show that the second law of thermodynamics requires α ≤ 0 that allows the screening mechanism to take place. We also show that the model gives rise to an entire class of new stable late-time solutions with H → ( Λ + 2 α ) / 3 as a → ∞ , where the new term is due to the constant effective inertial mass density that arises from EMLG contribution of dust, whereas H → Λ / 3 as a → ∞ in the Λ CDM model. We also show the existence of new interesting features and trajectories that are absent in Λ CDM with or without spatial curvature.

Anisotropic spacetimes in f(T, B) theory II: Kantowski–Sachs Universe

In the context of the modified teleparallel f(T, B) theory of gravity, we consider a homogeneous and anisotropic background geometry described by the Kantowski–Sachs line element. We derive the field equations and investigate the existence of exact solutions. Furthermore, the evolution of the trajectories for the field equations is studied by deriving the stationary points at the finite and infinite regimes. For the $$f(T,B)=T+F\left( B\right) $$ theory, we prove that for a specific limit of the function $$F\left( B\right) $$ , the anisotropic Universe has the expanding and isotropic Universe as an attractor with zero spatial curvature. We remark that there are no future attractors where the asymptotic solution describes a Universe with nonzero spatial curvature.

Bianchi I cosmological solutions in teleparallel gravity

In this paper, we discuss the dynamics of the anisotropic Bianchi I Universe filled with perfect fluid in teleparallel [Formula: see text]-gravity. Using both analytical and numerical approaches, we confirm the main results of the previous authors such as the presence of a future isotropic attractor, and find a number of new ones such as the existence of an essential number of bounce solutions and the possibility to construct late-time acceleration without any specific matter.

Anisotropic spacetimes in chiral scalar field cosmology

Study the behaviour and the evolution of the cosmological field equations in an homogeneous and anisotropic spacetime with two scalar fields coupled in the kinetic term. Specifically, the kinetic energy for the scalar field Lagrangian is that of the Chiral model and defines a two-dimensional maximally symmetric space with negative curvature. For the background space we assume the locally rotational spacetime which describes the Bianchi I, the Bianchi III and the Kantowski-Sachs anisotropic spaces. We work on the $H$% -normalization and we investigate the stationary points and their stability. For the exponential potential we find a new exact solution which describes an anisotropic inflationary solution. The anisotropic inflation is always unstable, while future attractors are the scaling inflationary solution or the hyperbolic inflation. For scalar field potential different from the exponential, the de Sitter universe exists.

Future attractors of Bianchi types II and V cosmologies with massless Vlasov matter

It is shown that the generalized Collins–Stewart radiation and Milne solutions are attractors of the massless Einstein–Vlasov system for Bianchi types II and V spacetimes, respectively. The proof is based on an energy method and bootstrap argument which are used to determine the decay rates of the perturbations away from the attractors.

Dynamical study of ellipsoidal universes

A Universe containing uniform magnetic fields, strings, or domain walls is shown to have an ellipsoidal expansion. This case is motivated from observational cosmology, especially the anomaly concerning the low quadrupole amplitude compared to the best-fit ΛCDM prediction in WMAP data. It is shown that a Universe with eccentricity at decoupling of order 10−2 can reduce the quadrupole amplitude without affecting higher multipoles of the angular power spectrum of the temperature anisotropy. The evolution of ellipsoidal Universes is studied using dynamical system techniques for the first time. The determined critical points vary between saddle and past attractors depending on dark energy state equation parameter w Λ, with no future attractors. Numerical integrations of this dynamical system done using several initial conditions are showing new important results. For instance, a tendency for high expansion differences between planar and perpendicular axes is observed which contradicts previous assumptions on the evolution behaviour of ellipsoid Universes.

Weak tracking in nonautonomous chaotic systems.

Previous studies have shown that rate-induced transitions can occur in pullback attractors of systems subject to "parameter shifts" between two asymptotically steady values of a system parameter. For cases where the attractors limit to equilibrium or periodic orbit in past and future limits of such an nonautonomous systems, these can occur as the parameter change passes through a critical rate. Such rate-induced transitions for attractors that limit to chaotic attractors in past or future limits has been less examined. In this paper, we identify a new phenomenon is associated with more complex attractors in the future limit: weak tracking, where a pullback attractor of the system limits to a proper subset of an attractor of the future limit system. We demonstrate weak tracking in a nonautonomous Rössler system, and argue there are infinitely many critical rates at each of which the pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor. We also state some necessary conditions that are needed for weak tracking.

Future Attractors in 2+1 Dimensional Λ Gravity

Any 2+1 dimensional Einsteinian spacetime with positive cosmological constant and compact hyperbolic spatial topology evolves in the expanding direction to a de Sitter model with homogeneous hyperbolic spatial geometry. In particular, the latter constitutes the uniform future attractor for this class of spacetimes for arbitrary large initial data. This is an example of large-data homogenization and demonstrates the ability of gravitation to cause late time homogenization in the evolution of cosmological models independently of their initial state.

Inhomogeneous spacetimes in Weyl integrable geometry with matter source

We investigate the existence of inhomogeneous exact solutions in Weyl Integrable theory with a matter source. In particular, we consider the existence of a dust fluid source while for the underlying geometry we assume a line element which belongs to the family of silent universes. We solve explicitly the field equations and we find the Szekeres spacetimes in Weyl Integrable theory. We show that only the isotropic family can describe inhomogeneous solutions where the LTB spacetimes are included. A detailed analysis of the dynamics of the field equations is given where the past and future attractors are determined. It is interesting that the Kasner spacetimes can be seen as past attractors for the gravitation models, while the unique future attractor describes the Milne universe similar with the behaviour of the gravitational model in the case of General Relativity.

Collins in Wonderland

What is the asymptotic future of a scalar-field model if the assumption of isotropy is relaxed in generic, homogeneous Universe models? This paper is a continuation of our previous work on Bianchi cosmologies with a p-form field (where p ∈ {1, 3})—or equivalently: an inhomogeneous, mass-less scalar gauge field with a homogeneous gradient. In this work we investigate such matter sector in general relativity, and restrict to space-times of the particular Bianchi types VI0 and , where = {h < 0⎸h ≠ −1/9 ∪ −1}. We show that the previously found fabric of exact solutions named Wonderland are future attractors in (VI0) and (), extending the Collins perfect-fluid equilibrium set to include a p-form (with p ∈ {1, 3}). We also write down the line-element corresponding to Wonderland in and give explicit expressions for the underlying gauge-potential ϕ(t, x) corresponding to this solution. Simulation of a path approaching Wonderland in Bianchi type I is also given.

Approaching Wonderland

Continuing previous work, we show the existence of stable, anisotropic future attractors in Bianchi invariant sets with a p -form field () and a perfect fluid. In particular, we consider the not previously investigated Bianchi invariant sets (II), (IV), (VII0) and (VIIh) and determine their asymptotic behaviour. We find that the isolated equilibrium set Wonderland is a future attractor on all of its existence () in all these sets except in (II), where the peculiar equilibrium sets Edge and Rope show up, taking over the stability for certain values of . In addition, in (IV) and (VIIh) plane gravitational wave solutions (with a non-zero p -form) serve as attractors whenever .

Dynamics of Einstein\u2013Aether scalar field cosmology

We study the cosmological evolution of the field equations in the context of Einstein–Aether cosmology by including a scalar field in a spatially flat Friedmann–Lemaître–Robertson–Walker spacetime. Our analysis is separated into two separate where a pressureless fluid source is included or absent. In particular, we determine the critical points of the field equations and we study the stability of the specific solutions. The limit of general relativity is fully recovered, while the dynamical system admits de Sitter solutions which can describe the past inflationary era and the future late-time attractor. Results for generic scalar field potentials are presented while some numerical behaviours are given for specific potential forms.