Homoclinic bifurcation -- Saddle-saddle connection, separatrix

[Deimantas]

Homework Statement

The equation:

where A is the parameter.
1) Find the value of parameter A at which the homoclinic bifurcation occurs.
2)Find the separatrices
3)Calculate the first integral and find the saddle-saddle connection equation.

Homework Equations

The Attempt at a Solution

First order system:

There are two saddles at (-1,0), (1,0) and a spiral at (0,0). When parameter A is decreased below 0, a stable limit cycle occurs. The limit cycle ceases to exist at approx. A=-0.34. The analytical result turns out to be A=-12/35.

However, I'm having trouble finding the separatrices. I'm using forward and backward integration to plot many of the solutions in MATLAB and it is possible to notice where the separatrix should be, but I don't know how to plot the exact separatrix. Still working on that.

Lastly, I tried computing the first integral of the system by dividing (dx/dt) / (dy/dt). After integrating and substituting the saddle values into the equation, I get that the constant of integration equals (-1/4). I then insert this value of the constant into the equation and get this equation:

which I'm pretty sure is wrong. Since saddle connecton should happen at around A=-(12/35), I insert this value, but the graphical plot looks like nonsense to me.

I kindly await any help with questions 2) and 3). Thank you

 

[mighty2000]

for reading and I hope we can figure this out together.
Thank you for your question! It seems like you are on the right track with your analysis. Here are some suggestions for finding the separatrices and calculating the first integral:

2) To find the separatrices, you can use the fact that they correspond to the solutions of the system that tend to the saddle points at (-1,0) and (1,0). This means that the separatrices will be the curves that pass through these two points. You can use the initial conditions x(-1,0)=(-1,0) and x(1,0)=(1,0) to numerically solve the system and plot the separatrices.

3) To calculate the first integral, you can use the method of separation of variables. This involves separating the variables x and y on opposite sides of the equation and then integrating both sides. This should give you an equation of the form F(x,y)=constant, where F is a function of x and y. To find the saddle-saddle connection equation, you can substitute the values of the saddle points (-1,0) and (1,0) into this equation and solve for the constant. This should give you the equation of the separatrix connecting the two saddle points.

I hope this helps and good luck with your analysis! Let me know if you have any further questions.