Why do phase trajectories point upwards and downwards in a quadratic potential?

[Lambda96]

Hi,

I am currently preparing for my exam and have just watched a video about motion in phase space.



From minute 4 a quadratic potential is introduced and then from minute 6 minute the phase trajectory.

Here are the pictures

quadratic potential

phase trajectory

Regarding phase trajectory on the left side, I understand that these are pointing downwards, but I don't understand why the phase trajectory on the right side is pointing upwards, I would say that these must also be pointing downwards, as the potential is unstable and no matter where I place a test particle, it will always accelerate downwards towards the potential.

 

[Frabjous]

Take a point on the curve with v₀<0, x₀>a. At the next instant in time, 0>v₁>v₀ and a<x₁<x₀ so the curve is ”pointed“ upwards.