# M

Fig. 5.159 Examples of possible forms for F (p) in a case where k (p) increases monotonically (redrawn from Welander, 1982).

Fig. 5.159 Examples of possible forms for F (p) in a case where k (p) increases monotonically (redrawn from Welander, 1982).

A crucial assumption made here is that K(p) is a positive function, monotonically increasing with p.

The steady state of the system is

— — aKTTi /3ksSa p = -aT + 0 S =-T-^- + = Fp) (5.287)

The solutions to this problem can be found graphically as the intersections between the dashed straight line p = p and the curve p = F (p) (Fig. 5.159). As shown in this figure, in cases (a) and (b) there is one steady state; in case (c) there are three steady states.

A simple case of oscillator is obtained by assuming that the mixing coefficient k takes two fixed values only:

k = ki, for p > —s and k0 is smaller than k1. Thus, there exist two "attractors" in the phase plane, as shown in Figure 5.160. An attractor in the phase space means that all trajectories in the vicinity are "attracted" to the attractor, i.e., they flow toward the attractors, but can never reach them. As soon as the system crosses the diagonal, p = 0, in the phase plane, the system is attracted by the attractor on the other side of the diagonal (Fig. 5.160); thus, the system can never reach a steady state, instead there is a limit cycle (Fig. 5.161). Introducing the following nondimensional variables:

 j