8.2: Problem Set

Exercise \(\PageIndex{1}\)

Consider the following autonomous vector fields on the plane depending on a scalar parameter \(\mu\). Verify that each vector field has a fixed point at (x, y) = (0, 0) for \(\mu = 0\). Determine the linearized stability of this fixed point. Determine the nature (i.e. stability and number) of the fixed points for μ in a neighborhood of zero. (In other words, carry out a bifurcation analysis.) Sketch the flow in a neighborhood of each fixed point for values of μ corresponding to changes in stability and/or numbers of fixed points.

1. \(\dot{x} = \mu+10x^2\),

   \(\dot{y} = x-5y\).

2. \(\dot{x} = \mu x+10x^2\),

   \(\dot{y} = x-2y\).

3. \(\dot{x} = \mu x+x^5\),

   \(\dot{y} = y\).