Almost linear systems with a behavior that is different from the linearized system

Instead of x(t), y(t) we will use polar coordinates r(t), phi (t) with x = r cos( phi ), y = r sin( phi ).

An example of a proper node for a linear system is given by

x' = -x
y' = -y

or, in polar coordinates,

r' = -r
' = 0

An almost linear system with the same linear part is

r' = -r
' = sin( ) / log(r)

In this case all trajectories except one approach the critical point from one direction ( phi

= 0), not all directions as for a proper node, or two directions as for a node or improper node.

An example of a center for a linear system is given by

x' = -y
y' = x

or, in polar coordinates,

r' = 0
' = 1

An almost linear system with the same linear part is

r' = r^2 sin(1/r)
' = 1

In this case there are infinitely many limit cycles at r = 1/

, 1/(2

), 1/(3

), ... which are approached by the trajectories in between.