Almost-invariant sets and invariant manifolds — Connecting probabilistic and geometric descriptions of coherent structures in flows

2009 ◽  
Vol 238 (16) ◽  
pp. 1507-1523 ◽  
Author(s):  
Gary Froyland ◽  
Kathrin Padberg
Keyword(s):  
Coherent Structures ◽  
Invariant Manifolds ◽  
Invariant Sets ◽  
Almost Invariant Sets

scholarly journals CROSS-CONSTRAINED VARIATIONAL PROBLEM AND THE NON-LINEAR KLEIN–GORDON EQUATIONS

2008 ◽  
Vol 50 (3) ◽  
pp. 467-481 ◽  
Author(s):  
ZAIHUI GAN ◽  
JIAN ZHANG
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Variational Method ◽  
Variational Problem ◽  
Invariant Manifolds ◽  
Invariant Sets ◽  
Sharp Threshold ◽  
Non Linear ◽  
Klein Gordon ◽  
Three Space

AbstractIn this paper, we put forward a cross-constrained variational method to study the non-linear Klein–Gordon equations with an inverse square potential in three space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish some new types of invariant sets for the equation and derive a sharp threshold of blowup and global existence for its solution. Finally, we give an answer to the question how small the initial data are for the global solution to exist.

scholarly journals From manifolds to Lagrangian coherent structures in galactic bar models

2018 ◽  
Vol 618 ◽  
pp. A72 ◽  
Author(s):  
P. Sánchez-Martín ◽  
J. J. Masdemont ◽  
M. Romero-Gómez
Keyword(s):  
Dynamical System ◽  
Fluid Dynamics ◽  
Coherent Structures ◽  
Invariant Manifolds ◽  
Lagrangian Coherent Structures ◽  
Galactic Dynamics ◽  
Secular Evolution ◽  
Autonomous Case ◽  
The Galaxy ◽  
Pattern Speed

We study the dynamics near the unstable Lagrangian points in galactic bar models using dynamical system tools in order to determine the global morphology of a barred galaxy. We aim at the case of non-autonomous models, in particular with secular evolution, by allowing the bar pattern speed to decrease with time. We have extended the concept of manifolds widely used in the autonomous problem to the Lagrangian coherent structures (LCS), widely used in fluid dynamics, which behave similar to the invariant manifolds driving the motion. After adapting the LCS computation code to the galactic dynamics problem, we apply it to both the autonomous and non-autonomous problems, relating the results with the manifolds and identifying the objects that best describe the motion in the non-autonomous case. We see that the strainlines coincide with the first intersection of the stable manifold when applied to the autonomous case, while, when the secular model is used, the strainlines still show the regions of maximal repulsion associated to both the corresponding stable manifolds and regions with a steep change of energy. The global morphology of the galaxy predicted by the autonomous problem remains unchanged.

Fuzzy Responses and Bifurcations of a Forced Duffing Oscillator with a Triple-Well Potential

2015 ◽  
Vol 25 (01) ◽  
pp. 1550005 ◽  
Author(s):  
Ling Hong ◽  
Jun Jiang ◽  
Jian-Qiao Sun
Keyword(s):  
Steady State ◽  
Master Equation ◽  
State Space ◽  
Invariant Manifolds ◽  
The State ◽  
Invariant Sets ◽  
Generalized Cell Mapping ◽  
Potential Wells ◽  
Potential System ◽  
Fuzzy Noise

Responses and bifurcations of a forced triple-well potential system with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established as a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min–max operator of fuzzy logic. A fuzzy response is characterized by its topology in the state space and its possibility measure of membership distribution functions (MDFs). A fuzzy bifurcation implies a sudden change both in the topology and in the MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The MDFs are determined by the quantitative analysis of the FGCM with the min–max calculations. With an increase of the intensity of fuzzy noise, noise-induced escape from each of the potential wells defines two types of bifurcations, namely catastrophe and explosion. This paper focuses on the evolution of transient and steady-state MDFs of the fuzzy response. As the intensity of fuzzy noise increases, steady-state MDFs cover a bigger area in the state space with higher membership values spreading out to a larger area. The previous conjectures are further confirmed that steady-state MDFs are dependent on initial possibility distributions due to the nonsmooth and nonlinear nature of the min–max operation. It is found that as time goes on, transient MDFs spread around three potential wells. The evolutionary orientation of transient MDFs aligns with unstable invariant manifolds leading to stable invariant sets. Two examples of additive and multiplicative fuzzy noise are given.

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