Question

Problem 1 (20 points) Given the following non-linear autonomous system, || x' = 2cy " || y' = 9-+ y2 : a) What are the equilibrium points? (2 points) b) Can you tell their stability via linearization? If you can, please determine their stability and if they are locally a sink, a port or a source. If you cannot, please explain why. (3 points) c) Please find a first integral of the form f (2,y) = xg (22 + y2) with f (1,0) = -1. Hint: What does the first integral has to verify by definition? (5 points) d) Please sketch the curves that correspond to f (x,y) = -1,0 and 1. What is the shape of the curves that correspond to f (x, y) = a for a € (-3,)? Please sketch the phase space with arrows along the orbits to indicate direction. (3 points) e) From your sketch of the phase space, what is the stability of the equilibrium points? (2 points) f) Find the solution that is at (0,0) at t= 0. What time interval is the solution defined on? Hint: Can the solution be indefinitely prolongated towards the present and the past? (3 points) g) Regarding the other orbits, can we tell from the orbits if the correspondent solutions indefinitely prolongated towards the present and the past? If so, why? Otherwise, why not? (2 points)

Answer 1

x = 2x4 y'=9-x²+42 1. X 2=0, y = -9 w (a) The equilibrium points are, x=0, y'=0 = 22y =o > Boys x'a 2= 0 on 4 = 0 5 :: = 0, T=9 -> x= 13 i points are (3,0) = (-3,0). b) Jo [ at Dal T roy 2x7 I do have any 7 -2 2y] blona ] : Jeu [33] (3,0) = 10- 61co 3670 o-xl 11=7867 :- (3,6) is the stable limit cycle.

重重重重重重重重重重重 (6 0A|| | so » 13 so A = + Fi? "(-3,0) is stable limit cycle. 影 - 278 = =-x+y^ > 24 影 - -x+y | let, i=str>] = 最 能 - - t三 4:1。 o-13 at - == = -1 > 子 - 三 = >三{(-1) -3-7 + S ?z=-9-7749x ?a= \ex-29 >y”= Crz ->- > }* = x - > (3749)= f(x,y) This is the 1st integral of the form whose, F(x,y)= Qqx 1 5478x-9 (too) = -1 > S-9= -1 ael = 0-1=8。(AA)