Stability of a continuous-time dynamical system

Question

In this problem we consider one-dimensional continuous-time dynamical system $x'=f(x)$ with fixed point $u$ at which $f(u)=0$. For each of the following systems, discuss the stability of the fixed point $u=0$.

1. $f(x)=x^2$.

2. $f(x)=-x^2$.

3. $f(x)=x^3$.

4. $f(x)=-x^3$.

Answer 1

You mention $\epsilon$ language, so I'm imagining that the class requires you to apply whatever method they teach. I think such things can often over-complicate these problems. Before you apply specific mathematical tools, I think you should use extremely simple logic to set expectations.

Particularly, why don't you just make a table posing the question of "what does the particle do when moving in X direction?" Let's look at this one for example:

$$x' = f(x) = -x^2$$

please excuse my substitution of left and right for negative and positive respectively

• Displaced right of origin $\rightarrow$ Is moving left
• Displaced left of origin $\rightarrow$ Is moving left

Is this stable? The system will return to $x=0$ in the case that it starts with $x>0$. It is right of origin and moving left. In the other case it is left of origin and moving left, so it is headed for $-\infty$. No, this is not stable.

Rinse and repeat.

Answer 2

To get some intuition in equilibria of 1-dimensional dynamical systems try this. You already know that $x=0$ is the (only) equilibrium. So set an $x$ and see if $x>0$, $x$ increases or decreases? If $x<0$?

To find the stability, you can draw $f$ and view this as the velocity of a particle, so given the starting position, you can find the trajectory.