Periodic-orbit theory of the blowout bifurcation

This paper presents a theory for characterization of the blowout bifurcation by periodic orbits. Blowout bifurcation in chaotic systems occurs when a chaotic attractor, lying in some symmetric invariant subspace, becomes transversely unstable. We present an analysis and numerical results that indicate that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor. There are two distinct groups of periodic orbits: one transversely stable and another transversely unstable. The bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced. Our results thus categorize the blowout bifurcation as a unique type of bifurcation that involves an infinite number of periodic orbits, in contrast to most previously known bifurcations that are mediated by only a finite number of periodic orbits.