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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. Disclaimers. It’s Strogatz’ fault. This is not serious sociology.

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mathematical models of love happiness

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

disclaimers
Disclaimers

It’s Strogatz’ fault

This is not serious sociology

I won’t get to the happiness part

the mathematics
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”

some romantic styles
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)
what about juliet
What about Juliet?

dJ/dt = cR + dJ

She has her own style

4 parameters with 3 choices

for each gives 81 different

romantic pairings

both out of touch with their own feelings
Both out of touch with their own feelings

0

  • 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

out of touch with their own feelings continued
Out of touch with their own feelings (continued)

b > 0, c > 0

b < 0, c < 0

b > 0, c < 0

War

Two lovers

Love fest (or war)

Two nerds

Unrequited love

Nerd + lover

Never-ending cycle

with self awareness and bc 0 nerd lover
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)

fire and water do opposites attract
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
peas in a pod are clones bored or blissful
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
romeo the robot how does juliet react
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
a love triangle
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
romeo s fate
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!
effect of nonlinearities
Effect of Nonlinearities

Replace ax with ax(1-|x|), etc. (logistic function)

ax(1 - |x|)

ax

x

possible further studies
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!

summary
Summary
  • Love and happiness are wonderful
  • So is mathematics
references
http://sprott.physics.wisc.edu/ lectures/love&hap/ (expanded version of this talk)

Steven H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994)

[email protected]

References
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