Exploring mathematical relationship bewteen time-series of three variables and their ratios

Question

I am trying to evaluate three parameters ($V_{1}$, $V_{2}$, and $V_{3}$) (3D of V1 V2 and V3) related to a physical phenomenon and trying to find a mathematical connection between them. Time-series of these variables are measured. These parameters form a closed looped pattern when plotted (Scatter plot of the time-series of three variables). Ratio of these variables can be defined as $R_{1}=\frac{V_{2}}{V_{1}}$, $R_{2}=\frac{V_{2}}{V_{3}}$, $R_{3}=\frac{V_{1}}{V_{3}}$, $R_{4}=\frac{V_{1}}{V_{2}} = \frac{1}{R_{1}}$, $R_{5}=\frac{V_{3}}{V_{1}} = \frac{1}{R_{3}}$, and $R_{6}=\frac{V_{3}}{V_{2}} = \frac{1}{R_{2}}$. The times-series of these ratios is being calculated as defined. The physical system continues in a cycle and has six states. The plot of Variables (V1, V2, and V3) shows that they (mostly) interplay with each other concerning the state of the system. Moreover, these form a closed loop pattern. However, the ratios $R_{1}$, $R_{2}$, ... , $R_{6}$ have a very defined pattern (Scatter plot of Ratios); though, for any specific data point, it is not possible to tell that at which state of the system the ratio belongs. So far, so good, beyond that, I am unable to find a way to go further to explore these relationships. I am not sure how to explain the data and plot mathematically. What tools /methods should I use to develop (or explain) the mathematical relationship between these variables.