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Bifurcations from robust homoclinic cycles
Driesse, R.
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Driesse, R. (2009). Bifurcations from robust homoclinic cycles
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Download date: 18 Jun 2017
Chapter 4
Magnetic dynamos in
rotating convection
In this chapter we discuss another example of an essentially asymptotically stable
network. It is motivated by a model for the self-sustained magnetic field of the
Earth. In the solution space of this model there will be a 12-dimensional invariant
subspace in which there is a so called magnetic dynamo. We can then further reduce
to an invariant 6-dimensional system in which there is a simple homoclinic network.
We can prove that this network is essentially asymptotically stable and from this
it follows that the magnetic dynamo also is essentially asymptotically stable. For
completeness, parts of Chossat et al. [1999] are also summarized in this chapter.
A fine review of dynamo theory by two experts in the field is given by Roberts &
Soward [1992].
4.1
Introduction
The Earth has a magnetic field which shields us from radiation from the Sun (i.e. the
solar wind). Once in a while the magnetic field reverses which can be observed from
volcanic rocks that are magnetized in different directions. These magnetic reversals
are believed to take place in a period of a few hundreds to a few thousands of years.
The time between two reversals is observed to vary largely. The magnetic field has
in the past maintained its orientation for tens of millions of years while also reversals
were observed within tens of thousands years. The last reversal was approximately
720 thousand years ago.
It is believed that the reversals can be explained by the dynamo theory of how
the geomagnetic field is generated. This field is generated by the motion of liquid
83
84
Chapter 4. Magnetic dynamos in rotating convection
metal in the Earth’s core. This fluid motion is driven by convection which leads
us to studying the underlying rotating magnetohydrodynamic equations which can
explain self-sustained magnetic fields in electrically conducting fluids.
Chossat et al. [1999] discuss a model offering a possible explanation for the
magnetic reversals in the Earth magnetic field. The main goal of this chapter is to
use the the methods from Chapter 3 to show that an essentially asymptotically stable
network exist in the space of solutions of this model. Furthermore, the existence of
an attractor containing magnetic solutions explains the existence of self-sustained
magnetic fields and could explain the magnetic fields in the Earth and the Sun.
In Section 4.2 we briefly discuss the deduction of the system of differential equations and the type of solutions that exist for different parameter values. See Chossat
et al. [1999] for a more complete overview. The system possesses a noncompact
symmetry group which will be reduced to a compact group by restricting to a class
of functions that is doubly spatially-periodic. For more details than given below we
refer to Chossat et al. [1999] and Roberts & Soward [1992]. For some parameter
values there is a homoclinic network Γ. The system can be reduced to a normal
hyperbolic center manifold of finite dimension and there is a 6-dimensional flow invariant subspace in which there is a simple homoclinic network Γ . If we let the
original symmetry group act on Γ we obtain the cycle Γ. In section 4.3 we study
this finite dimensional reduced system. We prove that the homoclinic network can
be essentially asymptotically stable.
4.2
Modelling the system
Since fluid motion in the Earth’s core is driven by convection, we consider simple
models of convecting fluids, following Chossat et al. [1999]. Consider an incompressible, uniformly rotating fluid between two infinite horizontal plates and suppose that
the lower plate is heated uniformly. At low temperatures, the fluid is in a purely
conductive, stationary state. Convection sets in at a critical temperature that is
determined by the linear stability of the conducting state. The departure from the
pure conduction state is governed by the Boussinesq equations for rotating convection. We suppose that the fluid is electrically conducting, so that the Boussinesq
equations are coupled to the magnetic induction equation by the Lorentz force.
Let Ω = R2 × [0, 1] denote the domain of the fluid. We let (x, z) denote points
in the domain Ω where x = (x1 , x2 ) ∈ R2 and z ∈ [0, 1]. Writing V : Ω → R3 for
the velocity field of the fluid, B : Ω → R3 for the magnetic field, p : Ω → R for the
pressure and θ : Ω → R for the deviation of the temperature from the conduction
state, the governing partial differential equation, see Chandrasekhar [1961], is given
4.2. Modelling the system
by:
∂V /∂t = −(V · ∇)V − ∇p +
85
√
T V × k + ΔV +
√
−1
∂θ/∂t = −(V · ∇)θ + Pr (Δθ + RV · k),
∂B/∂t = βΔB + curl(V × B),
√
Rθk − B × curlB,
(4.2.1)
divV = divB = 0,
B1 =
B2 = 0.
Ω
Ω
The constants R, T , Pr and β are the Rayleigh number, Taylor number, Prandtl
number and magnetic Prandtl number. Here k = (0, 0, 1)T . We impose the homogenous boundary conditions
∂V2
∂B1
∂B2
∂V1
=
=
=
= 0,
∂z
∂z
∂z
∂z
V3 = B3 = θ = 0,
for z = 0 and z = 1. Now, V can be written as the sum of a conservative vector
field and a divergence free vector field. The equation for ∂V /∂t splits up into two
components one of which can be solved for the pressure p. The remaining system of
equations (which we continue to label as (4.2.1) which we do not write out explicitly)
is an evolution equation in (V, θ, B) where V and B are divergence free vector fields
and θ is a scalar field. We are interested in finding a magnetic dynamo:
Definition (Magnetic dynamo). An ω-limit set in the space of functions (V, θ, B) :
R2 → R7 is called a magnetic dynamo if it contains points (V, θ, B) with B = 0.
Considering all possible perturbations in the infinite planar model would be problematic for the notion of stability. A standard remedy is to consider stability only
to perturbations with a certain kind of spatial periodicity.
In search of a magnetic dynamo by using bifurcation theory it follows that the
trivial solution (V, θ, B) = (0, 0, 0) cannot lose stability directly to a magnetic dynamo, see Chossat et al. [1999]. There does exist a bifurcation to stable rolls which
become unstable (Küppers-Lortz instability) if T exceeds some critical value, see
Küppers & Lortz [1969]. As shown by Busse and Heikes (Busse & Heikes [1980],
Heikes & Busse [1980]) there is a homoclinic cycle connecting rolls solutions to rolls
solutions inclined at an angle of 60 degrees to the old rolls. This phenomenon is
observed in fluid experiments. The network formed by these connections is called a
Busse-Heikes cycle. Chossat et al. [1999] analyze four different cases of bifurcations.
We will show that in the case of a steady-state bifurcation from the Busse-Heikes
cycle the Busse-Heikes cycle is essentially asymptotically stable and there is an
asymptotically stable magnetic dynamo.
Equation (4.2.1) is equivariant under the special Euclidean group SE(2) of rotations and translations in the plane. In addition there is an up-down symmetry τ
86
Chapter 4. Magnetic dynamos in rotating convection
which transforms z to 1 − z and a field-reversal symmetry ρ which transforms B to
−B. Altogether, we have an action of the group SE(2) × Z2 (τ ) × Z2 (ρ) on functions
(V, θ, B) : R2 × [0, 1] → R7 . There are technical problems in dealing with noncompact symmetry groups such as the one described. A standard method for avoiding
these difficulties is to restrict to a class of functions that is doubly spatially-periodic
in the plane, see Golubitsky et al. [1988]. For our purposes, it is sufficient to restrict
to the hexagonal lattice. Define the subgroup of translations L ⊂ SE(2) generated
by translations
1 √
(4.2.2)
l1 = c(0, 1), l2 = c( 3, 1),
2
where c is determined later. Then
L = {m1 l1 + m2 l2 | m1 , m2 ∈ Z} ≡ Z2
(4.2.3)
is a discrete subgroup of the group of translations. Consider the space Fix(L) of
functions that are spatially periodic with respect to L, that is, V (x + l, z) = V (x, z)
for all l ∈ L and similarly for θ and B. Leaving out the details, this leads to a
compact symmetry group Γ acting on Fix(L):
Γ = (Z6 T2 ) × Z2 (τ ) × Z2 (ρ).
(4.2.4)
As discussed in the introduction, the primary bifurcation from the trivial solution
(V, θ, B) = (0, 0, 0) takes place inside the flow-invariant subspace {B = 0} and
leads to pure convection solutions (V, θ, 0). The stability of the trivial solution,
the convective rolls and the Busse-Heikes cycle can be considered inside the space
{B = 0}. At first all eigenvalues of the linearization of the Boussinesq equations
around the pure conduction solution (V, θ) = (0, 0) have negative real part, and
thus this solution is asymptotically stable. As the Rayleigh number R is increased
for fixed values of P r > 1 and T , the initial loss of stability occurs via a steadystate bifurcation. That is, an eigenvalue of the linear terms passes through zero.
At criticality, the (complexified) kernel W0 of the linearized PDE is spanned by
eigenfunctions of the form d(z)eik·x , where the wave vectors k satisfy |k| = kc for
some fixed kc > 0 (the critical wavenumber). It follows
√ that the dimension of W0
is finite and depends on c. For the choice c = 4π/ 3kc we have dim W0 = 6.
We work in the abstract framework introduced in Golubitsky et al. [1988]. The
six-dimensional kernel is given by
W0 = {w1 e0 + w2 R2π/3 · e0 + w3 R4π/3 · e0 | w1 , w2 , w3 ∈ C} ≡ C3 ,
where e0 is an eigenfunction. The center manifold theorem reduces the underlying PDE, Equation (4.2.1), to a Γ-equivariant six-dimensional ordinary differential
equation on W0 while preserving the local dynamics close to criticality. The action
4.3. Modelling the system
87
of Γ on functions (V, θ, B) induces on W0 the action
φ = (φ1 , φ2 ) ∈ T2 : (w1 , w2 , w3 ) → (eiφ1 w1 , eiφ2 w2 , e−i(φ1 +φ2 ) w3 ),
Rπ/3 ∈ Z6 : (w1 , w2 , w3 ) → (w̄2 , w̄3 , w̄1 ),
τ : w → −w,
ρ : w → w.
There is generically a pitchfork bifurcation to a branch of rolls solutions ξ1 =
(α1 , 0, 0), α1 ∈ R. By equivariance, there is a group orbit Γξ1 of rolls solutions. The
continuous group orbit of translates working on ξ1 will also be denoted by ξ1 . The
discrete symmetry leads to groups of rolls solutions ξ2 = R2π/3 ξ1 and ξ3 = R4π/3 ξ1 .
When the Taylor number is large enough, we have the Küppers-Lortz instability
whereby the supercritical rolls are saddles in W0 with one neutral eigenvalue, three
stable eigenvalues and two unstable eigenvalues. There is a saddle-sink connection
from ξ1 to ξ2 inside the flow-invariant subspace {(α1 , α2 , 0) | α1 , α2 ∈ R}. By
symmetry, the result is a homoclinic cycle, the Busse-Heikes cycle, that is robust
to perturbations that preserve the symmetry. In addition the Busse-Heikes cycle
is asymptotically stable to perturbations that preserve the hexagonal lattice. See
Goldstein et al. [1990], Guckenheimer & Holmes [1988], Krupa [1997] for details.
For rolls of large enough amplitude, there may be a secondary instability to
magnetic perturbations as predicted in Childress & Soward [1972]. We assume that
such a magnetic instability occurs for suitable choices of the parameters in Equation
(4.2.1). There are two types of bifurcations, but we will restrict to the case where
there is a steady-state bifurcation from the Busse-Heikes cycle. Let Y0 denote the
critical eigen space of rolls in the ∂B/∂t component of Equation (4.2.1). It follows
that Y0 ≡ C2 and coordinates y can be chosen so that the action of Σrolls on Y0 is
given by
Rπ ∈ Z2 : y → ȳ, (0, φ2 ) ∈ T1 : y → einφ2 y,
τ̃ y = τ ◦ (π, π)y = y, ρy = −y,
for some positive integer n. We restrict to the case where n is odd, this is the case
where essential asymptotic stability occurs. Note that the bifurcating equilibria
at ξ1 , ξ2 and ξ3 each have a different critical eigen space, meaning that at each
equilibria there are two different unstable directions after the bifurcation. This gives
rise to a 12-dimensional normally hyperbolic center manifold that is attracting. The
coordinates are given by (w, y) ∈ C6 . The Busse-Heikes cycle is inside the fixed point
subspace {w ∈ R3 , y = 0}. We will show that the Busse-Heikes cycle is essentially
asymptotically stable inside the space R = {(w, y) ∈ C6 | wi , yi ∈ R6 }. The BusseHeikes cycle is part of an attractor A ⊂ R. If we let the rotations φ act on A we get
an attractor for the full system.
88
Chapter 4. Magnetic dynamos in rotating convection
4.3
The reduced system
For ε small, we consider the 12-dimensional system of differential equations
(ẇ, ẏ) = f (w, y; ε)
that results from the theory in the previous section. We write w = (w1 , w2 , w3 ) ∈ C3
and y = (y1 , y2 , y3 ) ∈ C3 . We assume that for ε < 0 close to zero the system
possesses a Busse-Heikes cycle and that there is a steady-state bifurcation from the
Busse-Heikes cycle to equilibrium dynamos at ε = 0. From now on we write Γ for
the Busse-Heikes cycle. After the bifurcation, there are two new equilibria created
at each equilibrium of Γ. There are local connections joining old equilibria to new
equilibria and due to fixed point subspaces there are also connections joining new
equilibria with old equilibria. This is similar to the network in Chapter 3 after the
bifurcation. We denote all the equilibria and the connections joining them by Γ .
Let G ∼
= (Z2 (Rπ/3 ) T2 ) × Z2 (τ ) × Z2 (ρ) be the group generated by
φ : (w, y) → (eiφ1 w1 , eiφ2 w2 , e−i(φ1 +φ2 ) w3 , einφ2 y1 , e−in(φ1 +φ2 ) y2 , einφ1 y3 ),
Rπ/3 : (w, y) → (w̄2 , w̄3 , w̄1 , ȳ2 , ȳ3 , ȳ1 ),
τ : (w, y) → (−w, y),
ρ : (w, y) → (w, −y).
Note that Rπ (w, y) = (w̄, ȳ) induces the 6-dimensional flow invariant subspace
R = Fix(Rπ ) = {(w, y) ∈ C6 | wi , yi ∈ R}
that contains the Busse-Heikes cycle. After the bifurcation it also contains the
network Γ . On R the symmetry group G is reduced to Ḡ which is generated by the
elements:
ρ, τ, (π, π), (π, 0), (0, π).
We will show that the Busse-Heikes cycle is essentially asymptotically stable within
the invariant subspace R.
Theorem 4.3.1 (Essential asymptotic stability of the Busse-Heikes cycle).
Suppose we have a Ḡ-equivariant system of differential equations depending on a
parameter ε so that for ε < 0 the system possesses a Busse-Heikes cycle and at
ε = 0 the equilibria of the Busse-Heikes cycle undergo a supercritical pitchfork bifurcation as described above. Then for λy2 < λw3 the Busse-Heikes cycle is essentially
asymptotically stable for ε small. Furthermore, Γ is an attracting magnetic dynamo.
4.3. The reduced system
89
Sketch of proof (following the proof of Theorem 3.1.1). The symmetry
ρτ (π, π) : (w1 , w2 , w3 , y1 , y2 , y3 ) → (w1 , w2 , −w3 , y1 , −y2 , y3 ),
induces the flow invariant subspace R1 = {w3 = y2 = 0}. Furthermore the symmetry
ρ : (w, y) → (w, −y) induces the flow invariant subspace R2 = {y = 0}. Just like the
example in Chapter 3, assuming nonresonance conditions up to order N , there is a
C k -equivalence with a locally linearized diagonal system that is still G-equivariant.
We now construct a first return map. We define Σin and Σout in the same way as
we did before. After rescaling we have
Σin = {|(w1 , w3 )| = 1; |y1 |, |y2 |, |y3 | < 1},
Σout = {|w2 | = 1; |w1 |, |w3 |, |y1 |, |y2 |, |y3 | < 1}.
We use the flow invariant subspaces R1 and R2 to determine some terms that will not
show up in the connecting diffeomorphism Πfar : Σout → Rπ/3 Σin . After identifying
Σin with Rπ/3 Σin through the twist h the first return map Π : Σin → Σin is at lowest
order given by
⎞
⎛
−λw
−λy
3
2
λw
λw
⎛
⎞
A1 w3 |y| 2 + A2 y2 |w2 | 2
⎟
⎜
w2
−λy
⎟
⎜
2
λw
⎟
⎜ y1 ⎟ Π ⎜
3
By
|w
|
2 2
⎟,
⎜
⎟→⎜
−λy
−λy
−λy
3
1
2
⎟
⎜
⎝ y2 ⎠
⎜ C1 y3 |w2 | λw2 + C2 y1 |w2 | λw2 + C3 y2 |w2 | λw2 ⎟
⎠
⎝
y3
−λy
−λy
−λy
3
1
2
D1 y3 |w2 | λw2 + D2 y1 |w2 | λw2 + D3 y2 |w2 | λw2
−λ
showing only the four important coordinates. It holds that λww3 > 1 and that λw2
2
and λy1 are the only positive eigenvalues. The terms that obstruct the contracting
property for the first return map are of the form
O(y1 |w2 |
−λy
1
λw
2
).
Therefore we define a cuspoidal region Cδ :
λy
1
Cδ = {|y1 | < δ|w2 | λw2 /M },
for each δ-neighborhood N (δ) of the origin in Σin . It is clear that for all points
x inside N (δ) − Cδ we now have the contracting property that |Π(x)| < |x|/2. To
conclude that the cycle is essentially asymptotically stable we only have to show
that
λy
1
Πy1 < δ|Πw2 | λw2 /M,
90
Chapter 4. Magnetic dynamos in rotating convection
or
By2 |w2 |
−λy
2
λw
3
< δ|(A1 w3 |y|
−λw
3
λw
2
+ A2 y2 |w2 |
−λy
2
λw
2
λy
1
)| λw2 /M.
Under the assumption that λy2 < λw3 this estimate applies for δ small and x ∈
N (δ).
Note that in a similar way we can show that Γ is an attractor (or magnetic
dynamo) within R. Except for a codimension one manifold all points are attracted
to the Busse-Heikes cycle. This is observed by the same reasoning as in the proof of
Theorem 3.1.1.
Define the 8-dimensional manifold R̃ = {φx | x ∈ R, φ ∈ T2 } and the network
Γ̃ = {φx | x ∈ Γ, φ ∈ T2 }. The network Γ̃ is essentially asymptotically stable within
R̃, because for each x ∈ R̃, there is a φ ∈ T2 so that x is inside the flow invariant
subspace φR = {φx | x ∈ R}, where the flow is equivalent to the flow of R.
Note that, within the 12-dimensional subspace, at each equilibrium there are
4 attracting directions transverse to the manifold. Defining cross-sections close to
the equilibria, the time for the flow spend close to the equilibria, where there is
attraction towards R̃, is long (exponentially) and the time away from the equilibria
is bounded from above. This implies attraction to the 8-dimensional manifold R̃.
And thus the network Γ̃ is essentially asymptotically stable within the full system.
By the same reasoning we find that à = {φx | x ∈ A, φ ∈ T2 } is an attractor.
In other words we have found a magnetic dynamo. Except for a codimension one
manifold all points are attracted to Γ̃.
We have shown that there can be an essentially asymptotically stable network
that corresponds to a magnetic dynamo. Of course this will only exist if the eigenvalue assumption λy2 < λw3 is met. It is unclear what the equivalent condition is
for the starting model and thus also if this condition can be satisfied. Then again,
this is only a simplified model and we don’t believe in those.