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Mathematical Models of Love & Happiness
Disclaimers It’s Strogatz’ fault This is not serious sociology I won’t get to the happiness part
The Mathematics R  is Romeo’s love for Juliet (or hate if negative) J  is Juliet’s love for Romeo dR/dt = aR + bJ a  and  ...
Some “Romantic Styles” <ul><li>dR/dt = aR + bJ </li></ul><ul><li>a =0 (out of touch with own feelings) </li></ul><ul><li>b...
What about Juliet? dJ/dt = cR + dJ She has her own style 4 parameters with 3 choices for each gives 81 different  romantic...
Both out of touch with their own feelings <ul><li>dR/dt = aR + bJ </li></ul><ul><li>dJ/dt = cR + dJ </li></ul><ul><li>Four...
Out of touch with their own feelings (continued) b  > 0,  c  > 0 b  < 0,  c  < 0 b  > 0,  c <  0 Two lovers Love fest (or ...
With Self-Awareness and  bc  < 0 (nerd + lover) a  +  d  < -2| bc | 1/2 a  +  d  < 0 a  +  d  > 0 Extremely cautious Rapid...
Fire and Water (Do opposites attract?) <ul><li>Take  c  = - b  and  d  = - a </li></ul><ul><li>Result depends on  a, c,  a...
Peas in a Pod (Are clones bored or blissful?) <ul><li>Take  c  =  b  and  d  =  a </li></ul><ul><li>Result depends on  a, ...
Romeo the Robot (How does Juliet react?) <ul><li>Take  a  =  b  = 0  ( dR/dt =  0) </li></ul><ul><li>dJ/dt = cR + dJ </li>...
A Love Triangle <ul><li>Romeo has a mistress, Guinevere </li></ul><ul><li>Guinevere and Juliet don’t know about one anothe...
Love Triangle Examples
Romeo’s Fate <ul><li>Averaged over all romantic styles (64 combinations of parameters) and 64 initial conditions: </li></u...
Effect of Nonlinearities Replace  ax  with  ax (1-| x |), etc. (logistic function) x ax ax (1 - | x |)
One Chaotic Solution of Nonlinear Love Triangle “ Strange attractor of love”
Possible Further Studies <ul><li>What happens if Guinevere and Juliet know about one another?  </li></ul><ul><li>What happ...
Summary <ul><li>Love and happiness are wonderful </li></ul><ul><li>So is mathematics </li></ul>
References <ul><li>http://sprott.physics.wisc.edu/ lectures/love&hap/   (expanded version of this talk) </li></ul><ul><li>...
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  1. 1. Mathematical Models of Love & Happiness
  2. 2. Disclaimers It’s Strogatz’ fault This is not serious sociology I won’t get to the happiness part
  3. 3. The Mathematics R is Romeo’s love for Juliet (or hate if negative) J is Juliet’s love for Romeo dR/dt = aR + bJ a and b describe Romeo’s “Romantic Style”
  4. 4. Some “Romantic Styles” <ul><li>dR/dt = aR + bJ </li></ul><ul><li>a =0 (out of touch with own feelings) </li></ul><ul><li>b =0 (oblivious to other’s feelings) </li></ul><ul><li>a >0, b >0 (eager beaver) </li></ul><ul><li>a >0, b <0 (narcissistic nerd) </li></ul><ul><li>a <0, b >0 (cautious lover) </li></ul><ul><li>a <0, b <0 (hermit) </li></ul>
  5. 5. What about Juliet? dJ/dt = cR + dJ She has her own style 4 parameters with 3 choices for each gives 81 different romantic pairings
  6. 6. Both out of touch with their own feelings <ul><li>dR/dt = aR + bJ </li></ul><ul><li>dJ/dt = cR + dJ </li></ul><ul><li>Four subclasses: </li></ul><ul><ul><li>b > 0, c > 0 (mutual love fest or war) </li></ul></ul><ul><ul><li>b > 0, c < 0 (never-ending cycle) </li></ul></ul><ul><ul><li>b < 0, c > 0 (never-ending cycle) </li></ul></ul><ul><ul><li>b < 0, c < 0 (unrequited love) </li></ul></ul>0 0
  7. 7. Out of touch with their own feelings (continued) b > 0, c > 0 b < 0, c < 0 b > 0, c < 0 Two lovers Love fest (or war) Two nerds Unrequited love Nerd + lover Never-ending cycle War
  8. 8. With Self-Awareness and bc < 0 (nerd + lover) a + d < -2| bc | 1/2 a + d < 0 a + d > 0 Extremely cautious Rapid apathy Somewhat cautious Eventual apathy Overly eager Growing volatility (The only equilibrium is apathy)
  9. 9. Fire and Water (Do opposites attract?) <ul><li>Take c = - b and d = - a </li></ul><ul><li>Result depends on a, c, and the initial conditions </li></ul><ul><li>Can end up in any quadrant </li></ul><ul><li>Or with a steady oscillation </li></ul><ul><li>But never apathy </li></ul>
  10. 10. Peas in a Pod (Are clones bored or blissful?) <ul><li>Take c = b and d = a </li></ul><ul><li>Result depends on a, b, and the initial conditions </li></ul><ul><li>Can end up in any quadrant </li></ul><ul><li>Or at the origin (boredom) </li></ul><ul><li>But no oscillations </li></ul>
  11. 11. Romeo the Robot (How does Juliet react?) <ul><li>Take a = b = 0 ( dR/dt = 0) </li></ul><ul><li>dJ/dt = cR + dJ </li></ul><ul><li>There is an equilibrium at J = -cR/d </li></ul><ul><li>Can be either love or hate depending on signs of R, c, and d </li></ul><ul><li>Stable if d < 0, unstable if d > 0 </li></ul><ul><li>Her feelings never die </li></ul><ul><li>No oscillations are possible </li></ul>
  12. 12. A Love Triangle <ul><li>Romeo has a mistress, Guinevere </li></ul><ul><li>Guinevere and Juliet don’t know about one another </li></ul><ul><li>Romeo responds to each with the same romantic style (same a and b ) </li></ul><ul><li>Guinevere’s hate has the same effect on his feelings for Juliet as does Juliet’s love, and vice versa </li></ul>
  13. 13. Love Triangle Examples
  14. 14. Romeo’s Fate <ul><li>Averaged over all romantic styles (64 combinations of parameters) and 64 initial conditions: </li></ul><ul><ul><li>37% loves Juliet & hates Guinevere </li></ul></ul><ul><ul><li>37% loves Guinevere & hates Juliet </li></ul></ul><ul><ul><li>6% loves both (2% everyone in love) </li></ul></ul><ul><ul><li>6% hates both (2% everyone in hate) </li></ul></ul><ul><ul><li>14% apathy (10% everyone apathetic) </li></ul></ul><ul><li>Anything can happen! </li></ul>
  15. 15. Effect of Nonlinearities Replace ax with ax (1-| x |), etc. (logistic function) x ax ax (1 - | x |)
  16. 16. One Chaotic Solution of Nonlinear Love Triangle “ Strange attractor of love”
  17. 17. Possible Further Studies <ul><li>What happens if Guinevere and Juliet know about one another? </li></ul><ul><li>What happens if only Guinevere knows about Juliet? </li></ul><ul><li>What happens if Juliet and/or Guinevere has another lover? </li></ul><ul><li>What are the dynamics of a free-love commune? </li></ul><ul><li>Is there an optimum pairing of romantic styles that encourages success or portends failure? </li></ul>If such problems interest you, let’s collaborate!
  18. 18. Summary <ul><li>Love and happiness are wonderful </li></ul><ul><li>So is mathematics </li></ul>
  19. 19. References <ul><li>http://sprott.physics.wisc.edu/ lectures/love&hap/ (expanded version of this talk) </li></ul><ul><li>Steven H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994) </li></ul><ul><li>[email_address] </li></ul>

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