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a we will stay there forever. However, if we receive a small perturbation off that branch, interesting things happen. If the perturbation (until otherwise notified, all perturbations are small) puts us between the two branches, then x(t) declines and we move towards the negative branch. If the perturbation puts us above the positive branch, then x(t) increases and we move away from the positive branch. So, in either case, a perturbation moves us away from the positive branch. We say that such a branch is dynamically unstable (or just unstable). A similar argument shows that perturbations from the negative branch return to it; we say that the negative branch is stable. What happens when a = 0? The differential equation becomes dx/dt = x2, so that x(t) is always increasing. Thus, if x(0) < 0, x(t) rises towards 0; however if x(0) > 0, x(t) moves away from 0. We say that such a point is marginally stable; we also say that the equation dx/dt = x2 — a is structurally unstable (these words may appear to be needlessly complex, but think about them and they make sense) when a = 0, because small changes of a from the value 0 lead to very different properties of the equation (in this case, either no steady states or two steady states). We also sometimes say that the stable steady state and unstable steady state coalesce and annihilate each other (kind of like matter and antimatter) when a = 0.

The next most complicated equation involves two parameters and a cubic in x:

dx i

where a and P are the parameters of interest. The steady states of this equation satisfy the cubic equation x3 — ax — P = 0. We will momentarily discuss geometric solutions of this equation, but now begin with a bit of algebra. A cubic equation has three solutions (by the fundamental theorem of algebra), of which one may be real and two complex, three may be real with two equal, or three may be real and unequal. Which case applies is determined by the value of the discriminant D(a, P) = (P2/4) — (a3/27). (You probably once learned this in high school algebra, but most likely don't remember it. This is a case where I ask that you trust me; of course you can also go and check the formula in a book.) If D(a, P) > 0, then there is one real solution; if D(a, P) = 0, then there are three real solutions, two of which are equal; if D < (a, P) then there are three real, unequal solutions. Thus, in some sense D(a, P) = 0 is a boundary. So, we need to think about the shape of P2 = 4a3/27, which is shown in Figure 2.12. This kind of equation (in which the independent variable appears as a 3/2 power) is called a cusp; hence this is called the cusp bifurcation or sometimes the cusp catastrophe (see Connections).

Figure 2.12. A plot of the equation p2 = 4a3/27, which is called a cusp. Along the curves, there are two real solutions of the cubic (and thus three steady states of Eq. (2.32)). Elsewhere, there are either one real solution or three real solutions.

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