x(0) - Politecnico di Milano-DEIB

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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 = 𝑓1 π‘₯1 , π‘₯2
π‘₯2 = 𝑓2 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 = 𝛽1 π‘₯2
π‘₯2 = −𝛽2 π‘₯1
linear oscillator
π‘₯2
π‘₯(0)
0
π‘₯1
Two criticisms:
1. Why x(0) ≠ 0 ?
2. Why the asymptotic behavior depends on
the intial conditions?
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
π‘₯1 = −𝐹1 (π‘₯1 , π‘₯2 ) + 𝐺1 (π‘₯1 , π‘₯2 )+ 𝐻1 π‘₯1 , 𝐴2
π‘₯2 = −𝐹2 (π‘₯1 , π‘₯2 ) + 𝐺2 (π‘₯1 , π‘₯2 )+ 𝐻2 (π‘₯2 , 𝐴1 )
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
π‘₯1 = −𝐹1 (π‘₯1 , π‘₯2 ) + 𝐺1 (π‘₯1 , π‘₯2 )+ 𝐻1 π‘₯1 , 𝐴2
π‘₯2 = −𝐹2 (π‘₯1 , π‘₯2 ) + 𝐺2 (π‘₯1 , π‘₯2 )+ 𝐻2 (π‘₯2 , 𝐴1 )
Oblivion
Typically
−𝐹1 π‘₯1 , π‘₯2 = −𝛼1 π‘₯1
π‘₯1
π‘₯1 0
π‘₯1 0 exp(−𝛼1 𝑑)
0
time
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
π‘₯1 = −𝐹1 (π‘₯1 , π‘₯2 ) + 𝐺1 (π‘₯1 , π‘₯2 )+ 𝐻1 π‘₯1 , 𝐴2
π‘₯2 = −𝐹2 (π‘₯1 , π‘₯2 ) + 𝐺2 (π‘₯1 , π‘₯2 )+ 𝐻2 (π‘₯2 , 𝐴1 )
Reaction to love
𝐺1
𝐺1
𝐺1
π‘₯1 = π‘π‘œπ‘›π‘ π‘‘
π‘₯1 = π‘π‘œπ‘›π‘ π‘‘
π‘₯1 = π‘π‘œπ‘›π‘ π‘‘
0
secure linear
π‘₯2
0
secure non-linear
secure ≡ 𝐺1 increasing w.r.t. π‘₯2
(typically 𝐺1 bounded and convex-concave)
π‘₯2
0
non-secure
π‘₯2
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
π‘₯1 = −𝐹1 (π‘₯1 , π‘₯2 ) + 𝐺1 (π‘₯1 , π‘₯2 )+ 𝐻1 π‘₯1 , 𝐴2
π‘₯2 = −𝐹2 (π‘₯1 , π‘₯2 ) + 𝐺2 (π‘₯1 , π‘₯2 )+ 𝐻2 (π‘₯2 , 𝐴1 )
Reaction to appeal
𝐻1
𝐻1
𝐻1
π‘₯1 = π‘π‘œπ‘›π‘ π‘‘
π‘₯1 = π‘π‘œπ‘›π‘ π‘‘
π‘₯1 = π‘π‘œπ‘›π‘ π‘‘
0
𝐴2
0
𝐴2
0
𝐴2
Classification
Oblivion
Reaction to love
Reaction to appeal
π‘₯1 = −𝛼1 π‘₯1 + 𝐺1 (π‘₯1 , π‘₯2 )+ 𝐻1 π‘₯1 , 𝐴2
π‘₯2 = ...
Secure individual
𝐺1 increasing w.r.t. π‘₯2
Non-synergic individual
𝐺1 and 𝐻1 independent on π‘₯1
Standard = Secure + Non-synergic
The standard linear model
1998
AMC 95, pp. 181-192
π‘₯1 = −𝛼1 π‘₯1 + 𝛽1 π‘₯2 + 𝛾1 𝐴2
π‘₯2 = −𝛼2 π‘₯2 + 𝛽2 π‘₯1 + 𝛾2 𝐴1
𝛼𝑖 , 𝛽𝑖 , 𝛾𝑖 > 0
If individuals are appealing (𝐴1 , 𝐴2 > 0), the following properties hold:
𝛼1 𝛼2 > 𝛽1 𝛽2 implies stability
The equilibrium is unique and strictly positive
The love story is monotonic (π‘₯𝑖 > 0)
An increase of the reactiveness to love (𝛽𝑖 ) and/or appeal (𝛾𝑖 ) of individual 𝑖 produces
an increase of the love of both individuals at equilibrium. Moreover, the relative
increase is higher for individual 𝑖
5. An increase of the appeal (𝐴𝑖 ) of individual 𝑖 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 𝛽𝑖
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 = −𝛼1 π‘₯1 + 𝐺1 (π‘₯2 ) + 𝐻1 (𝐴2 )
π‘₯2 = −𝛼2 π‘₯2 + 𝐺2 (π‘₯1 ) + 𝐻2 (𝐴1 )
𝐴1 , 𝐴2 > 0
A stable negative equilibrium can exist
Isocline π‘₯1 = 0
1
1
π‘₯1 = 𝐺1 π‘₯2 + 𝐻1 (𝐴2 )
𝛼1
𝛼1
π‘₯ ′ ≤ π‘₯ ′′ ≤ π‘₯′′′
π‘₯2
𝐴 1∗
π‘₯2
π‘₯′′
0
𝐺1 π‘₯2
𝛼1
π‘₯′′′
π‘₯1
𝐴 2∗
𝐴 2∗ =
1
𝐻 (𝐴 )
𝛼1 1 2
π‘₯′
Isocline π‘₯2 = 0
1
1
π‘₯2 =
𝐺2 π‘₯1 + 𝐻2 (𝐴1 )
𝛼2
𝛼2
π‘₯2
0
𝐴 2∗
π‘₯1
𝐺2 π‘₯1
𝛼2
𝐴 1∗
0
SMS
(Stable Manifold
of the Saddle)
π‘₯1
Standard non-linear couples
ROBUST
π‘₯2
0
FRAGILE
π‘₯2
π‘₯′′′
π‘₯′′′
0
π‘₯1
π‘₯1
π‘₯′
WITH FAVORABLE
EVOLUTION
π‘₯2
Problem: partition all
couples in equivalent sets
WITH UNFAVORABLE
EVOLUTION
π‘₯2
π‘₯′′′
π‘₯′′′
π‘₯′′
BIFURCATION
ANALYSIS
π‘₯′′
0
0
π‘₯1
π‘₯′
π‘₯′
SMS
π‘₯1
SMS
Catalogue of behaviors
π‘₯ ′ ≤ π‘₯ ′′ ≤ π‘₯′′′
π‘₯2
If 𝐴1 increases π‘₯′ and π‘₯′′ collide and disappear
π‘₯′′′
𝐴 1∗
π‘₯′′
π‘₯′
0
𝐴 2∗
π‘₯1
Catalogue of behaviors
π‘₯ ′ ≤ π‘₯ ′′ ≤ π‘₯′′′
π‘₯2
If 𝐴1 increases π‘₯′ and π‘₯′′ collide and disappear
If 𝐴2 increases π‘₯′ and π‘₯′′ collide and disappear
π‘₯′′′
If 𝐴1 decreases π‘₯′′ and π‘₯′′′ collide and disappear
If 𝐴2 decreases π‘₯′′ and π‘₯′′′ collide and disappear
𝐴 1∗
π‘₯′′
π‘₯′
0
𝐴 2∗
π‘₯1
SADDLE-NODE BIFURCATIONS
Catalogue of behaviors
π‘₯2
π‘₯′′
π‘₯′′′
0
π‘₯1
SMS
π‘₯′
at P the origin is on SMS
π‘₯2
π‘₯′′′
π‘₯′′
0
π‘₯′
If the origin is on the right of SMS then the
couple tends to π‘₯′′′ (favorable evolution).
Otherwise the evolution is unfavorable.
π‘₯1
SMS
P
Cyrano de Bergerac – 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 – without Christian
π‘₯2
π‘₯′′′
π‘₯2 = 0
𝐴 1∗
π‘₯′′
π‘₯′
𝐴 2∗
π‘₯1
0
SMS
π‘₯1 = 0
𝐴2 = π΄πΆπ‘¦π‘Ÿ
𝐴 1∗ = 𝐴1 /𝛼2
𝐴 2∗ = 𝐴2 /𝛼1
Roxane & Cyrano – with Christian
π‘₯2
π‘₯2 = 0
𝑃
𝐴 1∗
0
𝐴 2∗
π‘₯1
π‘₯1 = 0
𝐴2 = π΄πΆβ„Žπ‘Ÿ
𝐴 1∗ = 𝐴1 /𝛼2
𝐴 2∗ = 𝐴2 /𝛼1
Roxane in convent
π‘₯2
𝑃
𝑄
0
π‘₯1
Roxane & Cyrano – the expected evolution
π‘₯2
π‘₯2 = 0
π‘₯′′′
𝑄
𝐴 1∗
π‘₯′′
π‘₯′
𝐴 2∗
π‘₯1
0
SMS
π‘₯1 = 0
𝐴2 = π΄πΆπ‘¦π‘Ÿ
𝐴 1∗ = 𝐴1 /𝛼2
𝐴 2∗ = 𝐴2 /𝛼1
Roxane & Cyrano – the overall story
π‘₯2
π‘₯2 = 0
π‘₯′′′
𝑃
𝑄
𝐴 1∗
π‘₯′′
π‘₯′
𝐴 2∗
0
π‘₯1
𝐴 2∗
SMS
π‘₯1 = 0
𝐴2 = π΄πΆπ‘¦π‘Ÿ
𝐴 1∗ = 𝐴1 /𝛼2
𝐴 2∗ = 𝐴2 /𝛼1
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) – 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) – Margaret Mitchell (1900-1949)
Scarlett O'Hara
Rhett Butler
Vivien Leigh
Clark Gable
Gone with the Wind (1939) - Victor Fleming
Il Canzoniere (1366) – Francesco Petrarca (1304-1374)
Laura de Sade
Francesco Petrarca
Francesco’s love
1.5
1
0.5
0
-0.5
-1
Time [years]
0
5
10
15
20
Jules et Jim (1953) – Henri-Pierre Roché (1879-1959)
Jules
Kathe
Jim
Oskar Werner
Jeanne Moreau
Henri Serre
Jules et Jim (1962) - François Truffaut
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