Analysis of 1D ODE

$$ \dot x=f(x) $$

where $f$ is a smooth function of $x$ with no explicit time dependence ie autonomous system

Example 1

$$ \dot x=\sin x $$

• Solving this explicitly gives:

$$ t=\ln\left | \frac{\csc x_0+\cot x_0}{\csc x+ \cot x} \right | $$

🗒️ Note: This is hard to read so we will try a different approach

We plot the $\dot x$ against $x$

Here we easily see the following properties:

• If $\dot x > 0$ then $x$ increase (arrow to right)

• If $\dot x=0$ there is no flow: Fixed point (FP)

• Stable FP (filled dots): solution flows in towards them (attractors)

• If $\dot x<0$ then $x$ decreases (arrow to left)

• Unstable FP (open dots): solution flows away from them (repeller's)

If we go back to our analytical solution and plot it we see that we get the behaviour we expect:

• At $x=\pi$ on both plots we have a fixed point (ie an object that starts is this position doesn't move with time) and we can see it is stable because the neighbouring lines go towards it
• At $x=2\pi$ is an unstable fixed point and we can see this because a particle that starts exactly at $2\pi$ is stationary through time however the other initial positions go away from it so it is an unstable fixed point

🗒️ Note: Using the method above is useful to notice general behaviour however it cannot give us a quantitative analysis of a system

Edge case

💃 Example: if we have $\dot x=x^2$ we get the following diagram

• $x^*=0$ is a fixed point however it is only half-stable (attracting from the left, repelling from the right)
• Ie the particle coming from the left go to the stable equilibrium and stop while the particle to the right go to infinity

Linear stability analysis

⚽ Goal: lets try and find an analytical way (ie not graphical) to find if an FP is stable or not

• Consider $\dot x =f(x)$ and let $x^$ be a fixed point such that $f(x^)=0$