On the determination of nontrivial equilibrium configurations close to a bifurcation point

Bifurcation phenomena of equilibrium states occur in both standard and complex materials. In this paper we study the equilibrium configurations close to a bifurcation point. In particular the attention is focused on bifurcations of pitchfork type [S.H. Strogatz, Non Linear Dynamics and Chaos, Addison-Wesley Publishing Company, 1994]. This problem is usually solved by using the Signorini’s compatibility of the solution expansion in a neighborhood of the critical point. We show how the same results can be reached in another way which involves just the linear term of the solution expansion. As a test, we analyze two bifurcation phenomena: the buckling of an elastic beam under an axial load and the magnetic field-induced optical switch in nematic liquid crystals.