# Romeo & Juliet: Nonlinear Dynamics

“My only love sprung from my only hate!
Too early seen unknown, and known too late!
Prodigious birth of love it is to me
That I must love a loathèd enemy”.

Steven Strogatz first analyzed love affairs via differential equations.

Let’s begin with the linear model for two individuals. An additional constant represents the appeal if positive, the repulsion if negative.

$\frac{dR}{dt}= aR+bJ$

$\frac{dJ}{dt}= cR+dJ$

Specifically, R(t) is the love (or hate) of individual 1 (R= Romeo) for individual 2 (J=Juliet), at time t; J(t) is the love (or hate) of individual 2 for individual 1, at time t. The coefficient a and b specify  the “romantic style” of individual 1 ( Romeo), while c and d that of individual 2 ( Juliet). Some “Romantic Styles” :

1. $\frac{dR}{dt}= aR+bJ$
• 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)

2. $\frac{dJ}{dt}= cR+dJ$

Four subclasses:

•  c > 0, d > 0 (mutual love or hate)
•  c > 0, d < 0 (never-ending cycle)
•  c < 0, d > 0 (never-ending cycle)
•  c < 0, d < 0 (unrequited love)

Classification:

1. Eager Beaver: individual 1 is encouraged by his own feelings as well as that of individual 2;
2. Secure or Cautious  lover: individual 1 retreats from his own feelings but is encouraged by that of individual 2;
3. Hermit: individual 1 retreats from his own feelings and that of individual 2;
4. Narcissistic Nerd: individual 1 wants more of what he feels but retreat from the feelings of individual 2.

Model for the dynamics of love affairs

Romeo is in love with Juliet but, in our story, Juliet is a fickle lover.

We can model the oscillating emotions felt by Romeo and his fair Juliet: the more Romeo loves her, the more Juliet wants to run away. When Romeo gets discouraged, Juliet begins to find him attractive.

Let:

R(t)= Romeo’s love/hate for Juliet at time t;

J(t)= Juliet’s love/hate for Romeo at time t;

If R>0, J signifies love; negative values of R signifies hate. A dynamic model of Romeo and Juliet’s Relationship is:

$\frac{dR}{dt}=aJ$

$\frac{dJ}{dt}=-bJ$

where a,b>0.

The system has a center at (R,J)=(0,0).

Now consider the general linear system:

$\frac{dR}{dt}= aR+bJ$

$\frac{dJ}{dt}= cR+dJ$

where the parameters a,b,c,d may have either sign!

A choice of signs specifies the romantic style.

For example: what happens when two identically cautious lovers get together?

The system is:

$\frac{dR}{dt}= aR+bJ$

$\frac{dJ}{dt}= bR+aJ$

where a is the measure of cautiousness (a<0) and b represents responsiveness (b>0).

The corresponding matrix is:

a         b

b        a

which has: $\tau=2a<0$, $\Delta= a^2-b^2$, $\tau^2- 4\Delta=4b^2>0$

The fixed point ( R,J)=(0,0) is a saddle point if $a^2 and a stable node if $a^2>b^2$.

The eigenvalues and corresponding eigenvectors are:

$\lambda_1= a+b$ ; $v_1= (1,1)$ ; $\lambda_2= a-b$ ; $v_2=(1, -1)$

The two cases:

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References:

– Steven H. Strogatz. Love affairs and differential equations. Mathematics Magazine, 61:35, 1988

– Steven H. Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Westview Press, 2001