I am investigating the stability of periodic solutions of an autonomous, undamped dynamical system whose solutions over the interval of interest $(T,T+\epsilon)$ are of the form $x(t)=A(t-T)RA(T)x_0$ where $R=\mathrm{diag}(1,...,1,-1)$ and $T$ is a period of the motion induced by $x_0$: $x(T)=x_0$. $A$ is a matrix exponential, i.e. $A(t)=e^{tM}$ for some $M$.
To this end, I choose a Poincaré section $H$. Considering a perturbation $\delta x_0$ on the first return state in $H$, if follows that
\begin{align} x(T)+\delta x &= RA(T+\delta T)(x_0+\delta x_0) \\ & \approx RA(T)x_0 + RA(T)\delta x_0 + RA'(T)x_0\delta T \end{align} and since $x(T)=RA(T)x_0$: \begin{equation} \delta x = RA(t)\delta x_0 +RA'(T)x_0\delta T. \end{equation} But $\delta T$ can be computed from the fact that $x(T)+\delta x\in H$. Eliminating $\delta T$, I found an expression of the form
$$\delta x = B(x_0)\delta x_0.$$
As I understand, the eigenvalues of $B(x_0)$ determine the stability. Below are plotted one component of a periodic solution (in blue) stemming from some $x_0$ and the effect of a perturbation (in orange) after 1000 periods.
If the blue solution $x$ was asymptotically stable, then the orange curve would "match" the blue one. What appears here is that both curves are "very similar", but shifted. Here are the eigenvalues of $B(x_0)$:
$$\{1.,-0.6+0.8 i,-0.6-0.8 i,-0.98+0.19 i,-0.98-0.19 i,0.25\, +0.97 i,0.25\, -0.97 i,-0.41+0.91 i,-0.41-0.91 i,0.\}$$
and their respective modulus:
$$\{1.,1.,1.,1.,1.,1.,1.,1.,1.,0.\}$$
The spectral radius is $1$, so the solution is not asymptotically stable. If it was greater than one, it would be unstable. Here are my questions:
- does the spectral radius of $1$ imply neutral stability?
- is the phase shift observed in the graph typical of neutral stability?
- am I guaranteed that the perturbated solution will stay "close" from the periodic solution if the perturbation is small enough?
- does this have some implications on the behaviour of the periodic solution if a small permanent perturbation is applied to the system?