Bifurcations and Exact Solutions of the Equation of Barotropic FRW Cosmologies

2017 ◽  
Vol 27 (05) ◽  
pp. 1750080 ◽  
Author(s):  
Jibin Li ◽  
Tonghua Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Exact Solutions ◽  
Periodic Solutions ◽  
Phase Portraits ◽  
Planar Dynamical System ◽  
Heteroclinic Solutions ◽  
Unbounded Solutions ◽  
Level Curves ◽  
Friedmann Robertson Walker

In this paper, we study the equation of barotropic Friedmann–Robertson–Walker cosmologies. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system. Corresponding to different level curves, we derive exact explicit parametric representations of bounded and unbounded solutions, such as periodic solutions, periodic peakon solutions, homoclinic and heteroclinic solutions and compacton solutions.

Isochronous cosmological solutions of the Friedmann–Robertson–Walker model

2019 ◽  
Vol 34 (11) ◽  
pp. 1950062
Author(s):  
Aiyong Chen ◽  
Xiaokai He ◽  
Caixing Tian
Keyword(s):  
Dynamical System ◽  
Periodic Solutions ◽  
Cosmic Time ◽  
Time Variable ◽  
Conformal Time ◽  
Order Ordinary Differential Equation ◽  
Period Function ◽  
Planar Dynamical System ◽  
Isochronous Centers ◽  
Friedmann Robertson Walker

In this paper, the periodic solutions of the equation of Friedmann–Robertson–Walker cosmology with a cosmological constant are investigated. Using variable transformation, the original second-order ordinary differential equation is converted to a planar dynamical system with cosmic time t. Numerical simulations indicate that period function T(h) of this dynamical system is monotonically increasing. However, a new planar dynamical system could be deduced by using conformal time variable [Formula: see text]. We prove that the new planar dynamical system has two isochronous centers under certain parameter conditions by using Picard–Fuchs equation. Explicitly, we find that there exist two families of periodic solutions with equal period for the new planar dynamical system which is derived from the Friedmann–Robertson–Walker model.

Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity

2017 ◽  
Vol 27 (12) ◽  
pp. 1750188
Author(s):  
Yan Zhou ◽  
Jibin Li
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Optical Waveguides ◽  
Bifurcation Theory ◽  
Phase Portraits ◽  
Explicit Solutions ◽  
Theory Method ◽  
Planar Dynamical System ◽  
Soliton Model ◽  
Straight Lines

Raman soliton model in nanoscale optical waveguides, with metamaterials, having polynomial law nonlinearity is investigated by the method of dynamical systems. The functions [Formula: see text] in the solutions [Formula: see text] [Formula: see text] satisfy a singular planar dynamical system having two singular straight lines. By using the bifurcation theory method of dynamical systems to the equations of [Formula: see text], under 23 different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons and peakons as well as compacton solutions for this planar dynamical system are obtained. 92 exact explicit solutions of system (6) are derived.

Exact Solutions and Bifurcations in Invariant Manifolds for a Nonic Derivative Nonlinear Schrödinger Equation

2016 ◽  
Vol 26 (08) ◽  
pp. 1650136 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Invariant Manifolds ◽  
First Integrals ◽  
Phase Portraits ◽  
Planar Dynamical System ◽  
Nonlinear Schrödinger ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Dimensional Integral

Propagating modes in a class of nonic derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions [Formula: see text] and [Formula: see text] in the solutions [Formula: see text], [Formula: see text] satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude [Formula: see text] can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions [Formula: see text] and [Formula: see text] can be exactly obtained. Thirty six exact explicit solutions of equation are derived.

Index Evaluation for Dynamical Systems and Its Application to Locating All the Zeros of a Vector Function

1983 ◽  
Vol 50 (4a) ◽  
pp. 858-862 ◽  
Author(s):  
C. S. Hsu ◽  
R. S. Guttalu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Vector Function ◽  
Evaluation Method ◽  
Point Mapping ◽  
Strongly Nonlinear ◽  
Index Evaluation

An index evaluation method is discussed in this paper. It can also serve as the basis of a procedure to locate all the zeros of a vector function. An application of the procedure is made to a strongly nonlinear point-mapping dynamical system in order to locate all the periodic solutions of period one and period two, 41 in total number.

scholarly journals Periodic Peakon and Smooth Periodic Solutions for KP-MEW(3,2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Liping He
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Phase Portrait ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Wave System ◽  
Integral Constant ◽  
Straight Line

In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for c < 0 , 0 < c < 1 , and c > 1 is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.

scholarly journals Existence of random invariant periodic curves via random semiuniform ergodic theorem

2016 ◽  
Vol 17 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Kenneth Uda
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Dissipative Structure ◽  
Random Dynamical Systems ◽  
Random Dynamical System ◽  
Compact Set

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system on a cylinder [Formula: see text] has a dissipative structure, we proved that a random invariant compact set can be expressed as a union of finite of number of random periodic curves. The idea in this paper is closely related to the work recently considered by Zhao and Zheng [46].

scholarly journals Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials, Having Anti-cubic Nonlinearity

2021 ◽  
Author(s):  
Guoan Xu ◽  
Jibin Li ◽  
Yi Zhang
Keyword(s):  
Dynamical System ◽  
Optical Waveguides ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Wave Transformation ◽  
Cubic Nonlinearity ◽  
Planar Dynamical System ◽  
Soliton Model ◽  
Wave Solutions

Abstract This paper investigates Raman soliton model in optical metamaterials, having anti-cubic nonlinearity. By travelling wave transformation, the model is transformed into a singular planar dynamical system having three singular straight lines. Using the bifurcation theory method of dynamical systems, under different parameter conditions, bifurcations of phase portraits are studied. More than 30 exact explicit solutions of planar dynamical system are derived, such as exact periodic wave solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic peakons and peakons as well as compacton solutions. In more general parametric conditions, all possible solutions are found.

BIANCHI COSMOLOGIES AS DYNAMICAL SYSTEMS WITHOUT THE CONSTRAINT

2000 ◽  
Vol 15 (17) ◽  
pp. 2771-2791
Author(s):  
MAREK SZYDŁOWSKI ◽  
ADAM KRAWIEC
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Phase Portraits ◽  
Finite Domain ◽  
Hamiltonian Constraint ◽  
Two Dimensional ◽  
Class A ◽  
Dimensional Dynamical System ◽  
Analysis Of Dynamics

The Bianchi class A cosmology is treated as a nonlinear dynamical system. In the new variables in which Hamiltonian constraint is solved algebraically, the Bianchi class A model assumes the form of autonomous dynamical system in ℝ4 with polynomial form of vector field. It is proposed that the dimension of minimum reduced phase spaces of unconstrained autonomous systems be treated as a measure of generality of solution. The behavior of these models is studied in terms of qualitative analysis of differential equations. It is shown that the more general Bianchi IX and Bianchi VIII models (called Mixmaster models) can be presented as four-dimensional. We argue that the reduced Mixmaster dynamical systems are chaotic in the same sense as the original ones. The Bianchi I and Bianchi II world models are described by one-dimensional and two-dimensional systems, respectively. We also study dynamics of Bianchi VI0 and Bianchi VII0 models as a three-dimensional dynamical system. For two-dimensional dynamical system, the phase portraits are constructed with the Poincaré sphere which allows the analysis of dynamics both in finite domain and at infinity. For the last class of models we find an invariant submanifold on which systems are analyzed in details.

scholarly journals VII.—On the Adelphic Integral of the Differential Equations of Dynamics

1918 ◽  
Vol 37 ◽  
pp. 95-116 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Two Degrees Of Freedom ◽  
Image Position ◽  
Kinetic And Potential Energies

§ 1. Ordinary and singular periodic solutions of a dynamical system. — The present paper is concerned with the motion of dynamical systems which possess an integral of energy. To fix ideas, we shall suppose that the system has two degrees of freedom, so that the equations of motion in generalised co-ordinates may be written in Hamilton's formwhere (q1q2) are the generalised co-ordinates, (p1, p2) are the generalised momenta, and where H is a function of (q1, q2, p1, p2) which represents the sum of the kinetic and potential energies.

A YING–YANG THEORY IN NONLINEAR DISCRETE DYNAMICAL SYSTEMS

2010 ◽  
Vol 20 (04) ◽  
pp. 1085-1098 ◽  
Author(s):  
ALBERT C. J. LUO
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Hénon Map ◽  
Henon Map ◽  
Nonlinear Dynamical ◽  
Iterative Maps

This paper presents a Ying–Yang theory for nonlinear discrete dynamical systems considering both positive and negative iterations of discrete iterative maps. In the existing analysis, the solutions relative to "Yang" in nonlinear dynamical systems are extensively investigated. However, the solutions pertaining to "Ying" in nonlinear dynamical systems are investigated. A set of concepts on "Ying" and "Yang" in discrete dynamical systems are introduced to help one understand the hidden dynamics in nonlinear discrete dynamical systems. Based on the Ying–Yang theory, the periodic and chaotic solutions in nonlinear discrete dynamical system are discussed, and all possible, stable and unstable periodic solutions can be analytically predicted. A discrete dynamical system with the Henon map is investigated, as an example.

Sign in / Sign up

Export Citation Format

Share Document