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 = 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 = −π½2 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 = −πΉ1 (π₯1 , π₯2 ) + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2 π₯2 = −πΉ2 (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 = −πΉ1 (π₯1 , π₯2 ) + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2 π₯2 = −πΉ2 (π₯1 , π₯2 ) + πΊ2 (π₯1 , π₯2 )+ π»2 (π₯2 , π΄1 ) Oblivion Typically −πΉ1 π₯1 , π₯2 = −πΌ1 π₯1 π₯1 π₯1 0 π₯1 0 exp(−πΌ1 π‘) 0 time Three basic mechanisms Oblivion Reaction to love Reaction to appeal π₯1 = −πΉ1 (π₯1 , π₯2 ) + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2 π₯2 = −πΉ2 (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 ≡ 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 = −πΉ1 (π₯1 , π₯2 ) + πΊ1 (π₯1 , π₯2 )+ π»1 π₯1 , π΄2 π₯2 = −πΉ2 (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 = −πΌ1 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 = −πΌ1 π₯1 + π½1 π₯2 + πΎ1 π΄2 π₯2 = −πΌ2 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 = −πΌ1 π₯1 + πΊ1 (π₯2 ) + π»1 (π΄2 ) π₯2 = −πΌ2 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₯ ′ ≤ π₯ ′′ ≤ π₯′′′ π₯2 π΄ 1∗ π₯2 π₯′′ 0 πΊ1 π₯2 πΌ1 π₯′′′ π₯1 π΄ 2∗ π΄ 2∗ = 1 π» (π΄ ) πΌ1 1 2 π₯′ Isocline π₯2 = 0 1 1 π₯2 = πΊ2 π₯1 + π»2 (π΄1 ) πΌ2 πΌ2 π₯2 0 π΄ 2∗ π₯1 πΊ2 π₯1 πΌ2 π΄ 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 π΄1 increases π₯′ and π₯′′ collide and disappear π₯′′′ π΄ 1∗ π₯′′ π₯′ 0 π΄ 2∗ π₯1 Catalogue of behaviors π₯ ′ ≤ π₯ ′′ ≤ π₯′′′ π₯2 If π΄1 increases π₯′ and π₯′′ collide and disappear If π΄2 increases π₯′ and π₯′′ collide and disappear π₯′′′ If π΄1 decreases π₯′′ and π₯′′′ collide and disappear If π΄2 decreases π₯′′ and π₯′′′ collide and disappear π΄ 1∗ π₯′′ π₯′ 0 π΄ 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 π΄ 1∗ π₯′′ π₯′ π΄ 2∗ π₯1 0 SMS π₯1 = 0 π΄2 = π΄πΆπ¦π π΄ 1∗ = π΄1 /πΌ2 π΄ 2∗ = π΄2 /πΌ1 Roxane & Cyrano – with Christian π₯2 π₯2 = 0 π π΄ 1∗ 0 π΄ 2∗ π₯1 π₯1 = 0 π΄2 = π΄πΆβπ π΄ 1∗ = π΄1 /πΌ2 π΄ 2∗ = π΄2 /πΌ1 Roxane in convent π₯2 π π 0 π₯1 Roxane & Cyrano – the expected evolution π₯2 π₯2 = 0 π₯′′′ π π΄ 1∗ π₯′′ π₯′ π΄ 2∗ π₯1 0 SMS π₯1 = 0 π΄2 = π΄πΆπ¦π π΄ 1∗ = π΄1 /πΌ2 π΄ 2∗ = π΄2 /πΌ1 Roxane & Cyrano – the overall story π₯2 π₯2 = 0 π₯′′′ π π π΄ 1∗ π₯′′ π₯′ π΄ 2∗ 0 π₯1 π΄ 2∗ SMS π₯1 = 0 π΄2 = π΄πΆπ¦π π΄ 1∗ = π΄1 /πΌ2 π΄ 2∗ = 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) – 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
