Lorenz type attractors from codimension one bifurcation
type: Item
instance of: scholarly article
title: Lorenz type attractors from codimension one bifurcation
zbMATH Open document ID: 0839.34064
DOI: 10.1007/BF02219362
author: Valentin Afraimovich
author: Shui-Nee Chow
author: Weishi Liu
published in: Journal of Dynamics and Differential Equations
publication date: 1996-06-04T00:00:00Z
review text: The authors describe a class of vector fields (on \(\mathbb{R}^3\)) on the boundary of Morse-Smale systems which, under generic bifurcation, produce Lorenz-type attractors. A Lorenz-type attractor is similar to the well-known ``geometric Lorenz attractors'' [\textit{J. Guckenheimer}, The Hopf Bifurcation and its Applications, Springer, 368-381 (1976; Zbl 0346.58007)], but its stable foliation need only be continuous rather than smooth [\textit{V. Afraimovich}, \textit{V. Bykov} and \textit{L. Shil'nikov}, Trans. Moscow Math. Soc. 44, 153-216 (1983)]. The vector fields studied by the authors have a hyperbolic equilibrium \(O\) with two-dimensional stable manifold and one-dimensional unstable manifold, a saddle-node limit cycle \(L\) with an attracting Floquet exponent; one heteroclinic orbit from \(L\) to \(O\); and two from \(O\) to \(L\). The authors assume that the two-dimensional unstable set of \(L\) is transverse to the stable manifold of \(O\). Then, there are also orbits homoclinic to \(L\). The authors define invariants of these homoclinic orbits called ``conventional Floquet exponents''. One of their hypotheses is stated in terms of these exponents.
reviewed by: Stephen Schecter
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 800426
zbMATH Keywords: vector fields
zbMATH Keywords: Morse-Smale systems
zbMATH Keywords: bifurcation
zbMATH Keywords: Lorenz-type attractors
zbMATH Keywords: hyperbolic equilibrium
zbMATH Keywords: stable manifold
zbMATH Keywords: unstable manifold
zbMATH Keywords: saddle-node limit cycle
zbMATH Keywords: heteroclinic orbit
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 800426