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

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The discussion focuses on finding fixed points and analyzing the stability of a non-linear system of ordinary differential equations (ODEs) defined by X’ = x² - ay and Y’ = y² - y(a). It emphasizes the dependency of fixed points on the parameter 'a' and suggests using linearization for stability analysis, with alternative methods like vector field analysis if linearization is inconclusive. Additionally, it addresses the need to determine bifurcation values for 'a' and to create phase portraits illustrating fixed points and their stability characteristics. The request for professor contact information indicates that this inquiry may be part of a graded assignment.
ZiniaDuttaGupta
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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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