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computed here. Also, it is determined that for smaller val-
ues of the Prandtl number the primary transition may
become hysteretic. Similar dynamics are observed for all
mode interaction points.
In future work, we will extend the analysis beyond the
primary transition and study other types of flow such as
vacillation. However, the above analysis consisting of the
reduction to the normal form equations (2.16)-(2.19) is
based on weakly nonlinear theory and thus only predicts
the qualitative behavior of the full problem (2.4)-(2.8) for
parameter values and flow structures close to those of
the primary bifurcation points. Further away from these
points, we need to consider the fully nonlinear problem.
Moreover, in some scenarios, such as the one in which the
axisymmetric solution loses stability in a subcritical Hopf
bifurcation, the third-order coefficients that are computed
in this chapter are not sufficient to determine to what the
low will equilibrate, and the computation of the necessary
higher order terms is a monumental task.
The dynamical systems approach can still be applied
by implementing numerical continuation methods to the
fully nonlinear problem. Using such methods, we are
able to compute both stable and, in contrast to numeri-
cal and laboratory experiments, unstable special solutions
of dynamical systems, such as equilibria and travelling
waves. To conclude this chapter, we briefly describe the
methods that we are implementing to extend the results
discussed here to parameter values away from the primary
transition.
Continuation methods aim to track special solutions as
a parameter of the system is varied; that is, they aim to
compute
branches
of special solutions in the given param-
eter. In the cases described in Section 2.4, these solutions
are expected to be stationary (fixed points of type 1), peri-
odic (fixed points of types 2 and 3) or quasi-periodic (fixed
points of type 4). The method we use is called pseudo-
arclength continuation, where the qualifier
pseudo-
arclength
indicates that the computed branch of solutions
is parameterized by the approximate arc length, rather
than by the system parameter. The pseudo-arclength
approach can be explained independently of the type of
solution under consideration. Thus, we will describe it
in the context of steady solutions, and modifications for
computing periodic solutions will be mentioned later.
In the following, we will discuss the computation of
solutions to a spatial discretization of model equations
(2.4)-(2.8). The discretization can be the result of a finite-
difference approximation of the spatial derivatives, as
we used to compute the approximations presented in
Section 2.4, or a spectral decomposition of the variables.
In either case, the discretized model consists of a large set
of ordinary differential equations with a highly dissipa-
tive structure due to the diffusive and viscous terms in the
continuous model. The applications of pseudo-arclength
μ
2
ρ
2
μ
2
=-
μ
1/
b
ρ
1
μ
1
μ
2
=
c
μ
1
Figure 2.9.
Two-parameter bifurcation diagram for case V. See
caption of Figure 2.5 for description.
presented in Figure 2.9. In this case, the fixed point 4 exists
in the wedge defined by
μ
2
<
μ
1
/b
and
μ
2
> cμ
1
,where
the borders of the wedge given by
μ
2
=
−
μ
1
/b
and
μ
2
=
cμ
1
are in the upper left quadrant and lower left quad-
rant, respectively, of the parameter space. In (2.16)-(2.19),
a bifurcation from a periodic orbit to a 2-torus, i.e., a
Neimark-Sacker bifurcation, occurs as a parameter is var-
ied across these borders. A linear stability analysis of the
fixed points [
Guckenheimer and Holmes
, 1983;
Kuznetsov
,
2004] reveals that fixed point 3 is stable when it exists
inside this wedge and unstable outside while fixed points 2
and 4 are always unstable. Thus, the mixed-mode solution
corresponding to fixed point 4 would not be physically
observable.
For cases VIIb and VIb, see
Guckenheimer and Holmes
[1983] or
Kuznetsov
[2004] for a bifurcation diagram.
−
2.5. CONCLUSION AND FUTURE WORK
Using methods from dynamical systems theory, we
performed a detailed study of the effects of the Prandtl
number on the primary transition. The analysis indicates
that, for Prandtl number between approximately 4 and
13, bistability of rotating waves is observed, i.e., there
are regions within the wave regime where the wave num-
ber of the rotating wave depends on the initial conditions
and there is hysteresis in the transition between these
waves. As the Prandtl number is decreased, quasi-periodic
mixed azimuthal mode flows can be observed. In certain
cases, depending on the values of the higher order nor-
mal form coefficients, it may also be possible to observe
three-frequency flows. However, these coefficients are not
