The equations that fall in love!

Probably you are like me and you want to gift your girl friend/wife/other loved ones never ending and perpetual love. I will tell you what I did. So obviously I wanted some thing however far you go from it you come back and fall in love ? so as a romantic I symbolize love by the symbol of a heart ! That reduced my problem very much so I need an oscillator which has a limit cycle which looks like a heart.

Hmm. You might say I know the function that looks like heart but how would you make a differential equation which has a limit cycle which looks the same ? So here is the simple answer : ‘co-ordinate transformation’. You want your equations to be like love you better go to the world of love.

The method I am describing here is pretty general so you can create an oscillator that looks like a square, circle or anything circlish (A topological circle, after all you need a limit cycle) . What you need to do is the following.

1. Write down the equation that describe your favourite curve
2. Remember the fact that $\dot{r} = r(1-r)$ and $\dot{\theta}=1$ has a limit cycle at $r=1$
3. Therefore, replace $r$ by the equation of the favourite curve and $theta$ by the $ArcTan(x/y)$
4. Solve and get the differential equation.

An oscillator is a set of ODEs which gives oscillatory solutions (like a simple pendulum). A limit cycle is when for any nearby point in phase space(space where x and y represent the states and no time is in it) you come back to the same oscillating ‘circle’ in the phase space (not like a simple pendulum but like our solution here!)

So I did the same for heart equations. So what did I get? The answer is the following.

$\dot{y} = -(2 (x^{12} y+6 x^{10} y^3-6 x^{10} y+15 x^8 y^5-2 x^8 y^4-30 x^8 y^3+15 x^8 y+3 x^7+20 x^6 y^7-6 x^6 y^6-60 x^6 y^5+6 x^6 y^4+60 x^6 y^3-21 x^6 y+9 x^5 y^2-6 x^5+15 x^4 y^9-6 x^4 y^8-59 x^4 y^7+12 x^4 y^6+90 x^4 y^5-6 x^4 y^4-63 x^4 y^3+18 x^4 y+9 x^3 y^4-x^3 y^3-12 x^3 y^2+3 x^3+6 x^2 y^{11}-2 x^2 y^{10}-30 x^2 y^9+6 x^2 y^8+60 x^2 y^7-6 x^2 y^6-63 x^2 y^5+3 x^2 y^4+36 x^2 y^3-9 x^2 y+3 x y^6-x y^5-6 x y^4+3 x y^2+y^{13}-6 y^{11}+15 y^9-21 y^7+18 y^5-9 y^3+2 y)/(6 x^6+18 x^4 y^2-12 x^4+18 x^2 y^4-5 x^2 y^3-24 x^2 y^2+6 x^2+6 y^6-12 y^4+6 y^2$

and

$\dot{x} = -\frac{-\frac{3 x^2 y^2}{2 \sqrt{\left(x^2+y^2-1\right)^3-x^2 y^3}}+\frac{3 y \left(x^2+y^2-1\right)^2}{\sqrt{\left(x^2+y^2-1\right)^3-x^2 y^3}}+\frac{x \left(x^2 y^3-\left(x^2+y^2-1\right)^3+1\right) \sqrt{\left(x^2+y^2-1\right)^3-x^2 y^3}}{y^2 \left(\frac{x^2}{y^2}+1\right)}}{-\frac{x \left(\frac{3 x \left(x^2+y^2-1\right)^2}{\sqrt{\left(x^2+y^2-1\right)^3-x^2 y^3}}-\frac{x y^3}{\sqrt{\left(x^2+y^2-1\right)^3-x^2 y^3}}\right)}{y^2 \left(\frac{x^2}{y^2}+1\right)}-\frac{\frac{3 y \left(x^2+y^2-1\right)^2}{\sqrt{\left(x^2+y^2-1\right)^3-x^2 y^3}}-\frac{3 x^2 y^2}{2 \sqrt{\left(x^2+y^2-1\right)^3-x^2 y^3}}}{y \left(\frac{x^2}{y^2}+1\right)}}$

And the beautiful solution of these equations is given below. Can you notice the love in the phase space and all the trajectories that fall towards it ?

And of course what better to gift someone you love !! 🙂