Mathematical model of Love & Happiness
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Mathematical Models of Love & Happiness
J. C. SprottDepartment 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
It’s Strogatz’ fault
This is not serious sociology
I won’t get to the happiness part
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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”
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Some “Romantic Styles”
dR/dt = aR + bJ a=0 (out of touch with own feelings)
b=0 (oblivious to other’s feelings)
a>0, b>0 (eager beaver)
a>0, b<0 (narcissistic nerd)
a<0, b>0 (cautious lover)
a<0, b<0 (hermit)
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What about Juliet?
dJ/dt = cR + dJ
She has her own style
4 parameters with 3 choicesfor each gives 81 different romantic pairings
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Both out of touch with their own feelings
dR/dt = aR + bJ
dJ/dt = cR + dJ
Four subclasses:
b > 0, c > 0 (mutual love fest or war)
b > 0, c < 0 (never-ending cycle)
b < 0, c > 0 (never-ending cycle)
b < 0, c < 0 (unrequited love)
0
0
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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
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With Self-Awarenessand 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)
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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 < 0, unstable if d > 0 Her feelings never die No oscillations are possible
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A Love Triangle
Romeo has a mistress, Guinevere Guinevere and Juliet don’t know
about one another Romeo responds to each with the
same romantic style (same a and b) Guinevere’s hate has the same
effect on his feelings for Juliet as does Juliet’s love, and vice versa
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Love Triangle Examples
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Romeo’s 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 NonlinearitiesReplace ax with ax(1-|x|), etc. (logistic function)
x
axax(1 - |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, let’s 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/love&hap/
(expanded version of this talk)
Steven H. Strogatz, Nonlinear
Dynamics and Chaos (Addison-
Wesley, 1994)