Geometric desingularization of degenerate singularities in the presence of fast rotation: A new proof of known results for slow passage through Hopf bifurcations

Abstract

In this article, we present a new, geometric proof of known results for slow passage through Hopf bifurcations (Baer et al. (1989), Neishtadt (1987, 1988), Shishkova (1973)). The new proof employs integration along a suitable choice of contour in the complex time plane and then the method of geometric desingularization, which is also known as the blow-up method. The contour in the complex time plane is chosen so that the singularities in these slow passage problems become nilpotent, and the loss of hyperbolicity can then be analyzed using the desingularization method without the complication of the fast rotation.Besides being of interest in their own right, the new method and the new proof of the known results for the slow passage through Hopf bifurcation points are also of current interest, since the phenomena of delayed passage through bifurcations plays an increasingly important role in the analysis of folded singularities in higher-dimensional fast–slow systems with curves and manifolds of nonhyperbolic points. For some of these problems, it will be useful also to formulate the delayed stability loss results by choosing appropriate contours in the complex time plane to make the singularities nilpotent and then by applying the geometric desingularization method.