Abstract

CHAPTER 3. BIFURCATION ANALYSIS 50
(α − ¯r 2 ) > 0 =⇒ d¯r
> 0. So ¯r = 0 is an unstable equilibrium, since any small
dt
increase in ¯r will force ¯r to continue to move away from ¯r =0. Inthecaseofα>0,
if ¯r > √ α, then by a similar argument, d¯r
dt < 0. So if ¯r = 0,¯r approaches √ α for
α>0. So we have a stable closed orbit in this case.
If we start the parameter α as a negative value, then the origin is a stable equi-
librium. As we increase the parameter, the origin remains stable until α becomes
positive. Then the origin turns unstable and an oscillatory solution to the ODE
appears. This is what is known as a Hopf bifurcation. As shown in this example,
a Hopf bifurcation is characterized by a pair of complex-conjugate eigenvalues’ real
parts going to zero.
3.3 Continuation Methods
Continuation algorithms map out a system’s steady-state solution branches as a sys-
tem parameter is being changed. To do this numerically, we connected our simulator
to the Library of Continuation Algorithms (LOCA), a software library developed at
Sandia National Laboratories [33]. LOCA has three continuation algorithms available
for the user: a zero order continuation, a first order continuation, and a pseudo arc-
length continuation. LOCA also incorporates an eigensolver that checks the eigenval-
ues of the Jacobian matrix of the system to determine the stability of the steady-state
solution.
Numerically, we plan on generating a sequence of parameters {V j } along with