Connecting Orbits (Heteroclinic Orbits) Between Fixed Points on a Phase Portrait

Question

I was given a ODE system in which I'm needing to solve several things. The ODE is:

$v' = v(v-1) + \frac{5}{3}(e - \frac{3}{50}v)$

$e' = v[-\frac{(v-1)^2}{2} + e - \frac{3}{50} + \frac{1}{10}(v-1)]$

The questions I'm needing to solve for:

• Find and classify the fixed points ($\checkmark$) --> My fixed points at $(e,v)$ are:

  1. $(0,0)$ -- (unstable)

  2. $(\frac{3}{50},1)$ -- (saddle point)

  3. $(\frac{9}{50},\frac{3}{5})$ -- (source)

• Create the phase portrait ($\checkmark$) -- posted below
• Build a boundary value solver ($\checkmark$) --> we were required to not use a build-in solver, so we had to build our own. If needed/desired I can post the code for our solver.
• Solve for the connecting orbits

My phase portrait with the fixed points plotted.

My question lies with how to solve for the connecting orbits. I believe I understand the concept, however I want to explain it just in case I'm missing something.

Basically I want to plot a function that follows the phase portrait trends starting at one fixed point and ending at another (called a heteroclinic orbit I believe). This will require a boundary condition, an initial guess, and a function to define our guess in. I'll need to implement this in python, so I'm hoping for an explanation of how this is done, and a fair explanation of how to implement it (whether psuedo-code or another example). Also, some clarification on these questions would be awesome:

(Q1) I was told that a good initial guess as the function that would connect our fixed points would be $tan(x)$. Why is that? And how would I implement that in a solver?

(Q2) Is a boundary condition necessary so that our function knows where to start and stop to connect our orbits? Would it then be that our boundary conditions are:

• $Y_a$ = fixed pt. #1
• $Y_b$ = fixed pt. #2

and our solver would just run between those 2 points? Or am I mistaken on the role of the boundary functions?

(Q3) What exactly would be a good value to use as an initial guess? Just any value between the 2 fixed points? Or should I use a midpoint?

Thanks in advance for the help!