In a recent paper Moroz \cite{m02} considered a simplified version of the
third class of self-exciting Faraday-disk dynamo model, introduced by Hide
\cite{h97}, in the limit in which the Malkus-Robbins dynamo \cite{m72,r77}
results as a special case. In that study a linear series motor was incorporated
which led to an enriching of the range of possible behaviour that the
original Malkus-Robbins dynamo could support. In this paper, we replace the
linear motor by a nonlinear motor and consider the consequences on the dynamics
of the dynamo

Moroz, I. M.

English

Mathematical Institute Eprints Archive

September 2, 2004 14:14 01092
International Journal of Bifurcation and Chaos, Vol. 14, No. 8 (2004) 2885–2892
c© World Scientific Publishing Company
THE MALKUS ROBBINS DYNAMO
WITH A NONLINEAR SERIES MOTOR
IRENE M. MOROZ
Mathematical Institute, 24–29 St Giles’, Oxford OX1 3LB, UK
moroz@maths.ox.ac.uk
Received June 2, 2003; Revised July 28, 2003
In a recent paper Moroz [2003] considered a simplified version of third class self-exciting Faraday-
disk dynamo model, introduced by Hide [1997], in the limit in which leads to the Malkus–Robbins
dynamo [Malkus, 1972; Robbins, 1977] as a special case. In that study a linear series motor was
incorporated which led to an enriching range of possible behavior that the original Malkus–
Robbins dynamo could support. In this paper, we replace the linear motor by a nonlinear motor
and consider the consequences on the dynamics of the dynamo.
Keywords : Dynamos; bifurcation.
1. Introduction
In a recent paper, Moroz [2003] investigated the
behavior associated with the Malkus–Robbins dy-
namo [Malkus, 1972; Robbins, 1977], augmented to
include a linear series motor (hereafter denoted by
EMR). This model arises from a special simplifica-
tion of one of the self-exciting dynamo hierarchies,
discussed in [Hide, 1977]. The original Malkus–
Robbins dynamo reduces to the classic Lorenz equa-
tions under a linear transformation of the variables.
The linear and nonlinear behaviors observed to
occur in the EMR dynamo bear certain similari-
ties with another four-mode dynamo model [Hide
& Moroz, 1999; Moroz & Hide, 2000] in which an
azimuthal eddy current was incorporated into the
simple disc dynamo model of [Hide et al., 1996].
Moreover this azimuthal eddy current dynamo also
reduces to a transformed version of the Lorenz equa-
tions in the absence of the linear motor.
When the linear motor of [Hide et al., 1996]
is replaced by a nonlinear motor [Hide, 1997] all
oscillatory behavior disappears and the only possi-
ble nonlinear state is steady dynamo action. This
so-called “nonlinear quenching” arises because the
double-zero bifurcation, responsible for the onset of
oscillations, is absent. Moroz [2002] has shown that
when the parameter ε, introduced in [Hide, 1997]
to measure the degree of nonlinearity of the motor,
is increased from ε = 0 (linear motor) to ε = 1
(the nonlinear motor of [Hide, 1997]), the double-
zero bifurcation (and consequently the oscillatory
solutions) moves to infinity.
In the presence of an azimuthal eddy current,
Hide and Moroz [1999] have shown that partial
quenching of the oscillatory modes occurs for ε = 1.
Periodic and chaotic behavior is confined to a small
region near the subcritical Hopf bifurcation value
for the Lorenz limit of the dynamo equations.
The objective of the present study is to perform
for the EMR dynamo, the corresponding analysis
that was undertaken for the azimuthal eddy cur-
rent [Hide & Moroz, 1999] and simple self-exciting
dynamo models [Hide, 1997; Moroz, 2002]. A future
study will address the more complicated situation
which arises when 0 < ε < 1 (see also [Moroz &
Hide, 2000; Moroz, 2002]).
The paper is structured as follows. In Sec. 2
we present the EMR dynamo equations, modified
to include a nonlinear series motor, in their original
unscaled form. Then we apply the same rescaling as
2885
September 2, 2004 14:14 01092
2886 I. M. Moroz
used in [Moroz, 2003] to generate the nonlinear
model investigated in this paper. Section 3 con-
tains the bifurcation analysis, while Sec. 4 describes
selected numerical integrations of the four-mode
system. We summarize our results in Sec. 5,
indicating possible future research.
2. The Nonlinear Dynamo Equations
The EMR dynamo [Moroz, 2003] in the presence of
a nonlinear series motor takes the form
(1 + lr)x˙c + lrx˙a = xc(y − 1− rr)− rrxa
−βzcf(xc) , (1a)
x˙c = −xc − βzcf(xc) + raxa , (1b)
y˙ = α[1− xc(xc + xa)]− κy , (1c)
z˙c = xcf(xc)− λzc , (1d)
where
f(xc) = 1− ε + εxc (2)
and 0 ≤ ε ≤ 1. We use the same transformation as
in [Moroz, 2003], namely
y = AW − ra, xc + xa = BZ, xc = CY ,
zc = DU, t = Tτ ,
(3)
to obtain
W ′ = R− αZY − νW , (4a)
Z ′ = Y W − Z , (4b)
Y ′ = σ(Z − Y )− βˆUf(CY ) , (4c)
U ′ = Yf(CY )− ΛU , (4d)
where
A =
(1 + ra)(ra + rr)
ra
, C =
ra + rr√
lr
,
B =
A√
lr
, D =
√
lr ,
(5)
with
T =
lr
ra + rr
, α =
α
κ
, βˆ =
ν2β
κ2
,
f(CY ) = 1− ε + εCY .
(6)
The parameters appearing in (4) are, as before:
R =
ν2ra
σκ(ra + rr)
(ra + α), ν =
κlr
ra + rr
,
σ =
lr(1 + ra)
ra + rr
, Λ =
λlr
ra + rr
,
(7)
and derivatives are now with respect to τ .
When ε = 0, f(CY ) = 1 and we recover the
EMR dynamo with a linear series motor [Moroz,
2003]. Here we focus on the special case of ε = 1,
and compare our results with those of another four-
mode coupled system, namely the azimuthal eddy
current dynamo of [Hide & Moroz, 1999].
We note that, unlike the EMR dynamo of
[Moroz, 2003], Eqs. (4) are no longer invariant under
a rescaling by the parameter α, and so α is no longer
passive.
3. Bifurcation Analysis
When ε = 1, Eqs. (4) simplify to
W ′ = R− αZY − νW , (8a)
Z ′ = Y W − Z , (8b)
Y ′ = σ(Z − Y )− βˆCUY , (8c)
U ′ = CY 2 − ΛU . (8d)
If we replace (W, Z, Y, U) by (W, −Z, −Y, U),
then Eqs. (8) are unchanged. Equilibrium states are
given by
W0 = (W0, Z0, Y0, U0) =
(
R
ν
, 0, 0, 0
)
, (9)
as in [Moroz, 2003] and termed the trivial equilib-
rium state and
We =
(
We, WeYe, Ye,
CY 2
e
Λ
)
, (10)
which we term the nontrivial equilibria, where
Ye = ±
√
R− νWe
αWe
, (11a)
We =
[(
1− µν
α
)
+
√(
1− µν
α
)2
+
4µR
α
]/
2 ,
(11b)
where µ = βˆC2/σΛ, provided R > νWe.
As in [Moroz, 2003], we recover the equilib-
rium states of the original Malkus–Robbins dynamo
when βˆ = 0.
September 2, 2004 14:14 01092
The Malkus–Robbins Dynamo with a Nonlinear Series Motor 2887
3.1. Bifurcations from the trivial
state
If we consider the linear stability of W0 to expo-
nentially growing disturbances, then we obtain the
quadratic characteristic equation
m2 + (1 + σ)m + σ
(
1−R
ν
)
= 0 , (12)
together with the additional two roots m =
−ν, −Λ. The only bifurcation from the trivial equi-
librium state is therefore a steady bifurcation when
R = ν (c.f. [Hide & Moroz, 1999]).
3.2. Bifurcations from the
nontrivial states
Bifurcations from the nontrivial equilibrium states
We are determined from the nonseparable quartic
equation
M4 + C3M
3 + C2M
2 + C1M + C0 = 0 , (13)
where
C3 = 1 + ν + σWe + Λ , (14a)
C2 = Λ(1 + ν − 2σ) + σWe(3Λ + ν) + R/We ,
(14b)
C1 = σWe[2(Λ− ν) + 3νΛ]
+2σ[R− Λ(1 + ν)] + RΛ/We , (14c)
C0 = 2σΛ(2R − νWe −R/We) . (14d)
When R = ν, we have We = 1 and C0 = 0,
C1 = νΛ(1+σ). We can therefore have a steady bi-
furcation, corresponding to a pitchfork bifurcation
of the trivial equilibrium state, but not a double-
zero bifurcation, in contrast to the linear motor
problem of [Moroz, 2003] but similar to the non-
linear dynamo model with an azimuthal eddy cur-
rent (see [Hide & Moroz, 1999]). Any Hopf bifur-
cations that arise cannot terminate on the steady
bifurcation curve. When σ > 1 + ν, fulfilling the
criterion for the existence of the subcritical Hopf
bifurcation at Rc associated with the Lorenz limit
(β = 0) of the Malkus–Robbins dynamo, any Hopf
curves bifurcate from Rc. However when σ < 1 + ν,
we found that Hopf bifurcation curves are still pos-
sible, even though the subcritical Hopf bifurcation
condition fails. Such Hopf bifurcations must be com-
puted numerically and occur along the curve(s)
FHB = C21 − C1C2C3 + C0C23 = 0 , (15)
where Ω, the frequency of the bifurcating limit
cycle, is given by the common root of:
Ω2 =
C2
2
± 1
2
[C22 − 4C0]
1
2 =
C1
C3
. (16)
The procedure we adopted in calculating the Hopf
stability curves was similar to that described in
[Moroz & Hide, 2000]. We treated We and R as
the variables and recovered α via (7) and β by
rearranging the fixed point equation (11b). The
Hopf bifurcation criterion (15) yields a quintic equa-
tion for We as a function of α. We require only the
non-negative solutions to this polynomial since solv-
ing (11a) for We indicates that We ≥ 0.
3.3. Linear stability curves
The linear stability analysis for the EMR dy-
namo with linear series motor suggested four differ-
ent parameter regimes of interest, depending upon
whether Λ ≷ 1 and σ ≷ 1 + ν. Because of the
absence of the double-zero bifurcation point in the
ε = 1 problem, we have identified only two regimes
of interest: σ ≷ 1 + ν. However to illustrate the
effects of varying Λ and σ on these two cases, as
well as to facilitate comparisons with the ε = 0
dynamo, we chose ν = κ = 1 throughout, so that
again βˆ = β.
We present the linear stability curves in both
the (β, R)-plane as well as in the (β, α)-plane, so
that comparisons can also be made with the non-
linear dynamo model of [Hide, 1997] and the ε = 1
case of [Hide & Moroz, 1999]. In particular we are
interested in seeing if nonlinear quenching is also a
feature of our model. Recall that the fully-nonlinear
[Hide, 1997] model exhibited only steady-state so-
lutions, while the nonlinear [Hide & Moroz, 1999]
dynamo showed nonsteady solutions to be confined
to very small values of β but large values of α.
Figure 1 shows the linear stability boundaries
for Λ = 0.5 and σ = 5 in the (β, R)-plane, while
Fig. 2 shows the corresponding boundaries in the
(β, α)-plane. The onset of steady bifurcations oc-
curs for R = 1 or α = 2.25. In both figures (which
should be compared to Fig. 1 of [Moroz, 2003] for
the ε = 0 problem), the Hopf bifurcation again
occurs at R = 15 or α = 89.75 for β = 0, cor-
responding to the subcritical Lorenz Hopf bifurca-
tion. We can immediately see that the presence of
the subcritical Hopf bifurcation prevents total non-
linear quenching from occurring, even though there
is still a large region of parameter space below the
Hopf curve (the solid line) that could contain only
September 2, 2004 14:14 01092
2888 I. M. Moroz
Fig. 1. Linear stability curves for Λ = 0.5, σ = 5 and ν = 1
in the (β, R)-plane. The solid line denotes the curve of Hopf
bifurcations associated with the nontrivial equilibria, while
the faint line denotes the curve of steady bifurcations.
Fig. 2. As in Fig. 1, but in the (β, α)-plane.
steady solutions. Moreover the Hopf curve occurs
for much larger values of β than was the case in
[Hide & Moroz, 1999]. We shall see in Sec. 4 that
this boundary is subject to considerable hysteresis
effects.
Figures 3 and 4 show the corresponding linear
stability boundaries for Λ = 0.5 and σ = 1.5. We
chose σ = 1.5 instead of σ = 1.9 as in [Moroz, 2003]
in order to demonstrate that hysteresis is a feature
of the entire Hopf stability curve, including near
β = 0. We note that the absence of any intercepts
on the β = 0 axis, and so the absence of the subcriti-
Fig. 3. As in Fig. 1, but for σ = 1.5.
Fig. 4. As in Fig. 2, but for σ = 1.5.
cal Lorenz Hopf bifurcation, does not preclude the
existence of nonperiodic solutions nor does it favor
nonlinear quenching.
4. Numerical Investigations
We now present a selection of the numerical inte-
grations we conducted for Λ ≷ 1 and σ ≷ 1 + ν,
focusing upon the two cases described in Sec. 3.3.
We shall show phase portraits and/or time series
for certain parameter values, as well as plots of the
local maximum amplitude of W in the nonsteady
regimes as functions of β. As in [Moroz, 2003], we
found that such bifurcation transition sequences ex-
hibit notable hysteresis near to the Hopf bifurcation
September 2, 2004 14:14 01092
The Malkus–Robbins Dynamo with a Nonlinear Series Motor 2889
Fig. 5. Plot of the local maximum values of W (t) as an in-
creasing function of β for R = 50 when Λ = 0.5, σ = 5 and
ν = 1.
Fig. 6. As in Fig. 5 but for R = 100.
curve, depending upon whether β is increasing or
decreasing.
4.1. Λ < 1 and σ > 1 + ν
As in Sec. 3.3 above, we chose Λ = 0.5 and σ = 5.
When R = 50, chaotic solutions dominate the
regime from β = 0 to β ≈ 14, before the solution
loses stability to a steady state (see Fig. 5).
For R = 100, however, there are windows of pe-
riodic states, separating regions of chaos. Figure 6
shows the transition sequence when β is increased,
Fig. 7 showing the portion 10 ≤ β ≤ 30 in greater
Fig. 7. The portion of the bifurcation transition sequence of
Fig. 6 for 10 ≤ β ≤ 30.
Fig. 8. As in Fig. 6 but when β is decreased.
detail. In contrast, Fig. 8 shows how hysteresis
affects the transition sequence for R = 100 and
β decreasing. We therefore have the coexistence
of steady with chaotic or periodic solutions for
28 < β < 38.
Hysteresis also occurs when R is varied and β
is kept constant. When β = 0, the transition from
chaotic to steady state occurs at R ≈ 14.5 for R de-
creasing, while the transition from steady solution
to chaotic solution occurs at R ≈ 15.25 when R is
increased.
Figure 9 shows the phase portrait in the
(W, Z)-plane for β = 20 (near the middle of the
September 2, 2004 14:14 01092
2890 I. M. Moroz
Fig. 9. The phase portrait in the (W, Z)-plane for β = 20.
The remaining parameters are as in Fig. 6.
Fig. 10. The time series of W , corresponding to Fig. 9.
large periodic window), while Fig. 10 shows the cor-
responding time series for W . Figures 11 and 12
give the analogous plots for β = 10, in the chaotic
regime.
We close this discussion of the σ > ν case with a
few remarks concerning what happens when Λ > 1.
As in [Moroz, 2003], we chose Λ = 1.2 and σ = 5.
Although we have omitted the details of the linear
stability curves, the only difference we found was
that the Hopf curve (and so the nonsteady solu-
tions) covers more of parameter space. For exam-
ple when R = 100, the Hopf bifurcation occurs at
β ≈ 45. As for the nonlinear integrations, when
R = 50, we observed only chaotic states in the
Fig. 11. As in Fig. 9, but for β = 10.
Fig. 12. As in Fig. 10, but for β = 10.
transition sequence, while for R = 100, periodic
windows again arise. However their width is much
smaller than for the Λ = 0.5 case. For example,
Fig. 13 (c.f. Fig. 6) shows the bifurcation sequence
for 0 ≤ β ≤ 60 when β was increased.
4.2. Λ < 1 and 0 < σ < 1 + ν
To illustrate the behavior observed when Λ < 1
and 0 < σ < 1 + ν, we chose Λ = 0.5 as above and
σ = 1.5. For R = 50 and β increasing, the only
nonsteady behavior we found was chaotic within
2.7 < β < 18.3. When β is decreased, this range
September 2, 2004 14:14 01092
The Malkus–Robbins Dynamo with a Nonlinear Series Motor 2891
becomes 1 < β < 16.9 and again contains only
chaotic behavior. For R = 100 small windows of
periodic solutions now appear. Figure 14 shows the
transition sequence when β is increased. At either
end of the oscillatory regime, steady-state dynamo
action was observed. Figure 15 shows a magnifi-
cation of the transitions for 0 < β < 10. In con-
trast Fig. 16 shows this same region when β is
decreased. Note the additional nonsteady behavior
found in 0 < β < 2. At the other end of the chaotic
regime, when β is decreased, steady states persist
until β ≈ 56.1.
Fig. 13. Plot of the local maximum values of W (t) as an
increasing function of β for R = 100 and 0 ≤ β ≤ 60 when
Λ = 1.2 and σ = 5.
Fig. 14. Bifurcation transition sequence for R = 100, Λ =
0.5 and σ = 1.5, when β is increased.
Fig. 15. A magnification of the plot of Fig. 14 in the region
0 ≤ β ≤ 10.
Fig. 16. As in Fig. 15 but with β decreasing.
5. Discussion
In this paper we have considered the effects of a
fully nonlinear series motor (ε = 1) on the be-
havior of the Malkus–Robbins dynamo. In contrast
to the analogous version [Hide, 1997] of the self-
exciting Faraday-disk dynamo [Hide et al., 2002],
nonlinear quenching of oscillatory dynamo action
was never observed in any of the cases investi-
gated. Moreover the extent of the regime diagram
for which nonsteady behavior is possible is consid-
erably larger than for another four-mode nonlinear
dynamo model in which azimuthal eddy currents
are a feature [Hide & Moroz, 1999].
September 2, 2004 14:14 01092
2892 I. M. Moroz
As in the companion study of the EMR dynamo
with a linear series motor [Moroz, 2003], the β = 0
limit recovers the celebrated Lorenz equations. Also
as in [Moroz, 2003], the chaotic regime contains win-
dows of periodic behavior. The subcritical Hopf bi-
furcation curve, associated with instabilities of the
nontrivial equilibrium solutions, demonstrates clear
evidence of hysteresis, with coexistence of differ-
ent types of nonlinear dynamics. This was shown
in some of the bifurcation transition sequences in
the preceding section.
Yet to be investigated is the intermediate sce-
nario when 0 < ε < 1. Preliminary studies suggest
this to be a far more complicated problem than the
one considered here, especially in the Hopf bifurca-
tion analysis, and falls well outside the scope of the
present study.
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Hide, R. [1997] “The nonlinear differential equa-
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Hide, R. & Moroz, I. M. [1999] “Effects due to induced
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The Malkus-Robbins dynamo with a nonlinear motor

In a recent paper Moroz \cite{m02} considered a simplified version of the third class of self-exciting Faraday-disk dynamo model, introduced by Hide \cite{h97}, in the limit in which the Malkus-Robbins dynamo \cite{m72,r77} results as a special case. In that study a linear series motor was incorporated which led to an enriching of the range of possible behaviour that the original Malkus-Robbins dynamo could support. In this paper, we replace the linear motor by a nonlinear motor and consider the consequences on the dynamics of the dynamo

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The Malkus-Robbins dynamo with a nonlinear motor

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