The original system in (x,c,q):
The transformed system in (u,v,r):
The Jacobian of the transformed system (u,v,r):
Clearly J13=0 and J33=0.
The eigenvalues of (u,v,r):
The Phase Diagrams of the 2-D (u,v) system:
The phase diagram at this scale reveals little.
Zooming on the critical point: u=0, v=0 and placing the nullclines:
Zooming still further:
The 2-D approximation above assumes that r^3*u is small.
My understanding of this diagram is: the critical point (u=0, v=0) may be approached for u>0 and v>0, but not for u<0 and v<0. In terms of the original system, it means that both x and c must be rising and cannot be falling. Is this the correct interpretation?
The 3-D transformed system (u,v,r) exhibits 1 negative eigenvalue for 2 positive eigenvalues at the critical point (u=0,v=0,r=b), which points to a one-dimensional stable manifold onto which the critical point may be approached, for a given r(0)=0 and free u(0) and v(0). The simulation of the 3-D system does indeed yield a converging path, for a very small error tolerance, confirming the insights garnered from the 2-D approximation.
I simulated u(t), v(t), r(t), and then transformed back to x(t), c(t), q(t):
The following is the plot of x(t):
The following is the plot of c(t):
The following is the plot of q(t):
It is clear from the system that the only way, if any, to approach the critical point x=xs, c=cs, q=b is for q<b and that as the system is approaching the critical point, it is most likely that q will rise. The above shows q falling a little initially.
It seems that the critical/stationary point may be approached on a one-dimensional manifold, but only for r>rs and u>0 and v>0 or -- in terms of the original system, only for x>xs and both x and c rising during the transition. For the physical interpretation of my system, that's a little odd, so that's why I'm posting here to seek reassurance about my interpretation.
The system is tricky since the stationary/critical point is not actually defined as v->0 (i.e. xdot->0).
Any pointers will be greatly appreciated. Thanks.