  The Fedoriuk-Maslov catastrophe theory of caustics and turning points is
extended to solve the bifurcation problems by the improved stationary phase
method (ISPM). The trace formulas for the radial power-law (RPL) potentials are
presented by the ISPM based on the second- and third-order expansion of the
classical action near the stationary point. A considerable enhancement of
contributions of the two orbits (pair of consisting of the parent and newborn
orbits) at their bifurcation is shown. The ISPM trace formula is proposed for a
simple bifurcation scenario of Hamiltonian systems with continuous symmetries,
where the contributions of the bifurcating parent orbits vanish upon
approaching the bifurcation point due to the reduction of the end-point
manifold. This occurs since the contribution of the parent orbits is included
in the term corresponding to the family of the newborn daughter orbits. Taking
this feature into account, the ISPM level densities calculated for the RPL
potential model are shown to be in good agreement with the quantum results at
the bifurcations and asymptotically far from the bifurcation points.
