In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or by analysis. <br />
In a simulation, given a starting [[Array data structure|vector]] (list of the values of the variables), the progression of the system is calculated by solving the equations at each time-frame in small increments.<br />
In analysis, the proprietiesproperties of the equations are used to investigate the behavior of the system depending of the values of the parameters and variables. A system of differential equations can be represented as a [[vector field]], where each vector described the change (in concentration of two or more protein) determining where and how fast the trajectory (simulation) is heading. Vector fields can have several special points: a [[Fixed point (mathematics)|stable point]], called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an [[unstable point]], either a source or a [[saddle point]] which repels (forcing the concentrations to change away from a certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the concentrations oscillate).<br /> A better representation which can handle the large number of variables and parameters is called a [[bifurcation diagram]] ([[bifurcation theory]]): the presence of these special steady-state points at certain values of a parameter (e.g. mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the cell cycle has phases (partially corresponding to G1 and G2) in which mass, via a stable point, controls cyclin levels, and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a bifurcation event ([[cell cycle checkpoint]]), the system cannot go back to the previous levels since at the current mass the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a [[Hopf bifurcation]] and an [[infinite period bifurcation]].
