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. Write down the equation that describe your favourite curve Remember the fact that $ = r(1-r)$ and $ =1$ has a limit cycle at r = 1 Therefore, replace r by the equation of the favourite curve and theta by the ArcTan(x/y) 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. $ = -(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 $ = -{2 }+{}+}{y^2 \left({y^2}+1\right)}}{-{}-{}\right)}{y^2 \left({y^2}+1\right)}-{}-{2 }}{y \left({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 ?