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
• UW-Madison Math Club
• in Madison, Wisconsin
• on October 21, 2002

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Disclaimers
Its Strogatz fault
This is not serious psychology
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The Mathematics
R is Romeos love for Juliet (or hate if negative)
J is Juliets love for Romeo
The simplest linear model is
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?
She has her own style
dJ/dt cR dJ
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 Equations

• dRJ/dt aRJ b(J - G)
• dJ/dt cRJ dJ
• dRG/dt aRG b(G - J)
• dG/dt eRG fG
• System is 4D (4 variables)
• There are 6 parameters
• System is linear (no chaos)

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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) (logistic function)
ax(1 - x)
ax
x

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New kinds of Dynamics
New equilibrium points
Limit cycles
(but no chaos in 2D)
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One Chaotic Solution of Nonlinear Love Triangle
Strange attractor of love
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Simple 2D Linear Model

• dR/dt aR bJ
• dJ/dt cR dJ

• d2R/dt2 bdR/dt w2R 0
• b -a - d (damping)
• w2 ad - bc (frequency)

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Solutions of 2-D Linear System
Time
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Happiness Model

• d2x/dt2 bdx/dt w2x F(t)
• Happiness H dx/dt
• Habituation
• Acclimation
• Adaptation
• Only changes are perceived

Damping
Oscillation
External forces
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What is x?

• x integral of H
• x is what others perceive
• In the love model x is what the other feels
• H (your happiness) must average to zero (with
  positive damping)
• x does not average to zero

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Winning the Lottery
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Drug or Other Addiction
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Intermittent Reinforcement
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Random Events
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Real Life
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Parameter Space
b
w
2
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Nonlinearities

• There are many possibilities.
• Try bdx/dt --gt b(1 - x2)dx/dt
• This gives growth for small x and damping for
  large x.
• The resulting equation was studied by van der Pol
  (1926).
• Oscillations occur even without an external
  force.
• It has been used to model a variety of nonlinear
  oscillators.

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Bipolar Behavior
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Periodically Driven Chaos
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Happiness Attractor
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Autonomous Chaos
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Happiness Attractor 2
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Some Implications

• Constant happiness is an unrealistic goal.
• Others see less volatility in you and often
  wrongly conclude how you feel.
• Individuals can be categorized by their values of
  b and w.
• Bipolar disorders may correspond to negative or
  small b.
• Long prison terms may be ineffective.

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Some other happiness studies

• Brickman, Coates Janoff-Bulman (1978) report
  only small differences in life satisfaction
  between paraplegics, control subjects, and
  lottery winners.
• Lykken (1981) reports that religious people are
  not noticeably happier than freethinkers.
• Diener Diener (1996) review studies indicating
  that all American socioeconomic groups score
  above neutral in life satisfaction, as do people
  with severe disabilities.

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Studies involving disabilities

• Hellmich (1995) reports that 84 of individuals
  with extreme quadriplegia say that their life is
  average or above average.
• Delespaul DeVries (1987) report that people
  with chronic mental problems claim positive
  well-being.

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Studies of the dynamics

• Silver (1982) reports that individuals with
  spinal cord injuries are very unhappy immediately
  following their injury, but that 58 state that
  happiness is their strongest emotion by the third
  week after their injuries.
• Suh, Diener, Fujita (1996) report that good and
  bad events have almost no effect on happiness
  after 6 months.

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In Summary ...(Lykken 1999)

• There seem to be no permanent ups and downs
  natural selection has made us this way, because,
  by accommodating to both adversity and to good
  fortune in this fashion, we remain more
  productive, more adaptable to changing
  circumstances, and more likely to have viable
  offspring.

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Other Similar Qualities

• Sense of wealth
• Health
• Beauty
• Intelligence
• Spirituality
• Skills
• Senses
• hot/cold
• smell
• vision
• hearing ...

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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/ (This talk)
• Steven H. Strogatz, Nonlinear Dynamics and Chaos
  (Addison-Wesley, 1994)
• sprott_at_physics.wisc.edu
• Collaborations are welcome!