RESUMEN
Slowly varying conservative systems are analyzed in the case of a reverse subcritical pitchfork bifurcation in which two saddles and
a center coalesce. Before the bifurcation
there is a hyperbolic double-homoclinic orbit connecting a linear saddle point. At the
bifurcation a double nonhyperbolic homoclinic orbit connects to a nonlinear saddle point. Strongly
nonlinear oscillations obtained by the method of averaging
are not valid near unperturbed homoclinic orbits. In the case in which the solution slowly
passes through the nonhyperbolic homoclinic orbit associated with the subcritical pitchfork bifurcation, the solution consists
of a large sequence of nonhyperbolic
homoclinic orbits connecting autonomous nonlinear saddle approaches. Solutions are captured into the
left and right well. Phase
jumps and the boundaries of the basins
of attraction are computed. It is
shown that the change in action
in the slow passage through the nonhyperbolic homoclinic orbits is much larger
than the known change in action for the
slow crossing of hyperbolic homoclinic
orbits. Near the boundary of
the basin of attraction, where the energy
is particularly small, one of
the saddle approaches is governed
by the second Painlev\a'e transcendent, which is not
autonomous, and the solution may oscillate around the middle center
or approach the two saddles
created by the subcritical pitchfork bifurcation in addition to oscillating around the left
and right wells.