  We argue by way of examples that, as a nonlinear integral equation, the gap
equation can and does possess many physically distinct solutions for the
dressed-quark propagator. The examples are drawn from a class that is
successful in describing a broad range of hadron physics observables. We apply
the homotopy continuation method to each of our four exemplars and thereby find
all solutions that exist within the interesting domains of light current-quark
masses and interaction strengths; and simultaneously provide an explanation of
the nature and number of the solutions, many of which may be associated with
dynamical chiral symmetry breaking. Introducing a stability criterion based on
the scalar and pseudoscalar susceptibilities we demonstrate, however, that for
any nonzero current-quark mass only the regular Nambu solution of the gap
equation is stable against perturbations. This guarantees that the existence of
multiple solutions to the gap equation cannot complicate the description of
phenomena in hadron physics.
