By Carlos Simpson
This booklet issues the query of ways the answer of a process of ODE's varies while the differential equation varies. The target is to offer nonzero asymptotic expansions for the answer by way of a parameter expressing how a few coefficients visit infinity. a selected classof households of equations is taken into account, the place the reply shows a brand new form of habit no longer visible in so much paintings recognized earlier. The concepts contain Laplace remodel and the tactic of desk bound section, and a combinatorial process for estimating the contributions of phrases in an unlimited sequence growth for the answer. Addressed basically to researchers inalgebraic geometry, traditional differential equations and intricate research, the e-book can also be of curiosity to utilized mathematicians engaged on asymptotics of singular perturbations and numerical answer of ODE's.
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Extra resources for Asymptotic Behavior of Monodromy: Singularly Perturbed Differential Equations on a Riemann Surface
Along A we can make a further choice. Choose a continuous piecewise polynomial function : A × [0,1] h with fA(A,0) = A. Recall that r(z) was the first time when f~(z,~') E A. Let A(z) = fl(z, T(z)) E A. Under the hypotheses of the proposition, 7-(z) and A(z) are continuous piecewise linear functions of z. We may alter the definition of the flow for times ~ > r ( z ) i n the following way. F o r t < "r(z), keep f ( z , t ) = fl(z,t) as defined in the proposition. For t > ~(z), set f(z,$) = h ( A ( z ) , $ - "r(z)).
2 When composed with g, the flow goes approximately in the negative real direction, in other words Og(f(z,t)) & E s(+6) = : e E - 6, + 6]}. 3 Consequently, O(~g . 4 f(z, ~) is piecewise polynomial. In fact, it is piecewise linear in the region t _< r(z) and piecewise quadratic in the region t > r(z). 5 The flow decreases Ng by a specified amount L, unless A is encountered first. g(f(z, 1)) <_Ng(z)- L, or else z(z) _< 1/2. In the latter case there is a critical point l such that f(z, t) E A(l) for t _> r(z).
If 77. is a pro-chain then denote by ~/x the part supported on ZI, and let ~/. denote the chain ~][I1=-~/I on Z.. Occasionally we will drop the asterisk subscript for brevity. (~l) of Z. such that if a is a form which vanishes on the subset, then (r/,a) = 0. The support of a chain could be smaller than the union of the images of the polyhedra involved, because some cancellation might occur. Recall that two points P and Q are fixed in Z. Make the convention that if z = ( z l , . . , z,,) is a point in Z~, with 1II = n, then z0 denotes P and z,~+l denotes Q.