### 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?

0
time 0
time 0

0
time 0
time
0
time
time
From individuals to couples
1
2
0
time
2
time
0
0
time
1
Typical love stories
2
0
2
1
2
0
0
2
1
2
1
0
0
1
2
1
0
1
Second order models
1 = 1 1 , 2
2 = 2 1 , 2
1975
Etienne Guyon [I. Prigogine, J. Boullier]
1978
Steven Strogatz: term paper Sociology 212
The first model
1988
Math. Mag. 61, p.35
1 = 1 2
2 = −2 1
linear oscillator
2
(0)
0
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
1 = −1 (1 , 2 ) + 1 (1 , 2 )+ 1 1 , 2
2 = −2 (1 , 2 ) + 2 (1 , 2 )+ 2 (2 , 1 )
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
1 = −1 (1 , 2 ) + 1 (1 , 2 )+ 1 1 , 2
2 = −2 (1 , 2 ) + 2 (1 , 2 )+ 2 (2 , 1 )
Oblivion
Typically
−1 1 , 2 = −1 1
1
1 0
1 0 exp(−1 )
0
time
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
1 = −1 (1 , 2 ) + 1 (1 , 2 )+ 1 1 , 2
2 = −2 (1 , 2 ) + 2 (1 , 2 )+ 2 (2 , 1 )
Reaction to love
1
1
1
1 =
1 =
1 =
0
secure linear
2
0
secure non-linear
secure ≡ 1 increasing w.r.t. 2
(typically 1 bounded and convex-concave)
2
0
non-secure
2
Three basic mechanisms
Oblivion
Reaction to love
Reaction to appeal
1 = −1 (1 , 2 ) + 1 (1 , 2 )+ 1 1 , 2
2 = −2 (1 , 2 ) + 2 (1 , 2 )+ 2 (2 , 1 )
Reaction to appeal
1
1
1
1 =
1 =
1 =
0
2
0
2
0
2
Classification
Oblivion
Reaction to love
Reaction to appeal
1 = −1 1 + 1 (1 , 2 )+ 1 1 , 2
2 = ...
Secure individual
1 increasing w.r.t. 2
Non-synergic individual
1 and 1 independent on 1
Standard = Secure + Non-synergic
The standard linear model
1998
AMC 95, pp. 181-192
1 = −1 1 + 1 2 + 1 2
2 = −2 2 + 2 1 + 2 1
,  ,  > 0
If individuals are appealing (1 , 2 > 0), the following properties hold:
1 2 > 1 2 implies stability
The equilibrium is unique and strictly positive
The love story is monotonic ( > 0)
An increase of the reactiveness to love ( ) and/or appeal ( ) of individual  produces
an increase of the love of both individuals at equilibrium. Moreover, the relative
increase is higher for individual
5. An increase of the appeal ( ) of individual  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
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
1 = −1 1 + 1 (2 ) + 1 (2 )
2 = −2 2 + 2 (1 ) + 2 (1 )
1 , 2 > 0
A stable negative equilibrium can exist
Isocline 1 = 0
1
1
1 = 1 2 + 1 (2 )
1
1
′ ≤  ′′ ≤ ′′′
2
1∗
2
′′
0
1 2
1
′′′
1
2∗
2∗ =
1
( )
1 1 2
′
Isocline 2 = 0
1
1
2 =
2 1 + 2 (1 )
2
2
2
0
2∗
1
2 1
2
1∗
0
SMS
(Stable Manifold
1
Standard non-linear couples
ROBUST
2
0
FRAGILE
2
′′′
′′′
0
1
1
′
WITH FAVORABLE
EVOLUTION
2
Problem: partition all
couples in equivalent sets
WITH UNFAVORABLE
EVOLUTION
2
′′′
′′′
′′
BIFURCATION
ANALYSIS
′′
0
0
1
′
′
SMS
1
SMS
Catalogue of behaviors
′ ≤  ′′ ≤ ′′′
2
If 1 increases ′ and ′′ collide and disappear
′′′
1∗
′′
′
0
2∗
1
Catalogue of behaviors
′ ≤  ′′ ≤ ′′′
2
If 1 increases ′ and ′′ collide and disappear
If 2 increases ′ and ′′ collide and disappear
′′′
If 1 decreases ′′ and ′′′ collide and disappear
If 2 decreases ′′ and ′′′ collide and disappear
1∗
′′
′
0
2∗
1
Catalogue of behaviors
2
′′
′′′
0
1
SMS
′
at P the origin is on SMS
2
′′′
′′
0
′
If the origin is on the right of SMS then the
couple tends to ′′′ (favorable evolution).
Otherwise the evolution is unfavorable.
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
2
′′′
2 = 0
1∗
′′
′
2∗
1
0
SMS
1 = 0
2 =
1∗ = 1 /2
2∗ = 2 /1
Roxane & Cyrano – with Christian
2
2 = 0

1∗
0
2∗
1
1 = 0
2 = ℎ
1∗ = 1 /2
2∗ = 2 /1
Roxane in convent
2

0
1
Roxane & Cyrano – the expected evolution
2
2 = 0
′′′

1∗
′′
′
2∗
1
0
SMS
1 = 0
2 =
1∗ = 1 /2
2∗ = 2 /1
Roxane & Cyrano – the overall story
2
2 = 0
′′′

1∗
′′
′
2∗
0
1
2∗
SMS
1 = 0
2 =
1∗ = 1 /2
2∗ = 2 /1
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
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)
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
```