Question

3) Use the Jacobian to classify each of the equilibrium point(s). (spiral source, nodal sink, etc.) x' = (1 + x) sin(y) y' = 1- * - cos(y)

Please avoid the use of a plotter, but you can graph it by hand. Please explain how you got the equilibrium points in detail and thanks in advance! I've been stuck on this one for a few days.

Answer 1

у 1- (1 +1 ) sin(1) P(uy 8' = 1-4. — єфу eos (1 ) = 4 () Far fixed point plavy) =0 44 см, )=0 (1+1) *** *> siny yant . К k qw) =o -> 1-1- eos Fo =-І V0 1899 oth (= o- 4. f. = ПП =) ТІ во. =h since cosnt 1, neren L- с = 0 И ОН For neven for n Odd If у = 2. y=-=) aps = 2. = (e)soo P4

the critical points are for nevenas (0,07) & for nodd (2,07) Now Jony Pre Py Que Qy siny (1+) costs -1 sing conn) For n even ICOMT) ci 31A 10 NOW IA-13)=0 5 12 41=> dati so the critical point (0,n) , where nis even are centre a stable. ( eigenvalue are purely imaginary) For nodd (02,00) J(2,07) =B - 3 ☺ 3 ) 0 Now IB-d]1=0 ) ²-3=0=> d=13 so the critical point (2, 1), where his old are Saddle 4 unstable ,