Critical homoclinic orbits lead to snap-back repellers (Q634921)

For an expanding fixed point of a map, when does the first homoclinic orbit appear and when can other homoclinic explosions occur? By using the concept of a critical homoclinic orbit, the authors characterize these bifurcations for any kind of map. For an expanding fixed point \(p\) of a piecewise smooth noninvertible map \(f\), they show that in any neighborhood of a noncritical homoclinic orbit of \(p\), there exists an invariant Cantor like set on which \(f\) is chaotic. However, both for smooth and piecewise smooth maps, in a neighborhood of a critical homoclinic orbit, a chaotic set may exist or not exist. Several one-dimensional and higher-dimensional examples are considered to show that, in general, nothing can be stated about a critical homoclinic orbit a priori.