2003/04/28 by Bernold Fiedler, Stefan Liebscher
Mathematics · #math.DS #msc:34C23 #msc:34C29 #msc:34C37
published as Proceedings of the ICM, Beijing 2002, vol. 3, 305--316
arxiv created 2003/04/28 · arxiv updated 2009/11/30
Standard bifurcation theory is concerned with families of vector fields dx/dt = f(x,λ), x ∈ \Rn, involving one or several constant real parameters λ. Viewed as a differential equation for the pair (x,λ), we observe a foliation of the total phase space by constant λ. Frequently, the presence of a trivial stationary solution x=0 is also imposed: 0 = f(0,λ). Bifurcation without parameters, in contrast, discards the foliation by a constant parameter λ. Instead, we consider systems dx/dt = f(x,y), dy/dt = g(x,y). Standard bifurcation theory then corresponds to the special case y=λ, g=0. To preserve only the trivial solution x=0, instead, we only require 0 = f(0,y) = g(0,y) for all y. A rich dynamic phenomenology arises, when normal hyperbolicity of the trivial stationary manifold x=0 fails, due to zero or purely imaginary eigenvalues of the Jacobian fx(0,y). Specifically, we address the cases of failure of normal hyperbolicity due to a simple eigenvalue zero, a simple purely imaginary pair (Hopf bifurcation without parameters), a double eigenvalue zero (Takens-Bogdanov bifurcation without parameters), and due to a double eigenvalue zero with additional time reversal symmetries. The results are joint work with Andrei Afendikov, James C. Alexander, and Stefan Liebscher.