corrigé

6
(x, I)I=]0,+∞[
˙x(t) = f(t, x (t))
f(t, x) := 2xcos t
sin t.
f C1(]0,+∞[\πN)×R
(t∗, x∗)
t∗=kπ ∈πN>0
0 =2x∗(−1)k
x∗= 0 x∗6= 0
xcos
sin
Ik
1 1
Ik
1
sin2t→ −∞ t→π(k+ 1) α= 0
x]kπ, (k+ 2) π[
C1t∈Ik∪Ik+1
˙x(t)=2cos t
sin tx(t) = 2 cos t
exp −1
sin2t
sin t.
limu→01
uexp −1
u2= 0
lim
t→(k+1)π˙x(t) = 0 ,
x
(k+ 1) π x C1

(αk)k∈N∈RN
R>0
t∈πN7−→ 0
t∈Ik7−→ eαkexp −1
sin2t
C1R>0
Ck(x) = αkk x = 0
ck=eCk(x)k e 6= 0
||f(t, 0)|| ≤ M(t)||0|| = 0 0 = ˙x(t) = f(t, x (t)) = 0 x(t)=0
f C1(t∗,0)
0
ϕ(xj)1≤j≤nx
ϕ(t) = Pn
j=1 xj(t)2˙ϕ(t) = Pn j=1 2 ˙xj(t)xj(t) =
2h˙x(t)|x(t)i
|˙ϕ(t)|= 2 |hf(t, x (t)) |x(t)i| ≤ 2M(t)||x(t)|| ×
||x||
ϕ Imax t0||x(t0)|| = 0 x(t0)=0
x= 0 x6= 0
ϕ(t)>0
˙ϕ
ϕ(t)
≤2M(t)
t∗t≥t∗
ˆt
t∗
˙ϕ
ϕ(s) ds≤
ˆt
t∗
˙ϕ
ϕ(s) ds
≤ˆt
t∗
˙ϕ
ϕ(s)
ds≤2ˆt
t∗
M(s) ds .
t≤t∗
ˆt
t∗
˙ϕ
ϕ(s) ds≤
ˆt
t∗
˙ϕ
ϕ(s) ds
≤ˆt∗
t
˙ϕ
ϕ(s)
ds≤2ˆt∗
t
M(s) ds=−2ˆt
t∗
M(s) ds .
t∈Imax
ln ϕ(t)−ln ϕ(t∗)≤2
ˆt
t∗
M(s) ds .
exp
(x, Imax)Imax =]a, b[a b I I
a b I b ∈
I
K⊂I×Rnt0∈[t∗, b]⊂Imax x(t0)/∈K K :=
n(t, x) : t∗≤t≤b , ||x|| ≤ exp 
´t
t∗M(s) ds
oM[t∗, b]
t7→ exp 
´t
t∗M(s) ds
t0
Imax =I
˙x(t) = x2(t)
I:= Rx(t) := −1
t]0,+∞[

P∈GLn(C)P−1MP =: D M
D λj6= 0 z
`:= ln |z|+ i arg z z `j
λjdiag (`1, . . . , `n)M
exp diag (`1, . . . , `n) = diag (exp `1,...,exp `n) exp L=M M
ˆ
D:= diag (λ1, . . . , λn)λj∈2πZL
M L =Pdiag (`1, . . . , `n)P−1
Lˆ L:= Pˆ DP −1
exp L+ˆ L= exp Lexp ˆ L=MP exp ˆ DP−1=MP P −1=M
exp 0α
−α0=cos αsin α −sin αcos α  0α −α0∈
M2×2(R) iαRβ α
−α β 7→
β+ iα∈Cα∈2πZ
Akk A
A C1Ak
dA2
dt=˙
AA +A˙ A .
˙
A A ˙ A= exp A(A, Imax)
exp A=P∞
n=0 An
n!AdA2 dt= 2 ˙
AA
k
exp (−A) = P∞
n=0
(−A)n
n!
Mn×n(C)||exp A|| ≤ exp ||A|| Pn≥0
n(−˙ A)(−A)n−1
n!
˙
A
exp ||A|| Imax
˙
AAn
exp (−A)C1
d exp (−A)
dt=
∞
X
n=0
−˙
A(−A)n
n!=−˙
Aexp (−A).
˙
EA=−˙ Aexp (−A) = −Id ˙ A= exp A
EA(t) = exp (−A∗)−tId t∈R
EA(t)t A∗
P−1A∗P=: D∗P−1EA(t)P= exp (−D∗)−tId EA(t)
EA(t)
A(t)EAdet EA(t)6= 0 A(t)∈ M
EA(t)C1Rt7→ A(t) det (exp (−A∗)−tId) 6=
0texp (−A∗)