Page 107 - Numerical Methods for Chemical Engineering

96 2 Nonlinear algebraic systems

We can search for a bifurcation point along some path Θ(λ) in parameter space by solving
the augmented set of N + 1 equations for x s (Θ c ) and the value of λ c , Θ(λ c ) = Θ c ,at
which the bifurcation occurs,

f (x s ; Θ(λ)) = 0
|J(x s ; Θ(λ))| = 0 (2.146)

Computing the locations of bifurcation points allows us to carve up parameter space into
different regions, to find one in which one or more real solutions do indeed exist.

Example. Bifurcation points of a simple quadratic equation
As an example, consider the quadratic equation

2
f (x) = x + θx + 1 (2.147)
that has solutions at
√
2
−θ ± θ − 4
x = (2.148)
2
2
Obviously, this equation has two real solutions for θ > 4 and no real solutions within
−2 <θ < 2. There exist two bifurcation points at θ =±2 at which the two real solutions
are degenerate. Let us consider the bifurcation point at θ = 2 for which the solution is
x s =−1. The Jacobian of this system (here a 1×1 matrix, a scalar) is

∂ f
J(x,θ) = = 2x + θ (2.149)
∂x
At each bifurcation point θ =±2, the Jacobian is singular, as
√

2
−θ ± θ − 4
2
J(x,θ) = 2 + θ =± θ − 4 (2.150)
2
Here, we can compute the bifurcation point analytically, but let us see how we might do so
numerically (as we would have to do for more complex systems). Let us say that we wish
to look for bifurcation points along the path θ(λ) = 4λ, where we encompass in the range
0 ≤ λ ≤ 1 the points 0 ≤ θ(λ) ≤ 4. Obviously, there will be a bifurcation point at λ = 0.5,
but if we did not know this, we could try different values of λ as the initial guess and use
Newton’s method in (x,λ) space to solve the augmented system

2
f 1 (x,λ) = f (x) = x + θ(λ)x + 1 = 0
f 2 (x,λ) = det[J(x,λ)] = 2x + θ(λ) = 0 (2.151)

The Newton update rule is
[k+1] [k] [k]
x x x
= + (2.152)
λ [k+1] λ [k] λ [k]