<p>This paper is concerned with the dynamical behavior of a three-dimensional ordinary differential system, which is reduced from the Newell–Whitehead partial differential equations. We focus on the pitchfork bifurcation, Bogdanov–Takens bifurcation of equilibrium, and a brief discussion on the triple-zero bifurcation. The dynamics near nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated center manifold. The results show that the Newell–Whitehead system has complex dynamical behaviors and bifurcation phenomena under appropriate parameter perturbations, such as heteroclinic orbit, symmetric “figure-eight” homoclinic orbits, and three limit cycles enclosing two equilibria, etc. Furthermore, it is shown that the Bogdanov–Takens bifurcation of equilibria can be either of homoclinic or heteroclinic type, and the transition between these two types occurs by means of a triple-zero singularity. It implies that the Newell–Whitehead partial differential equations have periodic wave, solitary wave or kink wave solutions for some parameter values.</p>