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There is no simple love story.

If it’s simple, its not love.

If it’s love, its complicated.

How complicated can it get? Well let’s find out!

Let’s consider a love affair between Romeo and Juliet.

We shall use functions R(t) and J(t) to quantify their love:

R(t) –> Romeo’s love/hate for Juliet at time t

J(t) –> Juliet’s love/hate for Romeo at time t

These quantities are positive for love and negative for hate.

The simplest linear relationship that one can construct to describe their love affair is:

These set of primitive equations*** can be used to predict how their relationship is going to evolve with time.

Now if you had taken a course in Linear Algebra then you might know how to solve these system of linear differential equations and obtain the solution. Here’s a quick review:

Remember that we said that Romeo’s love for Juliet evolves with the equation:

Depending on the signs of the coefficients *a* and *b* , Romeo can exhibit one of four romantic styles:

1.** Eager beaver:** a > 0, b > 0 (Romeo is encouraged by his own

feelings as well as Juliet’s.)

2. **Narcissistic nerd: **a > 0, b < 0 (Romeo wants more of what

he feels but retreats from Juliet’s feelings.)

3. **Cautious (or secure) lover:** a < 0, b > 0 (Romeo retreats from

his own feelings but is encouraged by Juliet’s.) *–>Most people*

4. **Hermit:** a < 0, b < 0 (Romeo retreats from his own feelings

as well as Juliet’s.)

And based on the signs of the coefficients *c* and *d* Juliet can also exhibit one of four romantic styles as well.

If we know these coefficients for Romeo and Juliet, we can look compute the** trace (a+d)** and **determinant (ad-bc)**, and look up the Poincaré diagram to predict the future of the Romeo-Juliet relationship:

Let me illustrate with a special case where a = 0 , b = 1, c = (-1), d = 0:

**Trace** = a + d = 0

**Determinant** = ad – bc = 1

Looking up the Poincaré diagram we get a centered orbit:

And we can verify this by using solving this system of linear equation numerically using applications such as *pplane*:

* Plot of Romeo’s love for Juliet v/s Juliet’s love for Romeo*

What this plot implies is that the couple is stuck for eternity in an orbit of love and hate. Wherever the initial point of starting might be on the plane, they always end up orbiting round the origin.

They spend three quadrants of their cycle in a turmoil of love and hate but spend one entire quadrant (R > 0 & J > 0) in true and unconditional love.

*Happy Valentine’s day everybody!*

* This post was Inspired by Strogatz’s Book on Nonlinear Dynamics and Sprott’s paper titled ‘Dynamics of Love affairs’.

** Previous giftboxes: 2016 , 2017

*** This is the simplest system that one can analyze