Diff eq Phase Portrait

[pure quail]

Hi I've been self studying about differential equations and now I'm looking at the phase portraits of the trajectory vectors.
On the diagram attached, the arrows have lengths. Is there any significance in the length of the arrows? All the diagrams that I'm looking at have shorter arrows near the equilibrium points and longer arrows further the equilibrium points. So I've been wondering if the length of the arrows would represent something, but the book that I'm reading doesn't really explain on that. Please help!

[upper schooner]

Well yes, the length of the arrow represents the speed at which the system evolves at different points. The arrows near the equilibrium points are shorter because the system's velocity is lower (trajectories slow down) as they approach or remain near these points. Further from equilibrium, the arrows are longer, indicating a faster rate of change

[pure quail]

oh okay so basically it's the speed at which a particle is moving ie the magnitude of the vector at each point. Thank you!

[pure quail]

oh wait but now looking at some other diagrams I found online, some of them have the same length arrows everywhere. I'm confused now😭

[upper schooner]

Ahh maybe u r seeing a normalized vector fields w equal length arrows
It depends on the type of phase portrait. Some diagrams scale the arrow length to represent the actual speed of the system at different points (longer arrows = faster movement, shorter arrows = slower movement). However, some phase portraits normalize all arrows to the same length so that only the direction of movement is shown, this is often done for clarity so that the focus is on direction rather than speed

[pure quail]

omg okayyy that makes so much more sense thank you so much!!! Really appreciate your help🙏🏻👍🏻

[mellow forge]

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[pure quail]

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