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?
𝑥
𝑥
𝑥
0
time 0
time 0
𝑥
𝑥
𝑥
0
time 0
time
0
time
time
From individuals to couples
𝑥1
𝑥2
0
time
𝑥2
time
0
0
time
𝑥1
Typical love stories
𝑥2
0
𝑥2
𝑥1
𝑥2
0
0
𝑥2
𝑥1
𝑥2
𝑥1
0
0
𝑥1
𝑥2
𝑥1
0
𝑥1
Second order models
𝑥1 = 𝑓1 𝑥1 , 𝑥2
𝑥2 = 𝑓2 𝑥1 , 𝑥2
1975
Etienne Guyon [I. Prigogine, J. Boullier]
1978
Steven Strogatz: term paper Sociology 212
The first model
1988
Math. Mag. 61, p.35
𝑥1 = 𝛽1 𝑥2
𝑥2 = −𝛽2 𝑥1
linear oscillator
𝑥2
𝑥(0)
0
𝑥1
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
𝑥1 = −𝐹1 (𝑥1 , 𝑥2 ) + 𝐺1 (𝑥1 , 𝑥2 )+ 𝐻1 𝑥1 , 𝐴2
𝑥2 = −𝐹2 (𝑥1 , 𝑥2 ) + 𝐺2 (𝑥1 , 𝑥2 )+ 𝐻2 (𝑥2 , 𝐴1 )
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
𝑥1 = −𝐹1 (𝑥1 , 𝑥2 ) + 𝐺1 (𝑥1 , 𝑥2 )+ 𝐻1 𝑥1 , 𝐴2
𝑥2 = −𝐹2 (𝑥1 , 𝑥2 ) + 𝐺2 (𝑥1 , 𝑥2 )+ 𝐻2 (𝑥2 , 𝐴1 )
Oblivion
Typically
−𝐹1 𝑥1 , 𝑥2 = −𝛼1 𝑥1
𝑥1
𝑥1 0
𝑥1 0 exp(−𝛼1 𝑡)
0
time
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
𝑥1 = −𝐹1 (𝑥1 , 𝑥2 ) + 𝐺1 (𝑥1 , 𝑥2 )+ 𝐻1 𝑥1 , 𝐴2
𝑥2 = −𝐹2 (𝑥1 , 𝑥2 ) + 𝐺2 (𝑥1 , 𝑥2 )+ 𝐻2 (𝑥2 , 𝐴1 )
Reaction to love
𝐺1
𝐺1
𝐺1
𝑥1 = 𝑐𝑜𝑛𝑠𝑡
𝑥1 = 𝑐𝑜𝑛𝑠𝑡
𝑥1 = 𝑐𝑜𝑛𝑠𝑡
0
secure linear
𝑥2
0
secure non-linear
secure ≡ 𝐺1 increasing w.r.t. 𝑥2
(typically 𝐺1 bounded and convex-concave)
𝑥2
0
non-secure
𝑥2
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
𝑥1 = −𝐹1 (𝑥1 , 𝑥2 ) + 𝐺1 (𝑥1 , 𝑥2 )+ 𝐻1 𝑥1 , 𝐴2
𝑥2 = −𝐹2 (𝑥1 , 𝑥2 ) + 𝐺2 (𝑥1 , 𝑥2 )+ 𝐻2 (𝑥2 , 𝐴1 )
Reaction to appeal
𝐻1
𝐻1
𝐻1
𝑥1 = 𝑐𝑜𝑛𝑠𝑡
𝑥1 = 𝑐𝑜𝑛𝑠𝑡
𝑥1 = 𝑐𝑜𝑛𝑠𝑡
0
𝐴2
0
𝐴2
0
𝐴2
Classification
Oblivion
Reaction to love
Reaction to appeal
𝑥1 = −𝛼1 𝑥1 + 𝐺1 (𝑥1 , 𝑥2 )+ 𝐻1 𝑥1 , 𝐴2
𝑥2 = ...
Secure individual
𝐺1 increasing w.r.t. 𝑥2
Non-synergic individual
𝐺1 and 𝐻1 independent on 𝑥1
Standard = Secure + Non-synergic
The standard linear model
1998
AMC 95, pp. 181-192
𝑥1 = −𝛼1 𝑥1 + 𝛽1 𝑥2 + 𝛾1 𝐴2
𝑥2 = −𝛼2 𝑥2 + 𝛽2 𝑥1 + 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 (𝑥𝑖 > 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
𝑥1 = −𝛼1 𝑥1 + 𝐺1 (𝑥2 ) + 𝐻1 (𝐴2 )
𝑥2 = −𝛼2 𝑥2 + 𝐺2 (𝑥1 ) + 𝐻2 (𝐴1 )
𝐴1 , 𝐴2 > 0
A stable negative equilibrium can exist
Isocline 𝑥1 = 0
1
1
𝑥1 = 𝐺1 𝑥2 + 𝐻1 (𝐴2 )
𝛼1
𝛼1
𝑥 ′ ≤ 𝑥 ′′ ≤ 𝑥′′′
𝑥2
𝑥′′′
𝐴 1∗
𝑥2
𝑥2
0
𝑥′′
0
𝐺1 𝑥2
𝛼1
𝑥1
𝐴 2∗
𝐴 2∗ =
1
𝐻 (𝐴 )
𝛼1 1 2
𝑥′
Isocline 𝑥2 = 0
1
1
𝑥2 =
𝐺2 𝑥1 + 𝐻2 (𝐴1 )
𝛼2
𝛼2
𝐴 2∗
𝑥1
𝐺2 𝑥1
𝛼2
𝐴 1∗
0
SMS
(Stable Manifold
of the Saddle)
𝑥1
Standard non-linear couples
ROBUST
𝑥2
0
FRAGILE
𝑥2
𝑥′′′
𝑥′′′
0
𝑥1
𝑥1
𝑥′
WITH FAVORABLE
EVOLUTION
𝑥2
Problem: partition all
couples in equivalent sets
WITH UNFAVORABLE
EVOLUTION
𝑥2
𝑥′′′
𝑥′′′
𝑥′′
0
BIFURCATION
ANALYSIS
0
𝑥1
𝑥′′
𝑥′
𝑥′
SMS
𝑥1
SMS
Catalogue of behaviors
𝑥 ′ ≤ 𝑥 ′′ ≤ 𝑥′′′
𝑥2
If 𝐴1 increases 𝑥′ and 𝑥′′ collide and disappear
𝑥′′′
𝐴 1∗
𝑥′′
𝑥′
0
𝐴 2∗
𝑥1
Catalogue of behaviors
𝑥 ′ ≤ 𝑥 ′′ ≤ 𝑥′′′
𝑥2
If 𝐴1 increases 𝑥′ and 𝑥′′ collide and disappear
If 𝐴2 increases 𝑥′ and 𝑥′′ collide and disappear
𝑥′′′
If 𝐴1 decreases 𝑥′′ and 𝑥′′′ collide and disappear
If 𝐴2 decreases 𝑥′′ and 𝑥′′′ collide and disappear
𝐴 1∗
𝑥′′
𝑥′
0
𝐴 2∗
𝑥1
SADDLE-NODE BIFURCATIONS
Catalogue of behaviors
𝑥2
𝑥′′′
0
𝑥1
𝑥′′
SMS
𝑥′
at P the origin is on SMS
𝑥2
𝑥′′′
𝑥′′
0
𝑥′
If the origin is on the right of SMS then the
couple tends to 𝑥′′′ (favorable evolution).
Otherwise the evolution is unfavorable.
𝑥1
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
𝑥2
𝑥′′′
𝑥2 = 0
𝐴 1∗
𝑥′′
𝑥′
𝐴 2∗
𝑥1
0
SMS
𝑥1 = 0
𝐴2 = 𝐴𝐶𝑦𝑟
𝐴 1∗ = 𝐴1 /𝛼2
𝐴 2∗ = 𝐴2 /𝛼1
Roxane & Cyrano – with Christian
𝑥2
𝑥2 = 0
𝑃
𝐴 1∗
0
𝐴 2∗
𝑥1
𝑥1 = 0
𝐴2 = 𝐴𝐶ℎ𝑟
𝐴 1∗ = 𝐴1 /𝛼2
𝐴 2∗ = 𝐴2 /𝛼1
Roxane in convent
𝑥2
𝑃
𝑄
0
𝑥1
Roxane & Cyrano – the expected evolution
𝑥2
𝑥2 = 0
𝑥′′′
𝑄
𝐴 1∗
𝑥′′
𝑥′
𝐴 2∗
𝑥1
0
SMS
𝑥1 = 0
𝐴2 = 𝐴𝐶𝑦𝑟
𝐴 1∗ = 𝐴1 /𝛼2
𝐴 2∗ = 𝐴2 /𝛼1
Roxane & Cyrano – the overall story
𝑥2
𝑥2 = 0
𝑥′′′
𝑃
𝑄
𝐴 1∗
𝑥′′
𝑥′
𝐴 2∗
0
𝑥1
𝐴 2∗
SMS
𝑥1 = 0
𝐴2 = 𝐴𝐶𝑦𝑟
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