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We derive the global bifurcation diagram of a three-parameter family of cubic Li enard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications. In fact, it is the 3-jet of a three-dimensional slice of the universal unfolding of a doubly degenerate Bogdanov-Takens point, that is, of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form.

SUMMARYModels describing systems of coevolving populations often have asymptotically non-equilibrium dynamics (Red Queen dynamics (RQD)). We claim that if evolution is much slower than ecological changes, RQD arises due to either fast ecological processes, slow genetical processes, or to their interaction. The three corresponding generic types of RQD can be studied using singular perturbation theory and have very different properties and biological implications. We present simple examples of ecological, genetical, and ecogenetical RQD and describe how they may be recognized in natural populations. In particular, ecogenetical RQD often involve alternations of long epochs with radically different dynamics.