Vector fields in the vicinity of a circle of critical points

Abstract

In this paper the Cs-conjugacy between vector fields on R2 having a circle of critical points is studied. 0. Statement of the problem. We are interested in studying vector fields on R2 having a circle of critical points as well as diffeomorphisms on R2 having a circle of fixed points. We can produce examples of such vector fields (resp. diffeomorphisms) by blowing up in polar coordinates germs of C°° vector fields (resp. diffeomorphisms) on R2 at 0 having the form X(x,y) = (x2 + y2)k with b ,¿ 0 and k being a positive integer (resp. ip -Id +X).A natural question in the study of singularities is: "Under which conditions is a germ of a singularity Cs-determined by a finite jet and which finite jets are determining?" On R2 there is the following result due to F. Dumortier [1]:If a germ satisfies an inequality of Lojawiewicz and has a characteristic orbit, then it is C°-determined. (By "characteristic orbit" we mean an orbit which tends to the singularity or leaves the singularity with a well-defined direction. Observe that the field given above does not have a characteristic orbit.)Consider germs of vector fields in R2 at 0 having the form -Z(x,y) = rk-X(x,y), where r2 = x2 + y2, k is a positive integer and X is a germ of C°° vector fields at 0 with X(0) = 0 and the eigenvalues of X (0) are A = a±ib with a/0 and b ^ 0.Consider the germs of diffeomorphisms in the plane of the form y5 = Id +Z. Denote by Z and ip the respective blowing ups (in polar coordinates) of Z and ip.Call Xk and Dk the spaces of such Z and <p, respectively, both endowed with the Cs-topology (s > k).We recall that:(i) The number a -ab~x is a topological invariant for the structural stability in D2 (see for instance [6]).(ii) The diffeomorphism Tp is formally imbedded in a flow represented by a vector field H. Moreover the (k + l)-jet of H at 0 is Hk = (x2 + y2)Xo(x,y), where X0 is the linearized vector field of X (see [4]).

Hopf-zero bifurcations of reversible vector fields

“…The methods used in the paper come from [1,5,7,9,10,15]. More precisely, the methods used in [10,15] are to perform special changes of coordinates around the singularity.…”