Ordinary Differential Equations 16 | Periodic Solutions and Fixed Points

[@Ravi-ng3ee]

Ever considered uploading galois theory and algebraic geometry?

[@brightsideofmaths]

Yes :)

[@YouTube_username_not_found]

The 3rd type of solutions to the pendulum equation, is it really injective? 🤔 The pendulum willl return to the same position after some amount of turns. Thus, we could say that those solutions are periodic in some sense. Maybe what's happening here is due to the fact that I am changing the domain of the map V from R^2 to A*R where A is the set of equivalence classes of reals modulo 2pi .

[@YouTube_username_not_found]

Even the true phase portrait in this case would be different. One should transform the x1-axis into a circle in order for the visualization to be faithful.

[@IlTrojo]

@@KZbin_username_not_found Precisely. In the "periodic" solutions the pendulum actually goes through the same interval of angle values in a single (-pi,pi) interval, while in the "injective" solutions the angle never goes back and it spans multiple intervals. Hence the mathematical representation is indeed injective.

[@brightsideofmaths]

Exactly, we measure the distance from the start.

[@YouTube_username_not_found]

Yeah, 😃indeed! When one takes x1 to be the spanned distance (or the spanned angle, really they are the same, just divide by radius), everything adds up.

[@turtleomnom2673]

Every time I think I’m done with math, 15 more topics pop up.

[@brightsideofmaths]

Same for me :D

[@Agrover112]

+1