What are the fixed points and stability of a non-linear system of ODEs?

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The discussion focuses on analyzing a non-linear system of ordinary differential equations (ODEs) defined by X’ = x² - ay and Y’ = y² - y(a). Key tasks include finding the fixed points of the system, studying their stability through linearization, and using vector field analysis when linearization is inconclusive. Additionally, the discussion emphasizes determining bifurcation values for parameter 'a' and drawing phase portraits to illustrate the behavior of the system around these bifurcations.

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I need help with the following so please help me --

Consider the following non-linear system
X’ = x² - ay
Y’ = y² - y(a) Find the fixed points of this system. (depending on a, there may be different fixed points!)

(b) Study stability of each fixed point via linearization. In the case the linearization is inconclusive, use directions of vector field analysis (or any other information contained in the equations) to show stability/instability.

(c) Use the information above to determine the bifurcation values for a, and draw the phase portraits for the system before, at, and after each bifurcation. On phase portraits identify the fixed points as well as their stable/unstable manifolds (curves) where appropriate.
 
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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.
 

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