What kind of bifuraction/fixed point is this?

[retrofit81]

Homework Statement

Consider the ODE system:

dx/dt = x * (1 - x - d*y)
dy/dt = g * y * (1 - y)

where d and g are parameters.

I'm trying to find and classify the bifurcations of this system. The parameter g doesn't appear to influence any bifurcation, only the parameter d does. I have found there is a bifurcation when d = 1, but I'm having trouble classifying it. The attempt at a solution

If you graph the nullclines of the system there are always fixed points at (0,0) (a source) and (1,0) (a saddle). Additionally there are two more fixed points at (0,1) and (1 - d, 1).

When d < 1, (0,1) is a saddle and (1-d, 1) is a sink.

As d increases, the sink slowly moves to the left toward the saddle.

When d = 1, the two fixed points coalesce into one. I cannot determine what type of fixed point this is. The associated eigenvalues are 0 and g. Some trajectories go into the fixed point, and others seem to be repelled. Intuition says this is a stable line of equilibria, but I'm not sure.

When d > 1, there are again two fixed points. The former sink is now a saddle and the former saddle is now a sink, i.e. they've switched stability. As d increases, the saddle continues to move along the same line the sink did when d < 1.The fixed point has a bifurcation at d = 1 but it seems like it's a saddle-node (two FP become one) AND a pitchfork (stability switch). Is that possible? If you can also let me know what kind of fixed point (1,0) when d = 1, I would really appreciate it!

 

[mighty2000]

Thank you for sharing your work on this ODE system and your attempts at finding and classifying the bifurcations. From your description, it seems like you have a good understanding of the behavior of the system for different values of d.

To answer your question, it is indeed possible for a bifurcation to exhibit characteristics of both a saddle-node and a pitchfork bifurcation. This is known as a saddle-node bifurcation of limit cycles, where a limit cycle (in this case, the fixed point at (1,0)) disappears at the bifurcation point (d=1) and at the same time, the stability of the fixed point switches from a saddle to a sink. This type of bifurcation is commonly observed in systems with two parameters, where one parameter controls the existence of a limit cycle and the other controls its stability.

As for the type of fixed point at (1,0) when d=1, it is a stable fixed point with one eigenvalue at 0 and the other at g. This means that some trajectories will converge to this fixed point (those with initial conditions close to it) while others will be repelled (those with initial conditions farther away). This behavior is known as a saddle-focus fixed point.

I hope this helps clarify your understanding of the bifurcations in this system. Keep up the good work in your research!