I am interested in the study of the stability of a system via the Floquet multipliers. So, I understand that when I have a first-order ODEs system each of the variables is of interest. For instance, if we consider the Redox oscillation, the system is given by:
$\frac{dD_1}{dt}=p-aAD_1-dD_1$
$\frac{dD_2}{dt}=dD_1-eD_2$
$\frac{dR}{dt}=eD_2-qR$
$\frac{dA}{dt}=bIR-aAD_1$
Consequently, each Floquet multiplier can be plotted in the complex plane. So, each variable, i.e., $D_1, D_2, R$, and $A$ will have a corresponding Floquet multiplier to be assessed. Great!
However, I am a little bit confused about how to proceed with the assessment when I have a second-order ODEs system converted into a first-order ODEs system. For instance, let's suppose I have something like:
$\frac{d^2x_1}{dt^2} = a\frac{dx_1}{dt} + b\frac{dx_2}{dt} + c(t)x_1$
$\frac{d^2x_2}{dt^2} = d\frac{dx_2}{dt} + e\frac{dx_1}{dt} + f(t)x_2$
Then, I can introduce new variables (for instance: $z_1, z_2, z_3$, and $z_4$) and reduce the order of my system, but then I will have four equations and consequently four corresponding Floquet multipliers. So, my big question is: should I assess in the complex plane all four Floquet multipliers corresponding to ($z_1, z_2, z_3$, and $z_4$) or just the two corresponding back to ($x_1$, and $x_2$)? Does anyone know how to proceed in this case? Any reference (book, paper, etc.)? Most of the examples and works that I found are exploring first-order systems :/