ެཱ͸ͩͯ͜ະདྷେֶ 2020 ೥౓ γεςϜ৘ใՊֶ࣮श
άϧʔϓใࠂॻ
Future University-Hakodate 2020 System Information Science PracticeGroup ReportϓϩδΣΫτ໊ෳࡶ‫ܥ‬ͷ਺ཧͱγϛϡϨʔγϣϯ

Project Name
Mathematical and simulation of complex systemsϓϩδΣΫτ൪߸/Project No.

6ϓϩδΣΫτϦʔμ/Project Leader
Ֆࢁཽ໵

Tatsuya HanayamaάϧʔϓϦʔμ/Group Leader
Ֆࢁཽ໵

Tatsuya Hanayamaάϧʔϓϝϯό/Group MemberࡾӜྰԻReon Miuraླ໦ᠳే

Shoto Suzuki

౻ా३Atsushi Fujitaࢦಋ‫ڭ‬һ
઒‫ޱ‬૱

઒ӽහ࢘

੪౻ேً

Advisor
Satoshi Kawaguchi Toshiji Kawagoe Asaki Saitoఏग़೔
2021 ೥ 1 ݄ 14 ೔

Date of Submission
January 14, 2021֓ཁຊϓϩδΣΫτͰ͸ɺ೪‫ە‬ͷ֦ࢄɺࣶϗοέͷࣶ໛༷ͳͲɺੜ෺ͷ֦ࢄ΍໛༷͕൓Ԡ֦ࢄ‫Ͱܥ‬‫ܗ‬੒͞ΕΔύλʔϯΛɺϞσϧํఔࣜΛ༻͍ͯղੳ͢Δ͜ͱΛϓϩδΣΫτςʔϚͱ͍ͯ͠Δɻ൓Ԡ֦ࢄ‫ͱܥ‬͸Խֶ൓Ԡͱ൓Ԡ෺࣭͕ۭؒతʹ֦ࢄ͢Δͱ͍͏ೋͭͷϓϩηεʹΑͬͯ෺࣭ೱ౓͕࣌ؒతۭؒతʹมԽ͢Δ༷ࢠΛ‫ه‬ड़͢Δ਺ཧϞσϧͷ͜ͱͰ͋Δɻύλʔϯղੳͷํ๏ͱͯ͠ɺ൓Ԡ֦ࢄ‫ܥ‬ͷղͷৼΔ෣͍Λɺ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ɺγϛϡϨʔγϣϯΛߦ͏ɻ͜ͷ࣌‫ݱ‬ΕΔύλʔϯͷྫͱͯ͠ɺεύΠϥϧύλʔϯɺλʔήοτύλʔϯɺϦϯάঢ়ύλʔϯͳͲ͕͋ΔɻγϛϡϨʔγϣϯΛߦ͏ͨΊʹɺલ‫Ͱظޙͱظ‬΍Δ࡞‫ۀ‬Λલ‫ظ‬ͷૣ͍͏ͪʹ໌֬ʹ͢Δ͜ͱͰɺ‫ܭ‬ըతʹ࡞‫ۀ‬ΛਐΊͨɻલ‫ظ‬͸ɺ֤͕ࣗ༩͑ΒΕͨࢀߟจ‫ݙ‬Λྠಡ͠ɺຖिͷ՝͞ΕΔ՝୊Λࣗशͱ͍͏‫ͯ͠ͳ͜Ͱܗ‬͍͖ɺͦͷཌिͷϓϩδΣΫτֶशͷ࣌ؒͰɺzoom Λ௨ֶ֤͕ͯࣗ͠ΜͰ͖ͨ͜ͱΛൃද͢
Δ͜ͱͰɺ൓Ԡ֦ࢄ‫͚͓ʹܥ‬Δύλʔϯ‫ܗ‬੒ʹ͍ͭͯཧղΛਂΊͨɻ·ͨɺͦͷൃදΛ΋ͱʹ୲౰‫ڭ‬һ͔ΒΞυόΠε΍గਖ਼Λ΋Β͍ɺࣗश͚ͩͰ͸଍Γ͍ͯͳ͔ͬͨ෦෼ΛิͬͨΓɺؒҧֶͬͯशͨ͠෦෼Λ࠶֬ೝͨ͠Γͨ͠ɻ·ͨɺਐߦ೾ղ΍ύλʔϯͷछྨΛֶͿͱಉ࣌ʹɺ‫Ͱظޙ‬ඞཁͱͳΔ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͢ΔͨΊͷ‫ૅج‬஌ࣝΛֶΜͩɻ‫Ͱظޙ‬͸ɺલ‫Ͱظ‬਎ʹ෇͚ͨɺ਺஋‫ࢉܭ‬ϓϩάϥϜ࡞੒ͷ‫ૅج‬஌ࣝΛ‫͠࢖ۦ‬ɺલ‫ظ‬ಉ༷ͷຖि՝͞ΕΔ՝୊Λ͜ͳͨ͠ɻͦΕͱಉ࣌ʹɺ࠷ऴ੒Ռ෺Ͱ͋Δਐߦ೾ղͷৼΔ෣͍ͷมԽΛΞχϝʔγϣϯಈըͱͯ͠࡞੒͢ΔͨΊʹɺ਺஋‫͔ࢉܭ‬Βภඍ෼ํఔࣜͷղͷৼΔ෣͍Λௐ΂ɺ࣮ࡍʹ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ɺγϛϡϨʔγϣϯΛߦͬͨɻͦͷ࣌ʹ‫ݱ‬Εͨਐߦ೾ղΛ΋ͱʹɺͲͷΑ͏ͳ‫ܗ‬ͷύλʔϯ͕‫ܗ‬੒͞Ε͍ͯΔ͔Λ֬ೝͨ͠ɻΩʔϫʔυ൓Ԡ֦ࢄ‫ܥ‬ɺύλʔϯ‫ܗ‬੒ɺ਺஋‫ࢉܭ‬ʢ˞จ੹ : ࡾӜྰԻʣ-i-

Abstract
The theme of this project is to analyze the diﬀusion of living organisms and the patterns
formed by the reaction-diﬀusion system, such as the diﬀusion of slime molds and thestriped pattern of striped hockey, using model equations. A reaction-diﬀusion system isa mathematical model that describes how a substance concentration changes temporally
and spatially through two processes: a chemical reaction and the spatial diﬀusion of areactant. As a method of pattern analysis, a numerical calculation program is created
and a simulation is performed on the behavior of the solution of the reaction-diﬀusion
system. Examples of patterns that appear at this time include spiral patterns, target
patterns, and ring-shaped patterns. In order to carry out the simulation, we proceeded
systematically by clarifying the work to be done in the ﬁrst half and the second halfearly in the ﬁrst half.
In the ﬁrst semester, each person should read the given references, complete the tasks
assigned each week in the form of self-study, and announce what they have learned
through zoom during the next week’s project study time. So, I deepened my understanding of pattern formation in reaction-diﬀusion systems. In addition, based on thepresentation, we received advice and corrections from the instructor in charge, supplemented the parts that were not enough by self-study, and reconﬁrmed the parts thatwere learned incorrectly. At the same time as learning the types of traveling waves andpatterns, I also learned the basic knowledge for creating numerical calculation programs
that will be needed in the second half.
In the second semester, I made full use of the basic knowledge of creating numerical calculation programs that I acquired in the ﬁrst semester, and completed the same weekly
tasks as in the ﬁrst semester. At the same time, in order to create an animation movieof the change in the behavior of the traveling wave solution, which is the ﬁnal product,
the behavior of the solution of the partial diﬀerential equation was investigated fromthe numerical calculation, and a numerical calculation program was actually createdand simulated. .. Based on the traveling wave solution that appeared at that time, we
conﬁrmed what kind of pattern was formed.
KeywordReaction-diﬀusion system, pattern formation, numerical calculationʢ˞จ੹ : ࡾӜྰԻʣ- ii -

໨࣍
ୈ1ষ

എ‫ܠ‬

1

1.1

֘౰෼໺ͷ‫ݱ‬ঢ়ͱैདྷྫ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2‫ݱ‬ঢ়ʹ͓͚Δ໰୊఺ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.3

՝୊ͷ֓ཁ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

౸ୡ໨ඪ

2

ୈ2ষ

2.1

ຊϓϩδΣΫτʹ͓͚Δ໨త

2.1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . .

௨ৗͷत‫Ͱۀ‬͸ͳ͘ɺϓϩδΣΫτֶशͰߦ͏ར఺

2. . . . . . . . . . . .

2

2.2

۩ମతͳखॱɾ՝୊ઃఆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2.3

౸ୡϨϕϧ (໨ඪ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.4

՝୊ͷׂΓ౰ͯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

ୈ3ষ

՝୊ղܾͷϓϩηεͷ֓ཁ5

ୈ4ষ

՝୊ղܾͷϓϩηεͷৄࡉ7

4.1

֤ਓͷ՝୊ͷ֓ཁͱϓϩδΣΫτ಺ʹ͓͚ΔҐஔ͚ͮ

. . . . . . . . . . . . . . .

7

4.2୲౰՝୊ղܾաఔͷৄࡉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.2.1

Ֆࢁཽ໵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.2.2

ࡾӜྰԻ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.2.3

ླ໦ᠳే . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.2.4

౻ా३ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

ୈ5ষ

5.1

5.2

5.3

ୈ6ষ

݁Ռ

12

ϓϩδΣΫτͷ݁Ռ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

5.1.1

ั৯ऀʔඃ৯ऀ‫ ܥ‬. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

5.1.2

֦ࢄํఔࣜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

5.1.3

૒҆ఆ൓Ԡ֦ࢄํఔࣜ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

5.1.4

Ұ࣍‫ ݩ‬Fitzhugh-Nagumo ํఔࣜ . . . . . . . . . . . . . . . . . . . . . .

17

5.1.5

ೋ࣍‫ ݩ‬Fitzhugh-Nagumo ํఔࣜ . . . . . . . . . . . . . . . . . . . . . .

19੒ՌͷධՁ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

5.2.1

લ‫ظ‬ͷධՁ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

5.2.2

‫ظޙ‬ͷධՁ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20୲౰෼୲՝୊ͷධՁ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

5.3.1

Ֆࢁཽ໵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

5.3.2

ࡾӜྰԻ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

5.3.3

ླ໦ᠳే . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

5.3.4

౻ా३ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24ࠓ‫ޙ‬ͷ՝୊ͱల๬

25- iii -

ୈ7ষֶͼͱࠓ‫ޙ‬ͷ՝୊

26

7.0.1

Ֆࢁཽ໵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

7.0.2

ࡾӜྰԻ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

7.0.3

ླ໦ᠳే . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

7.0.4

౻ా३ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

ࢀߟจ‫ݙ‬

32- iv -

Mathematical and simulation of complex systemsୈ 1 ষ എ‫ܠ‬
1.1֘౰෼໺ͷ‫ݱ‬ঢ়ͱैདྷྫ

ຊϓϩδΣΫτͷ෼໺ͷ‫ݱ‬ঢ়ͱͯ͠ɺࣗવքʹ͓͚Δ༷ʑͳൃੜ‫ݱ‬৅ΛऔΓ্͛ͯ਺ཧϞσϧΛߏங͠ɺͦͷղੳʹΑΓ‫ܗ‬ଶ‫ܗ‬੒‫ߏػ‬Λղ໌͠ɺ਺ֶతղੳΛߦ͍ͬͯΔɻੜ෺ͷ໛༷΍ࡉ๔ӡಈͳͲʹΑͬͯൃੜ͢Δύλʔϯ‫ܗ‬੒Λ਺ཧϞσϧͱͯ͠ղੳ͢Δ͜ͱʹΑͬͯɺͦͷ৚݅Λܾఆ͢
Δ͜ͱ͕Ͱ͖Δɻྫ͑͹ɺώτ͕ոզͨ͠‫࣏ʹޙ‬༊͞ΕΔ‫ݱ‬৅ͳͲ͸ࡉ๔ूஂͷৼΔ෣͍͕ඞཁʹ
ͳΔɻࡉ๔ूஂͷৼΔ෣͍ʹΑͬͯ‫ܗ‬੒͞ΕΔύλʔϯͷ৚݅Λཧղ੍͠‫͢ޚ‬Δ͜ͱ͕Ͱ͖Ε͹ɺප‫ؾ‬ͷཧղ΍৽ͨͳ࣏ྍ๏ͷ։ൃʹ໾ཱͭ஌ࣝΛਂΊΔ͜ͱ΋ՄೳͰ͋Δɻ͜Ε͔Βͷύλʔϯ‫ܗ‬੒ͱͯ͠͸ɺਓؒͷൽෘͳͲͷύλʔϯΛղੳ͢Δ͜ͱΛ໨ࢦ͍ͯ͠Δɻൽෘͷύλʔϯ‫ܗ‬੒Λཧղ͢Δ͜ͱͰɺൽෘ͕ΜͳͲͷ࣬‫ױ‬ͷૣ‫ݟൃظ‬΍࣏ྍʹ‫͕ܨ‬Δ͜ͱ͕‫ظ‬଴͞Ε͍ͯΔɻ
ʢ˞จ੹ : ࡾӜྰԻʣ

1.2‫ݱ‬ঢ়ʹ͓͚Δ໰୊఺‫ݱ‬ঢ়Ͱ͸ɺύλʔϯ‫ܗ‬੒͢Δ͜ͱࣗମ͸Մೳ͕ɺ਺஋‫͢ࢉܭ‬Δ͏͑Ͱɺ͕ࣜෳࡶʹͳ͍͖ͬͯղ
ੳΛߦ͏͜ͱ͕΍΍ࠔ೉ʹͳͬͨΓɺΘ͔Γʹ͘͘ͳͬͨΓ͍ͯ͠Δɻͭ·Γɺύλʔϯ͕‫ܗ‬੒͞ΕΔৄࡉͳ৚݅Λܾఆ͢Δͷ͕೉͍͠ঢ়‫͍ͯͬͳʹگ‬ΔͷͰ͋Δɻͭ·ΓɺύλʔϯΛ‫ܗ‬੒ͤ͞Δ
਺ཧϞσϧʹର͢Δ஌ࣝΛਂΊ͔ͯΒɺৄࡉʹղੳ͍ͯ͘͜͠ͱ͕ɺύλʔϯ‫ܗ‬੒৚݅Λܾఆ͢Δ
ʹ͸େ੾Ͱ͋Δɻͦ͜Ͱɺঢ়‫ʹگ‬ΑͬͯɺΘ͔Γ΍͍͢ํ๏Ͱ਺ֶతղੳΛߦ͏͜ͱΛॏཁࢹͯ͠
͍Δɻ
ʢ˞จ੹ : ࡾӜྰԻʣ

1.3

՝୊ͷ֓ཁ

ຊϓϩδΣΫτͰ͸ɺϝϯόʔશһͰɺओʹੜ෺ֶʹ͓͚Δύλʔϯ‫ܗ‬੒ʹ͍ͭͯߟ͑ɺࢀߟจ
‫ݙ‬Λྠಡͯ͠ཧղΛਂΊΔ͜ͱ͔Β࢝Ίͨɻલ‫Ͱظ‬͸ɺओʹɺਐߦ೾ղʹ͍ͭͯͷཧղɺ਺஋‫ࢉܭ‬͍ͯ͘͠͏͑Ͱͷ‫ૅج‬஌ࣝͷशಘΛ՝୊ͱͨ͠ɻ‫Ͱظޙ‬͸ɺલ‫ʹظ‬਎ʹ෇͚ͨ஌ࣝͱٕज़Λ΋ͱʹϓϩάϥϜ࡞੒͠ɺ࣮ࡍʹγϛϡϨʔγϣϯΛߦ͏͜ͱͰɺΞχϝʔγϣϯಈըΛ࡞੒͢Δ͜ͱΛ՝୊ͱͨ͠ɻ
ʢ˞จ੹ : ࡾӜྰԻʣ

Group Report of 2020 SISP

-1-Group Number 6

Mathematical and simulation of complex systems

ୈ2ষ

౸ୡ໨ඪ

ຊϓϩδΣΫτʹ͓͚Δ໨త

2.1

൓Ԡ֦ࢄ‫਺͍ͯͭʹܥ‬஋‫ͱࢉܭ‬γϛϡϨʔγϣϯΛߦ͍ղੳ͢Δ͜ͱ͕՝୊ɻੜ෺ͷ֦ࢄ΍໛༷͕਺ཧϞσϧͰදͤΔ͜ͱΛࣔͨ͢Ίʹ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ɺ༷ʑͳύϥϝʔλͰγϛϡϨʔγϣϯΛߦ͍ɺಈըΛ࡞੒͢Δ͜ͱɻ
ʢ˞จ੹ : ླ໦ᠳేʣ

2.1.1

௨ৗͷत‫Ͱۀ‬͸ͳ͘ɺϓϩδΣΫτֶशͰߦ͏ར఺

ຊϓϩδΣΫτͰ͸·ͣɺ՝୊ͷղܾʹ޲͚ͯ࿦จΛಡΈɺඇઢ‫ܗ‬ภඍ෼ํఔࣜΛղͨ͘Ίͷ‫ج‬ૅ஌ࣝΛ਎ʹண͚Δɻ͜Ε·ͰͷߨٛͰ͸ೃછΈͷগͳ͔ͬͨඇઢ‫ܗ‬ඍ෼ํఔࣜͱ͍͏τϐοΫͷ
‫ૅج‬஌ࣝΛޮ཰తʹཧղ͢ΔͨΊʹҰͭͷࢀߟจ‫ݙ‬ΛϝϯόʔͰ෼୲Λͯ͠ಡΈɺཧղͰ͖ͳ͔ͬ
ͨ͜ͱΛ‫ࣜܗ͏͋͑ڭ‬Λऔͬͨɻ·ͨɺϓϩδΣΫτ಺Ͱ‫ڞ‬௨ͷ੍࡞෺Λ࡞੒͢ΔͨΊʹϝϯόʔؒͷ‫ૅج‬஌ࣝͷࠩҟΛͰ͖Δ͚ͩͳ͘͢ඞཁ͕͋ΔͨΊɺ‫͍͋͑ڭ‬ͷ‫ࣜܗ‬Λ࠾༻ͨ͠ɻ௨ৗͷत‫ۀ‬Ͱ͸‫ج‬ຊతʹ‫ݸ‬ਓͷ஌ࣝɾٕज़ʹ͍ͭͯߨٛɾԋश‫ߦͰࣜܗ‬ΘΕΔͨΊɺ‫ڞ‬௨ͷ੍࡞෺Λ࡞Δ͜ͱ
ʹؔͯ͠޲͔ͳ͍ɻ
ʢ˞จ੹ : ླ໦ᠳేʣ

2.2۩ମతͳखॱɾ՝୊ઃఆ

2.1 અͰड़΂ͨ໰୊ΛɺҎԼͷ੍໿৚݅ԼͰղܾ͢Δ͜ͱΛߟ͑ͨɻɾϝϯόʔؒͷ൓Ԡ֦ࢄ‫΁ܥ‬ͷཧղ౓ͷࠩΛ‫ݮ‬Β͢ɻɾҰ෦ͷϝϯόʔʹ՝୊͕ภΒͳ͍Α͏ʹ͢Δɻ
ͦͷ݁ՌɺҎԼͷ՝୊͕ఏҊ͞Εͨɻ

1. ༻‫ޠ‬ͷཧղ՝୊ɿࢀߟʹ͢Δ࿦จʹग़ͯ͘Δ਺ֶͷઐ໳༻‫ޠ‬Λཧղ͠άϧʔϓϝϯόʔʹཧղͯ͠΋Β͑ΔΑ͏ʹ͸આ໌͢Δɻ
ؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II

2. ൓Ԡ֦ࢄ‫ܥ‬ͷղੳ՝୊ɿ൓Ԡ֦ࢄ‫Ͳ͕ܥ‬ͷΑ͏ͳϞσϧͰ͋Δ͔Λղੳ͠ɺཧղ͢Δɻɹ
ؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश I‫ط‬ଘٕज़ɿߦྻࣜͷղ͖ํ (αϥεͷެࣜ)৽‫ن‬शಘٕज़ɿઢ‫҆ܗ‬ఆੑղੳɹ

3. ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղੳ՝୊ɿ൓Ԡ֦ࢄ‫͚͓ʹܥ‬Δั৯ऀʔඃ৯ऀ‫Ͳ͕ܥ‬ͷΑ͏ͳϞσϧͰ͋Δ͔Λղੳ͠ɺཧղ

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Mathematical and simulation of complex systems
͢Δɻؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II‫ط‬ଘٕज़ɿઢ‫҆ܗ‬ఆੑղੳ, ૬ฏ໘ղੳ
৽‫ن‬शಘٕज़ɿψϧΫϥΠϯʹΑΔྲྀΕਤɹɹɹ

4. ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղͷදࣔ՝୊ɿั৯ऀʔඃ৯ऀ‫ܥ‬ͷղ͕ͲͷΑ͏ͳάϥϑʹͳΔͷ͔දࣔ͢Δɻ
ؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II, ৘ใॲཧԋश I, ৘ใॲཧԋश II, ৘ใද‫ૅجݱ‬‫ط‬ଘٕज़ɿPython ʹΑΔϓϩάϥϛϯά
৽‫ن‬शಘٕज़ɿάϥϑ΍ΞχϝʔγϣϯΛग़ྗ͢Δϓϩάϥϛϯάɹɹɹ

5. ฏߧ఺पΓͷಈ͖ͷදࣔ՝୊ɿ‫ͨͬ·ٻ‬άϥϑͷฏߧ఺पΓͷಈ͖͕ͲͷΑ͏ʹͳ͍ͬͯΔ͔ͦΕͧΕ֬ೝ͠දࣔ
͢Δɻ
ؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II, ৘ใॲཧԋश I, ৘ใॲཧԋश II, ৘ใද‫ૅجݱ‬‫ط‬ଘٕज़ɿPython,C ‫ʹޠݴ‬ΑΔϓϩάϥϛϯάɹɹɹ

6. ั৯ऀʔඃ৯ऀ‫ܥ‬ͷ‫ܗ‬ঢ়͕Ұఆͳϑϩϯτਐߦ೾ղͷղੳ՝୊ɿั৯ऀʔඃ৯ऀ‫Ͳ͕ܥ‬ͷΑ͏ͳಛ௃Λ࣋ͭͷ͔ௐ΂ΔͨΊʹɺϑϩϯτਐߦ೾͕ଘࡏ
͠͏Δ৚݅Λ‫ٻ‬ΊΔɻؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II‫ط‬ଘٕज़ɿPython,C ‫ʹޠݴ‬ΑΔϓϩάϥϛϯά

7. ֦ࢄΛ‫ؚ‬ΊͨϞσϧͷղੳ՝୊ɿྟք఺Λ݁Ϳਐߦ೾͕ଘࡏ͢Δ͜ͱΛௐ΂Δɻؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II

8. ֦ࢄํఔࣜͷղੳ՝୊ɿ֦ࢄํఔࣜͷղͷৼΔ෣͍Λ਺஋‫ٻͰࢉܭ‬ΊΔɻؔ࿈ߨٛɿղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II‫ط‬ଘٕज़ɿPython,C ‫ʹޠݴ‬ΑΔϓϩάϥϛϯά
৽‫ن‬शಘٕज़ɿ਺஋‫ࢉܭ‬ΞϧΰϦζϜ (ࠩ෼๏)

9. ૒҆ఆ൓Ԡ֦ࢄํఔࣜͷղੳ՝୊ɿ૒҆ఆ൓Ԡ֦ࢄํఔࣜΛ༷ʑͳύϥϝʔλͰ਺஋‫ࢉܭ‬Λߦ͍ղͷৼΔ෣͍ΛಈըԽ
͢Δɻؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II‫ط‬ଘٕज़ɿPython,C ‫ʹޠݴ‬ΑΔϓϩάϥϛϯά

10. Ұ࣍‫ܥݩ‬ͷ Fitzhugh-Nagumo ํఔࣜͷղੳ՝୊ɿҰ࣍‫ܥݩ‬ͷ Fizthugh-Nagumo ํఔࣜΛύϥϝʔλΛௐ੔͠ͳ͕Βύϧεਐߦ೾͕‫ܗ‬੒͞ΕΔ͜ͱΛ֬ೝ͠ಈըԽ͢Δɻ
ؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II‫ط‬ଘٕज़ɿPython,C ‫ʹޠݴ‬ΑΔϓϩάϥϛϯά

11. ೋ࣍‫ܥݩ‬ͷ Fitzhugh-Nagumo ํఔࣜͷղੳ՝୊ɿҰ࣍‫ܥݩ‬ͷ Fitzhugh-Nagumo ํఔࣜΛೋ࣍‫֦ʹܥݩ‬ு͠ɺ࣌ؒൃలΛͤ͞ղͷৼΔ
෣͍Λ֬ೝ͢Δɻ

Group Report of 2020 SISP

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Mathematical and simulation of complex systemsؔ࿈ߨٛɿઢ‫ܗ‬୅਺ I, ઢ‫ܗ‬୅਺ II, ղੳֶ I, ղੳֶ II, ਺ֶ૯߹ԋश I, ਺ֶ૯߹ԋश II‫ط‬ଘٕज़ɿPython,C ‫ʹޠݴ‬ΑΔϓϩάϥϛϯά
ʢ˞จ੹ : ླ໦ᠳేʣ

2.3

౸ୡϨϕϧ (໨ඪ)ϓϩδΣΫτϝϯόʔؒͷ൓Ԡ֦ࢄ‫ܥ‬ͷཧղ౓ͷࠩҟΛ‫ݮ‬Βͭͭ͠ɺ਺஋‫ࢉܭ‬ϓϩάϥϜͷ࡞੒ͱγϛϡϨʔγϣϯͷ࣮ࢪʹΑͬͯΑΓΑ͍ಈըΛ࡞੒͢ΔͨΊʹҎԼͷ໨ඪΛઃఆͨ͠ɻ

• ൓Ԡ֦ࢄ‫͍ͯͭʹܥ‬ͷࢀߟจ‫ݙ‬Λྠಡ͢Δ࣌ؒΛ૿΍͠ɺଞͷϝϯόʔʹઆ໌Ͱ͖ΔϨϕϧ·Ͱཧղ͢Δɻ• ϓϩάϥϛϯά໘Ͱ͸ࣗ෼ͷಘҙͳϓϩάϥϛϯά‫ޠݴ‬Λ࢖༻ͭͭͦ͠ͷ‫͚͓ʹޠݴ‬Δ
Ξχϝʔγϣϯͷ࡞੒ʹ͍ͭͯษ‫͢ڧ‬Δɻ• ҰਓҰͭ͸਺஋‫ࢉܭ‬ΞϧΰϦζϜΛཧղ͠ɺϓϩάϥϜͰ਺஋‫ࢉܭ‬Λߦͬͯ݁ՌΛग़ྗ
͢Δɻʢ˞จ੹ : ླ໦ᠳేʣ

2.4

՝୊ͷׂΓ౰ͯ

൓Ԡ֦ࢄ‫ܥ‬ͷ‫ૅج‬ͷཧղΛਂΊΔͨΊʹϓϩδΣΫτϝϯόʔશһͰྠಡΛߦͬͨͨΊɺ༻‫ޠ‬ͷཧղɺ൓Ԡ֦ࢄ‫ܥ‬ͷղੳɺั৯ऀʔඃ৯ऀ‫ܥ‬ͷղੳʹؔͯ͠͸ϓϩδΣΫτϝϯόʔશһͰߦͬͨɻྠಡͰ͸֤ਓͷෛ୲͕‫ۉ‬౳ʹͳΔΑ͏ʹτϐοΫ͝ͱʹ෼୲͠ɺͦΕͧΕ͕୲౰ͨ͠‫ॴݸ‬ͷཧղΛਂΊͯɺϝϯόʔશһʹൃදͨ͠ɻั৯ऀʔඃ৯ऀ‫ܥ‬ͷղͷදࣔͱฏߧ఺पΓͷಈ͖ͷදࣔʹؔͯ͠͸਺ֶతͳΞϧΰϦζϜ΍࢖༻ͨ͠ϓϩάϥϛϯά‫ޠݴ‬ͷཧղ౓Λ‫ݒ‬೦ͯ͠౻ాͱླ໦
͕୲౰ͨ͠ɻั৯ऀʔඃ৯ऀ‫ܥ‬ͷ‫ܗ‬ঢ়͕Ұఆͳϑϩϯτਐߦ೾ղͷղੳ͸ࡾӜ͕୲౰ͨ͠ɻ·ͨɺ֦ࢄΛ‫ؚ‬ΊͨϞσϧͷղੳʹ͍ͭͯ͸Ֆࢁ͕୲౰ͨ͠ɻ֦ࢄํఔࣜͷղੳ͸ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղੳΛ‫͍ͯ͠ݧܦ‬Δ͜ͱ͔Β౻ాɺࡾӜɺླ໦͕୲౰ͨ͠ɻ֦ࢄํఔࣜͷൃల‫͋Ͱܥ‬Δ૒҆ఆ൓Ԡ֦ࢄํఔࣜͷղͷৼΔ෣͍ͷಈըԽ͸౻ా͕୲౰ͨ͠ɻҰ࣍‫ܥݩ‬ͷ Fitzhugh-Nagumo ํఔࣜ΋֦ࢄํఔࣜͷൃల‫ͳʹܥ‬ΔͷͰ֦ࢄํఔࣜͷղੳΛ‫͍ͯ͠ݧܦ‬Δླ໦͕୲౰ͨ͠ɻೋ࣍‫ܥݩ‬ͷ

Fitzhugh-Nagumo ํఔࣜ͸Ұ࣍‫ܥݩ‬ͷ֦ு‫͏͍ͱܥ‬ཧ༝Ͱླ໦͕୲౰ͨ͠ɻ
ʢ˞จ੹ : ླ໦ᠳేʣ

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Mathematical and simulation of complex systemsୈ 3 ষ ՝୊ղܾͷϓϩηεͷ֓ཁ
2.2 અͰ۩ମԽ֤ͨ͠খ՝୊ͷղܾͷϓϩηεͷ֓ཁΛɺ֤ʑ‫ه‬ड़͢Δɻ
1. ༻‫ޠ‬ͷཧղࢀߟจ‫ݙ‬ͷྠಡΛߦ͍ɺ֤ࣗཧղͨ͠෦෼ʹؔͯ͠͸άϧʔϓϝϯόʔʹઆ໌Ͱ͖ΔΑ͏ʹͨ͠ɻ·ͨཧղͰ͖ͳ͔ͬͨ෦෼ʹؔͯ͠͸ϝϯόʔؒͰͷ‫ڞ‬༗΍୲౰‫ڭ‬һʹҙ‫ݟ‬Λ‫ٻ‬ΊΔ౳Λͯ͠ཧղΛਂΊ͍ͯͬͨɻ

2. ൓Ԡ֦ࢄ‫ܥ‬ͷղੳ൓Ԡ֦ࢄ‫ܥ‬ͷ֤߲ʹରԠ͢Δ΋ͷͷҙຯ΍࢓૊ΈΛཧղ͠ɺ਺஋‫ࢉܭ‬Λߦ͍ͬͯͬͨɻ୲౰
͢ΔϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰
‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒղੳΛߦ͍ͬͯͬͨɻ

3. ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղੳั৯ऀʔඃ৯ऀ‫߲ࢄ֦ʹܥ‬ΛՃ͑ɺۭؒඇґଘతʹ‫ܥ‬Λղੳ͠ɺϞσϧͷཧղΛߦ͍ͬͯͬ
ͨɻ୲౰͢ΔϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒղੳΛߦ͍ͬͯͬͨɻ

4. ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղͷදࣔઢ‫҆ܗ‬ఆੑղੳΛ༻͍ͯɺ3 ͭͷฏߧ఺ͷ҆ఆੑΛ‫ٻ‬Ί͍͖ͯɺฏߧ఺पΓͷྲྀΕΛ֬ೝ͢
Δ͜ͱͰั৯ऀʔඃ৯ऀ‫ܥ‬ͷཧղΛਂΊ͍ͯͬͨɻ୲౰͢ΔϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒղͷදࣔΛߦ͍ͬͯͬͨɻ

5. ฏߧ఺पΓͷಈ͖ͷදࣔΦΠϥʔ๏Λ༻͍ͯ࣌ؒൃలΛάϥϑʹඳ͖ද͠ฏߧ఺पΓͷྲྀΕΛ͔֬Ίͨɻ୲౰͢ΔϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ཧղΛਂΊ͍ͯͬͨɻ

6. ั৯ऀʔඃ৯ऀ‫ܥ‬ͷ‫ܗ‬ঢ়͕Ұఆͳϑϩϯτਐߦ೾ղͷղੳ‫ݻ‬༗஋ํఔࣜΛ༻͍ͯɺϑϩϯτਐߦ೾͕ଘࡏ͢Δ͜ͱͷղੳΛߦ͍ͬͯͬͨɻϑϩϯτਐߦ೾ղΛΞχϝʔγϣϯԽ͠ͲͷΑ͏ͳৼΔ෣͍ʹͳΔ֬ೝͨ͠ɻ୲౰͍ͯͨ͠ϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒղੳΛߦ͍͖ͬͯɺϝϯόʔؒͰͷཧղ΋ਂΊ͍ͯͬͨɻ

7. ֦ࢄΛ‫ؚ‬ΊͨϞσϧͷղੳϩτΧʔϰΥϧςϥͷ‫ڝ‬૪‫ࢄ֦ʹܥ‬Λ‫ؚ‬ΊɺϑΟογϟʔʔίϧϞΰϩϑํఔࣜΛ༻͍ͯਐߦ೾ͷ଎౓ΛղੳΛߦ͍ͬͯͬͨɻ୲౰͢ΔϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒղੳΛߦ͍͖ͬͯɺϝϯόʔؒͰͷཧղ΋ਂΊ͍ͯͬͨɻ

8. ֦ࢄํఔࣜͷղੳϑʔϦΤม‫׵‬΍ࠩ෼๏Λ༻͍ͯɺ൓Ԡ֦ࢄํఔࣜͷ൓Ԡ߲͕̌ͷ࣌ͷղੳΛߦ͍ͬͯͬͨɻ
ϑʔϦΤม‫׵‬ɺࠩ෼๏ͱ΋ʹ਺஋‫ࢉܭ‬Λߦ͏ϓϩάϥϜͷ࡞੒Λߦ͍ɺ୲౰͢ΔϝϯόʔΛத৺ʹཧղΛਂΊ͍ͯͬͨɻཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒɺϝϯόʔؒͰͷཧղ΋ਂΊ͍ͯͬͨɻ

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9. ૒҆ఆ൓Ԡ֦ࢄํఔࣜͷղੳࠩ෼๏Λ༻͍ͯɺ૒҆ఆ൓Ԡ֦ࢄํఔࣜͷ༷ʑͳύϥϝʔλͰͷ਺஋ղੳΛߦ͍ɺղͷৼΔ
෣͍ΛΞχϝʔγϣϯԽ͢Δ͜ͱͰ֬ೝ͍ͯͬͨ͠ɻ୲౰͢ΔϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒղੳΛߦͬͨɻ

10. Ұ࣍‫ܥݩ‬ͷ Fitzhugh-Nagumo ํఔࣜͷղੳ
Fitzhugh-Nagumo ํఔࣜΛ༻͍ͯɺਐߦύϧεղͷৼΔ෣͍ʹ͍͔ͭͯ֬Ίͨɻ୲౰͢Δ
ϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺଞϝϯόʔʹ΋‫ڞ‬༗ͭͭ͠ɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒղੳΛߦ͍ͬͯͬͨɻ

11. ೋ࣍‫ܥݩ‬ͷ Fitzhugh-Nagumo ํఔࣜͷղੳೋ࣍‫֦ʹݩ‬ுͨ͠ Fitzhugh-Nagumo ํఔࣜΛ༻͍ͯɺ࣌ؒൃలΛͨ͠ࡍͷղͷৼΔ෣͍Λ͔֬Ίͨɻ୲౰͢ΔϝϯόʔΛத৺ʹཧղΛਂΊ͍͖ͯɺଞϝϯόʔʹ΋ཧղ಺༰Λ‫ڞ‬༗ͭͭ͠ɺཧղͷؒҧ͍΍Θ͔Βͳ͔ͬͨ෦෼ʹؔͯ͠͸ਵ࣌ɺ୲౰‫ڭ‬һʹमਖ਼ͯ͠΋Β͍ͳ͕ΒղੳΛߦ͍ͬͯͬͨɻ
ʢ˞จ੹ : Ֆࢁཽ໵ʣ

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-6-Group Number 6

Mathematical and simulation of complex systems

ୈ 4 ষ ՝୊ղܾͷϓϩηεͷৄࡉ
4.1

֤ਓͷ՝୊ͷ֓ཁͱϓϩδΣΫτ಺ʹ͓͚ΔҐஔ͚ͮલ‫ظ‬ͷ‫׆‬ಈͰ͸ɺ࢝Ίʹ൓Ԡ֦ࢄ‫͍ͯͭʹܥ‬ͷཧղΛਂΊΔͨΊʹจ‫[ ݙ‬1] Λ࢖ͬͯྠಡΛߦͬͨɻͦΕͧΕ୲౰ͱͳΔൣғΛܾΊɺࣄલʹ୲౰ʹͳͬͨൣғͷཁ໿Λߦͬͨɻͦͷ‫ޙ‬ɺ୲౰͞ΕͨൣғΛॱ൪ʹղઆ͠ɺଞͷϝϯόʔͱҙ‫ݟ‬ΛަΘ͢ͱ͍ͬͨྠಡΛߦͬͨɻ·ͨɺ֤ϝϯόʔׂ͕Γ౰ͯΒΕͨϖʔδʹग़ͯ͘Δ‫ܥ‬ͷղੳΛߦͬͨɻ‫ظޙ‬ͷ‫׆‬ಈͰ͸ɺ࢝Ίʹ֦ࢄ߲ͷΈͷ‫ܥ‬ͷղੳΛ֤ϝϯόʔʹׂΓ౰ͯΒΕͨख๏Ͱ਺஋‫ࢉܭ‬Λߦ͍άϥϑԽͨ͠ɻͦͷ‫ޙ‬͸ɺ૒҆ఆ൓Ԡ֦ࢄ‫ܥ‬΍ FitzHugh-Nagumo ํఔࣜͷղੳΛߦ͍ɺ਺஋
‫ࢉܭ‬΍ψϧΫϥΠϯɺϑϩϯτਐߦ೾ղɺύϧεղɺύλʔϯ‫ܗ‬੒Λϓϩοτ͢ΔϓϩάϥϜ࡞੒Λ֤ϝϯόʔͰ෼୲͠ɺ࠷ऴൃදʹ޲͚ͯ४උΛߦͬͨɻʢ˞จ੹ : ౻ా३ʣՖࢁཽ໵ͷ୲౰՝୊͸ҎԼͷͱ͓ΓͰ͋Δɻ

5݄

10pʙ13p ͷൣғͰྠಡͷ४උΛߦͬͨɻ

6݄ั৯ऀʔඃ৯ऀ‫͚͓ʹܥ‬Δɺۭؒඇґଘతͳ‫͍ͯͭʹܥ‬ཧղΛਂΊͨɻ

7݄ϑΟογϟʔʔίϧϞΰϩϑͷํఔ͔ࣜΒ֦ࢄ͕͋Δ৔߹ͷ‫ڝ‬૪ϞσϧΛղੳ͠ɺதؒൃද
ʹ޲͚ͯ४උΛߦͬͨɻ

8-9 ݄ લ‫ͱͨͬ͜ߦͰظ‬ͷ಺༰Λ෮श͠ɺ‫͚ͯ޲ʹظޙ‬ͷ֬ೝΛͨ͠ɻ
10 ݄ ֦ࢄํఔ͍͍ࣜͭͯ਺஋‫ࢉܭ‬Λߦͬͨɻ
11 ݄ Fitzhugh-Nagumo ํఔࣜͷղੳΛߦͬͨɻ
12 ݄ ࠷ऴൃදʹ޲͚ͯ४උΛߦͬͨɻʢ˞จ੹ : Ֆࢁཽ໵ʣ
ࡾӜྰԻͷ୲౰՝୊͸ҎԼͷͱ͓ΓͰ͋Δɻ

5݄

5pʙ7p ͷൣғͰྠಡͷ४උΛߦͬͨɻ6݄ϑϩϯτਐߦ೾ղʹ͍ͭͯͷཧղΛਂΊͨɻ

7݄ྠಡͷൃදΛߦͬͨɻ

8-9 ݄ લ‫ֶʹظ‬शͨ͜͠ͱͷ෮शͱɺ‫͚ͯ޲ʹظޙ‬ͷ֬ೝΛͨ͠ɻ
10 ݄ ֦ࢄํఔࣜʹ͍ͭͯ਺஋‫ࢉܭ‬Λߦͬͨɻ
11 ݄ FitzHugh-Nagumo ํఔࣜʹ͍ͭͯղੳΛߦͬͨɻ
12 ݄ ࠷ऴൃද༻ͷϙελʔΛ࡞੒ͨ͠ɻ
ʢ˞จ੹ : ࡾӜྰԻʣླ໦ᠳేͷ୲౰՝୊͸ҎԼͷͱ͓ΓͰ͋Δɻ5݄

8pʙ10p ͷൣғͰྠಡͷ४උΛߦͬͨɻ

6݄ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղΛ‫ٻ‬ΊΔͨΊϓϩάϥϜΛ࡞੒͠ग़ྗͨ͠ɻ

Group Report of 2020 SISP

-7-Group Number 6

Mathematical and simulation of complex systems
7݄ฏߧ఺पΓͷಈ͖Λ਺ֶతʹಋ͖ɺதؒൃද༻ͷࢿྉΛ࡞੒ͨ͠ɻ

8-9 ݄ લ‫Ͱظ‬΍͖ͬͯͨ͜ͱͷ෮शͱ‫Ͱظޙ‬΍Δ͜ͱͷ֬ೝΛͨ͠ɻ
10 ݄ ֦ࢄํఔࣜʹ͍ͭͯ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ղͷৼΔ෣͍Λ֬ೝͨ͠ɻ
11 ݄ Ұ࣍‫͓ܥݩ‬Αͼೋ࣍‫ܥݩ‬ͷ Fitzhugh-Nagumo ํఔࣜʹ͍ͭͯ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ղͷৼΔ෣͍ΛಈըԽͨ͠ɻ

12 ݄ ࠷ऴൃද༻ͷࢿྉΛ࡞੒ͨ͠ɻ
ʢ˞จ੹ : ླ໦ᠳేʣ
౻ా३ͷ୲౰՝୊͸ҎԼͷͱ͓ΓͰ͋Δɻ5݄

1pʙ5p ͷൣғͰྠಡͷ४උΛߦ͍ɺั৯ऀʔඃ৯ऀ‫ࢄ֦ʹܥ‬ΛՃ͑ͨ‫ֶ͍ͯͭʹܥ‬श͠ɺ
ൃදͨ͠ɻ

6݄

ั৯ऀʔඃ৯ऀ‫ࢄ֦ʹܥ‬ΛՃ͑ͨ‫͍ͯͭʹܥ‬ઢ‫҆ܗ‬ఆੑղੳʹΑͬͯࡾͭͷฏߧ఺ͷ҆ఆੑΛ൑ผͨ͠ɻ·ͨɺϦΞϓϊϑؔ਺ʹ͍ͭͯษ‫ͨ͠ڧ‬ɻ7݄ઢ‫҆ܗ‬ఆੑղੳʹΑͬͯ൑ผ͞Εͨ҆ఆ݁અ఺ɺ҆ఆӔঢ়఺ͷ৔߹ͷฏߧ఺पΓͷྲྀΕਤ΍ɺղͷ‫ي‬ಓΛΞχϝʔγϣϯԽ͢ΔϓϩάϥϜΛ࡞੒ͨ͠ɻ·ͨɺதؒൃද༻ͷεϥΠυ
࡞ΓΛͨ͠ɻ

8-9 ݄ ࠓ·Ͱֶशͨ͜͠ͱͷ෮शͱɺ‫ֶʹظޙ‬श͢Δ಺༰Λ֬ೝͨ͠ɻ
10 ݄ ൓Ԡ֦ࢄ‫ܥ‬ͷ֦ࢄ߲ͷΈͷํఔࣜΛࠩ෼๏Λ༻͍ͯղ͖άϥϑԽͨ͠ɻ
11 ݄ ϑ ϩ ϯ τ ਐ ߦ ೾ ղ ͷ ά ϥ ϑ ΍ ψ ϧ Ϋ ϥ Π ϯ Λ ϓ ϩ ο τ ͢ Δ ϓ ϩ ά ϥ Ϝ Λ ࡞ ੒ ͠ ͨ ɻ
FitzHugh-Nagumo ํఔࣜͷղੳΛߦ͍ɺψϧΫϥΠϯΛϓϩοτ͢ΔϓϩάϥϜΛ࡞੒ͨ͠ɻ

12 ݄ ࠷ऴൃදͷ४උΛߦͬͨɻʢ˞จ੹ : ౻ా३ʣ4.2

୲౰՝୊ղܾաఔͷৄࡉ

4.2.1

Ֆࢁཽ໵

5݄

ࣗ෼ͷ୲౰͢Δࢀߟจ‫ݙ‬ͷϖʔδͷಡΈࠐΈΛߦͬͨɻϩτΧʔϰΥϧςϥͷ‫ڝ‬૪‫ܥ‬΍ϑΟογϟʔʔίϧϞΰϩϑํఔࣜͱ͍ͬͨઐ໳తͳ਺ࣜ΍‫۟ޠ‬͸Ͱ͖Δ͚ͩࣗ෼͕ཧղͰ
ΔΑ͏ΠϯλʔωοτͰௐ΂ɺΘ͔Βͳ͍෦෼ʹؔͯ͠͸ϝϯόʔؒͰ‫ڞ‬༗Λߦ͍֬ೝ͠
߹ͬͨɻ

6݄

લ൒ʹ͸ɺั৯ऀʔඃ৯ऀ‫͚͓ʹܥ‬Δ֦ࢄ͕ͳ͍৔߹ʹ͍ͭͯߟ͍͖͑ͯɺ‫ޙ‬൒ʹ͸ɺࣗ෼ͷ୲౰͍ͯ͠Δϖʔδʹ͋ͨΔ֦ࢄ͕͋Δ৔߹ʹ͓͚Δั৯ऀʔඃ৯ऀ‫ܥ‬Ϟσϧʹ͍ͭͯͷղੳΛߦͬͨɻͦͷࡍɺϩτΧʔϰΥϧςϥͷ‫ڝ‬૪‫ܥ‬Ϟσϧʹ͍ͭͯਂ͘ௐ΂ཧղΛਂΊͯ
͍ͬͨɻ

7݄

‫ڝ‬૪‫͚͓ʹܥ‬Δ೾ͷ଎౓Λ‫ٻ‬ΊΔͨΊʹɺϑΟογϟʔʔίϧϞΰϩϑํఔࣜʹ͍ͭͯࢀߟ
จ‫ݙ‬΍ΠϯλʔωοτͰௐ΂ͨɻ‫ڝ‬૪‫ܥ‬ͷϑϩϯτ೾ͷ଎౓͸ϑΟογϟʔʔίϧϞΰϩϑ
ํఔࣜͷ೾ͷ࠷খ଎౓ʹ౳͘͠ͳΔ͔ɺ೾ͷ଎౓ΑΓେ͖͘ͳΔͱ͍͏ਪଌΛߦ͏͜ͱ͕Ͱ
͖ͨɻ

8-9 ݄ લ‫ͱͨͬ͜ߦͰظ‬ͷ෮शΛߦͬͨɻ·ͨ‫͚ͯ޲ʹظޙ‬ภඍ෼ํఔࣜͷ༧श΍൓Ԡ֦ࢄ‫ʹܥ‬Group Report of 2020 SISP

-8-Group Number 6

Mathematical and simulation of complex systems͍ͭͯվΊͯௐ΂ͨɻ

10 ݄ ൓Ԡ֦ࢄํఔࣜʹ͍ͭͯɺ൓Ԡ߲Λ̌ͱͨ࣌͠ͷ਺஋‫ࢉܭ‬Λߦ͍ɺ਺஋‫ࢉܭ‬Λߦ͏ϓϩάϥϜʹ͍ͭͯͷ՝୊ʹϝϯόʔؒͰऔΓ૊Μͩɻ·ͨɺ൓Ԡ֦ࢄํఔࣜͰ͸࣌ؒൃలΛ͍ͤͯ͘͜͞ͱͰۭؒతͳྔ͸ҰఆͳྔΛอ֦ͪͭͭࢄ͸ߦΘΕ͍ͯ͘͜ͱ͕֬ೝ͢Δ͜ͱ͕Ͱ͖ͨɻ

11 ݄ FitzHugh-Nagumo ํఔࣜʹ͍ͭͯ਺஋‫ࢉܭ‬Λߦͬͨɻ·ͨɺ਺஋‫ࢉܭ‬Λߦ͏ϓϩάϥϜ
ʹ͍ͭͯͷ՝୊ʹϝϯόʔؒͰऔΓ૊Μͩɻ਺஋‫ࢉܭ‬Λߦ͏͜ͱͰಘΒΕΔਐߦύϧεղͷ
ৼΔ෣͍ʹ͍ͭͯਪଌɺߟ࡯Λߦͬͨɻ·ͨɺ2 ࣍‫ࡍͨ͠ʹݩ‬ͷ FitzHugh-Nagumo ํఔࣜͷ਺஋‫ʹࢉܭ‬΋औΓ૊Μͩɻ

12 ݄ ࠷ऴൃදͷͨΊʹ४උΛߦͬͨɻʢ˞จ੹ : Ֆࢁཽ໵ʣ

4.2.2
5݄

ࡾӜྰԻ

ࢀߟਤॻͷࣗ෼ͷ୲౰͢ΔϖʔδΛಡΈࠐΜͩɻͦͯ͠ɺΘ͔Βͳ͍୯‫ޠ‬΍‫۟ޠ‬͸ௐ΂ͯཧղΛਂΊͨɻओʹਐߦ೾ʹֶ͍ͭͯΜͩɻ

6݄ั৯ऀʔඃ৯ऀ‫ࢄ֦ʹܥ‬ΛՃ͑ͨ‫ܥ‬Λۭؒඇґଘతͳৗඍ෼ํఔࣜͷ‫ʹܗ‬ม‫͠׵‬ɺ͞Βʹɺϑϩϯτਐߦ೾ղΛ‫ٻ‬ΊΔ͜ͱΛͨ͠ɻͦͷͨΊʹϑΟογϟʔʔίϧϞΰϩϑํఔࣜΛ࢖༻ͨ͠Γɺ‫ݻ‬༗ํఔࣜΛ‫ٻ‬ΊͨΓͯ͠ɺਐߦ೾ղʹ͍ͭͯཧղΛਂΊͨɻ

7݄

6 ݄ʹߦͬͨϑϩϯτਐߦ೾ղʹ͍ͭͯ͞Βʹ‫͠ٴݴ‬ɺύϧεղͱ͍͏΋ͷͱͷҧ͍ͳͲΛֶशͨ͠ɻ·ͨɺ৔߹෼͚ΛߦͬͨΓͯ͠ɺϑϩϯτਐߦ೾͕ͲͷΑ͏ͳ‫ܗ‬ঢ়ʹͳΔ͔Λ‫ܭ‬ࢉΛ΋ͱʹௐ΂ͨɻ

8-9 ݄ લ‫ͱͨͬ͜ߦͰظ‬ͷ෮शΛߦͬͨɻͦΕʹՃ͑ͯɺ‫ظޙ‬ͷֶशʹ޲͚ͨ४උ΋࢝Ίͨɻ
10 ݄ ൓Ԡ֦ࢄํఔࣜͷ֦ࢄ߲Λ 0 ͱఆٛͨ࣌͠ͷ਺஋‫ࢉܭ‬Λߦͬͨɻ͜ͷ࣌ࣗ෼͸ϑʔϦΤม‫׵‬Λ༻͍ͨ਺஋‫ࢉܭ‬ΛߦͬͨɻओͳϓϩάϥϜ࡞੒͸΄͔ͷਓ͕୲౰͠ɺͦΕΛ࠷ऴతʹ֬
ೝ͢Δ͜ͱͰɺ࣌ؒൃల͠ଓ͚͍ͯ͘͜ͱͰɺۭؒతͳྔ͸Ұఆʹ֦ࢄ͍ͯ͘͜͠ͱ͕෼
͔ͬͨɻ

11 ݄ FitzHugh-Nagumo ํఔࣜʹ͍ͭͯͷղੳΛߦͬͨɻ࣮ࡍʹ਺஋‫ࢉܭ‬Λߦ͍ɺਐߦύϧεղͷৼΔ෣͍Λߟ࡯ͨ͠ɻओͳϓϩάϥϜ࡞੒͸΄͔ͷਓ͕୲౰͠ɺͦΕΛ࠷ऴతʹ֬ೝ͠ཧղΛਂΊͨɻ

12 ݄ ࠷ऴൃද༻ͷϙελʔ࡞੒Λߦͬͨɻʢ˞จ੹ : ࡾӜྰԻʣ

4.2.3
5݄

ླ໦ᠳే

ࣗ෼͕୲౰͢ΔϖʔδΛಡΈࠐΜͰΘ͔Βͳ͍‫͕۟ޠ‬ग़͖ͯͨ৔߹ʹ͸ Web Ͱ‫ֶͯ͠ࡧݕ‬श͠ྠಡͷ४උΛߦͬͨɻ

6݄ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղΛ‫ٻ‬ΊΔͨΊʹ৚݅Λ਺௨Γઃఆͯ͠ python ͰϓϩάϥϜΛ࡞੒ͦ͠ΕͧΕͷ৚݅ʹ͓͚ΔղͷάϥϑΛग़ྗͨ͠ɻϓϩάϥϜͷ࡞੒ʹ͸৘ใද‫ૅجݱ‬ͷߨٛͰशಘͨ͠ python Ͱͷίʔυͷॻ͖ํΛ༻͍ͨɻͦͷࡍɺϥΠϒϥϦΛ༻͍ͯ਺ֶతͳάϥϑΛग़ྗ͢Δٕज़Λशಘͨ͠ɻ

Group Report of 2020 SISP

-9-Group Number 6

Mathematical and simulation of complex systems
7݄ฏߧ఺पΓͷಈ͖ΛψϧΫϥΠϯͷάϥϑΛखॻ͖Ͱॻ͖਺ֶతʹಋ͍ͨɻͦͷࡍɺղੳֶ
ͷߨٛͰशಘͨ͠߹੒ؔ਺ͷඍ෼΍ઢ‫ܗ‬୅਺ͷߨٛͰशಘͨ͠αϥεͷެࣜΛ༻͍ͯߦྻࣜ
Λղ͍ͨɻ·ͨɺதؒൃද༻ͷεϥΠυʹ‫ߘݪ‬Λ෇͚Δ࡞‫ۀ‬Λߦͬͨɻ

8-9 ݄ લ‫Ͱظ‬΍͖ͬͯͨ͜ͱͷ෮शΛߦ͍ɺ‫Ͱظޙ‬औΓѻ͏ภඍ෼ํఔࣜͷ༧शΛͨ͠ɻ
10 ݄ ൓Ԡ֦ࢄํఔࣜͷ൓Ԡ߲Λ 0 ͱఆٛ͠ɺ֦ࢄ߲͚ͩͷํఔࣜΛ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ύϥϝʔλΛઃఆͯ͠γϛϡϨʔγϣϯΛߦͬͨɻͦͷ݁ՌΛೋ࣍‫ݩ‬άϥϑͱࡾ࣍‫ݩ‬άϥϑͱͯ͠ग़ྗ͠ɺPython ͰΞχϝʔγϣϯΛ࡞੒ͨ͠ɻ

11 ݄ Ұ࣍‫ܥݩ‬ͷ Fitzhugh-Nagumo ํఔࣜΛ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒ͯ͠‫͢ࢉܭ‬ΔͨΊʹ਺஋‫ࢉܭ‬ΞϧΰϦζϜΛษ‫͠ڧ‬ɺϓϩάϥϜΛ࡞੒ͨ͠ɻಘΒΕͨਐߦύϧεղͷৼΔ෣͍Λप‫ظ‬త‫ڥ‬ք৚݅ͷ΋ͱಈըͰΘ͔ΔΑ͏ʹग़ྗ͠ɺͦͷ‫ޙ‬ɺFitzhugh-Nagumo ํఔࣜΛೋ࣍‫֦ʹܥݩ‬ுͯ࣌ؒ͠ൃలͰมԽ͢ΔղΛग़ྗͨ͠ɻ

12 ݄ ࠷ऴൃද༻ͷεϥΠυΛ࡞੒͠ɺ࿩͢಺༰ΛܾΊɺ‫ߘݪ‬Λ࡞੒ͨ͠ɻ
ʢ˞จ੹ : ླ໦ᠳేʣ

4.2.4
5݄

౻ా३

ࣗ෼ͷ୲౰ͨ͠ 1pʙ5p ͷൣғͰΘ͔Βͳ͍‫͍ͯͭʹ۟ޠ‬ௐ΂ͨΓɺั৯ऀʔඃ৯ऀ‫֦ʹܥ‬ࢄΛՃ͑ͨ‫͍ͯͭʹܥ‬ͷղੳΛߦ͍ྠಡͷൃදΛͨ͠ɻ

6݄ั৯ऀʔඃ৯ऀ‫ࢄ֦ʹܥ‬ΛՃ͑ͨ‫ܥ‬Λۭؒඇґଘతͳৗඍ෼ํఔࣜͷ‫ʹܗ‬ม‫͠׵‬ɺղੳΛߦͬͨɻͦͯ͠ࡾͭͷฏߧ఺ͷ҆ఆੑΛઢ‫҆ܗ‬ఆੑղੳʹΑͬͯ൑ผͨ͠ɻ·ͨɺ҆ఆঢ়ଶͷ৔߹ͷղͷ‫ي‬ಓΛΦΠϥʔ๏Λ͔ͭͬͯग़ྗ͢ΔϓϩάϥϜΛ࡞੒ͨ͠ɻ͜ͷϓϩάϥϜ
ʹΑͬͯɺ҆ఆ݁અ఺ͷ৔߹͸௚ઢతʹղʹऩଋ͠ɺ҆ఆӔঢ়఺ͷ৔߹͸ӔΛ‫͘ר‬Α͏ʹղʹऩଋ͢Δ͜ͱ͕Θ͔ͬͨɻ·ͨɺ͜ͷ‫͕ܥ‬ϦΞϓϊϑؔ਺Λ࣋ͭ͜ͱͷҙຯ΍ಋग़ͷ࢓ํΛֶशͨ͠ɻ

7݄ΦΠϥʔ๏Λ࢖ͬͨղͷ‫ي‬ಓͷϓϩάϥϜΛɺೋͭͷ৔߹ͷಉ࣌ग़ྗͱɺฏߧ఺΍ॳ‫ظ‬஋͕෼͔ΔΑ͏ʹվྑͨ͠ɻ·ͨɺ‫ܥ‬ͷψϧΫϥΠϯͱϕΫτϧ৔Λ૬ฏ໘্ʹϓϩοτ͠ղͷ‫ي‬ಓΛΞχϝʔγϣϯԽ͢ΔϓϩάϥϜΛ࡞੒ͨ͠ɻ͜ͷϓϩάϥϜʹΑͬͯฏߧ఺पΓͷϕΫτϧͷྲྀΕ͕Θ͔ͬͨɻ·ͨɺதؒൃද༻ͷεϥΠυΛ࡞੒ͨ͠ɻ

8-9 ݄ લ‫ֶͰظ‬श͖ͯͨ͜͠ͱͷ෮शͱ‫͏ߦʹظޙ‬ภඍ෼ํఔࣜͷղ͖ํʹ͍ͭͯௐ΂ͨɻ
10 ݄ ൓Ԡ֦ࢄ‫ܥ‬ͷ֦ࢄ߲ͷΈͷํఔࣜΛࠩ෼๏Λ༻͍ͯ਺஋‫͠ࢉܭ‬ɺϊΠϚϯ‫ڥ‬ք৚݅ͷ΋ͱͰ࣌ؒൃల͍༷ͯ͘͠ΛάϥϑԽͨ͠ɻ·ͨɺ࣌ؒൃలΛΑΓ෼͔Γ΍͘͢͢ΔͨΊʹࡾ࣍‫ݩ‬Խͨ͠άϥϑ΋ग़ྗͨ͠ɻ͜ͷάϥϑ͔Β࣌ؒൃల͢ΔʹͭΕۭͯؒ಺ͷྔ͸Ұఆͷ··֦ࢄ͍ͯ͘͜͠ͱ͕෼͔ͬͨɻ
ɻ

11 ݄ ൓Ԡ֦ࢄ‫ܥ‬ͷ൓Ԡ߲ͷΈͷํఔࣜͰψϧΫϥΠϯΛϓϩοτ͠ɺ҆ఆ఺ʹ޲͔͏ղͷྲྀΕΛௐ΂ͨɻ͜ͷ‫ܥ‬͸ɺ҆ఆ఺͕ೋͭ͋Δ͜ͱ͔Β૒҆ఆ‫͋Ͱܥ‬Δ͜ͱ͕෼͔ͬͨɻ·ͨɺ൓Ԡ֦ࢄ‫ܥ‬Λࠩ෼๏Λ༻͍ͯ਺஋‫͠ࢉܭ‬ɺϊΠϚϯ‫ڥ‬ք৚݅ͷ΋ͱͰϑϩϯτਐߦ೾ղ͕࣌ؒ
ൃల͍༷ͯ͘͠ΛάϥϑԽͨ͠ɻϑϩϯτਐߦ೾ղ͸ɺa=0.5 ͱ͖͸ɺఆৗঢ়ଶͱͳΓͦͷ৔͔Βಈ͔ͣɺ0.5 ΑΓখ͍͞৔߹͸ӈʹҠಈ͠ɺ0.5 ΑΓେ͖͍৔߹͸ࠨʹಈ͘͜ͱ͕֬ೝ
Ͱ͖ͨɻ·ͨɺFitzHugh-Nagumo ํఔࣜͷ਺஋‫ࢉܭ‬Λ͠ɺψϧΫϥΠϯΛϓϩοτ͢Δ͜
ͱͰɺ҆ఆ఺ʹ޲͔͏ղͷྲྀΕΛௐ΂ͨɻ͜ͷ‫ܥ‬͸ɺҰͭͷ҆ఆ఺ʹղ͕ऩଋ͍ͯ͠Δ͜ͱ͔Β୯҆ఆ‫͋Ͱܥ‬Δ͜ͱ͕෼͔ͬͨɻ

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Mathematical and simulation of complex systems
12 ݄ ࠷ऴൃදͷͨΊͷεϥΠυɺಈըΛ࡞੒ͨ͠ɻ
ʢ˞จ੹ : ౻ా३ʣGroup Report of 2020 SISP

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Mathematical and simulation of complex systems

ୈ5ষ

݁Ռ

5.1

ϓϩδΣΫτͷ݁Ռ

5.1.1

ั৯ऀʔඃ৯ऀ‫ܥ‬લ‫Ͱظ‬͸ɺԼ‫ه‬ͷั৯ऀʔඃ৯ऀ‫ࢄ֦ʹܥ‬ΛՃ͑ͨ‫ܥ‬ͷۭؒඇґଘతͳ‫ܥ‬ͷղੳΛߦͬͨɻ∂u= u(1 − u − v) ≡ f (u, v)∂t∂v= av(u − b) ≡ g(u, v)∂t·ͣɺ૬ฏ໘ղੳΛ͢Δ͜ͱʹΑͬͯɺ͜ͷ‫ʹܥ‬͸ࡾͭͷఆৗঢ়ଶ͕͋Δ͜ͱ͕෼͔ͬͨɻ͜ͷࡾͭͷఆৗঢ়ଶ͸ɺʮั৯ऀɺඃ৯ऀ͕ෆࡏͳঢ়ଶʯɺ
ʮั৯ऀ͕ෆࡏͰɺඃ৯ऀ਺͕‫ڥ؀‬ऩ༰ྗʹ౳͍͠ঢ়ଶʯɺ
ʮั৯ऀͱඃ৯ऀ͕‫ڞ‬ଘ͢Δঢ়ଶʯͰɺ͜ͷࡾͭͷঢ়ଶͷฏߧ఺͸ɺͦΕͧΕ (0,0)ɺ

(1,0)ɺ(b,1-b) Ͱ͋ͬͨɻͦͯ͠ɺࡾͭͷฏߧ఺ͷ҆ఆੑΛղੳ͢ΔͨΊʹઢ‫҆ܗ‬ఆੑղੳΛߦͬͨɻͦͷ݁Ռɺ(0,0)ɺ(1,0) ͷ৔߹ʹෆ҆ఆঢ়ଶɺ(b,1-b) ͷ৔߹ʹ҆ఆঢ়ଶͰ͋Δ͜ͱ͕൑ผͰ͖
ͨɻ·ͨɺ҆ఆঢ়ଶͷ৔߹ 4a ͕ b/1-b ΑΓେ͖͍৔߹͸҆ఆӔঢ়఺ɺখ͍͞৔߹͸҆ఆ݁અ఺ʹ൑ผ͞ΕΔ͜ͱ͕෼͔ͬͨɻ
࣍ʹɺ҆ఆঢ়ଶͷ৔߹ͷղͷ‫ي‬ಓ͕ͲͷΑ͏ʹͳΔ͔Λղੳ͢ΔͨΊʹΦΠϥʔ๏Λ࢖ͬͯ਺஋
‫ࢉܭ‬ΛߦͬͨɻΦΠϥʔ๏Λ࢖ͬͯ਺஋‫ͨ͠ࢉܭ‬ղΛσʔλϑΝΠϧʹอଘ͢ΔϓϩάϥϜΛ C‫࡞Ͱޠݴ‬੒͠ɺgnuplot Λ͔ͭͬͯग़ྗͨ͠ɻ

ਤ 5.1

҆ఆঢ়ଶ࣌ͷղͷ‫ي‬ಓ྘ઢ͸҆ఆӔঢ়఺ɺࢵઢ͸҆ఆ݁અ఺ɺॳ‫ظ‬஋͸੨ͷ఺ɺฏߧ఺͸ࠇͷ఺Ͱද͍ͯ͠Δɻ͜ͷάϥϑ͔Βɺ҆ఆӔঢ়఺ͷ৔߹͸ղ͕ӔΛ‫͘ר‬Α͏ʹऩଋ͓ͯ͠Γɺ҆ఆ݁અ఺ͷ৔߹͸ղ͕௚ઢత
ʹऩଋ͍ͯ͠Δ͜ͱ͕෼͔ͬͨɻ

Group Report of 2020 SISP

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Group Number 6

Mathematical and simulation of complex systems࣍ʹɺͦΕͧΕͷฏߧ఺पΓͷྲྀΕਤΛ‫ݟ‬ΔͨΊʹɺf,g ͷψϧΫϥΠϯͱϕΫτϧ৔Λ u,v ͷ૬ฏ໘্ʹग़ྗ͢ΔϓϩάϥϜΛ python Λ͔ͭͬͯ࡞੒ͨ͠ɻɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹ ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹ

ਤ 5.2

ਤ 5.3

҆ఆӔঢ়఺

҆ఆӔঢ়఺ (֦େਤ)͜Ε͸ a=4ɺb=0.7 ͷ৔߹ͷάϥϑͰ͋Δɻ͜ͷάϥϑ͔Βɺॳ‫ظ‬஋ (0.2,0.4) ͔Βฏߧ఺Ͱ͋Δ(0.7,0.3) ʹӔΛ‫͘ר‬Α͏ʹ௚ઢతʹղ͕ऩଋ͓ͯ͠Γɺฏߧ఺पΓͷϕΫτͷྲྀΕ΋ӔΛ‫͍͍ͯר‬Δ͜ͱ͕Θ͔ͬͨɻ

ਤ 5.4

ਤ 5.5

҆ఆ݁અ఺

҆ఆ݁અ఺ (֦େਤ)͜Ε͸ a=0.2ɺb=0.7 ͷ৔߹ͷάϥϑͰ͋Δɻ͜ͷάϥϑ͔Βɺਤ 5.2ɾ5.3 ͱҧͬͯॳ‫ظ‬஋͔Βฏߧ఺ʹ޲͔ͬͯ௚ઢతʹղ͕ऩଋ͓ͯ͠Γɺฏߧ఺पΓͷϕΫτϧͷྲྀΕ΋௚ઢతͳ΋ͷʹͳͬ

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Mathematical and simulation of complex systems
͍ͯΔ͜ͱ͕Θ͔ͬͨɻ࣍ʹɺ(u,v) ฏ໘ͷୈҰ৅‫͍͓ͯ͜ʹݶ‬ͷ‫ܥ‬͸v
u)]L(u, v) = a[u − b − bln( )] + [v − 1 + b − (1 − b)ln(b1−bͰ༩͑ΒΕΔϦΞϓϊϑؔ਺Λ࣋ͭɻϦΞϓϊϑؔ਺ͱ͸ɺ͋Δฏߧ఺ͷ҆ఆੑΛূ໌Ͱ͖Δؔ
਺Ͱ͋ΔɻL(b,1-b)=0 ͕੒Γཱ͓ͬͯΓɺ(b,1-b) Ҏ֎ͷୈҰ৅‫ݶ‬ͷ఺Ͱ͸ɺL(u, v) > 0 ͔ͭ

dL/dt < 0 ͕੒Γཱ͍ͬͯΔɻϦΞϓϊϑؔ਺͸‫ܥ‬ͷΤωϧΪʔΛද͢΋ͷͰ dL/dt ͕ 0 ΑΓখ͍͞ͱ͍͏͜ͱ͸࣌ؒ‫ܦ‬աͱͱ΋ʹΤωϧΪʔ͕୯ௐʹ‫ݮ‬গ͍ͯ͘͜͠ͱΛද͍ͯ͠Δɻฏߧ఺(b,1-b) ͔Β཭ΕΔ΄ͲΤωϧΪʔ͕େ͖͘ͳ͍͖ͬͯɺฏߧ఺ (b,1-b) Ͱ͸ΤωϧΪʔ͕ 0 ʹͳΔɻ͜ͷ͜ͱ͔Βฏߧ఺ (b,1-b) ͕҆ఆͰ͋Δͱ͍͏͜ͱ͕Θ͔ͬͨɻʢ˞จ੹ : ౻ా३ʣ

5.1.2

֦ࢄํఔࣜ

͜Ε·Ͱ͸൓Ԡ֦ࢄ‫ܥ‬ͷ֦ࢄ߲Λ 0 ͱఆٛͯ͠ɺۭؒతͳ֦ࢄ͕ൃੜ͠ͳ͍‫ܥ‬ͷղੳΛਐΊ͖ͯͨɻࠓ౓͸ҎԼʹࣔͨ͠Α͏ͳۭؒతͳ֦ࢄ͕ൃੜ͠ɺ෺࣭ͷ൓Ԡ͕ͳ͍‫͋Ͱܥ‬Δ֦ࢄํఔࣜΛղ
ੳͨ͠ɻ

∂2u
∂u
=d 2
∂t
∂x

͜ͷ‫͍ͯͭʹܥ‬ɺ਺஋‫ࢉܭ‬Λ͢ΔͨΊʹ·ͣςΠϥʔల։Λར༻ͯ͠཭ࢄԽͨ͠ɻ

un+1= uni +
idΔt n(u
− 2uni + uni−1 )Δx2 i+1ΞϧΰϦζϜͷཧղ͕༰қͳͨΊ਺஋‫ࢉܭ‬ϓϩάϥϜ͸ࠩ෼๏Λར༻ͯ͠࡞੒ͨ͠ɻ͜ͷ‫͍ͭʹܥ‬ͯ Python Λ࢖ͬͯ x,u,t ͷࡾ࣍‫ݩ‬άϥϑͱ x,u ͷೋ࣍‫ݩ‬άϥϑ͕࣌ؒൃలͰղͷৼΔ෣͍Λม͍͑ͯ͘Α͏ͳը૾Λग़ྗͨ͠ɻ‫ڥ‬ք৚݅͸ϊΠϚϯ‫ڥ‬ք৚݅ͱ͠ɺd=1,Δx = 1, Δt = 0.001 ͱύϥϝʔλΛઃఆ͠ɺॳ‫ظ‬৚݅Λ t=0 Ͱ u = eਤ 5.6Group Report of 2020 SISP−(x−500)2
200

Λ༩͑ͨɻ

֦ࢄํఔࣜͷ x,u ೋ࣍‫ݩ‬άϥϑ- 14 -

Group Number 6

Mathematical and simulation of complex systems

ਤ 5.7֦ࢄํఔࣜͷ x,u,t ࡾ࣍‫ݩ‬άϥϑ

ਤ 5.6,5.7 ͔Βॳ‫ظ‬৚݅Ͱ༩͑ΒΕͨ‫ܗ‬ͷࢁ͕࣌ؒൃలͱͱ΋ʹۭؒશମʹ޿͕͍͖ͬͯɺࢁͷߴ͕͞ͲΜͲΜ௿͘ͳ͍ͬͯ͘ͷ͕Θ͔Δɻ
ʢ˞จ੹ : ླ໦ᠳేʣ

5.1.3

૒҆ఆ൓Ԡ֦ࢄํఔࣜ

࣍ʹɺԼ‫ه‬ͷ૒҆ఆ൓Ԡ֦ࢄ‫ܥ‬ͷղੳΛߦͬͨɻ
Н

∂u
∂2u
= ε2 2 + f (u)∂t∂x

f (u) = u(1 − u)(u − a)·ͣɺ֦ࢄ߲Λ 0 ͱ͓͖ f(u) ͷΈͷৗඍ෼ํఔࣜͷ‫Ͱܗ‬ψϧΫϥΠϯΛग़ྗ͠ɺa=0.5 ͷͱ͖ͷ 4ͭͷൣғ͔ΒͷղͷྲྀΕΛϓϩοτͨ͠ɻ

ਤ 5.8

a=0.5 ͷ࣌ͷ 4 ͭͷൣғ͔ΒͷྲྀΕਤ

ਤ 5.8 ͷࠇ఺͕҆ఆ఺ɺന఺͕ෆ҆ఆ఺Ͱ͋Δɻu < 0,0 < u < 0.5 ͷ࣌͸ u=0 ʹऩଋ͠ɺ

0.5 < u < 1,1 < u ͷ࣌͸ u=1 ʹऩଋ͍ͯ͠Δ͜ͱ͕෼͔Δɻೋͭͷ҆ఆ఺ʹऩଋ͍ͯ͠Δ͜ͱ
Group Report of 2020 SISP

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Group Number 6

Mathematical and simulation of complex systems͔Βɺ͜ͷ‫͕ܥ‬૒҆ఆ‫͋Ͱܥ‬Δ͜ͱ͕֬ೝͰ͖ͨɻ࣍ʹɺ֦ࢄ߲΋Ճ͑ͨภඍ෼ํఔࣜͷ‫ܥͰܗ‬Λղੳͨ͠ɻઌ΄ͲͷࣜΛࠩ෼๏Λ࢖ͬͯ਺஋‫͢ࢉܭ‬Δͱɺ

un+1= uni +
i

1
Н

dtε2

1
1
[un − 2uni + uni−1 ] + (uni (1 − uni )(uni − a))(dx)2 i+1tͱͳΔɻͦͯ͠ɺ0 ≤ x ≤ 5, Н = 0.01, ε = 0.01, dt = 0.00001, dx = 0.005 ͱ͓͖ɺa ͷ஋͕

0.1,0.5,0.9 ͷࡾͭͷঢ়ଶͰ࣌ؒൃల͍ͯ͘͠ϑϩϯτ೾ղΛάϥϑԽͨ͠ɻਤ 5.9

a=0.5 ͷͱ͖ͷϑϩϯτ೾ղਤ 5.10a=0.5 ͷͱ͖ͷϑϩϯτ೾ղ (࣌ؒൃల‫)ޙ‬

ਤ 5.9,5.10 ͸ a=0.5 ͷͱ͖ͷϑϩϯτ೾ղͷ࣌ؒൃలΛάϥϑԽͨ͠΋ͷͰ͋Δɻa=0.5 ͷͱ͖͸ɺϑϩϯτ೾ղ͸ͦͷ৔͔Βಈ͔ͣք໘ཱ͚͍ͩͬͯ͘͜ͱ͔Βɺu-0 ͱ u=1 ͷ҆ఆੑ͕౳͠
͍ఆৗঢ়ଶͰ͋Δ͜ͱ͕֬ೝͰ͖ͨɻ

ਤ 5.11

a=0.1 ͷͱ͖ͷϑϩϯτ೾ղਤ 5.12a=0.1 ͷͱ͖ͷϑϩϯτ೾ղ (࣌ؒൃల‫)ޙ‬

ਤ 5.11,5.12 ͸ a=0.1 ͷͱ͖ͷϑϩϯτ೾ղͷ࣌ؒൃలΛάϥϑԽͨ͠΋ͷͰ͋Δɻa=0.1 ͷͱGroup Report of 2020 SISP

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Group Number 6

Mathematical and simulation of complex systems͖͸ɺϑϩϯτ೾ղ͸ӈʹਐߦ͠ɺu=1 ͷঢ়ଶ͕શମΛ෴͍͍ͭͯ͘͠Δ͜ͱ͔Βɺu=1 ͷ΄͏͕ΑΓ҆ఆঢ়ଶͰ͋Δ͜ͱ͕֬ೝͰ͖ͨɻ

ਤ 5.13a=0.9 ͷͱ͖ͷϑϩϯτ೾ղਤ 5.14a=0.9 ͷͱ͖ͷϑϩϯτ೾ղ (࣌ؒൃల‫)ޙ‬

ਤ 5.13,5.14 ͸ a=0.9 ͷͱ͖ͷϑϩϯτ೾ղͷ࣌ؒൃలΛάϥϑԽͨ͠΋ͷͰ͋Δɻa=0.9 ͷͱ͖͸ɺϑϩϯτ೾ղ͸ࠨʹਐߦ͠ɺu=0 ͷঢ়ଶ͕શମΛ෴͍͍ͭͯ͘͠Δ͜ͱ͔Βɺu=0 ͷ΄͏͕ΑΓ҆ఆঢ়ଶͰ͋Δ͜ͱ͕֬ೝͰ͖ͨɻ
ʢ˞จ੹ : ౻ా३ʣ5.1.4

Ұ࣍‫ ݩ‬Fitzhugh-Nagumo ํఔࣜ࣍ʹҎԼͰࣔͨ͠Α͏ͳҰ࣍‫ݩ‬ͷ Fitzhugh-Nagumo ํఔࣜΛ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒ͯ͠ɺ
ղͷৼΔ෣͍ΛಈըԽͨ͠ɻ

∂u
∂2u 1
= d 2 + f (u, v)
∂t
∂x

∂v
= g(u, v)∂tf (u, v) = u(u − α)(β − u) − v
g(u, v) = u − γv + σ͜ ͷ ‫ ܥ‬͸ 2 ม ਺ ؔ ਺ Ͱ ༩ ͑ Β Ε ɺd ͸ ֦ ࢄ ఆ ਺ ɺα, β, γ, σ,  ͸ ͦ Ε ͧ Ε ఆ ਺ Ͱ ͋ Δ ɻ· ͣ

f(u,v)=0,g(u,v)=0 Λղ͖ɺu,v ૬ฏ໘Ͱฏߧ఺෇ۙͷྲྀΕΛάϥϑԽͨ͠ɻGroup Report of 2020 SISP

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ਤ 5.15ฏߧ఺पΓͷྲྀΕਤ

ਤ 5.15 ͷྲྀΕਤ͔Β͜ͷ‫ܥ‬͸Ұͭͷ҆ఆ఺ʹ޲͔ͬͯྲྀΕ͍ͯΔ͜ͱ͕Θ͔Δɻ͜Ε͸͜ͷ‫ܥ‬͕୯҆ఆ‫͋Ͱܥ‬Δ͜ͱ͕‫͑ݴ‬Δɻ࣍ʹɺ͜ͷ‫ܥ‬Λࠩ෼๏Λ࢖ͬͯ਺஋‫͢ࢉܭ‬ΔͨΊʹҎԼͷΑ͏ʹࣜΛม‫ͨ͠ܗ‬ɻ

un+1= uni + d(uni+1 − 2uni + uni−1 ) +i

Δt n n(u (u − α)(β − uni ) − vin ) i i

vin+1 = vin + Δtuni − Δtγvin + Δtσ͜ͷࣜΛ࢖ͬͯ਺஋‫ࢉܭ‬Λߦ͍ಘΒΕͨղΛಈըԽͨ͠ɻ‫ڥ‬ք৚݅͸प‫ظ‬త‫ڥ‬ք৚݅ʹઃఆ͠ɺύ√ϥϝʔλΛ α = − 3, β =√3,  = 0.02, γ = 0.5, σ = 0.7, d = 1, Δt = 0.0025, Δx = 0.1875 ͱͨ͠ɻॳ‫ظ‬஋͸ۭؒҰ༷ʹ u=-1.41120000000000,v=-1.4224000238418 Λ༩͑ɺt=0 Ͱܹࢗͱͯ͠

x=0 ෇ۙʹ u = e−(x−1)2
0.5

Λ༩͑ͨɻ·ͨ x ͷൣғΛ 0 ͔Β 150 ʹ͠ɺt ͕ 0 ͔Β 45 ʹͳΔ·Ͱ‫ࢉܭ‬Λଓ͚ͨɻ

ਤ 5.16ܹࢗΛ༩͑ͨ௚‫ޙ‬ͷύϧεղਤ 5.17͕࣌ؒ‫ܦ‬աͨ͠ͱ͖ͷύϧεղάϥϑͷࠨ୺ͰܹࢗΛड͚ͨ‫ܥ‬͸ࠨ୺ͰύϧεղΛ‫ܗ‬੒͠ɺͦͷ‫ܦؒ࣌ޙ‬աͰঃʑʹӈ୺ʹਐߦ͍ͯ͘͠ͷ͕ਤ 5.16,5.17 ͔ΒΘ͔Δɻύϧεղ͕ӈ୺ʹͨͲΓண͍ͨΒप‫ظ‬త‫ڥ‬ք৚݅ʹΑͬͯ

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Mathematical and simulation of complex systems‫ܗ‬ঢ়Λอͬͨ··ࠨ୺͔Βग़ͯ͘Δ͜ͱ͕Θ͔ͬͨɻͦΕҎ߱΋࣌ؒൃలΛଓ͚ͯ΋ӈ୺ʹ޲͔ͬͯղ͕ਐߦ͍͚ͯͩ͘͠Ͱ͋ͬͨɻઌ΄ͲͷҰ࣍‫ ݩ‬Fitzhugh-Nagumo ํఔࣜ͸ࠓճͷύϥϝʔλΛ༩͑Δͱਐߦύϧεղ͕‫ܗ‬੒͞ΕΔ͜ͱ͕Θ͔ͬͨɻʢ˞จ੹ : ླ໦ᠳేʣ

5.1.5

ೋ࣍‫ ݩ‬Fitzhugh-Nagumo ํఔࣜ

Fitzhugh-Nagumo ํఔࣜΛೋ࣍‫֦ʹܥݩ‬ு͠ɺઢ্ͷܹࢗͱͯ͠Ұ࣍‫ݩ‬ͷ Fitzhugh-NagumoํఔࣜΛ਺஋‫ʹ͖ͱͨ͠ࢉܭ‬ಘΒΕͨύϧεղΛ y ࣠ํ޲ʹҰ༷ʹฒ΂ɺͦͷ‫ൃؒ࣌ޙ‬లΛͤͨ͞ɻͨͩ͠ɺύϧεղΛฒ΂Δൣғ͸ y ࣠ํ޲ͷ 0 ͔Β൒෼·Ͱͱ͠ɺ൒෼͔Β͸ॳ‫ظ‬஋ͱͯ͠

u=-1.41120000000000,v=-1.4224000238418 Λ༩͑ͨɻͦͯ͠ಘΒΕͨ݁ՌΛϓϩοτͨ͠ɻ

ਤ 5.18εύΠϥϧΛ‫ܗ‬੒͢Δղ͜ͷ‫ܥ‬͸ઢ্ͷܹࢗΛ༩͑Δͱ࣌ؒൃలͱͱ΋ʹ୺఺͕ࣦ଎͍͖ͯ͠ɺ͔ͦ͜Β‫ࠐ͖ר‬ΜͰ͍ͨ͘ΊঃʑʹεύΠϥϧύλʔϯΛ‫ܗ‬੒͍ͯ͘͜͠ͱ͕Θ͔ͬͨɻ࣍ʹ͜ͷ‫ܥ‬ͷத৺ʹ఺ঢ়ͷܹࢗΛ༩͑ɺ࣌ؒൃలͤͨ݁͞ՌΛϓϩοτͨ͠ɻ

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Mathematical and simulation of complex systems

ਤ 5.19ϦϯάΛ‫ܗ‬੒͢Δղઌ΄Ͳͷ‫ܥ‬ͷத৺ʹ఺ঢ়ͷܹࢗΛ༩͑࣌ؒൃలΛͤ͞Δͱঃʑʹ޿͕͍ͬͯ͘Ϧϯάঢ়ͷύλʔϯ͕ಘΒΕΔ͜ͱ͕Θ͔ͬͨɻʢ˞จ੹ : ླ໦ᠳేʣ

5.2

੒ՌͷධՁ

5.2.1

લ‫ظ‬ͷධՁ

૬ฏ໘ղੳʹΑͬͯఆৗ఺Λ‫ٻ‬Ίɺฏߧ఺ʹ͍ͭͯ͸ઢ‫҆ܗ‬ఆੑղੳͱϦΞϓϊϑؔ਺Λ༻͍ͯ҆ఆੑʹ͍ͭͯߟ࡯͠ɺั৯ऀʔඃ৯ऀ‫ܥ‬ͷղͱฏߧ఺पΓͷྲྀΕਤΛೋͭͷύλʔϯͰग़ྗͨ͠ɻ͜ΕʹΑΓ൓Ԡ֦ࢄ‫͍ͯͭʹܥ‬ཧղ͕ਂ·Γɺύλʔϯ‫ܗ‬੒΁ͷΞϓϩʔν͕Ͱ͖ͨɻ֦ࢄ߲
Λআ͍ͨঢ়ଶͰ͸൓Ԡ֦ࢄ‫ܥ‬Λղੳͨ͠ͱ͸‫͍ͳ͑ݴ‬ɻ࢒͞Εͨ՝୊ͱͯ͠͸ɺ֦ࢄ߲Λ࢒ͨ͠ঢ়ଶͰั৯ऀʔඃ৯ऀ‫ܥ‬Λղੳ͢Δ͜ͱͰ͋ΔɻͦΕʹΑͬͯਅʹ൓Ԡ֦ࢄ‫ܥ‬Λղੳͨ͠ͱ͍͏͜ͱ
͕Ͱ͖ɺੜ෺ֶͰ‫ݟ‬ΒΕΔύλʔϯ‫ܗ‬੒Λ֬ೝ͢Δ͜ͱʹ΋ͭͳ͕Δɻ
ʢ˞จ੹ : ླ໦ᠳేʣ

5.2.2‫ظޙ‬ͷධՁ

‫Ͱظޙ‬͸ɺ࠷ॳʹ֦ࢄ߲ͷΈͷ‫Ͱܥ‬ͷ਺஋‫ࢉܭ‬Λߦ͍ɺάϥϑԽ͢Δ͜ͱͰղ͕ͲͷΑ͏ʹ֦ࢄ͞Ε͍͔ͯ͘ཧղ͢Δ͜ͱ͕Ͱ͖ͨɻ࣍ʹɺ૒҆ఆ൓Ԡ֦ࢄ‫ܥ‬ͷղੳΛߦ͍ɺϑϩϯτਐߦ೾ղΛάϥϑԽ͢Δ͜ͱͰɺ൓Ԡ֦ࢄ‫ܥ‬ͷղ͖ํ΍ղͷಈ͖ʹ͍ͭͯཧղΛਂΊΔ͜ͱ͕Ͱ͖ͨɻ
࣍ʹɺFitzHugh-Nagumo ํఔࣜΛ਺஋‫͠ࢉܭ‬ɺ࣮ࡍʹύλʔϯ‫ܗ‬੒Λߦͬͨɻ‫ܗ‬੒͞Εͨύλʔϯ͸ɺԼਤͷΑ͏ͳ (a)BZ ൓Ԡʹ͓͍ͯੜ͡Δλʔήοτύλʔϯɺ(b) ࢼༀΛ؇΍͔ʹ᎟፩ͤͯ͞։࢝ͤͨ͞ཐટ೾ɺ(c) ΩΠϩλϚϗίϦΧϏ͕ൃੜͤ͞ΔεύΠϥϧύλʔϯʹࣅͨύλʔϯͱͳͬͨɻ

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ਤ 5.20ग़యɿࡾଜণହ૯‫؂‬म. ϚϨʔ਺ཧੜ෺ֶ Ԡ༻ฤ. ‫ؙ‬ળग़൛, 2016.

͜ͷ݁Ռ͔Βɺ͜ΕΒͷ (a),(b),(c) ͷΑ͏ͳ‫ݱ‬৅͸ɺFitzHugh-Nagumo ํఔࣜͰ‫ه‬ड़Ͱ͖Δ͜
ͱ͕ਪଌͰ͖Δɻ࢒͞Εͨ໰୊ͱͯ͠͸ɺλʔήοτύλʔϯ΍ࣶϗοέͳͲͷੜ෺ͷ໛༷Λ‫ܗ‬੒͢ΔύλʔϯΛ‫ܗ‬੒͢Δ͜ͱ͕Ͱ͖ͳ͔ͬͨ͜ͱͰ͋Δɻ͜Ε͸ FitzHugh-Nagumo ํఔࣜͱ͸ҧ͏਺ཧϞσϧͰද‫͢ݱ‬Δ͜ͱ͕ՄೳͰ͋ΔͱਪଌͰ͖Δɻҧ͏਺ཧϞσϧͷಋग़΍ղੳΛߦ͑ͳ͔ͬͨ͜ͱ͕໰
୊Ͱ͸͋Δ͕ɺϓϩδΣΫτ‫׆‬ಈΛ௨ͯ͠਺஋‫ࢉܭ‬ೳྗ΍ϓϩάϥϛϯάೳྗͷ޲্ʹ‫͕ͨͬܨ‬ͷͰྑ͔ͬͨɻ
ʢ˞จ੹ : ౻ా३ʣ5.3

୲౰෼୲՝୊ͷධՁ

5.3.1

Ֆࢁཽ໵

୲౰ϖʔδͷྠಡͷ४උ ઐ໳తͳ஌ࣝ΍‫͍ͯͭʹ۟ޠ‬͸Πϯλʔωοτ΍ࢀߟࢿྉΛ࢖͍ཧղ͸ਂΊ͕ͨɺଞͷϝϯόʔʹཧղͯ͠΋Β͏΄Ͳͷཧղʹ͸ࢸ͓ͬͯΒͣɺଞͷϝϯόʔͷൃද౳Ͱࣗ෼ͷཧղʹؒҧ͍͕ଟʑ͋Δ͜ͱʹ‫͍ͨ෇͕ؾ‬ͷͰɺ஌ࣝͷ‫ڞ‬༗Λ΋ͬͱੵ‫ۃ‬తʹߦͬͨΓɺ୲౰‫ڭ‬һʹ࣭໰Λੵ‫ۃ‬తʹ͢ΔͳͲɺཧղ͢Δ఺Ͱվળͷ༨஍͸େ͖͋ͬͨ͘ͱ
ߟ͑Δɻ‫ڝ‬૪‫͚͓ʹܥ‬Δ֦ࢄ͕͋Δ৔߹ͷϞσϧղੳ֦ࢄ͕ͳ͍৔߹ͷ‫ڝ‬૪Ϟσϧ‫ܥ‬ͷղੳΛߦ͔ͬͯΒɺ֦ࢄͷͳ͍৔߹ͷ‫ڝ‬૪ϞσϧͷղੳΛߦͬͨͨΊൺֱతεϜʔζʹਐΊΔ͜ͱ͸Ͱ͖͕ͨɺϩτΧʔϰΥϧςϥͷ‫ڝ‬૪‫ʹܥ‬ର͢Δ஌͕ࣝബ͗ͨͨ͢Ίͦͷ஌ࣝΛཧղ͢Δͷʹଟ͘ͷ࣌ؒΛඅ΍ͯ͠͠·ͬͨɻޮ཰తʹ਺஋‫ࢉܭ‬΍਺஋ղੳΛߦ͏͏͑Ͱɺఏࣔ͞ΕͨࢀߟจGroup Report of 2020 SISP

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Mathematical and simulation of complex systems‫ݙ‬Ҏ֎ʹ΋ຊΛ୳ͨ͠Γɺ୲౰‫ڭ‬һʹ࣭໰͢ΔͳͲཧղ͠ղੳ͢Δͱ͍͏఺Ͱվળͷ༨஍͸͋Δͱߟ͑Δɻ‫ڝ‬૪‫͚͓ʹܥ‬Δ೾ͷ଎౓ͷղੳϑΟογϟʔʔίϧϞΰϩϑํఔ͔ࣜΒ೾ͷ଎౓͕ํఔࣜͷ೾ͷ࠷খ଎౓ʹ౳͍͔͠ɺ࠷খ଎౓ΑΓ΋େ͖͘ͳΔͱਪ࿦͸Ͱ͖ͨɺ͕ͦͷਪ࿦ʹࢸΔ·Ͱʹ࣌ؒΛ͔͚͗ͨͨ͢Ίɺͦͷ‫ޙ‬ͷߟ࡯෦෼ʹ͋·Γ࣌ؒΛ͔͚Εͳ͔ͬͨͨΊɺਖ਼͘͠ཧղ
Ͱ͖͍ͯͳ͍෦෼΋͋Γɺ෮श࣌ؒ΋΄΅औΕͳ͔ͬͨɻ࣌ؒΛ͋·Γ͔͚ͣޮ཰తʹ೾ͷղੳΛߦ͍ɺϝϯόʔؒͰͷ‫ڞ‬༗Λߦ͍ͬͯΕ͹࣌ؒ୹ॖ΋Ͱ͖ͨͰ͋Ζ͏ɻͦͷ఺ʹ͓͍ͯվળͷ༨஍͸͋Δͱߟ͑Δɻ
֦ࢄํఔࣜʹ͍ͭͯͷ਺஋ղੳ֦ࢄํఔࣜͷ਺஋‫ࢉܭ‬Λߦ͏͏͑ͰϑʔϦΤม‫׵‬΍ࠩ෼๏ʹֶ͍ͭͯश͕ͨ͠ཧղʹ͕͔͔࣌ؒͬͯ͠·͍ɺ͋·Γޮ཰తʹ਺஋ղੳΛߦ͑ͳ͔ͬͨɻ༧श
࣌ؒ΋͋·ΓऔΕͳ͔͕ͬͨɺগͳ͘ͱ΋ϑʔϦΤม‫͔ࠩ׵‬෼๏ͷͲͪΒ͔ʹߜͬͯूதత
ʹऔΓ૊Ή΂͖Ͱ͋ͬͨɻ·ͨͦΕͰ΋ཧղ͕Ͱ͖ͳ͔ͬͨ৔߹͸ϝϯόʔؒͰ‫ͳ͏͋͑ڭ‬Ͳվળͷ༨஍͕͋Δͱߟ͑Δɻ

Fitzhugh-Nagumo ํఔࣜͷ਺஋ղੳ Fitzhugh-Nagumo ํఔࣜͷ਺஋‫ࢉܭ‬Λߦ͏͏͑Ͱɺภ
ඍ෼ํఔࣜͷ‫ʹࢉܭ‬ϓϩδΣΫτͰͷ࣌ؒΛଟׂ͍ͯ͘͠·ͬͨɻ͋Β͔͡Ί༧शΛ͍ͯͨ͠෦෼ʹ΋͔͔ΘΒͣɺ༧श͔Β͕࣌ؒͨͬͯ͠·ͬͨ͜ͱ΋͋Γɺ஌͕͍ࣝ͋·͍ʹͳΓ͍͗ͯͨ͢ɻ༧श͍͔ͯͨ͠Βʹ͸‫ܧ‬ଓͯ͠༧शΛଓ͚͍ͯ͘ͳͲɺվળͷ༨஍͕͋Δͱߟ͑Δɻ·ͨɺਐߦύϧεղΛ֬ೝ͔ͨͬͨ͠ͷ͕ͩ‫ݸ‬ਓͷྗͰ͸දࣔͤ֬͞ೝ͢Δ͜ͱ͕Ͱ͖ͳ͔ͬͨͨΊɺϓϩάϥϜͷ໘Ͱ΋஌ࣝෆ଍΍ٕज़ෆ଍͕໨ཱͬͨͷͰɺͦ͜ʹ΋վળͷ༨஍͸͋Δͱߟ͑Δɻ
࠷ऴൃදͷ४උ ࠷ऴൃද࣌ͷ؆қతͳൃදΛ୲౰ͨ͠ɻ౰೔ͷൃදͰ͸ɺ‫ߘݪ‬ͷಡΈࠐΈෆ଍ͰεϜʔζͳઆ໌͕Ͱ͖ͣɺ‫ݶ‬ΒΕͨ࣌ؒͷதͰ͋ͬͨʹ΋ؔΘΒͣ͋ͨ;ͨͯ͠͠·͏৔໘‫ݟ‬ΒΕͨɻ·࣭ͨٙԠ౴ͷࡍͰ΋ੵ‫ۃ‬తʹ࣭໰ʹճ౴Ͱ͖ͳ͔ͬͨͨΊɺམͪண͍ͯߟ͑ͨ
Γɺࣗ෼͔Βൃ‫͢ݴ‬ΔͳͲɺվળͷ༨஍͕͋Δͱߟ͑Δɻ
ʢ˞จ੹ : Ֆࢁཽ໵ʣ

5.3.2

ࡾӜྰԻ

୲౰ϖʔδͷྠಡͷ४උ ࣗ෼ͰΘ͔Βͳ͍୯‫ޠ‬΍‫۟ޠ‬Λௐ΂ͭ͜ͱͰཧղΛਂΊͨɻͨͩɺ͍͟
ൃද͢Δͱ͖ʹɺؒҧͬͨղऍΛ͍ͯͨ͠ͷͰɺཧղͷਂΊํΛվળ͢Δ༨஍͕͋Δͱߟ͑Δɻ
ϑϩϯτਐߦ೾ղͷղੳ ৗඍ෼ํఔ͔ࣜΒɺϑΟογϟʔʔίϧϞΰϩϑํఔࣜΛར༻͠ɺ‫ݻ‬༗ํఔࣜΛ‫ٻ‬ΊΔͱ͜Ζ·Ͱཧղ͢Δ͜ͱ͕Ͱ͖͕ͨɺϑϩϯτਐߦ೾͕ɺ৔߹ʹΑͬͯͲͷΑ͏ͳ‫ͲͰܗ‬ͷํ޲ʹಈ͘ͷ͔Λ‫׬‬શʹཧղ͢Δ͜ͱ͕Ͱ͖ͳ͔ͬͨͷͰɺͦͷ఺͸ͳ͓Ұ૚ཧղΛਂΊΔඞཁ͕͋Δͱߟ͑Δɻ
தؒൃදͷ४උ தؒൃද༻ͷϙελʔ࡞੒ʹऔΓ૊Μͩɻ୭͕‫ͯݟ‬΋‫ݟ‬΍͍͢ϨΠΞ΢τʹ͢Δ
Α͏౒Ίͨ΋ͷͷɺจࣈͷେ͖͞ͳͲͰɺ‫ͮݟ‬Β͍ͱͷࢦఠΛ΋ΒͬͨͷͰվળͷ༨஍͕͋
Δͱߟ͑Δɻ
֦ࢄํఔࣜʹ͍ͭͯͷ਺஋ղੳ ֦ࢄํఔࣜͷ਺஋‫ࢉܭ‬Λ͢Δʹ͋ͨΓɺϑʔϦΤม‫͔׵‬Β֦ࢄํఔࣜΛղ͘ษ‫ڧ‬Λͨ͠ɻ࠷ऴతͳղΛग़͠ɺ݁Ռ͸ग़ͤͨ΋ͷͷɺͦΕΛ΄ͱΜͲར༻ͤͣ
ऴΘͬͨ఺ʹؔͯ͠͸վળͷ༨஍͕͋Δͱߟ͑Δɻ

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Fitzhugh-Nagumo ํఔࣜͷ਺஋ղੳ Fitzhugh-Nagumo ํఔࣜΛ਺஋‫͢ࢉܭ‬Δ͏͑Ͱɺղ͖
ํΛཧղ͢Δͷʹ͕͔͔࣌ؒͬͯ͠·ͬͨ఺ʹؔͯ͠͸վળͷ༨஍͕͋Δͱߟ͑Δɻͨͩɺ࣌ؒΛඅ΍ͨ͠෼ɺύλʔϯ‫ܗ‬੒͢ΔϞσϧ͕୯҆ఆ‫ࣔ͞Ͱܥ‬ΕΔ͜ͱ͕ཧղͰ͖ͨɻ
࠷ऴൃදͷ४උ

࠷ऴൃද༻ͷϙελʔΛ୲౰ͨ͠ɻதؒൃදͷ࣌ͱ৭߹͍͸มߋͤͣʹɺதؒൃද‫ࢦʹޙ‬ఠ͞Ε͍ͯͨจࣈͷେ͖͞΍‫ݟ‬΍͍͢σβΠϯΛҙࣝͯ͠࡞੒ͨ͠ɻͨͩ͠ɺࡉ͔
͍࡞‫ۀ‬Λଵͬͯ͠·ͬͨ෦෼ʹؔͯ͠͸վળͷ༨஍͕͋Δͱߟ͑Δɻ
ʢ˞จ੹ : ࡾӜྰԻʣ

5.3.3

ླ໦ᠳే

୲౰ϖʔδͷྠಡͷ४උ ࣗ෼ͷΘ͔Βͳ͍‫Ͳͳ۟ޠ‬͸ Web Ͱ‫ͯ͠ࡧݕ‬ཧղΛਂΊɺྠಡͷ४උ͸Ͱ͖͕ͨ·ͩ‫׬‬શʹཧղͰ͖͍ͯͳ͔ͬͨՕॴ͕͋Γɺൃදͷࡍʹ͏·͘આ໌Ͱ͖ͳ͜ͱ
͕͋ͬͨɻͦͷ఺ʹؔͯ͠͸վળͷ༨஍͕͋Δͱߟ͑Δɻ
ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղΛදࣔ͢ΔϓϩάϥϜͷ࡞੒ ࣮ࡍʹϓϩάϥϜΛ࡞੒͠ɺղͷৼΔ෣͍Λάϥϑʹදࣔ͢Δ͜ͱ͕Ͱ͖͕ͨ·ͩ Python ͷϥΠϒϥϦΛଟ༻͢Δ͜ͱ͕ଟ͘ɺϓϩάϥϜͷߏ଄͕ཧղ͖͠Ε͍ͯͳ͍ͷͰɺͦͷ఺͸վળͷ༨஍͕͋Δͱߟ͑Δɻ
ฏߧ఺पΓͷಈ͖ͷಋग़ɺதؒൃදͷ४උั৯ऀʔඃ৯ऀ‫ܥ‬ͷղΛ‫ٻ‬Ίͨ͋ͱฏߧ఺पΓͷಈ͖ΛψϧΫϥΠϯͱϕΫτϧ৔ͷάϥϑͰखॻ͖Ͱදͨ͠ɻ͔͠͠ɺ࣮ࡍͷಈ͖ͱ͸ҧ͏͜ͱ͕͋ͬͨͷͰཧղ͕଍Γ͍ͯͳ͔ͬͨɻதؒൃද༻ͷࢿྉͷ࡞੒ʹ͍ͭͯ͸࣮ࡍͷൃදࢿྉͷ‫ߘݪ‬Λ࡞੒ͨ͠ɻ࡞੒ͨ͠‫ߘݪ‬ͷ෼ྔ͕ଟ͘ͳΓɺ‫ن‬ఆ࣌ؒΛগ͠௒͍͑ͯͨͨΊͦͷ఺ʹ͍ͭͯ͸վળͷ༨஍͕͋Δͱߟ͑Δɻ
֦ࢄํఔࣜͷ਺஋‫ํࢄ֦ ࢉܭ‬ఔࣜΛ਺஋‫͢ࢉܭ‬ΔͨΊʹࠩ෼๏ʹ͍ͭͯษ‫͠ڧ‬ɺ࣮ࡍʹ਺஋‫ࢉܭ‬Λߦ͍݁ՌΛग़ྗͨ͠ɻ࣮ࡍʹղ͕ม‫ࢠ༷͍ͯ͘͠ܗ‬ΛΞχϝʔγϣϯʹͰ͖ͨ͜ͱ͸Α͔͕ͬͨɺࠩ෼๏ͷΞϧΰϦζϜͱΞχϝʔγϣϯʹ͢Δٕज़Λཧղ͢Δͷʹ࣌ؒΛඅ΍ͯ͠͠·ͬͨ͜ͱʹ͍ͭͯ͸΋͏গ͠ษ‫ڧ‬ͷεϐʔυΛ্͛Δ΂͖ͩͱ‫ͨ͡ײ‬ɻ

Fitzhugh-Nagumo ํఔࣜͷ਺஋‫ࠓ ࢉܭ‬ճ൓Ԡ֦ࢄ‫ܥ‬ͷϞσϧͱͯ͠ೋม਺ͷภඍ෼ํఔࣜͰ
ද͞Ε͍ͯΔͱ͍͏ཧ༝Ͱ Fitzhugh-Nagumo ํఔࣜΛ਺஋‫͢ࢉܭ‬Δ͜ͱʹ͕ͨ͠ೋม਺ͷภඍ෼ํఔࣜΛ਺஋‫͢ࢉܭ‬Δͷʹ΋͕͔͔࣌ؒͬͯ͠·ͬͨͷͰͦ͜ʹ͍ͭͯ͸վળͷ༨஍
͕͋Δͱ‫͑ݴ‬Δɻग़ྗͨ݁͠Ռʹ͍ͭͯ͸౰ॳͷ໨తͰ͋Δੜ෺ֶతͳύλʔϯΛՄࢹԽ͢Δ͜ͱʹΞϓϩʔν͢Δ͜ͱ͕Ͱ͖ͨͱߟ͑Δɻ͔͠͠ύϧεղ͕ਐߦ͢ΔಈըͳͲ͸ϓϩάϥϜ͕ෆ‫׬‬શͳͨΊʹग़ྗʹ͕͔͔࣌ؒͬͨΓɺ‫ࢉܭ‬ΞϧΰϦζϜ͕ޮ཰తͳ΋ͷͰ͸ͳ͍ͷͰɺ਺஋‫ࢉܭ‬ͷΞϧΰϦζϜ΍ޮ཰తͳσʔλߏ଄ʹ͍ͭͯཧղ͢Δඞཁ͕͋Δͱߟ͑ͨɻ࠷ऴൃදͷ४උ

࠷ऴൃදͷΪϦΪϦ·Ͱ਺஋‫ࢉܭ‬ϓϩάϥϜ΍ಈըΛ࡞੒͍ͯͨ͠ͷ͕‫ݪ‬ҼͰൃදࢿྉͷ࡞੒ʹे෼ೲಘͰ͖Δ·Ͱ࣌ؒΛඅ΍͢͜ͱ͕Ͱ͖ͳ͔ͬͨͷͰͦͷ఺ʹ͍ͭͯ͸վળͷ༨஍͕͋Δͱߟ͑Δɻ
ʢ˞จ੹ : ླ໦ᠳేʣ

Group Report of 2020 SISP

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Mathematical and simulation of complex systems

5.3.4

౻ా३

୲౰ϖʔδͷྠಡͷ४උ ෼͔Βͳ͍‫۟ޠ‬Λௐ΂ͯཧղ͍ͯ͠Δͭ΋ΓͰ͍͕ͨɺઌੜʹࢦఠ͞Εͨͱ͖ʹ౴͑ΒΕͳ͍͜ͱ͕ଟ͔ͬͨͷͰɺཧղͷਂΊํʹվળͷ༨஍͕͋Δͱߟ͑Δɻ
ั৯ऀʔඃ৯ऀ‫ܥ‬ͷղͷ‫ي‬ಓ΍ྲྀΕਤͷϓϩάϥϜ࡞੒ ΦΠϥʔ๏Λ࢖ͬͨ਺஋‫ࢉܭ‬ϓϩάϥϜ
΍ྲྀΕਤɺղͷ‫ي‬ಓΛΞχϝʔγϣϯԽ͢ΔϓϩάϥϜΛ࡞੒͢Δ͜ͱ͕Ͱ͖͕ͨɺ‫ط‬ग़ͷํ๏Λ࢖͍ͬͯΔ͚ͩͰࣗ෼ͷϓϩάϥϛϯྗͷ޲্ʹ͸͋·Γ‫ͨͨ͡ײͱ͍ͳ͍͕ͯͬܨ‬Ίվળͷ༨஍͕͋Δͱߟ͑Δɻ
தؒൃදͷ४උ தؒൃද༻ͷεϥΠυ΍ಈը͕‫ͯݟ‬΋Β͏૬खʹ‫ڵ‬ຯΛ࣋ͨͤΔΑ͏ͳ޻෉͕Ͱ
͖ͳ͔ͬͨͱ‫ͨ͡ײ‬ͷͰվળͷ༨஍͕͋Δͱߟ͑Δɻ
֦ࢄ߲ͷΈͷৗඍ෼ํఔࣜͷ‫Ͱܗ‬ͷ਺஋‫ͱࢉܭ‬άϥϑ࡞੒ લ‫Ͱظ‬͸ɺ൓Ԡ߲ͷΈͷ‫ܥ‬ͷ਺஋‫ࢉܭ‬Λߦ͕ͬͨɺ‫ॳ࠷ظޙ‬͸ɺ֦ࢄ߲ͷΈͷ‫ܥ‬Λલ‫ظ‬ಉ༷ʹࠩ෼๏Λ༻͍ͯ਺஋‫ࢉܭ‬Λߦͬͨɻલ‫ظ‬ͷ਺஋‫ࢉܭ‬ͷख๏Λ‫Ͱظޙ‬΋ੜ͔͢͜ͱ͕Ͱ͖ͨ఺͸ྑ͔ͬͨɻάϥϑԽʹ͍ͭͯ΋ C‫ޠݴ‬Λ༻͍ͯೋ࣍‫ݩ‬ɾࡾ࣍‫ݩ‬ͷάϥϑΛϓϩοτ͢Δ͜ͱͰղͷ֦ࢄͷ࢓ํʹ͍ͭͯཧղΛਂΊΔ͜ͱ͕Ͱ͖ͨͷͰྑ͔ͬͨɻ
૒҆ఆ൓Ԡ֦ࢄ‫ܥ‬ͷղੳ૒҆ఆ൓Ԡ֦ࢄ‫ܥ‬Λࠩ෼๏Λ༻͍ͯ਺஋‫ࢉܭ‬Λߦ͍ɺψϧΫϥΠϯͱϑϩϯτਐߦ೾ղΛϓϩοτ͢Δ͜ͱ͕Ͱ͖͕ͨɺ਺஋‫ࢉܭ‬ͷख๏͕‫͕ྔࢉܭ‬ଟ͘ͳΓɺϓϩοτ͢Δͷʹ͕͔͔࣌ؒͬͯ͠·͍ͬͯͨɻࠩ෼๏ΑΓޮ཰తʹ਺஋‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖
Δख๏Λ໛ࡧ͢Ε͹ϓϩοτ࣌ؒͷ୹ॖʹ΋‫͕ܨ‬Δ͜ͱΛߟ͑Δͱɺ਺஋‫ࢉܭ‬ͷख๏ʹվળͷ༨஍͕͋Δͱߟ͑Δɻ

FitzHugh-Nagumo ํఔࣜͷղੳ FitzHugh-Nagumo ํఔࣜͷψϧΫϥΠϯΛ࡞੒͢Δ͜ͱͰɺύλʔϯΛ‫ܗ‬੒͢ΔΑ͏ͳϞσϧ͕୯҆ఆ‫͋Ͱܥ‬Δ͜ͱ͕ཧղͰ͖ͨɻ
࠷ऴൃදͷ४උ ࠷ऴൃද༻ͷεϥΠυͱಈը࡞੒Λ୲౰͕ͨ͠ɺ‫ݟ‬΍͍͢จࣈɾਤͷ഑ஔ΍੠ͷτʔϯͳͲ૬खʹ‫ڵ‬ຯΛ࣋ͨͤΔΑ͏ͳ޻෉͕͋ͬͨʹ΋ؔΘΒͣɺ࣮ફͰ͖ͳ͔ͬͨ͜ͱ
ʹվળͷ༨஍͕͋Δͱߟ͑Δɻ
ʢ˞จ੹ : ౻ా३ʣGroup Report of 2020 SISP

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Mathematical and simulation of complex systemsୈ 6 ষ ࠓ‫ޙ‬ͷ՝୊ͱల๬
લ‫Ͱظ‬͸൓Ԡ֦ࢄ‫͚͓ʹܥ‬Δั৯ऀʔඃ৯ऀ‫ܥ‬ͷ֦ࢄ߲Λ 0 ͱఆٛ͠ɺ൓Ԡ߲͚ͩͷࣜͱͯ͠਺
஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ɺγϛϡϨʔγϣϯΛߦ͍ղੳΛਐΊ͍ͯͬͨɻ͔֦͠͠ࢄ߲Λ 0 ͱఆ
ٛ͢Δͱั৯ऀʔඃ৯ऀ‫Ͱܥ‬΋ۭؒʹґଘ͠ͳ͍࣌ؒతͳมԽ͔͠‫ݟ‬ΒΕͣั৯ऀʔඃ৯ऀ‫ܥ‬Λ‫׬‬શʹղੳͰ͖͍ͯͳ͍ɻͦͷͨΊʹ·֦ͣࢄํఔࣜͷղੳΛ՝୊ͱ͠ɺ‫Ͱظޙ‬͸֦ࢄํఔࣜͷ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ɺγϛϡϨʔγϣϯΛߦ͍ɺղͷৼΔ෣͍Λ֬ೝͨ͠ɻࠓ‫ޙ‬͸֦ࢄ߲Λ 0ͱఆٛͤͣั৯ऀʔඃ৯ऀ‫ܥ‬ͷ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͜͠ͷ‫ܥ‬ͷղੳΛਐΊ͍͖͍ͯͨɻͦ͏
͢Δ͜ͱͰั৯ऀʔඃ৯ऀ‫ܥ‬ͷղ΋ۭؒʹґଘ͢Δ‫Ͱܗ‬ಘΔ͜ͱ͕Ͱ͖ɺ൓Ԡ֦ࢄ‫ܥ‬ͷղੳΛ͞ΒʹਐΊΔ͜ͱ͕Ͱ͖Δͱߟ͑Δɻ
‫Ͱظޙ‬͸૒҆ఆ൓Ԡ֦ࢄํఔࣜͱ Fitzhugh-Nagumo ํఔࣜͷ਺஋‫ࢉܭ‬Λߦ͍ɺղͷৼΔ෣͍ΛΞχϝʔγϣϯʹͨ͠ɻ൓Ԡ߲ͱ֦ࢄ߲͕ೖ͍ͬͯΔ‫ܥ‬Λղੳ͢ΔͨΊʹ਺஋‫ࢉܭ‬ͷΞϧΰϦζϜ
Λษ‫͕ͨ͠ڧ‬ɺ࣮ࡍʹϓϩάϥϜʹద༻Ͱ͖ͨΞϧΰϦζϜ͸গ͔͠͠ͳ͔ͬͨͨΊɺ൓Ԡ֦ࢄ‫ܥ‬ͷޮ཰తͰ҆ఆతͳղੳΛ͢ΔͨΊʹෳࡶͳ਺஋‫ࢉܭ‬ΞϧΰϦζϜΛษ‫͠ڧ‬ɺϓϩάϥϜʹద༻͢
Δ͜ͱ͕՝୊ʹͳΔɻ͜ͷ՝୊Λୡ੒͠ɺΑΓޮ཰తͰ҆ఆతͳ਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ɺ૒
҆ఆ൓Ԡ֦ࢄํఔࣜ΍ Fitzhugh-Nagumo ํఔࣜҎ֎ͷ൓Ԡ֦ࢄ‫ܥ‬ͷղੳΛਐΊɺੜ෺ͷ֦ࢄ΍໛༷ͷύλʔϯ‫ܗ‬੒ΛγϛϡϨʔγϣϯΛ௨ͯ֬͠ೝ͍͖͍ͯͨ͠ɻ
ʢ˞จ੹ : ླ໦ᠳేʣ

Group Report of 2020 SISP

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Mathematical and simulation of complex systems

ୈ7ষ
7.0.1

ֶͼͱࠓ‫ޙ‬ͷ՝୊Ֆࢁཽ໵ϓϩδΣΫτΛ࢝ΊΔʹ͋ͨͬͯϓϩδΣΫτϝϯόʔ͸ࠓճऔΓѻͬͨ൓Ԡ֦ࢄ‫ܥ‬ͷ஌͕ࣝ΄΅̌ͷঢ়‫͔گ‬Β࢝·ͬͨͨΊɺϝϯόʔશһͰͷࢀߟจ‫ݙ‬ͷྠಡ΍ղઆΛߦ͍ͳ͕Β஌ࣝΛ஝͍͑ͯͬͨɻͦͷࡍɺ༧උ஌͕ࣝͳ͘΄΅ख୳Γͷঢ়ଶ͔Β΍Δ΂͖͜ͱΛ໌֬ʹ΋Ͱ͖ͣ࠷ॳͷ՝୊ʹऔΓ૊Ή·Ͱʹ͔ͳΓͷ࣌ؒΛඅ΍ͯ͠͠·ͬͨɻ͕࣌ؒগͳ͔ͬͨ͜ͱ΋͋Γࢀߟจ‫ݙ‬ͷϖʔδ਺͝ͱʹ୲౰ՕॴΛ෼୲͠ɺΑΓਂ͍ཧղʹ౒Ί͕ͨɺࣗ෼͕୲౰ͨ͠Օॴ͸࣌ؒͷؔ܎΍ཧղ౓ʹ͋ͨͬͯϝϯόʔશһͰਂ͘औΓ૊Ή͜ͱ͸ͳ͔ͬͨɻͦΕʹՃ͑ɺ‫ݸ‬ਓతͳཧղ΋ઙͦ͘ͷ‫ޙ‬ͷϓϩδΣΫτͷ‫׆‬ಈʹ͏·͍͔ͤ͘ͳ͔ͬͨͱ‫͡ײ‬Δɻࣗ෼͕୲౰ͨ͠Օॴ͸ϝϯόʔؒͷதͰ࠷‫ͨ͋ʹޙ‬Δ෦෼ͩͬͨ͜ͱ΋͋Γɺલ൒Ͱ୲౰ʹͳͬͨϝϯόʔͷ୲౰෦෼ͷཧղ΍ղઆ΋ੵ‫ۃ‬తʹߦ͏΂͖Ͱ͋ͬͨͱߟ͑Δɻ·ͨɺࣗ෼Ͱ͸ཧղͰ͖ͳ͔ͬͨͱ͜Ζͷղઆ͸ੵ‫ۃ‬తʹ୲౰‫ڭ‬һ΍ϝϯόʔͷҙ‫ݟ‬Λ‫ٻ‬ΊΔ΂͖Ͱ͋ͬͨͱ΋ߟ͑ͨɻલ‫ظ‬ͷ಺༰ͱͯ͠͸ɺั৯ऀʔඃ৯ऀ‫ܥ‬ͷϞσϧʹؔͯ͠਺஋ղੳ΍ϓϩάϥϜͷ࡞੒Λߦͬͨɻั৯ऀʔඃ৯ऀͰ͸ੜ෺΍΢Πϧεͱ͍ͬͨ΋ͷͷ‫ݸ‬ମ਺ͷมಈΛද͢ϞσϧͰ͋Δɻࠓճ͸ั৯ऀͱඃ৯ऀͷยํ͕͍ͳ͍ύλʔϯ΍ྍ๏ͱ΋‫ڞ‬ଘ͍ͯ͠ΔύλʔϯͳͲෳ਺ͷύλʔϯʹ͍ͭͯߟ࡯ΛਐΊ͍ͯͬͨɻಛʹ҆ఆঢ়ଶʹ͋ͨΔ৔߹Λத৺తʹߟ͑ɺઢ‫҆ܗ‬ఆੑղੳ΍ϠίϏߦྻΛ༻͍ͯ҆ఆঢ়ଶͷੑ࣭Λ֬ೝͨ͠ɻ਺஋‫ʹࢉܭ‬͸͔ͳΓͷ͕͔͔࣌ؒͬͯ͠·ͬͨΓɺࣄલͷ஌ࣝ΋΄΅օແͷঢ়ଶ͔Β͸͡·ͬͨ͜ͱ΋͋Γɺؒҧͬͨ਺஋‫ࢉܭ‬΍ཧղͷ͔ͨ͠Λͯ͠͠·͏৔໘͕ଟ͋ͬͨ͘ɻ୲౰‫ڭ‬һʹϓϩδΣΫτ࣌ؒ֎Ͱͷ࣭໰͢ΔͳͲੵ‫ۃ‬తʹ՝୊ʹऔΓ૊Ήඞཁ͕͋ͬͨͱߟ͑ΔɻΠϯλʔωοτͳͲͰ༷ʑͳ஌ࣝΛ஝͍͕͑ͯͬͨ͏·͍͔ͤ͘ͳ͔ͬͨɻϓϩάϥϜ͸ޮ཰త͔ͭ؆୯ͳϓϩάϥϜΛ͔͚ΔΑ͏ʹͳΔ΂͖ͩͱ‫ͨ͡ײ‬ɻ·ͨɺதؒൃදͰ͸࢘ձਐߦΛ຿Ί͕ͨεϜʔζͳਐߦ͕Ͱ͖ͣ࿅शෆ଍Ͱ͋ͬͨͱࢥ͏ɻ࣭ٙԠ౴ͷࡍ͸͢ΜͳΓ౴͑ΕΔ৔໘΋͕͋ͬͨ͢΂ͯͷ࣭໰ʹରԠͰ͖ͨΘ͚Ͱ͸ͳ͔ͬͨͷͰ४උෆ଍Ͱ͋ͬͨͷͰࣄલʹ΋ͬͱ༷ʑͳ࣭໰Λ૝ఆͯ͠ൃදʹऔΓ૊Ήඞཁ͕͋ͬͨͰ͋Ζ͏ͱߟ͑Δɻ‫ظޙ‬ͷ಺༰ͱͯ͠͸ɺ֦ࢄํఔࣜ΍ Fitzhugh-Nagumo ํఔࣜͷ਺஋‫ࢉܭ‬΍ϞσϧղੳΛߦ͖͕ͬͯͨɺϓϩάϥϛϯά͕͏·͍͔ͣ͘पΓͷϝϯόʔͱͷϓϩάϥϜΛ‫ڞ‬༗͠՝୊ʹऔΓ૊Ή͜ͱ΋ଟʑ͋ͬͨɻ‫׳‬Εͳ͍΍ΓํΛͱͬͯ͠·ͬͨͨΊ࣌ؒޮ཰͸ྑ͍΋ͷͰ͸ͳ͔͕ͬͨɺࣗ෼
ͷ஌Βͳ͔ͬͨϓϩάϥϜͷ஌ࣝΛଟ͘஌Δ͍͍‫ػ‬ձʹͳͬͨͱ‫ͨ͡ײ‬ɻ͔͠͠ɺࣗ෼ͷ஌ࣝෆ଍΍ཧղྗ͸ͱͯ΋஗͍΋ͷͩͬͨͷͰɺࣗ෼ʹ༧උ஌͕ࣝͳ͍ͱ͖ͷ՝୊ʹऔΓ૊Ήࡍ͸ਓΑΓ΋࣌ؒΛ͔͚ͯ༧උ஌ࣝΛ஝͑ͨ͏͑Ͱ՝୊ʹऔΓ૊Ήඞཁ͕͋ͬͨͷͰɺࠓ‫ޙ‬ͷ՝୊ͱͯ͠͸ϝϯόʔʹ஗ΕΛऔΒͳ͍Α͏ʹཧղΛਂΊͯੵ‫ۃ‬తʹ՝୊ʹऔΓ૊Ή΂͖Ͱ͋Δɻ·ͨɺ‫ظ‬຤ൃදͰ͸౰೔ͷ؆қతͳൃදΛ୲౰͕͕ͨ࣌ؒ͠‫ݶ‬ΒΕ͍ͯΔͳ͔Ͱͷൃදʹ‫׳‬Ε͓ͯΒͣɺ᪴͘৔໘΋ଟ͋ͬͨ͘ͱ‫͡ײ‬ΔɻൃදΛ߇͍͑ͯΔͱ͖͸ࣗ෼͕ೲಘ͢Δ·Ͱ࿅शΛ‫܁‬Γฦͯ͠མͪண͍ͯ࿩
ͤΔඞཁ͕͋Δͱߟ͑Δɻ࣭ٙԠ౴ͷࡍ͸౴͑ΒΕͳ͔࣭ͬͨ໰΋͋Γͦͷ໘Ͱ΋४උෆ଍Ͱ͋ͬ
ͨͷͰɺࠓ‫ޙ‬͸४උΛଵΒͣߦ͏ඞཁ͕͋Δɻ
਺஋ղੳ΍ղͷৼΔ෣͍Λ֬ೝ͢Δ͏͑ͰɺϓϩάϥϛϯάΛ͢Δ࣌ؒ΋͔ͳΓͷ͕࣌ؒ͋ͬͨ
͕ɺϓϩάϥϛϯάͷ஌ࣝ΍ٕज़ʹؔͯ͠΋ྗෆ଍Λ௧‫ͨ͠ײ‬ɻେ൒ͷϓϩάϥϛϯά͸ PythonͰߦ͕ͬͨɺ̲‫͕ޠݴ‬ඞཁʹͳͬͨΓɺ༧උ஌ࣝ΋গͳ͍தɺࢼߦࡨ‫͕ͳ͠ޡ‬ΒϓϩάϥϜΛߦͬGroup Report of 2020 SISP

- 26 -

Group Number 6

Mathematical and simulation of complex systems͍͕ͯͬͨେ൒͸ωοτͰ‫ͨ͠ࡧݕ‬ϓϩάϥϜΛࢀߟʹ͠ͳ͕Βߟ͑ͨΓ͍ͯͨͨ͠Ίɺࣗ෼ͷதͷ஌ࣝͰϓϩάϥϜΛ૊ΈཱͯΔ౒ྗΛ΋ͬͱߦ͏΂͖ͩͱߟ͑ͨɻ਺஋‫ͯؔ͠ʹࢉܭ‬΋ࣗ෼ͷཧղྗͷ஗͞΍ɺํఔࣜͷҙຯΛ͸͖͕͍͋ͪ͑ͯͨΓͨ͠ͷͰᐆດͳ஌ࣝͰ͸ͳ͘ɺͪΌΜͱͨ͠஌ࣝΛ͚͍ͭͯͳ͚Ε͹ͳΒͳ͍ͱ‫ͨ͡ײ‬ɻ·ͨɺࠓճϓϩδΣΫτ಺Ͱ࡞੒ͨ͠Ξχϝʔγϣϯ΍ಈը͸ଞϝϯόʔʹཔΓ͖ͬΓʹͳͬͯ͠·ͬͨͨΊɺଞͷਓ͕Ͱ͖Δ՝୊ͩͱͯ͠΋ੵ‫ۃ‬తʹࣗ෼͔ΒऔΓ૊Έ‫ͯ͑ڭ‬΋Β͏ͳͲΛͯࣗ͠෼Λ஌ࣝΛ૿΍͍ͯ͘͠΂͖Ͱ͋ͬͨɻ
ϓϩδΣΫτϦʔμʔͱͯ͠͸࠷௿‫ݶ‬ͷ࢓ࣄ͸ߦ͍͕ͬͯͨɺ՝୊ʹର͢Δϝϯόʔͷཧղ౓ͷ֬ೝ΍஌ࣝ‫ڞ‬༗ɺ·ͨϝϯόʔ͚ؒͩͰͷձٞ΍ྠಡ࣌ͷԁ‫ͳ׈‬ਐߦ͸·ͩ·ͩվળͷ༨஍͕͋Δ
΋ͷʹͳͬͯ͠·ͬͨɻΦϯϥΠϯͰͷձٞʹ‫׳‬Ε͍ͯͳ͔ͬͨ͜ͱ΋͋Δ͕ɺਐߦ͢Δͱ͍͏໾ׂ΍͢΂͖͜ͱͷ༏ઌॱҐΛ͋Β͔͡ΊܾΊ͓ͯ͘͜ͱͷେ੾͞Λ࣮‫ͨ͠ײ‬ɻൃ‫ݴ‬Λଅͨ͠Γɺਐ
ߦ͢Δ͜ͱʹूத͗ͯ͢͠ɺࣗ෼ͷҙ‫ݟ‬Λ͔ͬ͠Γ࿩ͤͳ͔ͬͨɻ͔͠͠ϓϩδΣΫτϦʔμʔ͸ͱͯ΋͍͍‫ͨͬͳʹݧܦ‬ͷͰɺࠓ‫ͨ·ޙ‬ϦʔμʔΛ͢Δࡍ΍ϝϯόʔؒͰͷձٞΛߦ͏ࡍ͸ԁ‫͔׈‬ͭޮ཰ͷ͍͍ձٞΛߦ͏͜ͱ͕՝୊ʹͳͬͨɻʢ˞จ੹ : Ֆࢁཽ໵ʣ

7.0.2

ࡾӜྰԻ

ϓϩδΣΫτֶशΛ͍࢝͟ΊΔ࣌ɺϝϯόʔશһ͕൓Ԡ֦ࢄ‫͍ͯͭʹܥ‬શ͘஌ࣝͷͳ͍ঢ়ଶͰ
࢝·ͬͨɻͦͷΑ͏ͳঢ়ଶͰ͋ͬͨͨΊɺલ‫ظ‬ͷϓϩδΣΫτֶशͰ͸ɺࢀߟจ‫[ ݙ‬1] ͷྠಡ͔Βߦͬͨɻ஌͕ࣝͳ͔ͬͨͨΊɺ࠷ॳ͸εϩʔϖʔεͰߦ͏Α͏ʹ͠ɺ๯಄ͷ໿ 15 ϖʔδΛֶशͨ͠ɻͦͷࡍϝϯόʔશһͰಉ࣌ʹ 1 ͔ΒਐΊͯ͠·͏ͱലେͳ࣌ؒΛඅ΍ͯ͠͠·͏ͷͰɺ֤ࣗͰ
໾ׂ෼୲ͯ͠ɺ‫׬‬ᘳʹཧղ͢ΔՕॴΛܾΊɺཧղΛਂΊ͍ͯͬͨɻͦ͜Ͱɺ֤ʑΘ͔Βͳ͍୯‫͕ޠ‬͋ͬͨΒɺจ‫ݙ‬ΛಡΜͩΓɺωοτ΍ࣙॻͰ‫ͨ͠ࡧݕ‬Γͯ͠ɺϝϯόʔશһʹ‫ڞ‬༗ͨ͠ɻͦΕҎ֎ͷ֤ʑ͕ཧղΛਂΊͨՕॴΛຖिͷϓϩδΣΫτֶशͷ࣌ؒ಺Ͱൃද͠ɺϝϯόʔ಺Ͱͷཧղ΋ਂΊͨɻͦͯ͠ɺ‫ͯͬޡ‬஌ࣝΛ஝͍͑ͯͨ෦෼ʹؔͯ͠͸ɺ୲౰‫ڭ‬һ͔Βղઆ΍ΞυόΠεΛ௖͖ɺ஌ࣝΛ͞Βʹ஝͍͑ͯͬͨɻࣗ෼͕୲౰ͨ͠Օॴʹ͍ͭͯ͸ɺઢ‫ܗ‬୅਺ֶͰ΍ͬͨ‫ݻ‬༗ํఔࣜͳͲ
Λར༻ͯ͠‫͖͍ͯ͠ࢉܭ‬ɺਐߦ೾ղʹ͍ͭͯཧղΛਂΊͨɻਐߦ೾ղ͸‫Ͱظޙ‬ॏཁʹͳͬͯ͘ΔͨΊɺ‫ͨͬޡ‬஌ࣝΛ਎ʹ෇͚ͳ͍Α͏ʹ౒ྗͨ͠ɻ΋ͱ΋ͱͷ஌͕ࣝͳ͔ͬͨ෼ɺ୲౰‫ڭ‬һ͔ΒͷࢦఠΛଟ͘΋Βͬͯ͠·ͬͨͷ͸ཧղͷਂΊํʹ໰୊͕͋ͬͨͱߟ͑Δɻ·ͨɺϝϯόʔશһ͕୲౰ՕॴΛൃදͰ͖ͣʹલ‫ظ‬ͷϓϩδΣΫτֶश͕ऴΘͬͯ͠·ͬͨͷͰɺࣗ෼ͷ୲౰ՕॴΛ΋͏গ͠ૣ͘ऴΘΒͨ͢Ίͷ౒ྗ͕଍Γ͍ͯͳ͔ͬͨͱߟ͑Δɻࣗ෼ͰԿͱ͔͠Α͏ͱͯ͠ɺϝϯόʔ΍୲౰‫ڭ‬һʹཔΕͳ͔ͬͨͷ͕‫ݪ‬Ҽͩͱߟ͑Δɻ͞Βʹɺଞͷϝϯόʔͷ୲౰ՕॴͷৼΓฦΓΛ͍͟ߦͬͨ࣌ɺશવཧղͰ͖͍ͯͳ͍෦෼͕͋ͬͨͷͰɺࣗ෼ͷ୲౰Օॴʹ͹͔Γ໨͕͍͓ͬͯΓɺଞͷϝϯόʔͷ୲౰ՕॴͷൣғʹखΛ෇͚Ε͍ͯͳ͔ͬͨͷ͸ྑ͘ͳ͔ͬͨ఺Ͱ͋ΔɻΘ͔Βͳ͘ͳ͍ͬͯͨΒɺͦͷ࣌఺Ͱଞͷϝϯόʔʹฉ͚͹Α͔ͬͨ΋ͷͷɺ‫ޙ‬ճ͠ʹͯ͠͠·ͬͨͷ͸վળ͍ͯ͜͠͏ͱࢥ͏ɻͦ͏͍ͬͨ͜ͱ΋͋Γɺ՝୊Λ͓͜ͳ͏্Ͱͷ࣌ؒ഑෼ͱɺίϛϡχέʔγϣϯΛͱΔ͜ͱͷେ੾͞ΛֶΜͩɻ‫Ͱظޙ‬͸ɺ֦ࢄํఔࣜ΍ Fitzhugh-Nagumo ํఔࣜͷ਺஋‫ࢉܭ‬Λ͓͜ͳͬͨɻલ‫਺ʹظ‬஋‫ࢉܭ‬ͷ΍ΓํΛ͋Δఔ౓ֶश͍ͯͨ͠΋ͷͷɺ‫ظޙ‬ͷ༧श͕଍Γ͍ͯͳ͔ͬͨ͜ͱ΋͋Γɺ‫ظޙ‬ͷ͸͡Ί͸εϜʔζʹ࡞‫ۀ‬ΛਐΊΔ͜ͱ͕Ͱ͖ͳ͔ͬͨɻ֦ࢄํఔࣜͷ਺஋‫Ͱࢉܭ‬͸ɺϑʔϦΤม‫׵‬΍ࠩ෼๏༻͍ͯɺ࠷ॳ͸ϓϩάϥϜͳͲΛ࡞੒ͤͣʹ਺ֶతղੳͨ͠ɻϝϯόʔ಺Ͱ୲౰͢Δ΍ͭΛܾΊGroup Report of 2020 SISP

- 27 -

Group Number 6

Mathematical and simulation of complex systemsͯɺ࣮ࡍʹ‫ͨ͠ࢉܭ‬ɻࣗ෼͸ϑʔϦΤม‫׵‬Λ༻͍ͨ΋ͷΛ୲౰͕ͨ͠ɺ࠷ऴతʹ‫݁ࢉܭ‬ՌΛಋ͖ग़͠ɺ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖ͨ఺ʹؔͯ͠͸ྑ͔ͬͨͱߟ͑Δɻ͔͠͠ɺ࠷ऴతʹϓϩδΣΫτ಺Ͱ͸ࠩ෼๏Λओʹར༻ͨͨ͠Ίɺࠩ෼๏ͷཧղ͕Ͱ͖͍ͯͳ͔ͬͨ఺ʹ໰୊͕͋ͬͨͱߟ͑Δɻ
ຊϓϩδΣΫτͰͷ਺஋‫ࢉܭ‬ϓϩάϥϜͷ࡞੒Ͱ͸ɺ౤ʹ࢖༻͢Δ‫ޠݴ‬͸ఆΊͣʹߦͬͨɻ֤ʑ
Ͱ͖ͦ͏ͳ‫਺ʹࡍ࣮Ͱޠݴ‬஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͢Δ͜ͱΛࢼΈΔΑ͏ʹͨ͠ɻϓϩδΣΫτ಺Ͱओʹ࢖༻ͨ͠ͷ͸ɺC ‫ ͱޠݴ‬Python Ͱ͋Δɻࢲ͸ Python Λ࢖༻͍͕ͯͨ͠ɺPython Ͱߦ͖٧·ͬͯ͠·ͬͨ࣌ʹ͸ɺC ‫ͨࣅͰޠݴ‬Α͏ͳϓϩάϥϜΛ࡞੒͠ɺ࣮ߦ͢Δ͜ͱΛࢼΈͨɻψϧΫϥΠϯͷදࣔ΍ɺ؆୯ͳάϥϑͷදࣔ͸͋Δఔ౓Ͱ͖͍ͯͨͷ͕ͩɺ࠷ऴతͳΞχϝʔγϣϯಈըͷ࡞੒ʹ͸ۤ͠ΜͰ͠·͍ɺ΄͔ͷϝϯόʔʹཔͬͯ͠·ͬͨ఺ʹؔͯ͠ɺ΋͏গ͠౒ྗ͕ඞཁ
ͩͬͨͱ‫͡ײ‬Δɻ͔͠͠ɺຊདྷͷϓϩδΣΫτͷ໨తͰ͋Δɺʮ࣮ࡍʹ਺஋‫ࢉܭ‬Λ͓͜ͳ͍ɺϞσϦϯάೳྗɺϓϩάϥϛϯάೳྗΛ਎ʹ෇͚ͯɺղͷৼΔ෣͍Λௐ΂ͯɺͲͷΑ͏ͳύλʔϯ‫ܗ‬੒͕ग़དྷ͍ͯΔ͔Λ֬ೝ͢Δʯͱ͍͏΋ͷ͕͔ͬ͠ΓͱՌͨ͢͜ͱ͕Ͱ͖͍ͯͨͷͰ͸ͳ͍͔ͱࢥ
͏ɻ͞Βʹɺ‫ظޙ‬͸લ‫ʹظ‬ൺ΂ͯɺҰͭͷ՝୊ʹରͯ͠ϓϩδΣΫτϝϯόʔ಺શһͰҰͭͷ՝୊Λ͜ͳ͢‫ͨͬͩ͡ײ‬ͷͰɺϝϯόʔ಺Ͱ࿩͠߹͍Λ૿΍͠ɺίϛϡχέʔγϣϯΛͱΔ͜ͱ͕Ͱ͖
ͨͷ͸ͱͯ΋Α͔ͬͨͷͰ͸ͳ͍͔ͱߟ͑Δɻ
࠷ऴൃදͷ࣌ʹؔͯ͠ɺࢲ͕͓͜ͳͬͨ͜ͱͱͯ͠ɺ࠷ऴൃද༻ͷϙελʔ࡞੒ͱɺൃද࣌ͷ࢘ձਐߦΛ͓͜ͳͬͨɻϙελʔ࡞੒͸தؒൃදͷ࣌ʹ΋ߦ͕ͬͨɺͦͷ࣌͸ɺൃද͕ऴΘͬͯධՁγʔτΛ‫ʹ͖ͱͨݟ‬ɺͨ͘͞ΜͷࢦఠΛ͍͍ͨͩͨͷͰɺ࠷ऴ༻Ͱ͸ɺࢦఠ͞Ε͍ͯͨ͜ͱΛվળ
͢ΔΑ͏ʹ࡞੒ͨ͠ɻϙελʔ࡞੒ࣗମॳΊͯͰɺ࠷ॳ͸શવԿ΋Ͱ͖ͳ͔͕ͬͨɺϙελʔΛ‫ݟ‬ͯͩ͘͞Δਓ͕‫ݟ‬΍͍͢΋ͷΛ࡞Ζ͏ͱ͍͏ҙࣝͰ࡞੒ͨ͠Βɺࣗવʹྑ͍ϙελʔΛ࡞੒͢Δ͜
ͱ͕Ͱ͖ͨɻ·ͨɺൃද࣌ͷ࢘ձਐߦʹؔͯ͠͸ɺελʔτͨ͠Β͔ͬ͠Γͱ࢝ΊΔ͜ͱ͕Ͱ͖ɺ
࣭ٙԠ౴ʹؔͯ͠΋͔ͬ͠ΓͱਐΊΔ͜ͱ͕Ͱ͖ͨ఺͸ྑ͔ͬͨͱߟ͑Δɻ͔͠͠ɺ࣭໰͞Εͯɺͪ͜Βͷϝϯόʔ಺Ͱ౴͑ΒΕͳ͔ͬͨ࣌ʹ௜໧Λ࡞ͬͯ͠·͍ɺͦͷ৔໘Ͱɺগ͠Ͱ΋࿩͕Ͱ͖Ε͹ɺͳ͓੒ޭͤ͞Δ͜ͱ͕Ͱ͖ͨͷͰ͸ͳ͍͔ͱߟ͑Δɻ
ຊϓϩδΣΫτͰֶΜͩ͢΂ͯͷ͜ͱ͸ɺࠓ‫ޙ‬ͷଔ‫ڀݚۀ‬΍ब৬͔ͯ͠Βͷ࢓ࣄʹ΋ඞͣඞཁʹ
ͳΔ͜ͱ͹͔ΓͰ͋ΔɻಛʹίϛϡχέʔγϣϯΛͱΔ͜ͱ͸Ұ൪େࣄͰ͋Δͱ‫ͨ͡ײ‬ɻίϛϡχέʔγϣϯΛͱΒͳ͚Ε͹ɺ՝୊Λ͜ͳ͢εϐʔυ͕஗͘ͳΓɺϓϩδΣΫτશମ͕‫ޙ‬ΕΛͱͬͯ
͠·͏͜ͱΛֶΜͩɻ·ͨɺࠓ೥౓͸ɺ΄ͱΜͲΦϯϥΠϯͰ࡞‫ۀ‬Λ͓͜ͳ͍ͬͯͨͨΊɺ࣮ࡍʹର໘Ͱձ͍ͬͯͳ͍෼ɺԁ‫ۀ࡞ʹ׈‬ΛਐΊΔ͜ͱ͕Ͱ͖Δ৔໘Ͱ΋ɺϝϯόʔ಺Ͱߦ͖ҧ͍͕͋ͬͨΓͱɺଟগۤ࿑ͨ͠෦෼͕ଟ͔ͬͨɻͨͩɺΦϯϥΠϯͰ͔͋ͬͨΒͦ͜ͷֶͼͰ΋͋ΔͷͰɺͦΕΛࠓ‫ޙ‬ɺ༗ޮ‫͏ࢥͱ͍͖͍ͨͯ͠༻׆‬ɻʢ˞จ੹ : ࡾӜྰԻʣ

7.0.3

ླ໦ᠳే

લ‫ظ‬ͷϓϩδΣΫτͷ‫׆‬ಈͰ͸൓Ԡ֦ࢄ‫͍ͨͬ͏Ͳ͕ܥ‬΋ͷͳͷ͔ϝϯόʔશһͰจ‫ݙ‬Λྠಡ͠ͳ͕Βษ‫ڧ‬Λͨ͠ɻ·ͣɺϝϯόʔؒͰྠಡ͢ΔՕॴΛ෼୲͠୲౰ͨ͠‫ॴݸ‬ͷษ‫ڧ‬ΛਐΊɺϓϩδΣΫτͷ࣌ؒʹશһͷલͰൃද͢Δ‫ࣜܗ‬Λ࠾༻͠ਐΊ͍ͯͬͨɻࣗ෼͕୲౰ͨ͠‫Ͱॴݸ‬΋ࣗ෼ͷཧղྗ͕଍Γͣɺൃදͷ࣌ʹϝϯόʔ΍୲౰‫ڭ‬һʹे෼ʹઆ໌͢Δ͜ͱ͕Ͱ͖ͣɺϓϩδΣΫτͷ
͕࣌ؒԡͯ͠͠·͏͜ͱ͕͋ͬͨɻ͜ͷ఺ʹؔͯ͠͸Θ͔Βͳ͍ͱ͜Ζ͸ࣗ෼ҰਓͰௐ΂ͯղܾ͠Α͏ͱ͢ΔͷͰ͸ͳ͘ɺϝϯόʔ΍୲౰‫ڭ‬һʹΘ͔Βͳ͍ͱ͜Ζʹ͍࣭ͭͯ໰Λͯ͠ॿ‫ݴ‬Λ‫ٻ‬ΊΔ

Group Report of 2020 SISP

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Group Number 6

Mathematical and simulation of complex systems΂͖Ͱ͋ͬͨͱߟ͑Δɻϝϯόʔ͕ൃද͢Δͱ͖΋ࣗ෼͕ཧղͰ͖ͳ͍ͱ͜Ζ͕͋ͬͨΒͦͷ‫ॴݸ‬ʹ͍ͭͯଞͷϝϯόʔʹ࣭໰ͯ͠ೲಘͰ͖Δ·Ͱษ‫ͯͬ߹͖෇ʹڧ‬΋Β͏΂͖ͩͬͨͱߟ͑Δɻཧ
ղͰ͖͍ͯͳ͍Օॴ͕গͣͭ͠Ͱ΋͋Δͱͦͷ‫ޙ‬ͷ਺஋‫ࢉܭ‬ϓϩάϥϜͷ࡞੒΍ϓϩδΣΫτͷਐߦʹѱӨ‫ڹ‬Λ‫͕ͱ͜͢΅ٴ‬Θ͔ͬͨɻ࣮ࡍʹࣗ෼ͷ୲౰‫ॴݸ‬Λൃදͨ͠ͱ͖΋ϝϯόʔͱ୲౰‫ڭ‬һʹ͍·͍ͪཧղͯ͠΋Β͑ͣΘ͔Γ΍͢આ໌͕Ͱ͖͍ͯͳ͔ͬͨɻϝϯόʔ͕ޮ཰తʹ୲౰‫ॴݸ‬Ҏ֎ͷ෦෼ΛཧղͰ͖ΔΑ͏ʹ͢ΔͨΊʹ΋ɺ·ͣࣗ෼͕ͦͷ‫ॴݸ‬Λ‫׬‬શʹཧղ͢Δ͜ͱ͕ඞཁͰ͋
ΔͱΘ͔ͬͨɻͦͷͨΊʹ΋จষதʹग़͖ͯͨฉ͖ͳ͡Έͷͳ͍୯‫ޠ‬ΛΠϯλʔωοτͰௐ΂ͨ··આ໌͢ΔͷͰ͸ͳ͘ɺ୭ʹͰ΋ཧղͰ͖ΔΑ͏ͳઆ໌΋Ұॹʹ‫͕ͱ͜͏ݴ‬ඞཁͩͬͨɻଞͷਓͷ໨ઢͰൃදΛߟ͑ɺͦΕʹ޲͚ͯษ‫͠ڧ‬ɺཧղͰ͖ͳ͍ͱ͜Ζ͕͋ͬͨΒਓʹ૬ஊ͢Δ͜ͱ͕େࣄ
ͱ͍͏͜ͱ͕ൃදʹؔͯ͠૯ͯ͡‫͑ݴ‬Δ͜ͱͰ͋Δͱߟ͑Δɻ
͜ͷϓϩδΣΫτ͸‫ج‬ຊతʹ Python ΍ C ‫Ͳͳޠݴ‬Λ௨ͯ͡਺஋‫ࢉܭ‬ϓϩάϥϜΛ࡞੒͠ɺάϥϑΛग़ྗ͢Δ͜ͱ͕ओͳ‫׆‬ಈͷҰͭͰ͋ΔͨΊ਺஋‫ࢉܭ‬ͷΞϧΰϦζϜͷཧղ΍֤ϓϩάϥϛϯ
ά‫͚͓ʹޠݴ‬Δάϥϑ΍ಈըͷग़ྗ͕େࣄͩͱߟ͑Δɻࣗ෼͸ࠓ·Ͱ਺஋‫ࢉܭ‬ͷΞϧΰϦζϜΛ
࡞੒ͨ͜͠ͱ͕ͳ͘ɺϓϩδΣΫτͷਐߦͱ࣌ؒͷ౎߹্ɺൺֱతΞϧΰϦζϜͷཧղ͕؆୯ͳ‫ܭ‬ࢉख๏Λ࠾༻͠ϓϩάϥϜΛ࡞੒͕ͨ͠ɺ҆ఆੑ΍‫ࢉܭ‬ͷޮ཰͕ଞͷΞϧΰϦζϜͱൺ΂ͯམͪ
ͯ͠·ͬͨɻͦͷ݁ՌɺύϥϝʔλΛ҆ఆੑ͕औΕΔൣғͰ͔͠ઃఆ͢Δ͜ͱ͔͠Ͱ͖ͳ͘ɺύϥϝʔλΛมߋͯ͠γϛϡϨʔγϣϯΛ͢Δͷʹଟ͘ͷ࣌ؒΛඅ΍ͯ͠͠·ͬͨɻ‫ͯؔ͠ʹ཰ޮࢉܭ‬΋ଞͷΞϧΰϦζϜʹൺ΂ͯޮ཰͕ѱ͘ɺ‫ࣗࢉܭ‬ମʹ࣌ؒΛඅ΍ͯ͠͠·ͬͨɻ͜ͷ఺ʹؔͯ͠͸ࣗ෼͕ଞͷ਺஋‫ࢉܭ‬ΞϧΰϦζϜΛཧղ͠ɺϓϩάϥϜʹ૊ΈࠐΉ͜ͱ͕Ͱ͖Ε͹‫཰ޮࢉܭ‬Λ্͛Δ͜ͱ͕Ͱ͖ɺଞͷ൓Ԡ֦ࢄ‫ʹܥ‬΋खΛग़͢͜ͱ͕Ͱ͖ͨͱߟ͑Δɻࠓ‫ޙ‬ͷ՝୊ͱͯ͠͸਺छྨͷ਺஋‫ࢉܭ‬ΞϧΰϦζϜΛษ‫͠ڧ‬ɺ‫׬‬શʹཧղͨ͠͏͑ͰϓϩάϥϜΛ࡞੒͠ɺγϛϡϨʔγϣϯΛߦ͏͜ͱͰ͋Δɻϓϩάϥϛϯά‫ޠݴ‬Λ௨ͨ͡άϥϑ΍Ξχϝʔγϣϯͷ࡞੒ʹؔͯ͠͸ɺࣗ෼͸ࠓճ‫ج‬ຊతʹ Python Λ௨ͯ͡ϓϩάϥϜΛ࡞੒͠ɺάϥϑ΍ΞχϝʔγϣϯΛ࡞੒͕ͨ͠ɺࣗ
෼ͷ Python ͷཧղྗ͕଍ΓͣάϥϑԽ΍ΞχϝԽʹ͕͔͔࣌ؒͬͯ͠·ͬͨɻֶߍͷߨ͚ٛͩͰ
͸ Python ͸ཧղͰ͖ͣࣗ෼Ͱ΋ϓϩδΣΫτͱฒߦͯ͠ษ‫͕͍ͦͨͯ͠ڧ‬ΕͰ΋ษ‫͕ڧ‬଍ΓͣϓϩάϥϜ࡞੒࣌ʹख͕ࢭ·͍ͬͯͨͷ͕‫ݪ‬ҼͰ͋ΔͨΊɺ΋ͬͱ Python ͷษ‫ؒ࣌ڧ‬ΛऔΔ΂͖Ͱ͋ͬͨͱߟ͑Δɻࠓ‫ޙ‬ͷ՝୊ͱͯ͠ Python Λ΋ͬͱษ‫͠ڧ‬ɺϥΠϒϥϦΛ্ख͘࢖ͬͯ‫ݟ‬΍͍͢άϥϑ΍ΞχϝʔγϣϯΛ୹࣌ؒͰ࡞੒͢Δ͜ͱ͕͋͛ΒΕΔɻ
‫ظޙ‬ͷϓϩδΣΫτͷ‫׆‬ಈͰ͸֦ࢄํఔࣜͱ Fitzhugh-Nagumo ํఔࣜͷ਺஋‫ࢉܭ‬ϓϩάϥϜΛ
࡞੒͠ɺάϥϑͱΞχϝʔγϣϯΛ࡞੒͕ͨ͠ɺͦͷ్தͰϝϯόʔͱҰॹʹ‫ڞ‬௨ͷϓϩάϥϜΛ
࡞੒͢Δ‫ػ‬ձ͕͋ͬͨɻࠓ·Ͱଞͷਓͱ‫ڞ‬௨ͷϓϩάϥϜΛ࡞੒͢Δ͜ͱ͕ͳ͔ͬͨͷͰͦͷ࣌ʹ࣌ؒΛඅ΍ͯ͠͠·ͬͨɻϓϩάϥϜͷ։ൃΛԁ‫ʹ׈‬ਐΊΔͨΊʹ֎෦αʔϏεͷར༻ํ๏Λ΋ͬͱֶͿ΂͖ͩͱࢥͬͨɻͦ͏͢Δ͜ͱʹΑͬͯϓϩάϥϜΛޮ཰తʹ࡞੒͢Δ͜ͱ͕Ͱ͖ɺϓϩδΣΫτશମͰ΋ଞͷ՝୊ʹऔΓ૊ΊͨͷͰͦͷ఺ʹؔͯ͠͸ࣗ෼ͷษ‫ڧ‬ෆ଍ͩͬͨͱߟ͑Δɻࠓ‫ޙ‬ͷ՝୊ͱͯ͠͸֎෦αʔϏεʹ͍ͭͯଞͷਓʹԿΛ࢖͍ͬͯΔ͔ͳͲΛฉ͖ɺࣗ෼ͷ։ൃʹ΋ͦΕΛऔΓೖΕ͍ͯ͘͜ͱ͕͋͛ΒΕΔɻ
·ͨϝϯόʔؒͰձٞΛߦͬͨͱ͖ʹ͸ϓϩάϥϜͷ಺༰΍֤খ՝୊ͷׂΓৼΓͷ࿩Λ͍͕ͯͨ͠ɺΦϯϥΠϯͱ͍͏͜ͱ΋͋Γࣗ෼͔Βͳ͔ͳ͔੠Λग़͢͜ͱ͕Ͱ͖ͣʹ͚͕࣌ؒͩա͍͗ͯ͘
͜ͱ͕࣌ʑ͋ͬͨͷͰϦʔμʔͰͳ͘ͱ΋ࣗ෼͔ΒձٞΛਐΊΔ͜ͱΛͰ͖ΔΑ͏ʹ͍ͨ͠ͱߟ͑
ͨɻখ՝୊ͷׂΓৼΓ΋ࣗ෼͕΍Γ͍ͨ՝୊͕͋ͬͯ΋ͦΕΛϝϯόʔʹ఻͑ΒΕͣଞͷ՝୊ʹऔΓֻ͔Δ͜ͱ͕͋ͬͨͷͰࣗ෼͔ΒਐΜͰ੠Λग़͍ͨ͠ͱߟ͑ͨɺࠓ‫ޙ‬ͷ՝୊ͱͯ͠͸ϝϯόʔͰू·ͬͨͱ͖ʹ͸ࣗ෼͔Β੠Λग़͠ɺձٞΛԁ‫ʹ׈‬ਐΊΔ͜ͱ͕͋͛ΒΕΔɻ

Group Report of 2020 SISP

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Group Number 6

Mathematical and simulation of complex systemsʢ˞จ੹ : ླ໦ᠳేʣ

7.0.4

౻ా३

ϓϩδΣΫτΛ࢝ΊΔʹ͋ͨͬͯϓϩδΣΫτϝϯόʔશһ͕൓Ԡ֦ࢄ‫͍ͯͭʹܥ‬༧උ஌ࣝͷͳ͍ঢ়ଶͰ͜ͷϓϩδΣΫτ͸࢝·ͬͨɻ·ͣɺ୲౰‫ڭ‬һ͕༻ҙͯ͘͠Εͨจ‫ݙ‬ΛϓϩδΣΫτϝϯ
όʔͰ෼୲͠ɺ൓Ԡ֦ࢄ‫ͯؔ͠ʹܥ‬ͷ஌ࣝΛਂΊΔͨΊʹྠಡΛߦͬͨɻจ‫ʹݙ‬ग़ͯ͘Δ‫۟ޠ‬͸ɺͲΕ΋ॳΊͯฉ͘΋ͷ͹͔ΓͰ࠷ॳ͸ۤઓ͠ͳ͕Β΋ωοτͰ෼͔Βͳ͍‫۟ޠ‬Λ‫͕ͳ͠ࡧݕ‬Β୲౰ͨ͠಺༰ΛεϥΠυʹ·ͱΊͯൃද͠ɺ֤ϓϩδΣΫτϝϯόʔͱ஌ࣝΛ‫ڞ‬༗ͨ͠ɻ͔͠͠ɺൃද
ͷ৔໘Ͱ͸ࣗ෼͕୲౰ͨ͠಺༰ʹରͯ͠ͷཧղ͕௿͘ɺ୲౰‫ڭ‬һͷ࣭໰ʹ౴ΒΕͳ͍͜ͱ΍ɺଞͷϓϩδΣΫτϝϯόʔʹ΋ཧղ͠΍͍͢Α͏ͳൃද͕Ͱ͖͍ͯͳ͔ͬͨͱ‫ͨ͡ײ‬ͷͰɺࣄલʹ୲౰ͨ͠಺༰ͷཧղΛΑΓਂΊɺ୭͕ฉ͍ͯ΋Θ͔ΔΑ͏ͳઆ໌Λ৺ֻ͚Δ΂͖ͩͬͨͱߟ͑Δɻ·ͨɺࣗ෼ͷ୲౰ͨ͠಺༰ʹ‫͍·ͯࣥ͠͠ݻ‬ɺଞͷϓϩδΣΫτϝϯόʔ͕୲౰ͨ͠಺༰ͷൃදʹ࣭ؔͯ͠໰Λ͢Δ͜ͱ΋ͳ͘ཧղͨ͠‫͕͍ͨͯͬͳʹؾ‬ɺ‫ৼͰޙ‬Γฦͬͨࡍʹ͍֮͑ͯͳ͔ͬͨΓɺཧղͰ͖͍ͯͳ͔ͬͨ͜ͱ͕͋ͬͨͷͰɺϝϯόʔͷؒͰ΋෼͔Βͳ͍ͱ͜Ζ͸࣭໰͋ͬͯ͠ΑΓཧղΛਂΊΔ౒ྗΛ͢Δ΂͖ͩͬͨͱߟ͑Δɻ·ͨɺωοτͰௐ΂ͨ͜ͱΛӏವΈʹ͍͗ͯ͢͠Δ
෦෼͕͋ͬͨͷͰɺ಺༰Λۛຯͯ͠‫͢༻׆‬Δ΂͖ͩͬͨͱߟ͑Δɻ·ͨɺࣅͨ಺༰ͷ࿦จͳͲͷจ‫ݙ‬ΛಡΉ͜ͱ͕ͳ͔ͬͨͷͰɺଞͷจ‫ݙ‬΋ಡΈਐΊͳ͕Β൓Ԡ֦ࢄ‫ͯؔ͠ʹܥ‬ͷ஌ࣝΛΑΓਂΊΔ΂͖ͩͬͨͱ‫͡ײ‬Δɻ࣍ʹɺ͜ͷϓϩδΣΫτ͸ɺओʹ୲౰ͨ͠಺༰ʹग़ͯ͘Δ൓Ԡ֦ࢄ‫਺͍ͯͭʹܥ‬஋‫ࢉܭ‬ΛߦͬͨΓɺάϥϑΛϓϩοτ͢ΔϓϩάϥϜͷ࡞੒Λߦͬͨɻ਺஋‫͍ͯͭʹࢉܭ‬͸ɺղ͖ํ͕Ұ൪୯७ͩͬͨࠩ෼๏͹͔Γ࢖ͬͯ͠·͍ɺάϥϑΛϓϩοτ͢ΔࡍͷϓϩάϥϜ಺ͷ‫͕ྔࢉܭ‬ଟ͘ͳͬͯ
͠·͍ॲཧʹ͕͔͔࣌ؒͬͯ͠·͍ɺޮ཰తͱ͸‫͔ͨͬͳ͑ݴ‬ͷͰɺΑΓ‫͕ྔࢉܭ‬গͳ͍਺஋‫ࢉܭ‬ͷख๏Λ໛ࡧ͢Δ΂͖ͩͬͨɻϓϩάϥϜʹؔͯ͠͸ɺωοτͰௐ΂ͨϓϩάϥϜΛࢀߟʹ͗͢͠
ͯ͠·͍ɺίʔυͷཧղʹ͕͔͔࣌ؒͬͯ͠·͏͜ͱ͕͋ͬͨͷͰɺࣗ෼Ͱߟ͑ͯίʔυΛॻ͘͜ͱ͕େࣄͩͱࢥͬͨɻ·ͨɺ਺஋‫ࢉܭ‬΍ϓϩάϥϜΛ࡞੒͢Δʹ͋ͨͬͯɺϝϯόʔؒͰͷ‫͕ྗڠ‬গͳ͔ͬͨͱ‫ͨ͡ײ‬ͷͰɺࠔͬͨͱ͖ʹ͸΋ͬͱཔΔ΂͖ͩͬͨɻલ‫ظ‬͸ɺશମతʹϝϯόʔؒͰͷ‫͕ྗڠ‬଍Γͳ͘ɺ੒Ռ΋͋·Γ্͛ΒΕͳ͔ͬͨͷͰɺϝϯόʔؒͰͷަྲྀΛਂΊɺ཰ઌతʹ‫ڠ‬ྗΛ‫͚͔ͼݺ‬Δ΂͖ͩͬͨͱߟ͑Δɻલ‫ظ‬͸ɺ·ͣั৯ऀʔඃ৯ऀϞσϧʹؔͯ͠ղੳΛߦͬͨɻ͜ͷϞσϧ͸ɺิ৭ʔඃ৯ؔ܎ʹΑΔ‫ݸ‬ମ਺ͷมಈΛද‫͢ݱ‬Δ਺ཧϞσϧͰɺओʹั৯ऀͱඃ৯ऀ͕྆ํଘࡏ͍ͯ͠Δঢ়ଶʹ͍ͭͯղੳΛߦͬͨɻ͜ͷঢ়ଶΛ҆ఆঢ়ଶͱ‫͍ݴ‬ɺઢ‫ܗ‬ੑ҆ఆੑղੳ΍ϠίϏߦྻΛ࢖͏͜ͱʹΑͬͯͲͷΑ͏ͳ҆ఆঢ়ଶͰ͋Δ͔ௐ΂ΒΕΔ͜ͱΛֶΜͩɻ·ͨɺීஈͷ਺ֶͰ͸ߦΘͳ͍਺஋‫ࢉܭ‬ͷख๏Ͱ͋ΔΦΠϥʔ๏ΛֶΜͩɻ਺஋‫ࢉܭ‬ΞϧΰϦζϜ͸άϥϑΛϓϩοτ͢ΔࡍʹϓϩάϥϜʹऔΓࠐΜ͕ͩɺ্ख͍͔͘ͳ͍͜ͱ͕ଟ͘ɺԿճ΋վྑ͠ͳ͕Βؒҧ͍Λ‫ݮ‬Β͢౒ྗΛߦͬͨɻࠓ‫ޙ‬ͷ՝୊ͱͯ͠ΑΓޮ཰తͳ਺஋‫ࢉܭ‬ΞϧΰϦζϜΛ໛ࡧ͢Δ͜ͱͰɺϓϩάϥϛϯάʹؔͯ͠ͷཧղ΋ਂ·Δͱ‫ͨ͡ײ‬ɻ‫ظޙ‬͸ɺภඍ෼ํఔࣜͷ‫Ͱܗ‬൓Ԡ֦ࢄ‫ܥ‬ΛղੳΛߦͬͨɻภඍ෼ํఔࣜͷ਺஋‫ࢉܭ‬Λߦ͏ʹ͋ͨͬͯࠩ෼๏ͱ͍͏ख๏ΛֶΜͩɻࠩ෼๏ͷΞϧΰϦζϜΛϓϩάϥϜʹऔΓࠐΉͱɺॲཧ͕࣌ؒͱͯͭ΋ͳ͘௕͘ͳͬͯ͠·͍࡞‫͕ۀ‬ਐ·ͳ͔ͬͨ͜ͱ͔ΒɺΑΓॲཧ͕࣌ؒ୹͘
ͳΔΑ͏ͳ਺஋‫ࢉܭ‬ͷख๏Λௐ΂Δ΂͖ͩͬͨͱ‫ͨ͡ײ‬ɻൃදͷ৔Ͱ͸ɺΦϯϥΠϯͰߦ͏ʹ͋ͨͬͯ੒Ռൃද༻ͷಈըΛࡱӨͨ͠ɻൃද༻ͷεϥΠυʹ
਺ࣜ͹͔Γࡌͤͯ͠·͍ɺ൓Ԡ֦ࢄ‫ʹܥ‬ཧղͷແ͍ਓ͕‫ݟ‬Δʹ͸೉ͭ͘͠·Βͳ͍಺༰ʹͳͬͯ͠Group Report of 2020 SISP

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Mathematical and simulation of complex systems
·͍ͬͯͨͷͰɺ୭͕‫ͯݟ‬΋Θ͔Γ΍͘͢‫ڵ‬ຯΛ࣋ͬͯ΋Β͑ΔΑ͏ͳ޻෉Λߟ͑Δ΂͖ͩͬͨɻ
·ͨɺಈըͰͷൃද΋੠͕ฉ͖औΓͮΒ͔ͬͨΓɺૣ‫͜ͱͳޱ‬Ζ΋͋ͬͨͷͰɺ૬ख͕ฉ͖΍͍͢޻෉Λ͢Δ΂͖ͩͬͨͱߟ͑Δɻ࣭ٙԠ౴Ͱ͸ɺӶ͍಺༰ͷ࣭໰ʹ౴͑Δࡍʹɺ͕͔͔࣌ؒͬͯ͠
·ͬͨΓɺ౴͑ΒΕͳ͍ͱ͍͏͜ͱ͕͋ͬͨɻͲΜͳ࣭໰ʹ΋౴͑ΒΕΔΑ͏ʹΑΓ४උΛ͢Δ΂͖ͩͬͨʹ΋͔͔ΘΒͣɺͦͷ౒ྗΛଵͬͯ͠·͍ͬͯͨ͜ͱ͸໰୊఺ͱͯ͋͛͠ΒΕΔɻ
ϓϩδΣΫτΛ௨ֶͯ͠Μͩ͜ͱ΍໰୊఺͸ࠓ‫ޙ‬ͷଔ‫ʹڀݚۀ‬΋ੜ͔͞ΕΔ͜ͱͩͱࢥ͏ͷͰɺ໰୊఺͸ղܾ͠ɺೋ౓ͱಉ͡ϛεΛ͠ͳ͍Α͏৺ֻ͚ɺϓϩάϥϛϯά΍਺஋‫ࢉܭ‬ͷख๏ͳͲͷֶΜͩ͜ͱ͸ɺଔ‫ʹڀݚۀ‬ੜ͔ͤΔΑ͏ʹ౒ྗ͠ͳ͕Β‫׆‬ಈ͍͖͍ͯͨ͠ͱߟ͍͑ͯΔɻ
ʢ˞จ੹ : ౻ా३ʣGroup Report of 2020 SISP

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Group Number 6

Mathematical and simulation of complex systemsࢀߟจ‫ݙ‬[1] ࡾଜণହ૯‫؂‬म. ϚϨʔ਺ཧੜ෺ֶ Ԡ༻ฤ. ‫ؙ‬ળग़൛, 2016.
[2] ౻ຊ෢จ. ‫ڵ‬ฃ‫͚͓ʹܥ‬Δύλʔϯͷ੍‫ޚ‬. ‫۝‬भେֶେֶӃ૯߹ཧ޻ֶ෎ ྔࢠϓϩηεཧ޻ֶઐ߈ ඇઢ‫ܗ‬෺ੑֶߨ࠲, 2003.

Group Report of 2020 SISP

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Group Number 6