Let
f
:
R
×
R
m
¯
×
R
→
R
m
¯
,
f
=
f
(
ε
,
x
,
t
)
f:R \times {R^{\overline m }} \times R \to {R^{\overline m }},f = f(\varepsilon ,x,t)
be a
C
2
{C^2}
-mapping
1
1
-periodic in
t
t
having the form
f
(
0
,
x
,
t
)
=
A
x
+
o
(
|
x
|
)
f(0,x,t) = Ax + o(|x|)
as
x
→
0
x \to 0
where
A
∈
L
(
R
m
¯
)
A \in \mathcal {L}({R^{\overline m }})
has no eigenvalues with zero real parts. We study the relation between local stable manifolds of the equation
\[
x
′
=
ε
f
(
ε
,
x
,
t
)
,
ε
>
0
is
small
x’ = \varepsilon f(\varepsilon ,x,t),\varepsilon > 0{\text {is}}\;{\text {small}}
\]
and of its discretization
\[
x
n
+
1
=
x
n
+
(
ε
/
m
)
f
(
ε
,
x
n
,
t
n
)
,
t
n
+
1
=
t
n
+
1
/
m
,
{x_{n + 1}} = {x_n} + (\varepsilon /m)f(\varepsilon ,{x_n},{t_n}),{t_{n + 1}} = {t_n} + 1/m,
\]
where
m
∈
{
1
,
2
,
…
}
=
N
m \in \{ 1,2, \ldots \} = \mathcal {N}
. We show behavior of these manifolds of the discretization for the following cases: (a)
m
→
∞
,
ε
→
ε
¯
>
0
m \to \infty ,\varepsilon \to \overline \varepsilon > 0
, (b)
m
→
∞
,
ε
→
0
m \to \infty ,\varepsilon \to 0
, (c)
m
→
k
∈
N
,
ε
→
0
m \to k \in \mathcal {N},\varepsilon \to 0
.