I am having some difficulties figuring out how to approach "Test stability problems".
I usually test the linearization of the system (since it is very straightforward and easy), and if that doesn't work - I construct a Lyapunov function.
After I go for the obvious functinos (2nd degree polynomials, or energy\hamiltonian functions) - I feel rather lost.
For example:
Given $n\in \mathbb N$ and the system: $$\begin{aligned} & \dot x = v \\ & \dot v =-x^n \end{aligned}$$ Test the stability of $(0,0)$.
The energy function (as a Lyapunov function) shows that for odd $n$: $(0,0)$ is stable (and cannot be asympotitcally stable).
For even $n$ i want to argue that the flow is unstable. I can draw the phase portrait and write down the general solution to this type of equations, but I want to approach it from a more insightful perpsective.
- Hints and overall suggestions to such problems (and this problem in particular) are welcome.
Also, reading Bertrand's Theorem, that states that newtonian systems of central force potentials, the only systems with the property that bounded orbits are also closed - are the ones with $n=1,-2$.
- Does this imply that the dynamics of the systems with $n\in \mathbb N_\text{odd}$ do not close? This is, also, unclear.