x(0) - Politecnico di Milano-DEIB
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Transcript x(0) - Politecnico di Milano-DEIB
ROMANTIC RELATIONSHIPS
IN STANDARD COUPLES
Sergio Rinaldi
DEI, Politecnico di Milano, Milano, Italy
EEP IIASA, Laxenburg, Austria
Can we graphically describe the evolution of a love story?
π₯
π₯
π₯
0
time 0
time 0
π₯
π₯
π₯
0
time 0
time
0
time
time
From individuals to couples
π₯1
π₯2
0
time
π₯2
time
0
0
time
π₯1
Typical love stories
π₯2
0
π₯2
π₯1
π₯2
0
0
π₯2
π₯1
π₯2
π₯1
0
0
π₯1
π₯2
π₯1
0
π₯1
Second order models
π₯1 = π1 π₯1 , π₯2
π₯2 = π2 π₯1 , π₯2
1975
Etienne Guyon [I. Prigogine, J. Boullier]
1978
Steven Strogatz: term paper Sociology 212
The first model
1988
Math. Mag. 61, p.35
π₯1 = π½1 π₯2
π₯2 = βπ½2 π₯1
linear oscillator
π₯2
π₯(0)
0
π₯1
Two criticisms:
1. Why x(0) β 0 ?
2. Why the asymptotic behavior depends on
the intial conditions?
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
π₯1 = βπΉ1 (π₯1 , π₯2 ) + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2
π₯2 = βπΉ2 (π₯1 , π₯2 ) + πΊ2 (π₯1 , π₯2 )+ π»2 (π₯2 , π΄1 )
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
π₯1 = βπΉ1 (π₯1 , π₯2 ) + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2
π₯2 = βπΉ2 (π₯1 , π₯2 ) + πΊ2 (π₯1 , π₯2 )+ π»2 (π₯2 , π΄1 )
Oblivion
Typically
βπΉ1 π₯1 , π₯2 = βπΌ1 π₯1
π₯1
π₯1 0
π₯1 0 exp(βπΌ1 π‘)
0
time
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
π₯1 = βπΉ1 (π₯1 , π₯2 ) + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2
π₯2 = βπΉ2 (π₯1 , π₯2 ) + πΊ2 (π₯1 , π₯2 )+ π»2 (π₯2 , π΄1 )
Reaction to love
πΊ1
πΊ1
πΊ1
π₯1 = ππππ π‘
π₯1 = ππππ π‘
π₯1 = ππππ π‘
0
secure linear
π₯2
0
secure non-linear
secure β‘ πΊ1 increasing w.r.t. π₯2
(typically πΊ1 bounded and convex-concave)
π₯2
0
non-secure
π₯2
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
π₯1 = βπΉ1 (π₯1 , π₯2 ) + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2
π₯2 = βπΉ2 (π₯1 , π₯2 ) + πΊ2 (π₯1 , π₯2 )+ π»2 (π₯2 , π΄1 )
Reaction to appeal
π»1
π»1
π»1
π₯1 = ππππ π‘
π₯1 = ππππ π‘
π₯1 = ππππ π‘
0
π΄2
0
π΄2
0
π΄2
Classification
Oblivion
Reaction to love
Reaction to appeal
π₯1 = βπΌ1 π₯1 + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2
π₯2 = ...
Secure individual
πΊ1 increasing w.r.t. π₯2
Non-synergic individual
πΊ1 and π»1 independent on π₯1
Standard = Secure + Non-synergic
The standard linear model
1998
AMC 95, pp. 181-192
π₯1 = βπΌ1 π₯1 + π½1 π₯2 + πΎ1 π΄2
π₯2 = βπΌ2 π₯2 + π½2 π₯1 + πΎ2 π΄1
πΌπ , π½π , πΎπ > 0
If individuals are appealing (π΄1 , π΄2 > 0), the following properties hold:
πΌ1 πΌ2 > π½1 π½2 implies stability
The equilibrium is unique and strictly positive
The love story is monotonic (π₯π > 0)
An increase of the reactiveness to love (π½π ) and/or appeal (πΎπ ) of individual π produces
an increase of the love of both individuals at equilibrium. Moreover, the relative
increase is higher for individual π
5. An increase of the appeal (π΄π ) of individual π produces an increase in the love of both
individuals at equilibrium. Moreover, the relative increase is higher for the partner
6. The dominant time constant increases with π½π
7. In a community of N+N individuals there is no tendency to exchange the partner if and
only if the i-th most attractive woman is coupled with the i-th most attractive man
1.
2.
3.
4.
Standard non-linear couples
1998
NDPLS 2, pp. 283-301
π₯1 = βπΌ1 π₯1 + πΊ1 (π₯2 ) + π»1 (π΄2 )
π₯2 = βπΌ2 π₯2 + πΊ2 (π₯1 ) + π»2 (π΄1 )
π΄1 , π΄2 > 0
A stable negative equilibrium can exist
Isocline π₯1 = 0
1
1
π₯1 = πΊ1 π₯2 + π»1 (π΄2 )
πΌ1
πΌ1
π₯ β² β€ π₯ β²β² β€ π₯β²β²β²
π₯2
π₯β²β²β²
π΄ 1β
π₯2
π₯2
0
π₯β²β²
0
πΊ1 π₯2
πΌ1
π₯1
π΄ 2β
π΄ 2β =
1
π» (π΄ )
πΌ1 1 2
π₯β²
Isocline π₯2 = 0
1
1
π₯2 =
πΊ2 π₯1 + π»2 (π΄1 )
πΌ2
πΌ2
π΄ 2β
π₯1
πΊ2 π₯1
πΌ2
π΄ 1β
0
SMS
(Stable Manifold
of the Saddle)
π₯1
Standard non-linear couples
ROBUST
π₯2
0
FRAGILE
π₯2
π₯β²β²β²
π₯β²β²β²
0
π₯1
π₯1
π₯β²
WITH FAVORABLE
EVOLUTION
π₯2
Problem: partition all
couples in equivalent sets
WITH UNFAVORABLE
EVOLUTION
π₯2
π₯β²β²β²
π₯β²β²β²
π₯β²β²
0
BIFURCATION
ANALYSIS
0
π₯1
π₯β²β²
π₯β²
π₯β²
SMS
π₯1
SMS
Catalogue of behaviors
π₯ β² β€ π₯ β²β² β€ π₯β²β²β²
π₯2
If π΄1 increases π₯β² and π₯β²β² collide and disappear
π₯β²β²β²
π΄ 1β
π₯β²β²
π₯β²
0
π΄ 2β
π₯1
Catalogue of behaviors
π₯ β² β€ π₯ β²β² β€ π₯β²β²β²
π₯2
If π΄1 increases π₯β² and π₯β²β² collide and disappear
If π΄2 increases π₯β² and π₯β²β² collide and disappear
π₯β²β²β²
If π΄1 decreases π₯β²β² and π₯β²β²β² collide and disappear
If π΄2 decreases π₯β²β² and π₯β²β²β² collide and disappear
π΄ 1β
π₯β²β²
π₯β²
0
π΄ 2β
π₯1
SADDLE-NODE BIFURCATIONS
Catalogue of behaviors
π₯2
π₯β²β²β²
0
π₯1
π₯β²β²
SMS
π₯β²
at P the origin is on SMS
π₯2
π₯β²β²β²
π₯β²β²
0
π₯β²
If the origin is on the right of SMS then the
couple tends to π₯β²β²β² (favorable evolution).
Otherwise the evolution is unfavorable.
π₯1
SMS
P
Cyrano de Bergerac β Edmond Rostand (1868-1918)
Cyrano de Bergerac
Roxane
ACyr = - 2
ARox = 0.6
Gérard Depardieu
Anne Brochet
Christian de Neuvillette
AChr = 1
Vincent Perez
Cyrano de Bergerac (1990) - Jean-Paul Rappeneau
Roxane & Cyrano β without Christian
π₯2
π₯β²β²β²
π₯2 = 0
π΄ 1β
π₯β²β²
π₯β²
π΄ 2β
π₯1
0
SMS
π₯1 = 0
π΄2 = π΄πΆπ¦π
π΄ 1β = π΄1 /πΌ2
π΄ 2β = π΄2 /πΌ1
Roxane & Cyrano β with Christian
π₯2
π₯2 = 0
π
π΄ 1β
0
π΄ 2β
π₯1
π₯1 = 0
π΄2 = π΄πΆβπ
π΄ 1β = π΄1 /πΌ2
π΄ 2β = π΄2 /πΌ1
Roxane in convent
π₯2
π
π
0
π₯1
Roxane & Cyrano β the expected evolution
π₯2
π₯2 = 0
π₯β²β²β²
π
π΄ 1β
π₯β²β²
π₯β²
π΄ 2β
π₯1
0
SMS
π₯1 = 0
π΄2 = π΄πΆπ¦π
π΄ 1β = π΄1 /πΌ2
π΄ 2β = π΄2 /πΌ1
Roxane & Cyrano β the overall story
π₯2
π₯2 = 0
π₯β²β²β²
π
π
π΄ 1β
π₯β²β²
π₯β²
π΄ 2β
0
π₯1
π΄ 2β
SMS
π₯1 = 0
π΄2 = π΄πΆπ¦π
π΄ 1β = π΄1 /πΌ2
π΄ 2β = π΄2 /πΌ1
OTHER STANDARD
COUPLES
La belle et la bête (1740)
Jeanne-Marie Leprince de Beaumont (1711-1780)
Beauty and the Beast (1991) - Walt Disney
Pride and Prejudice (1813) β Jane Austen (1775-1817)
Elizabeth Bennet
Fitzwilliam Darcy
Keira Knightley
Matthew Macfadyen
Pride & Prejudice (2005) - Joe Wright
NON-STANDARD
COUPLES
Gone with the Wind (1936) β Margaret Mitchell (1900-1949)
Scarlett O'Hara
Rhett Butler
Vivien Leigh
Clark Gable
Gone with the Wind (1939) - Victor Fleming
Il Canzoniere (1366) β Francesco Petrarca (1304-1374)
Laura de Sade
Francesco Petrarca
Francescoβs love
1.5
1
0.5
0
-0.5
-1
Time [years]
0
5
10
15
20
Jules et Jim (1953) β Henri-Pierre Roché (1879-1959)
Jules
Kathe
Jim
Oskar Werner
Jeanne Moreau
Henri Serre
Jules et Jim (1962) - François Truffaut