Conditions for the abrupt bifurcation to chaotic scattering

One of the generic ways in which chaotic scattering can come about as a system parameter is varied is the so-called "abrupt bifurcation" in which the scattering is nonchaotic on one side of the bifurcation and is chaotic and hyperbolic on the other side, Previous work demonstrating the abrupt bifurcation [S. Bleher et al., Phys. Rev. Lett. 63, 919 (1989); Physica D 46, 87 (1990)] was primarily for the case where the scattering potential had maxima ("hilltops") which had locally circular isopotential contours. Here we extend these considerations to the more general case of locally elliptically shaped isopotential contours at the hilltops. It turns out that the conditions for the abrupt bifurcation change drastically as soon as even a small amount of noncircularity is included (i.e., the circular case is singular). The illustrative case of scattering from three isolated potential hills is dealt with in detail. One interesting result is a simple geometrical sufficient condition for an abrupt bifurcation in the case of large enough ellipticity of the hill with lowest potential at its hilltop.