Quadratic models for generic local 3-parameter bifurcations on the plane

Abstract

Abstract. The first chapter deals with singularities occurring in quadratic planar vector fields. We make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated, or Hamiltonian, or integrable, or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers. In the second chapter we provide quadratic models for all the known versal fc-parameter unfoldings with k = 1, 2, 3 , except for the nilpotent focus which cannot occur as a quadratic system. We finally show that a certain type of elliptic points of codimension 4 does not have a quadratic versal unfolding.

Differential Equations and Dynamical Systems

“…It was shown in [46] and in [61] that the bifurcation diagram for the system (9), which has a Takens-Bogdanov bifurcation of co dimension 3 at I' = 0, is a cone with its vertex at the origin of the three-dimensional parameter space (/1-1, /1-2, /1-3)' The intersection of this cone with any small sphere centered at the origin can be projected on the plane and, as in [46] and [61], this results in the bifurcation diagram (or bifurcation set) for the system (9) or for the system (10) shown in Figure 1 above. The bifurcation diagram in a neighborhood of either of the T B2 points is shown in detail in Figure 3 of Section 4.13.…”

“…The next theorem, which follows from Theorem 3.9 in [60], describes the codimension-3, Takens-Bogdanov bifurcation that occurs at the origin of the system (1), which, according to the results in [61], is a cusp of codimension 3; cf. Remarks 1 and 2 in Section 2.13.…”

Some results on homoclinic and heteroclinic connections in planar systems

“…From a computational point of view it is much simpler the case where all the numbers implied in their obtention are rational and then C k (x, y) and D k (x, y) are in Q[x, y]. It is easy to see that this happens when b and the eigenvalues of the saddle point given in (7) are rational numbers. From now one we will particularize our study to case n = 1/4, that is m 2 = 1/2, although clearly our approach can be adapted to all values of n such that √ n is a rational number.…”

“…Notice that the aim of our work is in the spirit of what Coppel proposed in his well-known paper [5]: "Ideally one might hope to characterize the phase portraits of quadratic systems by means of algebraic inequalities on the coefficients", taking into account the results of [7] where the authors proved that there are bifurcation curves in quadratic systems which are neither algebraic nor analytic. Since there is no hope to find analytic or algebraic expressions of the bifurcation curves, we try to sandwich them between two algebraic curves.…”

Geometrical and topological investigation of some families of quadratic differential systems possessing saddle-nodes or invariant ellipses

“…On the other hand, in 1991, Dumortier and Fiddelaers (1991) showed that, starting with the quadratic systems (and following with all the higher-degree polynomial systems), there exist geometric and topological phenomena in phase portraits of such systems whose determination cannot be established by means of algebraic expressions. More specifically, most part of the connections among separatrices and the occurrence of double or semistable limit cycles cannot be algebraically determined.…”