We consider two-parameter bifurcation of equilibrium states of an elastic rod on a deformable foundation. Our main theorem shows that bifurcation occurs if and only if the linearization of our problem has nontrivial solutions. In fact our proof, based on the concept of the Brouwer degree, gives more, namely that from each bifurcation point there branches off a continuum of solutions.

We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classi cations are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from R^n to R^n and proper gradient maps from R^2 to R^2 with the Brouwer degree greater or equal to zero.

We study bifurcation of equilibrium states of an elastic rod on a two-parameter Winkler foundation. In the article "Bifurcation of equilibrium forms of an elastic rod on a two-parameter Winkler foundation" [Nonlinear Anal., Real World Appl. 39 (2018) 451-463] the existence of simple bifurcation points was proved by the use of the Crandall-Rabinowitz theorem. In this paper we want to present an alternative proof of this fact based...