scholarly journals Lyapunov function for cosmological dynamical system

2017 ◽  
Vol 50 (1) ◽  
pp. 51-55
Author(s):  
Marek Szydłowski ◽  
Adam Krawiec
Keyword(s):  
Dynamical System ◽  
Lyapunov Function ◽  
Stable Solution ◽  
First Integral ◽  
Asymptotically Stable ◽  
Exact Form ◽  
De Sitter ◽  
Sitter Solution ◽  
Friedmann Robertson Walker ◽  
Asymptotically Stable Solution

Abstract We prove the asymptotic global stability of the de Sitter solution in the Friedmann-Robertson-Walker conservative and dissipative cosmology. In the proof we construct a Lyapunov function in an exact form and establish its relationship with the first integral of dynamical system determining evolution of the flat Universe. Our result is that de-Sitter solution is asymptotically stable solution for general form of equation of state p = (ρ, H), where dependence on the Hubble function H means that the effect of dissipation are included.

A DYNAMICAL SYSTEM APPROACH TO THE GBIG COSMOLOGY

2011 ◽  
Vol 08 (06) ◽  
pp. 1179-1188 ◽  
Author(s):  
KOUROSH NOZARI ◽  
F. KIANI
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Cosmological Constant ◽  
System Approach ◽  
Stable Solution ◽  
Late Time ◽  
Curvature Effect ◽  
De Sitter ◽  
Dynamical System Approach ◽  
Bonnet Term

We study the phase space of an extension of the normal DGP cosmology with a cosmological constant on the brane and curvature effect that is incorporated via the Gauss–Bonnet term in the bulk action. We study late-time cosmological dynamics of this scenario within a dynamical system approach. We show that the stable solution of the cosmological dynamics in this model is a de Sitter phase.

scholarly journals Asymptotically Stable Solutions of a Generalized Fractional Quadratic Functional-Integral Equation of Erdélyi-Kober Type

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohamed Abdalla Darwish ◽  
Beata Rzepka
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Stable Solution ◽  
Functional Integral ◽  
Quadratic Functional ◽  
Asymptotically Stable ◽  
Stable Solutions ◽  
Asymptotically Stable Solution

We study a generalized fractional quadratic functional-integral equation of Erdélyi-Kober type in the Banach spaceBC(ℝ+). We show that this equation has at least one asymptotically stable solution.

scholarly journals Asymptotic Stability for Some Types of Nonlinear Fractional Order Differential-Algebraic Control Systems

2021 ◽  
pp. 623-638
Author(s):  
Sameer Qasim Hasan
Keyword(s):  
Feedback Control ◽  
Asymptotic Stability ◽  
Control Systems ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Stable Solution ◽  
Asymptotically Stable ◽  
Fractional Differential ◽  
Necessary And Sufficient ◽  
Asymptotically Stable Solution

The aim of this paper is to study the asymptotically stable solution of nonlinear single and multi fractional differential-algebraic control systems, involving feedback control inputs, by an effective approach that depends on necessary and sufficient conditions.

scholarly journals Autonomous dynamical system description of de Sitter evolution in scalar assisted f(R) − ϕ gravity

2018 ◽  
Vol 15 (12) ◽  
pp. 1850212 ◽  
Author(s):  
K. Kleidis ◽  
V. K. Oikonomou
Keyword(s):  
Dynamical System ◽  
Numerical Analysis ◽  
Scalar Field ◽  
Analysis Approach ◽  
Single Variable ◽  
De Sitter ◽  
Exponential Potential ◽  
System Description ◽  
Variable Formula ◽  
Autonomous Dynamical System

In this paper we will study the cosmological dynamical system of an [Formula: see text] gravity in the presence of a canonical scalar field [Formula: see text] with an exponential potential by constructing the dynamical system in a way that it is rendered autonomous. This feature is controlled by a single variable [Formula: see text], which when it is constant, the dynamical system is autonomous. We focus on the [Formula: see text] case which, as we demonstrate by using a numerical analysis approach, leads to an unstable de Sitter attractor, which occurs after [Formula: see text] [Formula: see text]-foldings. This instability can be viewed as a graceful exit from inflation, which is inherent to the dynamics of de Sitter attractors.

scholarly journals DUALITY AND SELF-DUAL TRIPLET SOLUTIONS IN EUCLIDEAN GRAVITY

1996 ◽  
Vol 11 (34) ◽  
pp. 2669-2679
Author(s):  
SWAPNA MAHAPATRA
Keyword(s):  
String Theory ◽  
The Self ◽  
Target Space ◽  
De Sitter ◽  
Gravitational Instanton ◽  
Duality Symmetry ◽  
Sitter Solution

Starting from the self-dual “triplet” of gravitational instanton solutions in Euclidean gravity, we obtain the corresponding instanton solutions in string theory by making use of the target space duality symmetry. We show that these dual triplet solutions can be obtained from the general dual Taub-NUT de Sitter solution through some limiting procedure as in the Euclidean gravity case. The dual gravitational instanton solutions obtained here are self-dual for some cases, with respect to certain isometries, but not always.

scholarly journals On asymptotically stable sets

1970 ◽  
Vol 17 (2) ◽  
pp. 181-186 ◽  
Author(s):  
D. Desbrow
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Metric Space ◽  
Physical System ◽  
Asymptotically Stable ◽  
Stable Sets ◽  
Compact Closure ◽  
First Order ◽  
Closed Sets ◽  
The Future

In this paper we study closed sets having a neighbourhood with compact closure which are positively asymptotically stable under a flow on a metric space X. For an understanding of this and the rest of the introduction it is sufficient for the reader to have in mind as an example of a flow a system of first order, autonomous ordinary differential equations describing mathematically a time-independent physical system; in short a dynamical system. In a flow a set M is positively stable if the trajectories through all points sufficiently close to M remain in the future in a given neighbourhood of M. The set M is positively asymptotically stable if it is positively stable and, in addition, trajectories through all points of some neighbourhood of M approach M in the future.

scholarly journals Energy conditions in non-local gravity

2019 ◽  
Vol 16 (10) ◽  
pp. 1950149 ◽  
Author(s):  
M. Ilyas
Keyword(s):  
Power Law ◽  
Arbitrary Function ◽  
Energy Conditions ◽  
De Sitter ◽  
Sitter Solution ◽  
Local Gravity ◽  
Non Local

We investigate the different energy conditions in non-local gravity, which is obtained by adding an arbitrary function of d’Alembertian operator, [Formula: see text], to the Hilbert–Einstein action. We analyze the validity of four different energy conditions and illustrate the different constraints over parameters of the power-law solution as well as de Sitter solution.

Dynamical system approach for the modified Brans–Dicke theory

2019 ◽  
Vol 28 (10) ◽  
pp. 1950132 ◽  
Author(s):  
Jianbo Lu ◽  
Xin Zhao ◽  
Shining Yang ◽  
Jiachun Li ◽  
Molin Liu
Keyword(s):  
Dynamical System ◽  
Dark Energy ◽  
Critical Point ◽  
System Approach ◽  
State Parameter ◽  
Kinetic Term ◽  
De Sitter ◽  
Dynamical System Approach ◽  
De Sitter Universe ◽  
Manifold Theory

A modified Brans–Dicke theory (abbreviated as GBD) is proposed by generalizing the Ricci scalar [Formula: see text] to an arbitrary function [Formula: see text] in the original BD action. It can be found that the GBD theory has some interesting properties, such as solving the problem of PPN value without introducing the so-called chameleon mechanism (comparing with the [Formula: see text] modified gravity), making the state parameter to crossover the phantom boundary: [Formula: see text] without introducing the negative kinetic term (comparing with the quintom model). In the GBD theory, the gravitational field equation and the cosmological evolutional equations have been derived. In the framework of cosmology, we apply the dynamical system approach to investigate the stability of the GBD model. A five-variable cosmological dynamical system and three critical points ([Formula: see text], [Formula: see text], [Formula: see text]) are obtained in the GBD model. After calculation, it is shown that the critical point [Formula: see text] corresponds to the radiation dominated universe and it is unstable. The critical point [Formula: see text] is unstable, which corresponds to the geometrical dark energy dominated universe. While for case of [Formula: see text], according to the center manifold theory, this critical point is stable, and it corresponds to geometrical dark energy dominated de Sitter universe ([Formula: see text]).

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