Contemporary Mathematics

Volume

373, 2005

A singularly perturbed Riccati Equation

Sadjia Ait-Mokhtar

ABSTRACT. We are interested in the type, the position and the move of sin-

gularities formed by solutions of a singulary perturbed differential equation

eu1

=

P(z, u, a, e) with respect to the parameter a. We show in the case of

a Riccati equation that the overstability values are logarithmic singularities

of the multi valued function "Indicatrice des poles ": which for each value of

the parameter associates the poles of a specific solution called " distinguished

solution ". This means that after surrounding an overstability value, the poles

move exchanging their positions.

1.

Introduction and general results

Let c be a positive infinitesimal number and V

C

C be a simply connected

domain. We consider a Riccati equation of the form

(1.1)

cu'

= p2

(z, c)- u2

+

ca

=:

I(z, u, a, c)

where

p

is a standard analytic function in a simply connected domain S1

C

C

2

,

and

a

is a complex parameter. We assume that for each

z

E

V,

(z,

0)

E

0.

For a fixed value a0

(=

0) and c small, the vector field associated to (1.1) is

structured by two limited analytic curves (where the vector field is limited) called

slow curves

and defined by

I(z,u0 (z),a0 ,0)

=

0

(u0 (z)

:=

±p(z,O)).

We suppose that the function

8I

(1.2)

f(z)

=

au (z, uo(z), ao, 0)

has a unique zero z0

,

which is simple. We denote

(1.3)

F(z)

=

t

f(s)ds

}zo

and we consider the landscape associated the

slow curve u0

(1.4)

R(z)

=

~(F(z)).

The landscape (1.4) is composed with two mountains and two valleys. We have the

following results

((3] ,{4])

1991

Mathematics Subject Classification.

Primary 54C40, 14E20; Secondary 46E25, 20C20.

Key words and phrases.

Singularly perturbation, Transseries, Asymptotics, Stokes

Phenomenon. ©

2005

American Mathematical Society

http://dx.doi.org/10.1090/conm/373/06912