Splitting of a dynamical system in the neighborhood of a stable invariant manifold

1978 ◽  
Vol 29 (4) ◽  
pp. 427-431 ◽  
Author(s):  
A. M. Samoilenko ◽  
A. V. Dvorak
Keyword(s):  
Dynamical System ◽  
Invariant Manifold ◽  
Stable Invariant

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

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.

FRACTAL DIMENSION FOR POINCARÉ RECURRENCES AS AN INDICATOR OF SYNCHRONIZED CHAOTIC REGIMES

2000 ◽  
Vol 10 (10) ◽  
pp. 2323-2337 ◽  
Author(s):  
VALENTIN S. AFRAIMOVICH ◽  
WEN-WEI LIN ◽  
NIKOLAI F. RULKOV
Keyword(s):  
Fractal Dimension ◽  
Invariant Manifold ◽  
Frequency Ratio ◽  
Chaos Synchronization ◽  
Invariant Set ◽  
Chaotic Oscillations ◽  
New Approach ◽  
Stable Invariant

The studies of the phenomenon of chaos synchronization are usually based upon the analysis of the existence of transversely stable invariant manifold that contains an invariant set of trajectories corresponding to synchronous motions. In this paper we develop a new approach that relies on the notions of topological synchronization and the dimension for Poincaré recurrences. We show that the dimension of Poincaré recurrences may serve as an indicator for the onset of synchronized chaotic oscillations. This indicator is capable of detecting the regimes of chaos synchronization characterized by the frequency ratio p:q.

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.

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