The Utility of an Invariant Manifold Description of the Evolution of a Dynamical System

1989 ◽  
Vol 20 (6) ◽  
pp. 1447-1458 ◽  
Author(s):  
A. J. Roberts
Keyword(s):  
Dynamical System ◽  
Invariant Manifold

scholarly journals Structure Results for Semilinear Elliptic Equations with Hardy Potentials

2018 ◽  
Vol 18 (1) ◽  
pp. 65-85 ◽  
Author(s):  
Matteo Franca ◽  
Maurizio Garrione
Keyword(s):  
Dynamical System ◽  
Invariant Manifold ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Complete Picture ◽  
Radial Solutions ◽  
Semilinear Elliptic ◽  
Slow Decay ◽  
Manifold Theory ◽  
Semilinear Problem

AbstractWe prove structure results for the radial solutions of the semilinear problem\Delta u+\frac{\lambda(|x|)}{|x|^{2}}u+f(u(x),|x|)=0,where λ is afunctionandfis superlinear in theu-variable. As particular cases, we are able to deal with Matukuma potentials and with nonlinearitiesfhaving different polynomial behaviors at zero and at infinity. We give the complete picture for the subcritical, critical and supercritical cases. The technique relies on the Fowler transformation, allowing to deal with a dynamical system in{{\mathbb{R}}^{3}}, for which elementary invariant manifold theory allows to draw the conclusions involving regular/singular and fast/slow-decay solutions.

scholarly journals Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems

1989 ◽  
Vol 31 (1) ◽  
pp. 48-75 ◽  
Author(s):  
A. J. Roberts
Keyword(s):  
Dynamical System ◽  
Invariant Manifold ◽  
Initial Conditions ◽  
Long Term Evolution ◽  
Equation Model ◽  
Starting Position ◽  
System A ◽  
Term Evolution ◽  
Shear Dispersion

AbstractA centre manifold or invariant manifold description of the evolution of a dynamical system provides a simplified view of the long term evolution of the system. In this work, I describe a procedure to estimate the appropriate starting position on the manifold which best matches an initial condition off the manifold. I apply the procedure to three examples: a simple dynamical system, a five-equation model of quasi-geostrophic flow, and shear dispersion in a channel. The analysis is also relevant to determining how best to account, within the invariant manifold description, for a small forcing in the full system.

Dynamical system analysis of Hessence scalar field in teleparallel gravity: Invariant manifold technique

2019 ◽  
Vol 34 (28) ◽  
pp. 1950156 ◽  
Author(s):  
Subhajyoti Pal ◽  
Subenoy Chakraborty
Keyword(s):  
Dynamical System ◽  
Scalar Field ◽  
Invariant Manifold ◽  
Critical Points ◽  
System Analysis ◽  
Teleparallel Gravity ◽  
Center Manifolds ◽  
Field Equations ◽  
Stability Diagrams ◽  
Nonlinear Autonomous System

This paper investigates the cosmological dynamics of the Hessence scalar field coupled with the dark matter in the background of the teleparallel gravity. We have assumed that the potential of the scalar field is exponential in nature whereas the [Formula: see text] appearing in the teleparallel theory has the form [Formula: see text]. The field equations of this system reduce to a nonlinear autonomous system and dynamical system analysis is then performed. Due to the nonlinearity and the existence of multiple zero eigenvalues, the traditional procedures of analysis break down. So some novel technique is required. One of the latest such techniques is the invariant manifold theory. By the application of this theory, one projects the variables linked with the zero eigenvalues onto the variables linked with the nonzero eigenvalues to compute the center manifolds and the reduced systems associated with the critical points. These reduced systems reflect the nature of the whole dynamical systems. They also have less dimension and are often simple in nature. Hence, it is possible to solve them directly. In this paper, we work exactly in this spirit and find the center manifolds and solve the corresponding reduced system for some of the critical points associated with the dynamical system. We discover some interesting results namely that there are certain bounds on the interaction term [Formula: see text] which asserts the stability of the systems. We also present various stability diagrams of the reduced systems. An asymptotic analysis is then done for the critical points at infinity. Finally, we discuss the cosmological interpretation of our results.

Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

2015 ◽  
Vol 25 (06) ◽  
pp. 1550091 ◽  
Author(s):  
Kie Van Ivanky Saputra
Keyword(s):  
Dynamical System ◽  
Invariant Manifold ◽  
Normal Forms ◽  
Volterra Model ◽  
Infection Model ◽  
Global Bifurcations ◽  
Codimension Two ◽  
Codimension One ◽  
Variation Of Parameters ◽  
Local And Global Bifurcations

We investigate a dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analyzed and yield the transcritical bifurcation as the codimension-one bifurcation while the saddle-node–transcritical interaction and the Hopf–transcritical interactions as the codimension-two bifurcations. The unfolding of this degeneracy is also analyzed and reveal global bifurcations such as homoclinic and heteroclinic bifurcations. We apply our results to a modified Lotka–Volterra model and to an infection model in HIV diseases.

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