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FICHA CATALOGRÁFICA
Scarabello, Marluce da Cruz.
S296e
Estudo de Sistemas Lineares por Partes com três Zonas e Aplicação na
Análise de um Circuito Elétrico Envolvendo um Memristor / Marluce da Cruz
Scarabello. - Presidente Prudente : [s.n], 2012
105 f. : il.
Orientador: Marcelo Messias
Dissertação (mestrado) - Universidade Estadual Paulista, Faculdade de
Ciências e Tecnologia
Inclui bibliografia
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1
+
ǫ
2
x
˙
2
=
ǫ
1
ϕ
1
(x)
−
ǫ
1
xa
1
ǫ
2
+
ya
1
ǫ
2
+
ǫ
2
ϕ
2
(x) +
a
2
y
ǫ
2
−
ǫ
1
x
ǫ
2
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
1
a
1
ǫ
2
+
a
2
x
+
y
ǫ
1
a
1
ǫ
2
+
a
2
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
3
x
+
yǫ
3
,
ǫ
3
=
ǫ
1
a
1
ǫ
2
+
a
2
ǫ
1
, ǫ
2
4+
⎧
⎨
⎩
˙
x
=
ϕ
1
(x)
−
ǫ
1
xa
1
ǫ
2
+
ya
1
ǫ
2
,
˙
y
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
3
x
+
yǫ
3
.
1
•
x
≤ −
1
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x) =
ǫ
1
(e
1
x
+
e
1
−
c
1
) +
ǫ
2
(e
2
x
+
e
2
−
c
2
)
=
ǫ
1
e
1
x
+
ǫ
1
e
1
−
ǫ
1
c
1
+
ǫ
2
e
2
x
+
ǫ
2
e
2
−
ǫ
2
c
2
= (ǫ
1
e
1
+
ǫ
2
e
2
)x
+ (ǫ
1
e
1
+
ǫ
2
e
2
)
−
(ǫ
1
c
1
+
ǫ
2
c
2
)
.
•
−
1
< x <
1
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x) =
ǫ
1
(c
1
x) +
ǫ
2
(c
2
x)
= (ǫ
1
c
1
+
ǫ
2
c
2
)x .
•
x
≥
1
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x) =
ǫ
1
(e
1
x
−
e
1
+
c
1
) +
ǫ
2
(e
2
x
−
e
2
+
c
2
)
=
ǫ
1
e
1
x
−
ǫ
1
e
1
+
ǫ
1
c
1
+
ǫ
2
e
2
x
−
ǫ
2
e
2
+
ǫ
2
c
2
= (ǫ
1
e
1
+
ǫ
2
e
2
)x
−
(ǫ
1
e
1
+
ǫ
2
e
2
) + (ǫ
1
c
1
+
ǫ
2
c
2
)
.
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x) =
⎧
⎪
⎨
⎪
⎩
(ǫ
1
e
1
+
ǫ
2
e
2
)x
+ (ǫ
1
e
1
+
ǫ
2
e
2
)
−
(ǫ
1
c
1
+
ǫ
2
c
2
),
x
≤ −
1,
(ǫ
1
e
1
+
ǫ
2
e
2
)x,
|
x
|
<
1,
(ǫ
1
e
1
+
ǫ
2
e
2
)x
−
(ǫ
1
e
1
+
ǫ
2
e
2
) + (ǫ
1
c
1
+
ǫ
2
c
2
),
1
≤
x.
ǫ
1
e
1
+
ǫ
2
e
2
=
ǫ
1
c
1
+
ǫ
2
c
2
ǫ
2
(e
2
−
c
2
) =
ǫ
1
(e
1
−
c
1
)
=
ǫ
1
(e
1
+
a
1
−
a
1
−
c
1
)
=
ǫ
1
(T
E
−
T
C
),
a
1
= 0
˙
x
=
F
(x)
−
y,
˙
y
=
g
(x),
F
(x) =
⎧
⎪
⎨
⎪
⎩
T
E
x
+ (T
E
−
T
C
),
x
≤ −
1,
T
C
x,
−
1
< x <
1,
g
(x) =
⎧
⎪
⎨
⎪
⎩
D
E
x
+ (D
E
−
D
C
),
x
≤ −
1,
D
C
x,
−
1
< x <
1,
D
E
x
−
(D
E
−
D
C
),
x
≥
1.
ǫ
1
=
a
2
ǫ
2
=
−
a
1
•
x
≤ −
1
˙
x
=
ϕ
1
(x)
−
ǫ
1
a
2
x
ǫ
2
+
a
1
x
ǫ
2
=
e
1
x
+ (e
1
−
c
1
)
−
a
1
a
2
x
−
a
1
+
a
1
y
−
a
1
=
e
1
x
+ (e
1
−
c
1
) +
a
2
x
−
y
= (e
1
+
a
2
)x
+ (e
1
−
c
1
)
−
y
=
T
E
x
+ (T
E
−
T
C
)
−
y .
˙
y
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
3
x
+
ǫ
3
y
=
a
2
(e
1
x
+ (e
1
−
c
1
)) + (
−
a
1
)(e
2
x
+ (e
2
−
c
2
))
−
a
2
(a
1
a
2
+ (
−
a
1
)a
2
)x
−
a
1
+(a
1
a
2
+ (
−
a
1
)a
2
)y
=
a
2
e
1
x
+
a
2
(e
1
−
c
1
)
−
a
1
e
2
x
−
a
1
(e
2
−
c
2
)
= (a
2
e
1
−
a
1
e
2
)x
+
a
2
e
1
−
a
2
c
1
−
a
1
e
2
+
a
1
c
2
=
D
E
x
+ (D
E
−
D
C
)
.
•
−
1
< x <
1
˙
x
=
ϕ
1
(x)
−
ǫ
1
a
2
x
ǫ
2
+
a
1
x
ǫ
2
=
c
1
x
−
a
1
a
2
x
−
a
1
y
=
c
1
x
+
a
2
x
−
y
=
T
C
x
−
y .
˙
y
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
3
x
+
ǫ
3
y
=
a
2
c
1
x
+
−
a
1
c
2
x
=
D
C
x .
•
x
≥
1
˙
x
=
ϕ
1
(x)
−
ǫ
1
a
2
x
ǫ
2
+
a
1
x
ǫ
2
=
e
1
x
−
(e
1
−
c
1
)
−
a
1
a
2
x
−
a
1
+
a
1
y
˙
y
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
3
x
+
ǫ
3
y
=
a
2
(e
1
x
−
(e
1
−
c
1
)) + (
−
a
1
)(e
2
x
−
(e
2
−
c
2
))
−
a
2
(a
1
a
2
+ (
−
a
1
)a
2
)x
−
a
1
+(a
1
a
2
+ (
−
a
1
)a
2
)y
=
a
2
e
1
x
−
a
2
(e
1
−
c
1
)
−
a
1
e
2
x
+
a
1
(e
2
−
c
2
)
= (a
2
e
1
−
a
1
e
2
)x
−
a
2
e
1
+
a
2
c
1
+
a
1
e
2
−
a
1
c
2
=
D
E
x
−
(D
E
−
D
C
)
.
T
E
=
T
C
a
1
= 0
˙
x
=
f(x)
−
y,
˙
y
=
ρx
−
by,
f
(x) =
⎧
⎪
⎨
⎪
⎩
m
E
x
+ (m
E
−
m
C
),
x
≤ −
1,
m
C
x,
−
1
< x <
1,
m
E
x
−
(m
E
−
m
C
),
x
≥
1,
b
=
−
D
C
−
D
E
T
C
−
T
E
, m
E
=
T
E
+
b, m
C
=
T
C
+
b
ρ
=
m
C
−
m
E
2
b
+
D
C
+
D
E
2
.
ǫ
1
=
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
ǫ
2
=
−
a
1
,
•
x
≤ −
1
˙
x
=
e
1
x
+ (e
1
−
c
1
)
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
x
−
y
=
e
1
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
x
−
y
+ (e
1
−
c
1
)
=
e
1
+
a
2
−
a
2
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
x
−
y
+ (e
1
−
c
1
)
=
e
1
+
a
2
−
a
2
(T
E
−
T
C
) +
a
1
(c
2
−
e
2
)
T
E
−
T
C
x
−
y
+ (e
1
−
c
1
)
= (e
1
+
a
2
)x
−
a
2
e
1
−
a
2
c
1
+
a
1
c
2
−
a
1
e
2
T
E
−
T
C
x
−
y
+ (e
1
−
c
1
)
= (e
1
+
a
2
)x
−
D
C
−
D
E
T
E
−
T
C
x
−
y
+ (e
1
−
c
1
)
=
e
1
+
a
2
−
D
C
−
D
E
T
E
−
T
C
x
−
y
+ (e
1
−
c
1
)
˙
y
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
3
x
+
ǫ
3
y
=
−
a
1
T
(c
2
−
e
2
)
E
−
T
C
(e
1
x
+ (e
1
−
c
1
)) + (
−
a
1
)(e
2
x
+ (e
2
−
c
2
))
−
−
a
1
T
(c
2
−
e
2
)
E
−
T
C
a
2
+
a
1
(c
2
−
e
2
)
T
E
−
T
C
x
+
a
2
+
a
1
(c
2
−
e
2
)
T
E
−
T
C
y
=
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
e
1
x
−
a
1
(c
2
−
e
2
)(e
1
−
c
1
)
e
1
−
c
1
−
a
1
e
2
x
+
a
1
(c
2
−
e
2
)+
+
a
1
a
2
(c
2
−
e
2
)
T
E
−
T
C
x
+
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
2
x
−
by
= (b
+
a
2
)e
1
x
−
a
1
e
2
x
−
a
2
bx
−
a
2
2
x
+
b
2
x
+ 2ba
2
x
+
a
2
2
x
−
by
=
be
1
x
+
a
2
e
1
x
−
a
1
e
2
x
+
a
2
bxb
2
x
−
by
= [(e
1
+
a
2
)b
+
b
2
+ (a
2
e
1
−
a
1
e
2
)]x
−
by
= (T
E
b
+
b
2
+
D
E
)x
−
by
= (T
E
+
b)x
+
D
E
x
−
by.
•
−
1
< x <
1
˙
x
=
c
1
x
+
−
D
T
C
−
D
E
C
−
T
E
+
a
2
x
−
y
=
c
1
+
a
2
−
D
C
−
D
E
T
C
−
T
E
x
−
y
= (T
C
+
b)x
−
y
=
m
C
x
−
y .
˙
y
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
3
x
+
ǫ
3
y
=
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
c
1
x
+ (
−
a
1
)c
2
x
−
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
a
2
+
a
1
(c
2
−
e
2
)
T
E
−
T
C
x
+
a
2
+
a
1
(c
2
−
e
2
)
T
E
−
T
C
y
= (b
+
a
2
)c
1
x
−
a
1
c
2
x
−
a
2
(b
+
a
2
)x
+ (b
+
a
2
)
2
x
−
by
=
bc
1
x
+
a
2
c
1
x
−
a
1
c
2
x
−
a
2
bx
−
a
2
2
x
+
b
2
x
+ 2ba
2
x
+
a
2
2
x
−
by
=
bc
1
x
+
a
2
c
1
x
−
a
1
c
2
x
+
ba
2
x
−
by
= (c
1
+
a
2
)bx
+ (a
2
c
1
−
a
1
c
2
)x
−
by
=
T
C
x
+
D
C
x
−
by.
•
x
≥
1
˙
x
=
e
1
x
+
−
D
T
C
−
D
E
C
−
T
E
+
a
2
x
−
(e
1
−
c
1
)
−
y
=
e
1
+
a
2
−
D
C
−
D
E
T
C
−
T
E
x
−
(e
1
−
c
1
)
−
y
˙
y
=
ǫ
1
ϕ
1
(x) +
ǫ
2
ϕ
2
(x)
−
ǫ
1
ǫ
3
x
+
ǫ
3
y
=
−
a
1
T
(c
2
−
e
2
)
E
−
T
C
(e
1
x
−
(e
1
−
c
1
)) + (
−
a
1
)(e
2
x
−
(e
2
−
c
2
))+
−
−
a
1
T
(c
2
−
e
2
)
E
−
T
C
a
2
+
a
1
(c
2
−
e
2
)
T
E
−
T
C
x
+
a
2
+
a
1
(c
2
−
e
2
)
T
E
−
T
C
y
=
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
e
1
x
+
a
1
(c
2
−
e
2
)(e
1
−
c
1
)
e
1
−
c
1
−
a
1
e
2
x
+
a
1
(c
2
−
e
2
)+
+
a
1
a
2
(c
2
−
e
2
)
T
E
−
T
C
x
+
−
a
1
(c
2
−
e
2
)
T
E
−
T
C
2
x
−
by
= (b
+
a
2
)e
1
x
−
a
1
e
2
x
−
a
2
b
+
x
−
a
2
2
x
+
b
2
x
+ 2ba
2
x
+
a
2
2
x
−
by
=
be
1
x
+
a
2
e
1
x
−
a
1
e
2
x
+
a
2
bx
+
b
2
x
−
by
= [(e
1
+
a
2
)b
+
b
2
+ (a
2
e
1
−
a
1
e
2
)]x
−
by
= (T
E
b
+
b
2
+
D
E
)x
−
by
= (T
E
+
b)x
+
D
E
x
−
by.
T
E
T
C
<
0
˙
x
=
φ(x)
−
y,
˙
y
=
x
−
βy,
φ(x) =
⎧
⎪
⎨
⎪
⎩
μ
E
x
+ (μ
E
−
μ
C
),
x
≤ −
1,
μ
C
x,
−
1
< x <
1,
μ
E
x
−
(μ
E
−
μ
C
),
x
≥
1
β, μ
E
μ
C
ρ >
0
a
1
= 0
T
C
T
E
<
0
T
E
=
T
C
! "
˙
x
=
f(x)
−
y,
˙
y
=
ρx
−
by.
!
Γ
# $ %℄
Γ
x
ρ
≤
0
x
! ! " # $ % & 'λ
1,2
=
m
E
−
b
±
(m
E
+
b)
2
−
4ρ
2
(m
+
b)
2
−
4ρ
≥
0
⇔
ρ
≤
0.
x
!!"ρ
≤
0
( " " )
Γ
* (ρ
≤
0
+ $x
=
−
1
x
= 1
,Γ
)! "Γ
ρ
= 0
y
= 0
! * "Γ
!ρ <
0
+ $ "
-,
ρ >
0
'% & .
x
→
x
y
→
√
ρy,
t
→
t
√
ρ
,dx
dt/
√
ρ
=
√
ρ
dx
dt
=
√
ρ
(f
(x)
−
y(
√
ρ)
⇒
dx
dt
=
f
(x)
√
ρ
−
y,
f
(x)
√
ρ
=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
μ
E
√
ρ
x
+
(μ
E
√
−
μ
C
)
ρ
,
x
≤ −
1,
μ
C
√
ρ
x,
−
1
< x <
1,
μ
E
√
ρ
x
−
(μ
E
√
−
μ
C
)
ρ
,
x
≥
1,
dy
√
ρ
dt/
√
ρ
= (
√
ρ)
2
dy
dt
=
ρx
−
by
√
ρ
⇒
dy
dt
=
x
−
b
√
ρ
y .
- ' . /
β
=
√
b
ρ
, μ
C
=
m
C
√
ρ
μ
E
=
m
E
√
ρ
! " " #
$ % ! &'℄ %
D
= 0
D
!D
C
D
E
A
A
E
A
C
T
=
(A)
)
*
⎛
⎜
⎝
x(t)
∓
β(μ
C
−
μ)
D
y(t)
∓
μ
C
−
μ
D
⎞
⎟
⎠
=
e
At
⎛
⎜
⎝
x(0)
∓
β(μ
C
−
μ)
D
y(0)
∓
μ
C
−
μ
D
⎞
⎟
⎠
,
+e
At
=
e
T
2
t
C(t)
μ
!μ
E
μ
C
"% )
C(t)
Δ = (μ
−
β)
2
−
4(
−
μβ
+ 1)
= (μ
+
β)
2
−
4
=
Δ
E
,
|
x
| ≥
1,
Δ
C
,
|
x
|
<
1,
, ! "
-
D >
0
Δ
<
0
.// -
D >
0
Δ
≥
0
0 "
ω
=
|
Δ
|
2
=
ω
E
,
|
x
| ≥
1,
ω
C
,
|
x
|
<
1
Δ
Δ
E
Δ
C
ω
ω
E
ω
C
/ $C(t)
1
Δ
<
0
C(t) =
⎛
⎜
⎝
cos(ωt) +
(β
+
μ)sen(ωt)
2ω
−
sen(ωt)
ω
sen(ωt)
ω
cos(ωt)
−
(β
+
μ)sen(ωt)
2ω
Δ
>
0
C(t) =
⎛
⎜
⎝
cosh(ωt) +
(β
+
μ)senh(ωt)
2ω
−
senh(ωt)
ω
senh(ωt)
ω
cosh(ωt)
−
(β
+
μ)senh(ωt)
2ω
⎞
⎟
⎠
.
Δ = 0
C(t) =
1 +
t(μ
−
T /2)
−
t
t
1
−
t(μ
−
T /2)
.
D
= 1
−
μβ
= 0
β
= 0
μ
= 1/β
D
= 0
T
= 0
μ
= 1/β
0
=
|
β
|
= 1
! "x(t)
y(t)
=
1
T
⎛
⎜
⎝
1
β
e
T t
−
β
1
−
e
T t
e
T t
−
1
1
β
−
βe
T t
⎞
⎟
⎠
x(0)
y(0)
∓
D
C
t
⎛
⎝
β
+
1
β
e
T t
1 +
e
T t
⎞
⎠
,
T
= (1
−
β
2
)/β
#
$ % %
μ
=
β
=
±
1
D
=
T
= 0
&'( )
x(t)
y(t)
=
1 +
βt
−
t
t
1
−
βt
x(0)
y(0)
∓
T
C
t
1
β
.
* + , -.℄ 0 !
Δ
= 0
γ
E
=
μ
E
−
β
2ω
E
=
T
E
2ω
E
,
γ
C
=
μ
C
−
β
2ω
C
=
T
C
2ω
C
1'
Δ
>
0
D
= 0
γ
2
−
1 =
D/ω
2
|
γ
|
<
1
|
γ
|
>
1
γ
=
γ
C
γ
=
γ
E
ω
=
ω
C
ω
=
ω
E
2% 3
x
˙
= ˙
y
44
D
E
D
C
≥
0
D
C
= 0
D
E
= 0
D
C
= 0
β
= 0
y
=
x/β
|
x
| ≤
1
D
E
=
D
C
= 0
β
= 0
y
=
x/β
!
T
E
=
T
C
μ
E
=
μ
C
"
D
E
D
C
≥
0
D
C
=
0
D
C
>
0
! " #
T
C
T
E
<
0
$%T
C
>
0
$%T
C
<
0
% & '(""
T
C
= 0
T
E
= 0
) % & '(""
T
C
T
E
≥
0
T
C
= 0
* % % ! % & '+" '+""
D
C
<
0
% % ! % & '+""# $ $ %
"
&
D
C
>
0
$&
T
C
>
0
"'( (( ) '* $
F
(x) =
−
F
(
−
x), g(x) =
−
g(
−
x)
xg(x)
>
0
x
= 0
