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 Incoming links: Time reversal of an autonomous differential equation

Past exam of the differential equations 1 course of the University of Oxford / 2021 / 3 / b / ii / Solution 2026-10-25

Applying time reversal of an autonomous differential equation to the system from part (i), and renaming its coordinates

(x, y)

, gives

X (x, y) = - (y + 1), Y (x, y) = x^{2} - 1.

The equilibrium points remain

(1, - 1)

and

(- 1, - 1)

. Time reversal of an autonomous differential equation preserves a saddle equilibrium, interchanging its stable and unstable directions; it also reverses the direction around a centre equilibrium. Therefore this system has precisely the required counterclockwise centre at $(1, - 1)$ and saddle at $(- 1, - 1)$ .

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