Global invariant manifolds in the transition to preturbulence in the Lorenz system
type: Item
instance of: scholarly article
title: Global invariant manifolds in the transition to preturbulence in the Lorenz system
zbMATH Open document ID: 1246.37037
author: Eusebius J. Doedel
author: Bernd Krauskopf
author: Hinke M. Osinga
published in: Indagationes Mathematicae. New Series
publication date: 2011-12-14T00:00:00Z
review text: The authors study the first homoclinic bifurcation of the Lorenz system, where two primary periodic orbits of saddle type bifurcate from a symmetric pair of homoclinic loops. The transition through this homoclinic explosion point results in a dramatic change of the topological structure of the basins of the two attracting equilibria. A chaotic saddle is created in a tubular neighborhood of the two homoclinic loops and this invariant hyperbolic set gives rise to preturbulence. By means of numerical methods, the authors show how the two-dimensional stable manifolds of the origin and of the primary periodic orbits intersect a suitable sphere in the phase space. The computations reveal the bifurcating Cantor structure of invariant manifolds and basins, which explain where in phase space long transients -- the characterizing feature of preturbulence -- can be found.
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 5988385
zbMATH Keywords: Lorenz system
zbMATH Keywords: homoclinic explosion
zbMATH Keywords: invariant manifolds
zbMATH Keywords: preturbulence
zbMATH Keywords: Cantor structure
Wikidata QID: Q114476360
reviewed by: Kwok-wai Chung
describes a project that uses: AUTO
describes a project that uses: AUTO-07P
MaRDI profile type: Publication
full work available at URL: j.indag.2011.10.007
OpenAlex ID: W2118884582
cites work: Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields
cites work: Dynamical systems and chaos
cites work: Lorenz Equations Part I: Existence and Nonexistence of Homoclinic Orbits
cites work: Lorenz equations. II: ''Randomly'' rotated homoclinic orbits and chaotic trajectories
cites work: Knaster-like continua and complex dynamics
cites work: Q5424359
cites work: Global bifurcations of the Lorenz manifold
cites work: Computer assisted proof of chaos in the Lorenz equations
cites work: Local and global behavior near homoclinic orbits
cites work: Structural stability of Lorenz attractors
cites work: On compact spaces which are locally Cantor bundles
cites work: Q3673666
cites work: Existence of a Homoclinic Orbit of the Lorenz System by Precise Shooting
cites work: A computer proof that the Lorenz equations have ``chaotic'' solutions
cites work: A shooting approach to the Lorenz equations
cites work: Global aspects of homoclinic bifurcations of vector fields
cites work: Q3099080
cites work: The Lorenz System: I. The Global Structure of its Stable Manifolds
cites work: The Lorenz System: II. The Homoclinic Convolution of the Stable Manifolds
cites work: Preturbulence: A regime observed in a fluid flow model of Lorenz
cites work: Q4316891
cites work: Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields
cites work: Q5424362
cites work: A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS
cites work: Deterministic Nonperiodic Flow
cites work: Q5444070
cites work: Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system
cites work: Chaos in the Lorenz equations: a computer-assisted proof
cites work: Chaos in the Lorenz equations: A computer assisted proof. Part II: Details
cites work: Chaos in the Lorenz equations: A computer assisted proof. III: Classical parameter values
cites work: Q5606531
cites work: Q4003408
cites work: Q3903749
cites work: The topological classification of Lorenz attractors
cites work: On the nature of turbulence
cites work: The Lorenz equations: bifurcations, chaos, and strange attractors
cites work: The Lorenz attractor exists
cites work: What's new on Lorenz strange attractors?
cites work: The structure of Lorenz attractors
DOI: 10.1016/J.INDAG.2011.10.007
description: scientific article; zbMATH DE number 5988385