Question

Question 410 marks Consider the nonlinear system ェ=(1-y)2(4-12), ỳ=(1-z)y(y2-4) (0<x<2, o<y<2), which has a single fixed point at (1,1) (a) Show that the following expression for K(x, y) is a constant of motion for this system: K(x, y)- 2 ln(ry) + Inl( 2)(y- 2)]-3In(2)(y+2)]. (b) Use the constant of motion to show that the fixed point is a centre of the nonlinear system.

Answer 1

(a) k = 2 ln (x y) + ln [(x - 2) (y - 2)] - 3 ln [(x + 2) (y + 2)] To show this is constant of motion, we have to show dk/dt = 0 [where z = time] d k/dt = 2/x y (xy + yx) + x(y - 2) + y(x - 2)/(x - 2)(y - 2) - 3[x(y + 2) + y(x + 2)/(x + 2) (y + 2) = 2/xy [(1 - y) 4(4 - x^2) y + (1- x) y (y^2 - 4) x] + (1 - y) x (4 - x^2) (y - 2) + (x - 2) (1 - x) y(y^2 - 4)/(x - 2) (y - 2) -3 (x - y) (y + 2) x (4 - x^2) + (1 - x) y (y^2 - 4) (y + 2)/(x + 2) (y + 2) [by inserting given value of x and y] = 2[(1 - y) (4 - x^2) + (1 - x)(y^2 - 4)] + x(1 - y) (4 - x^2)/(x - 2) + (1 - x) y9y^2 - 4)/(y - 2) -3 (1 - y) x (4 - x^2)/(x + 2) - 3(1 - x) y (y^2 - 4)/(y + 2) = (1 - y)