the post-Newtonian limit, we find the sequence of the equilibrium solutions [*bifurcates*]{} to three branches at some critical central density. One branch consists of solutions that are identical to neutron stars in general relativity, and the other two consist of solutions that deviate significantly from neutron stars in general relativity. The general relativistic branch is secularly unstable in agreement with the result obtained by a perturbation study [@harada1997], while the non-general-relativistic branches are secularly stable.

This paper is organized as follows. In Sec. II, we summarize the field equations of scalar-tensor theory and the equations determining equilibrium solutions of neutron stars in this gravitational theory. In Sec. III, we present stability criteria on the grounds of the turning point method. In Sec. IV, the stability criteria are applied to equilibrium solutions of neutron stars in scalar-tensor theory and some consequences of catastrophe theory are discussed. Section V is devoted to conclusions. We use units in which $c=1$. The Greek indices run from 0 to 3. We follow the Misner-Thorne-Wheeler [@mtw1973] sign conventions for curvature quantities.

Basic equations
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Here we consider a class of scalar-tensor theories in which gravity is mediated by not only a metric tensor but also a 