https://youtu.be/Mjohcie9zOc

1. ਐԽήʔϜཧ࿦ೖ໳
ୈճɿ઴ۙ҆ఆͳฏߧ఺

2. ໨࣍
w લճ
ϨϓϦέʔλಈֶͷఆٛ
w લճ
ฏߧ఺ͷఆٛ
w ฏߧ఺͕҆ఆ
w ฏߧ఺͕઴ۙ҆ఆ
w ৼΓࢠͷΠϝʔδͰཧղ
w ϦΞϓϊϑͷఆཧ
w ࣍ճɿ઴ۙ҆ఆͳฏߧ఺ͱ44

3. ϨϓϦέʔλಈֶͷఆٛ
ఆٛɿϨϓϦέʔλಈֶ
ҎԼͷඍ෼ํఔࣜΛ ্ͷϨϓϦέʔλಈֶͱ͍͏
ઓུ ͷ‫ظ‬଴རಘ ͕
छͱͯ͠ͷฏ‫ۉ‬རಘ ΑΓߴ͚Ε͹
ͦͷ‫ݸ‬ମͷzׂ߹͸૿͑Δ
Δn−1
·
xi = xi(eiAxt
− xAxt
) i = 1, 2, ⋯, n
i eiAxt
xAxt
‫ظ‬଴རಘ ฏ‫ۉ‬རಘ

4. ฏߧ఺ͷఆٛ
ఆٛɿฏߧ఺
ͱͳΔΑ͏ͳ෼෍ Λฏߧ఺ͱ͍͏
λΧϋτήʔϜʹ͓͚Δฏߧ఺
λΧઓུͷ෼෍Λ
ϋτઓུͷ෼෍Λ ͱ͢Δ
·
xi = 0 i = 1, 2, ⋯, n x ∈ Δn−1
x1 1 − x1
·
x1 = x1(e1Axt
− xAxt
) = x1(e1 − x)Axt
= x1(
(1,0) − (x1,1 − x1)
)(
−1 4
0 2) (
x1
1 − x1)
= x1(1 − x1, x1 − 1)
(
−1 4
0 2) (
x1
1 − x1)
= x1(1 − x1)(1, − 1)
(
−1 4
0 2) (
x1
1 − x1)
= x1(1 − x1)(−1,2)
(
x1
1 − x1)
= x1(1 − x1)(2 − 3x1)
λΧʢ)ʣ ϋτʢ%ʣ
λΧ
ϋτ
x1
1 − x1
ฏߧ఺͸ͭ
ͦͷ͏ͪ ͸44΍φογϡ‫ͱߧۉ‬Ұக
·
x1 = 0 ⇔ x1(1 − x1)(2 − 3x1) = 0 ⇒ x1 = 0,
2
3
, 1
x1 =
2
3
1
0
2
3
·
x1(t) 0 if x1
2
3
·
x1(t) 0 if x1
2
3
ฏߧ఺͸φογϡ‫ߧۉ‬ΑΓऑ͍

5. ฏߧ఺͕҆ఆ
ఆٛɿฏߧ఺͕҆ఆ
ฏߧ఺ ͕҆ఆͰ͋Δͱ͸
ͷ೚ҙͷۙ๣ ʹରͯ͠
ʹ͓͚Δ ͷద౰ͳۙ๣ ͕ଘࡏͯ͠
ॳ‫఺ظ‬ ͕ ʹ‫·ؚ‬ΕΔ೚ҙͷղ ͸
͢΂ͯͷ ʹରͯ͠ఆٛ͞Ε
͕ ʹ‫·ؚ‬Ε͍ͯΔ͜ͱΛ͍͏
x ∈ W x U
U x U1 x(0) U1 x(t)
t ≥ 0 x(t) U
U
U1
x x(0)
U
U1
x
x(1)
x(0)
U
U1
x
x(1)
x(t)
ɾɾɾ
ฏߧ఺ͷۙ͘ͷ఺͸͕࣌ؒ‫ͯͬܦ‬΋͋Δൣғʹཹ·Δ
͸ ʹ‫·ؚ‬ΕΔ
x(0) U1
x(0)

6. ฏߧ఺͕઴ۙ҆ఆ
ఆٛɿฏߧ఺͕઴ۙ҆ఆ
ฏߧ఺ ͕઴ۙ҆ఆͰ͋Δͱ͸ ͕҆ఆͰ͋Δ͜ͱʹՃ͑ ʹର͠
x ∈ W x U1
lim
t→∞
x(t) = x
U
U1
x x(0)
U
U1
x
x(1)
U
U1
x
x(t)
ɾɾɾ
ฏߧ఺ͷۙ͘ͷ఺͸͕࣌ؒ‫ͱͭܦ‬ฏߧ఺ʹऩଋ
͸ ʹ‫·ؚ‬ΕΔ
x(0) U1
x(0)
ɾɾɾ
t → ∞

7. ৼΓࢠͷΠϝʔδͰཧղ
ฏߧ఺ ҆ఆͳฏߧ఺ ઴ۙ҆ఆͳฏߧ఺
͕࣌ؒ‫ͯͬܦ‬΋͋Δൣғʹཹ·Δ ͕࣌ؒ‫ͱͭܦ‬ฏߧ఺ʹऩଋ

8. ϦΞϓϊϑͷఆཧ
ఆٛɿϦΞϓϊϑͷఆཧ
ฏߧ఺ ͱ͢Δ
ͷ͋Δۙ๣ ্ͷ࿈ଓؔ਺ ͕ Ͱඍ෼ՄೳͰ͋Γ
࣍ͷ̎৚݅Λຬͨ͢ͱ͢Δ
͔ͭ ʹରͯ͠
্Ͱ
͜͜Ͱඍ෼ํఔࣜ ͜ͷͱ͖
͸҆ఆͳฏߧ఺Ͱ͋Δ
্Ͱ
͕ຬͨ͞ΕΔͱ͖
͸઴ۙ҆ఆͳฏߧ఺Ͱ͋Δ
Λຬͨ͢ ΛϦΞϓϊϑؔ਺
΋ຬͨ͢ͱ‫ڱ‬ٛͷϦΞϓϊϑؔ਺
x ∈ W
x U V : U → ℜ U − {x}
V(x) = 0 y ≠ x V(y) 0
U − {x}
·
V(x) ≡
n
∑
i=1
∂V
∂xi
fi(x) ≤ 0.
·
xi = fi(x) x
U − {x}
·
V(x) 0 x
V : U → ℜ

9. ϦΞϓϊϑͷఆཧ
ฏߧ఺ ͱ͢Δ
ͷ͋Δۙ๣ ্ͷ࿈ଓؔ਺ ͕ Ͱඍ෼Մೳ
͔ͭ ʹରͯ͠
ฏߧ఺͸ۙ๣ʹͯ࠷খ
x ∈ W
x U V : U → ℜ U − {x}
V(x) = 0 y ≠ x V(y) 0
y ≠ x
U
U
x
V
x
ʢ ʣ
y
y

10. ϦΞϓϊϑͷఆཧ
ฏߧ఺ ͱ͢Δ
ͷ͋Δۙ๣ ্ͷ࿈ଓؔ਺ ͕ Ͱඍ෼Մೳ
্Ͱ
x ∈ W
x U V : U → ℜ U − {x}
U − {x}
·
V(x) ≡
n
∑
i=1
∂V
∂xi
fi(x) ≤ 0.
U
x
ʢ ʣ
U1
U1
x(0) U
V
x
ʢ ʣ
x(0)
B
͕࣌ؒ‫Ͳ΄ͭܦ‬ ͸খ͘͞ͳΔɾ·ͨ͸ಉ͡஋ΛͱΔ
V(x)
C
͕খ͘͞ͳΔํ޲·ͨ͸ಉ͡஋ʹͳΔΑ͏ʹ ͸ಈ͘
V(x) x

11. ϦΞϓϊϑͷఆཧ
ฏߧ఺ ͱ͢Δ
ͷ͋Δۙ๣ ্ͷ࿈ଓؔ਺ ͕ Ͱඍ෼Մೳ
্Ͱ
͕ຬͨ͞ΕΔͱ͖
͸઴ۙ҆ఆͳฏߧ఺Ͱ͋Δ
x ∈ W
x U V : U → ℜ U − {x}
U − {x}
·
V(x) 0 x
U
x
ʢ ʣ
U1
U1
x(0)
U
V
x
ʢ ʣ
x(0)
B
͕࣌ؒ‫Ͳ΄ͭܦ‬ ͸খ͘͞ͳΔ
V(x)
C
͕খ͘͞ͳΔํ޲ʹ ͸ಈ্͘
͍ͣΕ͸ඞͣ ʹऩଋ͢Δ
V(x) x x

12. ࣍ճɿ઴ۙ҆ఆͳฏߧ఺ͱ44
44ʼ઴ۙ҆ఆͳฏߧ఺ʼφογϡ‫ߧۉ‬ʼฏߧ఺
φογϡ‫ߧۉ‬ͷਫ਼៛Խͷ̍ͭ44ͱ઴ۙ҆ఆͳฏߧ఺ͷؔ܎Λ֬ೝ
803%ɿ઴ۙ҆ఆͳฏߧ఺
ٕज़ඪ४ͷબ୒ήʔϜ
1
0
2
3
·
x1(t) 0 if x1
2
3
·
x1(t) 0 if x1
2
3

13. ਐԽήʔϜཧ࿦ೖ໳
ୈճɿ઴ۙ҆ఆͳฏߧ఺
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