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Mathematical Models of Love & Happiness J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to the Chaos and Complex Systems Seminar in Madison, Wisconsin on February 6, 2001

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Outline n Love model - Inspired by Steve Strogatz (Cornell University) n Happiness model - In collaboration with Keith Warren (Ohio State Univ)

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Simple Linear Model n dR/dt = aR + bJ n dJ/dt = cR + dJ n where u R is Romeos love for Juliet u J is Juliets love for Romeo u (or hate if negative) u a, b, c, d are constants that determine the Romantic styles

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Limitations of Model n Its difficult to quantify and measure love and hate. n Love is not a scalar (different types). n Parameters change in time and with the situation. n Parameters may be different for love and hate. n There are always other variables.

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Some Romantic Styles dR/dt = aR + bJ n a=0 (out of touch with own feelings) n b=0 (oblivious to others feelings) n a>0, b>0 (eager beaver) n a>0, b<0 (narcissistic nerd) n a 0 (cautious lover) n a<0, b<0 (hermit)

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Number of Pairings n 6 styles for Romeo X 6 styles for Juliet = 36 different pairings. n Only 21 give unique dynamics (because of R/J symmetry) but… Its actually worse than that: n 4 parameters with 3 choices (-,0,+) for each gives 3 4 = 81 combinations of which 45 are unique n And there are subclasses depending on values and initial conditions.

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Both out of touch with their own feelings n dR/dt = aR + bJ n dJ/dt = cR + dJ n Four subclasses: u b > 0, c > 0 (mutual love fest or war) u b > 0, c < 0 (never-ending cycle) u b 0 (never-ending cycle) u b < 0, c < 0 (unrequited love) 0 0

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Out of touch with their own feelings (continued) b > 0, c > 0b < 0, c < 0b > 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-Awareness and bc < 0 (nerd + lover) a + d < -2|bc| 1/2 a + d < 0a + 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?) n Take c = -b and d = -a n Result depends on a, c, and the initial conditions n Can end up in any quadrant n Or with a steady oscillation n But never apathy

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Peas in a Pod (Are clones bored or blissful?) n Take c = b and d = a n Result depends on a, b, and the initial conditions n Can end up in any quadrant n Or at the origin (boredom) n But no oscillations

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Romeo the Robot (How does Juliet react?) n Take a = b = 0 (dR/dt = 0) n dJ/dt = cR + dJ n There is an equilibrium at J = -cR/d n Can be either love or hate depending on signs of R, c, and d n Stable if d 0 n Her feelings never die n No oscillations are possible

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

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New kinds of Dynamics New equilibrium pointsLimit cycles (but no chaos in 2D)

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A Love Triangle n Romeo has a mistress, Guinevere n Guinevere and Juliet dont know about one another n Romeo responds to each with the same romantic style (same a and b) n 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 n dR J /dt = aR J + b(J - G) n dJ/dt = cR J + dJ n dR G /dt = aR G + b(G - J) n dG/dt = eR G + fG n System is 4D (4 variables) n There are 6 parameters n System is linear (no chaos)

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Linear Love Triangle Examples

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Romeos Fate n Averaged over all romantic styles (combinations of parameters) and initial conditions: u 37% loves Juliet & hates Guinevere u 37% loves Guinevere & hates Juliet u 6% loves both (2% everyone in love) u 6% hates both (2% everyone in hate) u 14% apathy (10% everyone apathetic) n Anything can happen!

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One Chaotic Solution of Nonlinear Love Triangle a,b,c,f > 0; d,e < 0 (Romeo is an eager beaver) Strange attractor of love

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Possible Further Studies n What happens if Guinevere and Juliet know about one another? (6D system) n What happens if only Guinevere knows about Juliet? (5D system, asymmetric) n What happens if Juliet and/or Guinevere has another lover? (6D or 8D system) n What are the dynamics of a free-love commune? (large-D system) n 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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Simple 2D Linear Model n dR/dt = aR + bJ n dJ/dt = cR + dJ d 2 R/dt 2 + dR/dt + 2 R = 0 = -a - d (damping) 2 = ad - bc (frequency)

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Solutions of 2-D Linear System

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d 2 x/dt 2 + dx/dt + 2 x = F(t) n Happiness: H = dx/dt u Habituation u Acclimation u Adaptation n Only changes are perceived Happiness Model DampingOscillationExternal forces

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What is x ? n x = integral of H n x is what others perceive n H (your happiness) must average to zero (with positive damping) n 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 2

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Some Implications n Constant happiness is an unrealistic goal. n Others see less volatility in you and often wrongly conclude how you feel. Individuals can be categorized by their values of and. Manic depression may correspond to ~ 0. n Long prison terms may be ineffective.

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A few other happiness studies n Brickman, Coates & Janoff-Bulman (1978) report only small differences in life satisfaction between paraplegics, control subjects, and lottery winners. n Lykken (1981) reports that religious people are not noticeably happier than freethinkers. n 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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What disabilities, you ask? n Hellmich (1995) reports that 84% of individuals with extreme quadriplegia say that their life is average or above average. n Delespaul & DeVries (1987) report that people with chronic mental problems claim positive well-being.

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As for the dynamics n 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. n 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) n 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 n Sense of wealth n Health n Intelligence n Skills n Senses u hot/cold u smell u vision u hearing...

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Summary n Love and happiness are wonderful n So is mathematics

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References n lectures/love&hap/ (This talk) lectures/love&hap/ n Steven H. Strogatz, Nonlinear Dynamics and Chaos (Addison- Wesley, 1994) n

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