The last two chapters provide means how to grapple with some real systems with variables and parameters that keep on changing. Although these could be mathematized, there exist approaches that require few more notions, types, definitions, theorems etc. There are many for theorizing, in some more depth, ideas on bifurcation, and catastrophes. We begin with a resume of what we have learnt in the earlier chapters. We take bifurcation first, then catastrophes and after that, interweaving one with the other to the extent they facilitate both in understanding of and pushing further concepts. But at every stage, we reckon non-linearity whose effect is to bend the solution branches, which we have already met in the example -u″ (x) = λ ∥u∥2u(x), 0 < x < π. We can call this as a mildly non-linear problem, in the sense, that ∥u∥2u does not reflect a genuine non-linearity (or pseudo non-linearity) so well represented by, say, f(xu)u where f(u) is nothing but a vector, which turns the vector to a new direction in the space.

Resumé on Bifurcations

Let us recall that our objective has been not merely to solve the system of equations in variables x = (x1, x2, …, xn)∈ Rn of the form

but also to investigate the behaviour of solution of (1) as the real scalar parameter λ varies; F = (F1, F2, …, Fm) is a non-linear mapping with the values in Rm.