Question

2. Use the nonlinear system below to answer each part. 5.2 x = y-1, y = -y + 1 -2 (a) Plot the nullclines and indicate the direction of the orbits in each region sepa- rated by the nullclines (do not use software, do this by hand). (b) Find all equilibria for the system. (c) Determine the Jacobian matrix for the system and use it to classify the local stability properties of each equilibrium in part (b).

Answer 1

Solution | an given x = 4.x = frany) y=-9 + S x² = g(ny). 22 4+ m2 (a) ana - Nulleling - given when NY=0 frany)=0 anol y- nullcling given when gmy)=0

-4 +5m? -0 =45m? ho yan Itsna 4+n? 4422 (0100) ar g(nay) as onco nullicling an direction of trajectories known as are given in table. f(and) >0 s nay)=0 / f (my) to glnasho 9(ny) 20 - genay) to

(6) fos finding equilibria point put f(ayl=0 grny)=0 and tolue them. y.n=0 = 4 and y t ro (8) put my in we get = *+5m2 = 0 = (x?s n+4)=0 =) = 1, 4.0 equilibrium points are (0.0), (11) and (4,4). as we already shown in the graph. (C) Ceneral, Jacobian at point P = (ny) (full) fy (f) I n J (1.91 - is given Il Now, P= (0,0) 569. - . :) Real, oqual and an eigenvalug ar I and al, le nogestive in sign,

. (0,0) in asymptotically stable Nude. and s= (11) are real and on solving we get ergenually as 10+ 4 sio which of opponite sign. (1.1) in a saddle point. 10 = (4,4) J1 (44) = slona :) eigenvalue are lot auto they are Real, negatius and unequal : (4,4) is an anymptotically stable node. The graph in past (a) can be. y=542 yn