Lorenz type attractors from codimension one bifurcation
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.
scientific article; zbMATH DE number 800426
scientific article