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Ordinary Differential Equations 16 | Periodic Solutions and Fixed Points

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Опубликовано:

11 июн 2024

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Комментарии : 9

@Ravi-ng3ee 20 дней назад

Ever considered uploading galois theory and algebraic geometry?

@brightsideofmaths 20 дней назад

Yes :)

@turtleomnom2673 19 дней назад

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

@brightsideofmaths 18 дней назад

Same for me :D

@YouTube_username_not_found 20 дней назад

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 20 дней назад

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 20 дней назад

@@RU-vid_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 20 дней назад

Exactly, we measure the distance from the start.

@YouTube_username_not_found 20 дней назад

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.