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Published on November 27, 2008

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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 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 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 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-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 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 & 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) 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/love&hap/ (expanded version of this talk) Steven H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, 1994) sprott@physics.wisc.edu

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