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On the formation of singularities of solutions of nonlinear differential systems in antistokes directions
We determine the position and the type of spontaneous singularities of
solutions of generic analytic nonlinear differential systems in the complex
plane, arising along antistokes directions towards irregular singular points of
the system. Placing the singularity of the system at infinity we look at
equations of the form with
analytic in a neighborhood of , with genericity
assumptions; is then a rank one singular point. We analyze the
singularities of those solutions which tend to zero for in some sectorial region, on the edges of the maximal region (also
described) with this property. After standard normalization of the differential
system, it is shown that singularities occuring in antistokes directions are
grouped in nearly periodical arrays of similar singularities as ,
the location of the array depending on the solution while the (near-) period
and type of singularity are determined by the form of the differential system.Comment: 61
Analytic linearization of nonlinear perturbations of Fuchsian systems
Nonlinear perturbation of Fuchsian systems are studied in regions including
two singularities. Such systems are not necessarily analytically equivalent to
their linear part (they are not linearizable). Nevertheless, it is shown that
in the case when the linear part has commuting monodromy, and the eigenvalues
have positive real parts, there exists a unique correction function of the
nonlinear part so that the corrected system becomes analytically linearizable
Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials
Differential systems with a Fuchsian linear part are studied in regions
including all the singularities in the complex plane of these equations. Such
systems are not necessarily analytically equivalent to their linear part (they
are not linearizable) and obstructions are found as a unique nonlinear
correction after which the system becomes formally linearizable.
More generally, normal forms are found.
The corrections and the normal forms are found constructively. Expansions in
multiple orthogonal polynomials and their generalization to matrix-valued
polynomials are instrumental to these constructions.Comment: 24 page
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