  We investigate neutron stars in scalar-tensor theories. We examine their
secular stability against spherically symmetric perturbations by use of a
turning point method. For some choices of the coupling function contained in
the theories, the number of the stable equilibrium solutions changes and the
realized equilibrium may change discontinuously as the asymptotic value of the
scalar field or total baryon number is changed continuously. The behaviour of
the stable equilibrium solutions is explained by fold and cusp catastrophes.
Whether the cusp catastrophe appears or not depends on the choices of the
coupling function. These types of the catastrophes are structurally stable.
Recently discovered spontaneous scalarization, which is non-perturbative
strong-field phenomenon due to the presence of the gravitational scalar field,
is well described in terms of the cusp catastrophe.
