5.11 Graphing Bifurcations: Numerical Continuation
Techniques
We consider fixed point solutions (i.e. static equilibriums) of the dynamic system
(5.1), but with only a single bifurcation parameter, i.e. k = 1 and l = {a}, so that a
is the only bifurcation parameter. The task is then to trace a curve x = x(a) (or some
norm ||x||) with points (a, x), which are solutions to the zero problem:
fðx; aÞ ¼ 0;
ð5:34Þ
where ðf; x; aÞ 2 R
n
 R
n
 R: Since (5.34) only implicitly defines x as a function
of a, this is typically not a trivial task, and will typically require numerical techniques. But it can be made less difficult by using continuation techniques, i.e. by
proceeding from one calculated curve point to the next in small steps, rather than
starting anew for every curve point. Here we provide a brief introduction to
numerical continuation, and refer to the specialized literature for more methods and
detail (e.g., Dankowicz and Schilder 2013; Keller 1986; Krauskopf et al. 2007;
Nayfeh and Balachandran 1995; Seydel 2010).
There are several techniques in use for numerical continuation of fixed points or
periodic solutions. These include sequential or natural parameter continuation,
simplicial or piecewise linear continuation, (Davidenko-)Gauss-Newton continuation, and (pseudo-)arclength continuation. If a good approximation to a point at the
solution curve is available, it is usually not difficult to quickly improve the accuracy
of this approximation using, e.g., Newton-Raphson iteration. So the main problem
is typically to provide initial approximations close enough for that to work, in
particular near turning points, branches, or other singularities. The main difference
among numerical continuation methods is how to accomplish coming close enough.
In this introductory presentation we focus only on single-parameter sequential
continuation and pseudo-arclength continuation of fixed points; this will highlight
the main ideas, and prepare for other methods, if necessary.
5.11.1 Sequential Continuation
Sequential continuation, or natural parameter continuation, is usually straightforward to apply. As illustrated by Fig. 5.11(a) it just involves proceeding from a
solution point (a 0 , x 0 ) at the solution curve to the next, by incrementing a a
reasonably small step Da, i.e. a 1 = a 0 + Da, and assuming as a first approximation
that x is unchanged, i.e. the prediction is (a 1 , x 1 ) = (a 1 , x 0 ). The prediction is then
corrected by solving f(x 2 , a 1 ) = 0 for x 2 (using e.g. Newton-Raphson iteration or
bisection or brute force) to find the next solution point (a 2 , x 2 ) = (a 1 , x 2 ). This
procedure is iterated, with the solution point (a 2 , x 2 ) now serving as a new starting
point (a 0 , x 0 ).
298
5 Bifurcation Analysis
Techniques
We consider fixed point solutions (i.e. static equilibriums) of the dynamic system
(5.1), but with only a single bifurcation parameter, i.e. k = 1 and l = {a}, so that a
is the only bifurcation parameter. The task is then to trace a curve x = x(a) (or some
norm ||x||) with points (a, x), which are solutions to the zero problem:
fðx; aÞ ¼ 0;
ð5:34Þ
where ðf; x; aÞ 2 R
n
 R
n
 R: Since (5.34) only implicitly defines x as a function
of a, this is typically not a trivial task, and will typically require numerical techniques. But it can be made less difficult by using continuation techniques, i.e. by
proceeding from one calculated curve point to the next in small steps, rather than
starting anew for every curve point. Here we provide a brief introduction to
numerical continuation, and refer to the specialized literature for more methods and
detail (e.g., Dankowicz and Schilder 2013; Keller 1986; Krauskopf et al. 2007;
Nayfeh and Balachandran 1995; Seydel 2010).
There are several techniques in use for numerical continuation of fixed points or
periodic solutions. These include sequential or natural parameter continuation,
simplicial or piecewise linear continuation, (Davidenko-)Gauss-Newton continuation, and (pseudo-)arclength continuation. If a good approximation to a point at the
solution curve is available, it is usually not difficult to quickly improve the accuracy
of this approximation using, e.g., Newton-Raphson iteration. So the main problem
is typically to provide initial approximations close enough for that to work, in
particular near turning points, branches, or other singularities. The main difference
among numerical continuation methods is how to accomplish coming close enough.
In this introductory presentation we focus only on single-parameter sequential
continuation and pseudo-arclength continuation of fixed points; this will highlight
the main ideas, and prepare for other methods, if necessary.
5.11.1 Sequential Continuation
Sequential continuation, or natural parameter continuation, is usually straightforward to apply. As illustrated by Fig. 5.11(a) it just involves proceeding from a
solution point (a 0 , x 0 ) at the solution curve to the next, by incrementing a a
reasonably small step Da, i.e. a 1 = a 0 + Da, and assuming as a first approximation
that x is unchanged, i.e. the prediction is (a 1 , x 1 ) = (a 1 , x 0 ). The prediction is then
corrected by solving f(x 2 , a 1 ) = 0 for x 2 (using e.g. Newton-Raphson iteration or
bisection or brute force) to find the next solution point (a 2 , x 2 ) = (a 1 , x 2 ). This
procedure is iterated, with the solution point (a 2 , x 2 ) now serving as a new starting
point (a 0 , x 0 ).
298
5 Bifurcation Analysis