kiwanis

Mathematical Models of Love & Happiness: 

Mathematical Models of Love andamp; 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) aandgt;0, bandgt;0 (eager beaver) aandgt;0, bandlt;0 (narcissistic nerd) aandlt;0, bandgt;0 (cautious lover) aandlt;0, bandlt;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 dR/dt = aR + bJ dJ/dt = cR + dJ Four subclasses: b andgt; 0, c andgt; 0 (mutual love fest or war) b andgt; 0, c andlt; 0 (never-ending cycle) b andlt; 0, c andgt; 0 (never-ending cycle) b andlt; 0, c andlt; 0 (unrequited love) 0 0

Out of touch with their own feelings (continued): 

Out of touch with their own feelings (continued) b andgt; 0, c andgt; 0 b andlt; 0, c andlt; 0 b andgt; 0, c andlt; 0 Two lovers Love fest (or war) Two nerds Unrequited love Nerd + lover Never-ending cycle War

With Self-Awarenessand bc < 0 (nerd + lover): 

With Self-Awareness and bc andlt; 0 (nerd + lover) a + d andlt; -2|bc|1/2 a + d andlt; 0 a + d andgt; 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 andlt; 0, unstable if d andgt; 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

Love Triangle Examples: 

Love Triangle Examples

Romeo’s Fate: 

Romeo’s Fate Averaged over all romantic styles (64 combinations of parameters) and 64 initial conditions: 37% loves Juliet andamp; hates Guinevere 37% loves Guinevere andamp; 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) x ax ax(1 - |x|)

One Chaotic Solution of Nonlinear Love Triangle: 

One Chaotic Solution of Nonlinear Love Triangle 'Strange attractor of love'

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: 

References http://sprott.physics.wisc.edu/ lectures/loveandamp;hap/ (expanded version of this talk) Steven H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994) sprott@physics.wisc.edu