Bifurcation Analysis for the ODE x' = \mux - x2 + x4

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The discussion focuses on analyzing the bifurcation points of the ordinary differential equation x' = μx - x² + x⁴. Participants emphasize the need to find equilibria, calculate the Jacobian, and assess the stability of these equilibria to identify bifurcation points. The initial equilibrium found is x = 0, but there is uncertainty about how to locate additional equilibria. The importance of sketching a bifurcation diagram to illustrate stability and bifurcation points is highlighted, along with the necessity of verifying conditions from bifurcation theorems. Overall, the conversation revolves around the standard methods for conducting bifurcation analysis in this context.
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Homework Statement


Consider the ODE
x' = \mux - x2 + x4
where x \in R and \mu \in R is a parameter.
Find and identify all bifurcation points for this equation. Sketch a bifurcation diagram, showing clearly the stability of all equilibria and the location of the bifurcation points.
You may identify any bifurcations you find from the bifurcation diagram but you must also check the conditions from any bifurcation theorems.

Homework Equations





The Attempt at a Solution


Is it just the same-old way.
1) Find equilibria and the Jacobian and from the Jacobian find stability of equilbria etc and if not what do I do.
 
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x = 0 is an equilibria but how do I find the others.
 

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