MHB Stability, phase portrait, bifurcations

[ZiniaDuttaGupta]

I am stuck with another one --

Assume that f(x) has the following graph: (for graph please see the attachment)
Consider the (1-dimensional) ODE:

X’ = f(x):

(a) Find all the xed points, and study their stability.

(b) Draw the phase portrait of the system, as well as the graphs of the solutions in all relevant cases.

(c) Study the bifurcations of the ODE

X’ = f(x) + α ; α € R - a parameter.

In particular, determine all the bifurcation values of α , and describe the change in behaviour before during and after each bifurcation. Make sure to draw the appropriate graphs

 

[MarkFL]

I have reason to believe this is part of a graded assignment. Please contact me with your professor's contact information so that I can verify that it is okay for you to receive outside help with this question.

Best Regards,

Mark.