Mathematical Models of Love

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Mathematical Models of Love Happiness

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented to the
• Kiwanis Club of Downtown Madison
• in Madison, Wisconsin
• on June 3, 2002

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Disclaimers
Its Strogatz fault
This is not serious sociology
I wont get to the happiness part
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The Mathematics
R is Romeos love for Juliet (or hate if negative)
J is Juliets love for Romeo
dR/dt aR bJ
a and b describe Romeos Romantic Style
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Some Romantic Styles

• dR/dt aR bJ
• a0 (out of touch with own feelings)
• b0 (oblivious to others feelings)
• agt0, bgt0 (eager beaver)
• agt0, blt0 (narcissistic nerd)
• alt0, bgt0 (cautious lover)
• alt0, blt0 (hermit)

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What about Juliet?
dJ/dt cR dJ
She has her own style
4 parameters with 3 choices for each gives 81
different romantic pairings
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Both out of touch with their own feelings
0

• dR/dt aR bJ
• dJ/dt cR dJ
• Four subclasses
• b gt 0, c gt 0 (mutual love fest or war)
• b gt 0, c lt 0 (never-ending cycle)
• b lt 0, c gt 0 (never-ending cycle)
• b lt 0, c lt 0 (unrequited love)

0
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Out of touch with their own feelings (continued)
b gt 0, c gt 0
b lt 0, c lt 0
b gt 0, c lt 0
War
Two lovers Love fest (or war)
Two nerds Unrequited love
Nerd lover Never-ending cycle
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With Self-Awarenessand bc lt 0 (nerd lover)
a d lt -2bc1/2
a d lt 0
a d gt 0
Extremely cautious Rapid apathy
Somewhat cautious Eventual apathy
Overly eager Growing volatility
(The only equilibrium is apathy)
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Fire and Water(Do opposites attract?)

• Take c -b and d -a
• Result depends on a, c, and the initial
  conditions
• Can end up in any quadrant
• Or with a steady oscillation
• But never apathy

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Peas in a Pod(Are clones bored or blissful?)

• Take c b and d a
• Result depends on a, b, and the initial
  conditions
• Can end up in any quadrant
• Or at the origin (boredom)
• But no oscillations

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Romeo the Robot(How does Juliet react?)

• Take a b 0 (dR/dt 0)
• dJ/dt cR dJ
• There is an equilibrium at J -cR/d
• Can be either love or hate depending on signs of
  R, c, and d
• Stable if d lt 0, unstable if d gt 0
• Her feelings never die
• No oscillations are possible

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A Love Triangle

• Romeo has a mistress, Guinevere
• Guinevere and Juliet dont know about one another
• Romeo responds to each with the same romantic
  style (same a and b)
• Guineveres hate has the same effect on his
  feelings for Juliet as does Juliets love, and
  vice versa

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Love Triangle Examples
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Romeos Fate

• Averaged over all romantic styles (64
  combinations of parameters) and 64 initial
  conditions
• 37 loves Juliet hates Guinevere
• 37 loves Guinevere hates Juliet
• 6 loves both (2 everyone in love)
• 6 hates both (2 everyone in hate)
• 14 apathy (10 everyone apathetic)
• Anything can happen!

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Effect of Nonlinearities
Replace ax with ax(1-x), etc. (logistic
function)
ax(1 - x)
ax
x

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One Chaotic Solution of Nonlinear Love Triangle
Strange attractor of love
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Possible Further Studies

• What happens if Guinevere and Juliet know about
  one another?
• What happens if only Guinevere knows about
  Juliet?
• What happens if Juliet and/or Guinevere has
  another lover?
• What are the dynamics of a free-love commune?
• Is there an optimum pairing of romantic styles
  that encourages success or portends failure?

If such problems interest you, lets collaborate!
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Summary

• Love and happiness are wonderful
• So is mathematics

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References

• http//sprott.physics.wisc.edu/
  lectures/lovehap/ (expanded version of this
  talk)
• Steven H. Strogatz, Nonlinear Dynamics and Chaos
  (Addison-Wesley, 1994)
• sprott_at_physics.wisc.edu