For a non linear system obeying the equations
finding the fixed points amounts to solving the simultaneous equations
In general there may be none, one or many fixed points, each of which can be investigated individually to ascertain it's nature.
Example: Find the fixed points of the system
Putting the second of these equal to 0 and factorising gives
From this we deduce
or
Substitutinginto the first, which we set equal to 0, gives
Hence
and
are fixed points of the system.
Substituting
into the first and setting it equal to 0 gives
Hence
and
are fixed points of the system
Example: Determine the fixed points of the system
Put
and the system becomes
Putting the second equal to 0 givesand substituting this into the second, which we set equal to 0 gives
The fixed point areand