figure). This point will generally not be a solution point, but with small Ds it will be
close to. It can then be corrected back to the solution curve along a normal to the
tangent vector, e.g. by Newton-Raphson iteration, to a new solution point (point 2
in the figure). The process then repeats, with the new solution point (2) serving as a
starting point (0) for yet a tangent prediction.
To employ pseudo-arclength continuation to trace solutions (a, x) to the zero
problem (5.34), the algorithm can be summarized as follows:
1. Starting point, (a 0 , x 0 ):
1:1. Initially: Use any method (e.g. Newton-Raphson iteration from an
approximate guess or bisection or brute force) to determine a solution
point, preferably in a region where the system is expected to behave
trivially (e.g. close to linearly).
1:2. Subsequently: Use the last corrected point (a 2 , x 2 ) obtained during the
continuation process.
2. Direction for the continuation, i.e. determine the sign of a′:
sgnða
0
Þ ¼ sgn
da
ds
¼
þ 1 for a increasing with s
À1 for a decreasing with s
&
:
ð5:41Þ
2:1. Initially: Just choose arbitrarily.
2:2. Subsequently: Keep sign unchanged, until a turning point has been
passed. At a turning point the Jacobian of f is singular,
4 i.e. |f x | = 0.
Thus the passage of a turning point can be detected a shift of sign in
|f x |, i.e. by the fulfillment of the condition |f x (x 2 , a 2 ) ||f x (x 0 , a 0 )| < 0.
3. Prediction step, to (a 1 , x 1 ):
3:1. Solve the linear algebraic equations (5.36), evaluated at the starting
point, for z 0 :
f x ðx 0 ; a 0 Þz 0 ¼ Àf a ðx 0 ; a 0 Þ:
ð5:42Þ
3:2. Compute a
0
0 from (5.40) with (5.37) inserted, and using (5.41) to
choose among the positive/negative solution:
a
0
0 ¼ sgnða
0
0 Þ
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ z T
0 z 0
q
:
ð5:43Þ
4
At a turning point a does not change with s (cf. Fig. 5.11(b)), i.e. da/ds = a′ = 0. Inserting this
into (5.35) gives f x x′ = 0, which since x′ 6 ¼ 0 implies that f x must be singular at this point.
5.11 Graphing Bifurcations: Numerical Continuation Techniques
301
close to. It can then be corrected back to the solution curve along a normal to the
tangent vector, e.g. by Newton-Raphson iteration, to a new solution point (point 2
in the figure). The process then repeats, with the new solution point (2) serving as a
starting point (0) for yet a tangent prediction.
To employ pseudo-arclength continuation to trace solutions (a, x) to the zero
problem (5.34), the algorithm can be summarized as follows:
1. Starting point, (a 0 , x 0 ):
1:1. Initially: Use any method (e.g. Newton-Raphson iteration from an
approximate guess or bisection or brute force) to determine a solution
point, preferably in a region where the system is expected to behave
trivially (e.g. close to linearly).
1:2. Subsequently: Use the last corrected point (a 2 , x 2 ) obtained during the
continuation process.
2. Direction for the continuation, i.e. determine the sign of a′:
sgnða
0
Þ ¼ sgn
da
ds
¼
þ 1 for a increasing with s
À1 for a decreasing with s
&
:
ð5:41Þ
2:1. Initially: Just choose arbitrarily.
2:2. Subsequently: Keep sign unchanged, until a turning point has been
passed. At a turning point the Jacobian of f is singular,
4 i.e. |f x | = 0.
Thus the passage of a turning point can be detected a shift of sign in
|f x |, i.e. by the fulfillment of the condition |f x (x 2 , a 2 ) ||f x (x 0 , a 0 )| < 0.
3. Prediction step, to (a 1 , x 1 ):
3:1. Solve the linear algebraic equations (5.36), evaluated at the starting
point, for z 0 :
f x ðx 0 ; a 0 Þz 0 ¼ Àf a ðx 0 ; a 0 Þ:
ð5:42Þ
3:2. Compute a
0
0 from (5.40) with (5.37) inserted, and using (5.41) to
choose among the positive/negative solution:
a
0
0 ¼ sgnða
0
0 Þ
0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ z T
0 z 0
q
:
ð5:43Þ
4
At a turning point a does not change with s (cf. Fig. 5.11(b)), i.e. da/ds = a′ = 0. Inserting this
into (5.35) gives f x x′ = 0, which since x′ 6 ¼ 0 implies that f x must be singular at this point.
5.11 Graphing Bifurcations: Numerical Continuation Techniques
301