One-dimesional system non-existence fixed points

[lahanadar]

Homework Statement

First things first, this is not a HW but a coursework question. I try to understand a concept.

Assume we have a one-dimensional dynamic system with:

x'=f(x)=rx-x^3

Homework Equations

Fixed points are simply calculated by setting f(x)=0.

The Attempt at a Solution

If I compute f(x)=0:

f(x)=x(r-x^2)=0 and so x*={0, -sqrt(r), +sqrt(r)}

If r<0, then -sqrt(r), +sqrt(r) becomes obsolete since they become imaginary.

What if I only come up with only imaginary fixed points for another system? How would the system behave in terms of stability?

 

[mfb]

What if I only come up with only imaginary fixed points for another system? How would the system behave in terms of stability?

Simple: there would be no stable position.
x'=1 is probably the easiest example of such a system. x'=x²+1 if you want imaginary solutions.