Catastrophe theory in Dulac unfoldings

Abstract

In this paper we study analytic properties of compensator and Dulac expansions in a single variable. A Dulac expansion is formed by monomial terms that may contain a specific logarithmic factor. In compensator expansions this logarithmic factor is deformed. We first consider Dulac expansions when the power of the logarithm is either 0 or 1. Here we construct an explicit exponential scaling in the space of coefficients which in an exponentially narrow horn, up to rescaling and division, leads to a polynomial expansion. A similar result holds for the compensator case. This result is applied to the bifurcation theory of limit cycles in planar vector fields. The setting consists of families that unfold a given Hamiltonian in a dissipative way. This leading part is of Morse type, which leads to the following three cases. The first concerns a Hamiltonian function that is regular on an annulus, and the second a Hamiltonian function with a non-degenerate minimum defined on a disc. In the third case the Hamiltonian function has a non-degenerate saddle point with a saddle connection. The first case, by an appropriate scaling, recovers the generic theory of the saddle node of limit cycles and its cuspoid degeneracies, while the second case similarly recovers the generic theory of limit cycles subordinate to the codimension $k$ Hopf bifurcations $k = 1,2,\dotsc$. The third case enables a novel study of generic bifurcations of limit cycles subordinate to homoclinic bifurcations. We then describe how the above analytic result is applied to bifurcations of limit cycles. For appropriate one-dimensional Poincaré maps, the fixed points correspond to the limit cycles. The fixed point sets (or zero-sets of the associated displacement functions) are studied by contact equivalence singularity theory. The cases where the Hamiltonian is defined on an annulus or a disc can be reduced directly to catastrophe theory. In the third case, the displacement functions are known to have compensator expansions, whose first approximations are Dulac expansions. Application of our analytic result implies that, in an exponentially narrow horn near a homoclinic loop, the bifucation theory of limit cycles again reduces to catastrophe theory.