By Carlos Simpson
This publication matters the query of the way the answer of a approach of ODE's varies whilst the differential equation varies. The objective is to provide nonzero asymptotic expansions for the answer by way of a parameter expressing how a few coefficients visit infinity. a specific classof households of equations is taken into account, the place the reply shows a brand new form of habit now not noticeable in such a lot paintings recognized formerly. The recommendations contain Laplace rework and the tactic of desk bound part, and a combinatorial strategy for estimating the contributions of phrases in an enormous sequence enlargement for the answer. Addressed basically to researchers inalgebraic geometry, usual differential equations and complicated research, the ebook may also be of curiosity to utilized mathematicians engaged on asymptotics of singular perturbations and numerical answer of ODE's.
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Extra info for Asymptotic Behavior of Monodromy: Singularly Perturbed Differential Equations on a Riemann Surface
Z,,) is a point in Z~, with 1II = n, then z0 denotes P and z,~+l denotes Q. Each space Zx is endowed with a holomorphic function gz defined by g ( ~ , - - . , ~ . ) = -g,o(~O) + g,o,~(~) + . . +~) (recall that g~j(z) = g~(z) - gj(z)). It is sometimes useful to think of ZI or the union Z. as a space relative to C, using the function g. )) C C. There is also an integrand bi assigned to each component ZI. If [I[ = n, then b1 is a holomorphic matrix-valued n-form on ZI, given by bl(zi,. . ,z,~) = ei,,iobi, i,,_l (z,~) .
There are less t h a n 6",,(,+1)-,,~(o) choices for this sequence. T h u s the total n u m b e r of choices of the ki for all i >__ r e ( l ) is less t h a n C ' . For i < r e ( l ) (or all i if there are no re(a)) we again have ki-1 = li ~ ki, and 0 < ki <_ n + r. Therefore there are less t h a n 6 '"+" choices there. T h u s the total n u m b e r of choices of ki for all i is less t h a n 6""+'. T h e s e choices d e t e r m i n e the li for i 5d re(a), so the n u m b e r of choices of the sequences satisfying t h e constraints is b o u n d e d by 6"~+'.
We may assume that the path ~, is linear with respect to the linear structure of Z, so/3. is actually piecewise linear. The formula for the monodromy is re(t) = ]~. be'~. 43 The formula for the Laplace transform is f(¢) = . g - ¢" These formulas imply sums over the components corresponding to the different indices 1. 1 The integral f(¢) is well defined and convergent for large values o/1¢1. Proof: Recall that ftj+l • . • ~- ~ aij with the conventions to = 0 and t,~+l = 1. , ~'(t,~))[ _< sup h,°a~l.
Asymptotic Behavior of Monodromy: Singularly Perturbed Differential Equations on a Riemann Surface by Carlos Simpson