(System of ODEs) Autonomous nonlinear systems Consider the following nonlinear s

Question

(System of ODEs) Autonomous nonlinear systems

Consider the following nonlinear system:

a) Find all the critical points of this system

b) Linearize the system about each of its critical point(s). Disscuss whether the linearised system can be used to approximate the behaviour of the non-linear system.

c) For the critical point in the third quadrant:

i) Determine the general solution of the linearised system using eigenvalues and eigenvectors.

iii. What is the type and stability of the critical point?

Explanation / Answer

a> For the critical points we'll have to solve dx/dt = 0 and dy/dt = 0 simultaneously

x -y - x^2 + xy = 0 ----->(1)
-x^2 - y = 0 ------->(2)

from (1) and (2)

x + xy = 0
x(1+y) = 0

=> x= 0 and y = -1

when x= 0 ,y = 0

and when y = -1 , x = +-1

=> the xcritical points are : (0,0) , (1,-1) and (-1,-1)

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