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 I won’t get to the happiness part

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” dR/dt = aR + bJ n a=0 (out of touch with own feelings) n b=0 (oblivious to other’s 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)

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 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

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

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)

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

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

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

A Love Triangle n Romeo has a mistress, Guinevere n Guinevere and Juliet don’t know about one another n Romeo responds to each with the same romantic style (same a and b) n Guinevere’s hate has the same effect on his feelings for Juliet as does Juliet’s love, and vice versa

Love Triangle Examples

Romeo’s Fate n Averaged over all romantic styles (64 combinations of parameters) and 64 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!

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

One Chaotic Solution of Nonlinear Love Triangle “Strange attractor of love”

Possible Further Studies n What happens if Guinevere and Juliet know about one another? n What happens if only Guinevere knows about Juliet? n What happens if Juliet and/or Guinevere has another lover? n What are the dynamics of a free-love commune? n Is there an optimum pairing of romantic styles that encourages success or portends failure? If such problems interest you, let’s collaborate!

Summary n Love and happiness are wonderful n So is mathematics

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