 Solution

ID: past-exam-of-the-flows-fluctuations-and-complexity-and-symmetry-and-relativity-paper-of-part-b-of-the-physics-course-of-the-university-of-oxford/2011/3/2/solution

y = - z

x = a z

, and

a z^{2} - cz + a = 0

. Hence

z_{\pm} = \frac{c \pm c ^{2} - 4 a ^{2}}{2 a}, (x_{\pm}, y_{\pm}, z_{\pm}) = (a z_{\pm}, - z_{\pm}, z_{\pm}) .

There are two distinct real equilibrium points only if

c > 2 a

; they merge if

c = 2 a

and there are none if

c < 2 a

. Positivity of

a, c

alone does not guarantee two. Under the later limits, the nearer one is

(x_{-}, y_{-}, z_{-}) ≃ (a^{2} / c, - a / c, a / c)

, while

(x_{+}, y_{+}, z_{+}) ≃ (c, - c / a, c / a)

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