The anisotropic BCS gap equation is applied to a tight-binding model of layered high-temperature superconductors. The possible solutions have s-, d-, and mixed s- and d-wave symmetries using nearest-neighbor intralayer singlet pairing interaction of the same strength in the x and y directions. The transitions from d- or s- to mixed s- and d-wave solutions result from pitchfork bifurcations. In the case of slightly different pairing strength in the x and y directions, perturbed pitchfork bifurcations emerge, leading to a dramatic change in the physical properties of the superconducting state.