 Solution

ID: past-exam-of-the-nonlinear-systems-course-of-the-university-of-oxford/2023/2/a/solution

x = μ y

and

0 = - μx + μ y + y^{3} = y (y^{2} + μ - μ^{2}) .

Thus the origin exists for every real

μ

. The nonzero fixed points are

(x, y) = (\pm μ μ (μ - 1), \pm μ (μ - 1)), μ < 0 or μ > 1,

with the same sign in both positions. At

μ = 0

μ = 1

these expressions coalesce with the origin and do not give additional fixed points. There are no nonzero real fixed points for

0 < μ < 1

 New to topics? Read the docs here!