Stable and fixed points Can you help me understand the difference between a stable and a fixed point of an ODE $x' = f(x)$ ? 
Let me tell you what I think. 
Let $x$ be a stable point. Then that means that $f(x)=x$. 
Now let $x$ be a fixed point. Then $f(x)=o$. 
This is what i've understood so far but I can't really tell the difference and I have the feeling that the definitions I gave above might be correct in a reverse order. If someone can clear it up for me I'd really appreciate it.
 A: You are comparing discrete dynamics
$$
x_{n+1}=F(x_n)
$$
and continuous dynamics
$$
y'(t)=f(y(t))
$$
In both cases the dynamical behavior is stationary if the solution sequence resp. function are constants $x_n\equiv x^*$ resp. $y(t)\equiv y^*$. 
For the discrete system this means that $x^*=F(x^*)$ which in other words describes a fixed point of $F$
In the continuous case the derivative of a constant is zero, so one has to look for points with $0=f(y^*)$, that is the roots of $f$.
A: Let’s consider a continuous-time dynamical system $\dot{x} = f(x)$ and a discrete-time dynamical system $x_{t+1} = F(x_t)$.
Then:


*

*A state $x$ is a fixed point, if it does not evolve to another state under the given dynamics. This is equivalent to $f(x)=0$ and $F(x)=x$, respectively.

*A fixed point is stable, if it is attracting all states in its vicinity, i.e., those states converge towards the fixed point over time.
This is equivalent to the Jacobian of $f$ having only eigenvalues with negative real parts or $\left| F'(x)\right| < 1$ (for sufficiently smooth $F$).
A typical illustration of this are the states of a ball residing on some geography.
If the ball is not moving, it is located at a fixed point.
However:


*

*If the ball rests in a valley, this is a stable fixed point: It will always roll back into the valley if slightly perturbed.

*If the ball resides on the top of a hill, this fixed point is not stable: It will roll away from the fixed point when slightly perturbed.


