By Carlos Simpson

This e-book issues the query of ways the answer of a approach of ODE's varies while the differential equation varies. The aim is to offer nonzero asymptotic expansions for the answer when it comes to a parameter expressing how a few coefficients visit infinity. a specific classof households of equations is taken into account, the place the reply indicates a brand new form of habit no longer obvious in so much paintings recognized beforehand. The innovations contain Laplace rework and the tactic of desk bound section, and a combinatorial procedure for estimating the contributions of phrases in an enormous sequence enlargement for the answer. Addressed basically to researchers inalgebraic geometry, usual differential equations and intricate research, the booklet can be of curiosity to utilized mathematicians engaged on asymptotics of singular perturbations and numerical resolution of ODE's.

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**Example text**

T h u s we m a y assume lm(~) are fixed. Fix an a. For re(a) < i < m(a + 1) the constraints on the choice of ki are as follows, k~(~) _> l,,(~) by the second condition, and k, < l,,~(,,) + m(a + 1 ) - re(a) by the third condition. F u r t h e r m o r e for re(a) < i < m ( a + 1) we have ki-1 = li < ki (the equality is by the first condition and the definition of t h e re(a) and the inequality is by the second condition). Therefore the ki for m(a) <_ i < m(a + 1) are an increasing sequence of m(a + 1) - re(a) n u m b e r s in a range of length m(a + 1) - re(a).

Re(q) d e n o t e the indices such t h a t 1,~ < / ~ - 1 . ) T h e re(i) are an increasing sequence between 1 and r, so there are less t h a n C" possibilities. T h u s we m a y assume t h a t the re(i) are fixed. If i > re(a) t h e n ki-1 (_ Im(,) + i - re(a). In p a r t i c u l a r / , _(Ira(,) + i - re(a), so m(a + 1) ___ re(a). Therefore the sequence {Im(~)- re(a)} is increasing. F u r t h e r m o r e - r _< l m ( , ) re(a) (_ n + r so the n u m b e r of choices of the sequence Im(~) is b o u n d e d by C "+'.

V~) points in the direction of the sector S(+&). We may assume (using the extra freedom of 6/2) that in any piece adjacent to A, the vector field points towards A. Any edge not in A is an ~rt - out edge, because on either side, the vector field points along the negative gradient of Ng, and the edge is sufficiently transverse to this gradient. Similarly, at the triple vertices, the picture is the one shown previously. At vertices in A the picture is like the other one shown above. The local hypotheses of the proposition are satisfied.