I'm beginning to look into stability analysis of limit cycles. Whilst looking through a limit cycle continuation I noticed that one of the Floquet multipliers was staying at value 1 throughout. When I tried to read around to find a reason why I came across the following sentence: "Among the Floquet multipliers there is always one equal to unity, which reflects neutral stability to a shift along the periodic orbit.”
I don't fully understand this sentence or how it might relate to my 5-dimensional system, in which I'm interested in periodic behaviour of one of the state variables. What is the significance of having 5 multipliers, with one of them constant at 1 as the limit cycle is continued? I have a loose idea of what the Floquet multipliers are, but not the nuance of how they're derived.
It is as is said in your quote, if $\phi(t;x_0)$ is the flow of $\dot x=f(x)$, the solution with $\phi(0;x_0)=x_0$, then the situation with a periodic solution is that $\phi(T;x_0)=x_0$ for $T>0$. By the transitivity of the flow, also $$\phi(T;\phi(s;x_0))=\phi(T+s;x_0)=\phi(s;x_0).$$ The Floquet multipliers characterize the linearization of the map $x\mapsto \phi(T;x)$ for $x\approx x_0$, and for $x=\phi(s;x_0)$, $s\approx 0$, you get the identity, thus the eigenvalue $1$ with eigenvector $v=f(x_0)$ in the linearization.