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the bifurcation point, the equations governing the dynam-
ics on the manifold are topologically equivalent to those
of the full PDEs. In particular, the existence and stabil-
ity of solutions of the PDEs can be determined from
the center manifold equations, the dimension of which
is the dimension of E c , in this case 4. Practically, the
assumptions allow us to express of (2.14) on the cen-
ter manifold in terms of the center space variables, i.e.,
the complex wave amplitudes z j ; thus the equations on
the center manifold are decoupled from the stable space
equations. Because we are interested in results close to
the bifurcation point, it is convenient to work in Taylor
series for the nonlinear part N (U) and the function that
describes the center manifold, the coefficients of which
can be computed order by order. The underlying assump-
tion is that the contribution of the nonlinear terms are
small but important. The analysis is sometimes called
weakly nonlinear and is similar to the method of sepa-
ration of time scales.
The four-dimensional ODE for the complex wave
amplitudes z j that describe the dynamics on the center
manifold is
where
G 11
G 12
G 21
G 11
G 22
G 11
a =
=
±
1, b =
, c =
,
G 22
G 22
d =
=
±
1,
(2.20)
4 ) terms are ignored in
and λ j = μ j + j .The O(
|
ρ 1 , ρ 2 |
2 ) terms are ignored
in the θ j equations. Ignoring these terms does not affect
the local dynamics, except for fine details of the dynamics.
Thus, given m 1 and m 2 , the coefficients of the scaled
normal form equations a , b , c , d can be written in terms
of the following functions: the axisymmetric solution,
the eigenfunctions and adjoint eigenfunctions, and cer-
tain Taylor coefficients of the center manifold function,
which are all only functions of two spatial variables r
and z . The eigenfunctions are found from the eigenvalue
problem (2.12), and the coefficients of the center mani-
fold function are found from linear PDEs that are derived
using certain properties of the center manifold. See Has-
sard et al. [1981] for a general discussion; the equations for
this problem are written out by Lewis and Nagata [2003].
The axisymmetric solution, eigenfunctions, and coef-
ficients of the center manifold cannot be computed
analytically, and therefore we use numerical approxima-
tions. Upon discretization of the spatial derivatives using
second-order finite differences, the axisymmetric solution
is found from a nonlinear system of equations; the linear
stability, including the eigenfunctions, is found from a gen-
eralized matrix eigenvalue problem; and the coefficients
of the center manifold are found from systems of linear
equations. A nonuniform grid is used in order to better
resolve the boundary layers that occur in the flow. The dis-
cretization of the PDEs leads to large, sparse systems, for
which sparse matrix computations are required in order
to achieve sufficient grid resolution. For more details and
a discussion of the implementation and convergence, see
Lewis and Nagata [2003] and Lewis [2010]. In particular, in
[ Lewis , 2010], computations with grid size up to N = 150
are performed, where N is the number of grid points in
each of the independent variables. It is shown that N = 50,
the value used in the results presented below, is sufficient
resolution to obtain a good approximation of the normal
form coefficients.
Given the values of the coefficients a , b , c ,and d and
the value of A = ad
ρ j equations, and the O( | ρ 1 , ρ 2 |
˙
the
z 1 = λ 1 z 1 + G 11 z 1 z 1 + G 12 z 1 z 2 z 2 + O( 4 ) ,
˙
˙
(2.15)
z 2 = λ 2 z 2 + G 21 z 1 z 1 z 2 + G 22 z 2 z 2 + O( 4 ) ,
where λ j = λ j ( , T) , O( 4 ) indicates terms of order 4 and
higher, and the normal form coefficients G kl are com-
plex numbers that depend on the nonlinear part N (U) of
(2.10), the eigenfunctions, the adjoint eigenfunctions, and
certain Taylor coefficients of the center manifold function.
Equations (2.15) are often referred to as the amplitude
equations. The details are suppressed here for space con-
siderations; however, all details, in the present context,
can be found in the literature [ Lewis and Nagata , 2003].
This normal form requires the nonresonance condition
that the imaginary parts of the eigenvalues, ω 1 and ω 2 ,
satisfy n 1 ω 1 + n 2 ω 2
= 0 for all integers n 1 and n 2 with
|
n 1 |
+
|
n 2 |≤
4 at the critical parameter values = 0 and
T = T 0 .
We write z 1 = ρ 1 e 1 / G 11 and z 2 = ρ 2 e 2 / G 22 ,
where G ij is the real part of the normal form coeffi-
cients G ij and substitute these expressions into (2.15). In
these scaled polar coordinates, the truncated normal form
equations are
bc , the dynamics close to the mode
interaction points can be determined by an analysis of the
truncated normal form equations (2.16)-(2.19). Because
ρ j and θ j correspond to a scaled amplitude and phase,
respectively, of the complex amplitude z j , these can be
reinterpreted in terms of the full PDEs through (2.14).
The analysis of (2.16)-(2.19) in a general context is
described in detail by Guckenheimer and Holmes [1983]
ρ 1 = ρ 1 μ 1 + 1 + 2 ,
˙
(2.16)
ρ 2 = ρ 2 μ 2 + 1 + 2 ,
˙
(2.17)
θ 1 = ω 1 ,
(2.18)
θ 2 = ω 2 ,
(2.19)
 
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