A new method to compute normal forms of vector-field singularities is proposed. Normal forms for some degenerate singularities of vector fields are computed. These normal forms are simpler than those known as Arnold-Takens normal form. Parameters in the normal forms are uniquely determined from the original singularity in the category of (jets of) coordinate transformations.The first step in studying the singularities of vector fields is to reduce the singularities to simpler forms which are called normal forms. Normal form theory has been one of the most important tools for the study of singularities and the theory of bifurcations. There have been plenty of results and applications in this field: H. Poincar› [22] [23], G. Birkhoff [6], L. C. Siegel [25], S. Sternberg [26] [27], F. Takens [28] [29] [30], V. I. Arnold [2] [3] [4] [5], J. Guckenheimer [11], W. F. Langford [20] [21], G. Iooss and D. D. Joseph [19], etc ....As far as the author knows, however, nobody has executed the complete computation of normal forms of vector field singularities, not even for the second order, which should be implied by the general theory, i.e., the computation of orbits of jets of germs of vector fields under the action of the group of coordinate transformations.Our theory leads us, if we restrict ourselves to jets of finite order, to a complete classification of orbits, modulo higher order terms, in the space ofjets of vector field singularities under the action ofjets of local diffeomorphisms.The result is surprising. Most of the degenerate singularities are much simpler than what was known. These jets can be transformed, by a polynomial coordinate transformation, into a vector field whose jet is in the list stated below.It is known that most degenerate singularities are of infinite codimension in the category of C~%equivalence. For example, F. Takens [28] showed the existence of non-stabilizable jets of vector fields. F. Ichikawa [16] [17] gave algebraic criteria for a jet to be wild. According to his result, wildness of singularities is a very common property of degenerate vector fields.

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In this letter we report the numerical observation of the co-existence of three distinct chaotic attractors for at least one choice of parameters (α = 15.60, β = 28.58, m0 = −1/7, m1 = 2/7) in Chua's circuit. This new phenomenon may have escaped earlier detection because the three attractors are located very close to each other and requires the "zooming" ability of our confinor approach to uncover them.