This paper investigates generic bifurcations from a D-m-invariant equilibrium of a D-m-symmetric dynamical system, for m = 3 or 4, near points of codimension-2 steady-state mode interactions. The center manifold is isomorphic to R-3 or R-4 and is non-irreducible. Depending on the group representation, in the unfolding of the linearization we find: symmetry-breaking bifurcations to primary branches, secondary steady-state and Hopf bifurcations; three symmetry-breaking bifurcations of the Bogdanov-Takens type (one transcritical for D-3 and two Z(2)-symmetric for D-4); and a structurally stable heteroclinic cycle in a three dimensional representation of D-4.

The aim of this paper is to present a flexible approach for the efficient computation of the mixed volume of a tuple of polytopes. In order to compute the mixed volume, a mixed subdivision of the tuple of polytopes is needed, which can be obtained by embedding the polytopes in a higher-dimensional space, i.e., by lifting them. Dynamic lifting is opposed to the static approach. This means that one considers one point at a time and only fixes the value of the lifting function when the point really influences the mixed volume. Conservative lifting functions have been developed for this purpose. This provides us with a deterministic manipulation of the lifting for computing mixed volumes, which rules out randomness conditions. Cost estimates for the algorithm are given. The implications of dynamic lifting on polyhedral homotopy methods for the solution of polynomial systems are investigated and applications are presented.

The aim of this paper is to compute all isolated solutions to symmetric polynomial systems. Recently, it has been proved that modelling the sparse structure of the system by its Newton polytopes leads to a computational breakthrough in solving the system. In this paper, it will be shown how the Lifting Algorithm, proposed by Huber and Sturmfels, can be applied to symmetric Newton polytopes. This symmetric version of the Lifting Algorithm enables the efficient construction of the symmetric subdivision, giving rise to a symmetric homotopy, so that only the generating solutions have to be computed. Efficiency is obtained by combination with the product homotopy. Applications illustrate the practical significance of the presented approach.

In two-parameter systems two symmetry breaking bifurcation points of different types coalesce generically within one point. This causes secondary bifurcation points to exist. The aim of this paper is to understand this phenomenon with group theory and the inner-connectivity of irreducible representations of supergroup and subgroups. Colored pictures of examples are included.