Bifurcations of Stable Sets in Noninvertible Planar Maps

Abstract

Many applications give rise to systems that can be described by maps that do not have a unique inverse. We consider here the case of a planar noninvertible map. Such a map folds the phase plane, so that there are regions with different numbers of preimages. The locus where the number of pre-images changes is made up of so-called critical curves, which are defined as the images of the locus where the Jacobian is singular. A typical critical curve corresponds to a fold under the map, so that the number of pre-images typically changes by two.We consider the question of how the stable set of a hyperbolic saddle of a planar noninvertible map changes when a parameter is varied. The stable set is the generalization of the stable manifold for the case of an invertible map. Owing to the changing number of pre-images, the stable set of a noninvertible map may consist of finitely or even infinitely many disjoint branches. It is now possible to compute stable sets with the Search Circle algorithm that we developed recently.We take a bifurcation theory point of view and consider the two basic codimensionone interactions of the stable set with a critical curve, which we call the outer-fold and the inner-fold bifurcations. By taking into account how the stable set is organized globally, these two bifurcations allow one to classify the different possible changes to the structure of a basin of attraction that are reported in the literature. The fundamental difference between the stable set and the unstable manifold is discussed. The results are motivated and illustrated with a single example of a two-parameter family of planar noninvertible maps.

Chaos and Wild Chaos in Lorenz-Type Systems

“…The map (6) has fixed points and periodic points, which correspond to periodic orbits of the associated vector field. If they are saddles then these points have stable and unstable invariant sets, which are the generalisations of stable and unstable manifolds to the context of noninvertible maps; see, for example, [16,17,32] for more details. Points on the stable set W s ( p) of a saddle periodic point p converge to p under iteration of f k where k is the (minimal) period of p; note that k = 1 if p is a fixed point.…”

Bifurcations of the Global Stable Set of a Planar Endomorphism Near a Cusp Singularity

“…They are also powerful when used, for example, for systems with symmetry [Golubitsky & Shaeffer, 1985, Golubitsky et al, 1988. Our work is very much in the spirit of the papers [England et al, 2005 on planar endomorphisms, but it is also close in spirit to bifurcation theory in other contexts. For example, piecewisesmooth systems are characterised by a critical locus where the flow or map changes in a non-smooth or discontinuous way.…”

“…Both bifurcations are tangencies with J 1 that change the geometry of the stable set locally near J 0 . The different global manifestations of these bifurcations account for the different phenomena that have been observed in the literature; see [England et al, 2005, England et al, 2004 for details.…”

“…As reported in [England et al, 2005], from the point of view of bifurcation theory and singularity theory there are generically only two global bifurcations of codimension one. As a parameter is varied, part of the (one-dimensional) stable set either protrudes from a region with k preimages into a region with k + 2 preimages, leading to a disjoint closed curve, or a segment that lies in a region with k preimages retracts so that it lies entirely in a region with k + 2 preimages, which causes a reorganisation of connected components.…”

“…The noninvertibility of the map can give rise to an unusually complex basin and there are many case studies reporting on different phenomena; recent examples are [Agliari, 2000], [Agliari et al, 2003], [Bischi et al, 2006], [England et al, 2004], [England et al, 2005], and [López-Ruiz & Fournier-Prunaret, 2003]. Indeed, smooth noninvertible maps (endomorphisms) on the plane typically fold the phase space to give regions that correspond to different numbers of preimages.…”