Read sections 24 and 25 of Terrell\'s notes. Run pplane (or the software of your

Question

Read sections 24 and 25 of Terrell's notes. Run pplane (or the software of your choice) on the system x' = sin(x + y) y' = sin(xy) How many saddles and spirals can you find, visually? Try to sketch a phase plane which contains two saddles. You are not asked for any formulas, but just to think about what such a thing could look like. Remember the uniqueness theorem, that solutions cannot run into each other. What's rong with this? According to this lecture, if you have a system of 3 or more variables you can get chaos. And, according to Lecture 15, if you have a system of 2 spring-masses, you get 2 Newton's laws or a system of 4 first-order equations. Therefore 2 spring-masses are always chaotic.

Explanation / Answer

1. for the saddle point we'll solve

y' = 0

both the differential equations together

sin(x+y) = 0

sin(xy) = 0

solving the above equations

=> x+y = 0

and xy = 0

x = -y

=> -x^2 = 0

so x = 0

and when x = 0 , y = 0

hence the saddle point is (0,0)

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