By M. Bocher
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Extra info for An Introduction to the Study of Integral Equations
E. 0; 0; 0/ > 0. The first condition is our structural assumption, (iii,v) are non-degeneracy assumptions which are fulfilled generically, and (ii,iv) describe our bifurcation point. Signs in (iii,v) are chosen without loss of generality, by switching signs of y and , if necessary. 0; 0; 0/ D 0 at the origin, only. 0; y. /; / Á 0 along a curve. Without loss of generality, we took this curve to be the -axis. e. under small perturbations of F respecting (i) there is a point near the origin satisfying (ii–v) for the perturbed system.
Regular perturbation theory can be applied to obtain results for ' 0, describing the dynamics in a neighborhood of x D 0. 6 Rescaling and Blow Up 19 Good choices for the scaling ˛ are given by the Newton polyhedron. 13) ˇ2Nn be the Taylor expansion of the vector field. 0; : : : ; 0; 1; 0; : : : ; 0/ is the k-th unit vector. 11). Every outer facet F of N , facing the origin, yields a viable scaling ˛ ? 15) The time rescaling ˛ is then given by the distance of F from the origin. 12, D 0) contains exactly the monomials to points of N in F .
Every outer facet F of N , facing the origin, yields a viable scaling ˛ ? 15) The time rescaling ˛ is then given by the distance of F from the origin. 12, D 0) contains exactly the monomials to points of N in F . 12) for 0 < 1 and kxk Q D1 as spherical coordinates near the origin. Then the boundary D 0 is the blow up of the singularity x D 0 to a sphere kxk Q D 1. 12, D 0/ on this sphere is expected to be less singular then the original vector field at the origin . Recursive blow ups can (and have been) used to further desingularize the vector field.
An Introduction to the Study of Integral Equations by M. Bocher