Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations
- Wasow, Wolfgang
DOIAbstract
AbstractLet A(x,ε) be an n×n matrix function holomorphic for |x|⩽x0, 0<ε⩽ε0, and possessing, uniformly in x, an asymptotic expansion A(x,ε)∾Σ∞r=0Ar(x) εr, as ε→0+. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion P(x,ε)∾Σ∞r=0Pr(x)εr, as ε→0+, is constructed, such that the transformation y = P(x,ε)z takes the differential equation εhdydx = A(x,ε)y,h a positive integer, into εhdzdx = B(x,ε)z, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold
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