From Problem Set 11: Exercise 1.1#
First we must set up the system of how the set/system of ODEs looks like, for a system that is globally non-linear. Let us consider exclusively the case of the homogenous non-linear 2x2 system. We have, when dealing with an xy phase plane,
, where both F(x,y) and G(x,y) are systems of both independent variables of x and y, which are both defined as some functions of t (independent variable). Let us also consider that there is an isolated point P with the coordinates of 
and 
. This P is shown as 
.
Given the condition that the provided system is locally twice differentiable at the point P, and we see the necessary condition that system remains continuous under this twice differentiation action, we can find the linearization of the system for that point P in the phase plane.
This non-linear system is defined as almost linear or locally linear at this isolated point P if 
is a critical point and if its linearization through the Jacobian matrix shows us the condition of continuity, along with the origin of the coordinates.
The linearization of the system for the isolated critical point at the origin occurs if and only if both the eigenvalues are shown to be non-zero for that point.