Phase plane representation of local stabilities

The concentrations a and P vary in time as they approach or move away from any particular stationary state. Often it is convenient to visualize the time-dependent behaviour another way, by plotting the variation of one concentration against that of the other, in what is known as the a-/ phase plane. 

In the phase plane we are really considering the solutions of the differential equation 

As a and ft vary in time, or equivalently as eqn (3.54) is solved from some given initial set of conditions, the a-/ locus marks out a trajectory on the phase plane. 

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

Representations of the different singular points in the concentration phase plane (a) stable node, sn (b) stable focus, sf (c) unstable focus uf (d) unstable node, un (e) saddle. point, sp. 

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