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
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