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One-dimesional system non-existence fixed points

  1. Jan 28, 2015 #1
    1. The problem statement, all variables and given/known data
    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

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

    3. 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?
     
  2. jcsd
  3. Jan 28, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    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.
     
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