# Solar dynamo

Post-publication activity

Curator: Guenther Ruediger

## The solar activity cycle

The solar dynamo yields the theoretical explanation of the basic solar activity cycle which is mainly (but not exclusively) manifested in the periodic appearance of sunspots. Its mean period is 11 yrs (i.e. 22 yrs for the cycle time of the full magnetic oscillation) but this value varies between 8 and 14 yrs. The variability of the activity period can be expressed by the quality $$\omega_{\rm cyc}/\delta_{\omega_{\rm cyc}}$$ which for the Sun has the rather high value of 5.

Figure 1: The butterfly diagram of sunspot statistics (Maunder 1904). Note in particular that the spots only exist very close to the equator; only a few spots appear at latitudes higher than (say) 30o. There is a well-established (but not perfect) symmetry of the phenomenon with respect to the equator.

Only the antisymmetric (dipolar) parts of the solar magnetic field oscillate and reverses its sign during the 22-yrs cycle (Stenflo & Vogel 1986). The magnetic reversal happens at the poles during the maximum of the sunspot activity. As the latter is located rather close to the equator one can be sure that the solar activity phenomenon is a global phenomenon rather than a local one.

### The Maunder minimum

The amplitude of the cycle is far from being constant. The most prominent activity drop known was the Maunder minimum between 1670 and 1715. The latitudinal distribution of the few sunspots observed during the end of the Maunder minimum was highly asymmetric with respect to the equator. All spots except two or three in this period appeared on the southern hemisphere (Spörer 1887). The first cycle after the minimum, with maximum in 1706, existed almost exclusively in one hemisphere. There is empirical evidence that the magnetic cycles persisted through the solar Maunder minimum, as first found by Wittmann (1978, old sunspot data), Schröder (1992, auroral activity) and more recently by Beer et al. (1998, Be data in ice cores).

Figure 2: The time series of the sunspot numbers indicate a highly nonlinear behavior. Both cycle amplitude and length are obvious functions of time. From 1670--1715 a grand minimum ('Maunder minimum') is seen.

Charbonneau (2001) reports an 'odd-even effect' of the cycle amplitudes. The odd-numbered cycles are stronger than their neighboring cycles or -- which is the same -- the sunspot number of the complete 22-year cycle does not fluctuate too much from cycle to cycle.

### Cycle length variations

Measurements of 14C abundances in sediments and long-lived trees provide much longer time series than sunspot datasets. Vos et al. (1997) found a secular periodicity of 80...90 yrs as well as a long-duration period of about 210 yrs. The measurements of atmospheric 14C abundances suggested a periodicity of 2400 yrs which is also associated with a long-term variation of solar activity. The variety of frequencies found in solar activity may even indicate chaotic behavior as discussed by Knobloch & Landsberg (1996) on the basis of nonlinear mode-mixing models (see Brandenburg et al. 1989, Schmitt & Schüssler 1989).

Figure 3: Cycle length (left) and cycle amplitudes (right) for 11 cycles separated for northern (green) and southern (blue) hemispheres.

Another question concerns a possible systematic north-south asymmetry of the cycle length. If it is true that the grand minima result from simultaneous nonlinear oscillation of modes with equatorial symmetry and antisymmetry then systematic differences of the cycle length in the two hemispheres must appear. Both cycle lengths and amplitudes in the dataset of Hathaway et al. (2003) are given in Figure 3. There seems to be a slight anticorrelation between cycle length and cycle amplitude in the data: short cycles are followed by stronger cycles (Hoyng 1993, Solanki et al. 2002).

Any single cycle is asymmetric in time. The strong cycles are faster to reach their activity maximum than the weak cycles. This 'Waldmeier rule' can be used to produce successful predictions of the strength and duration of future activity cycles (Schüssler 2007).

### Solar-stellar relations

The activity cycle of the Sun is not exceptional among the stars. The observation of chromospheric Ca-emission of solar-type stars yields activity periods between 3 and 30 yrs. Up to 15% of the solar-type stars, however, do not show any significant activity. This suggests that even the existence of the grand minima is a typical property of cool main sequence stars like the Sun. From ROSAT X-ray data Hempelmann et al. (1996) find that up to 70% of the stars with a constant level of activity exhibit a rather low level of coronal X-ray emission.

The dependence of the cycle frequency on the rotation rate $$\Omega$$ is rather weak. Saar & Brandenburg (1999) report for the relation $\tag{1} \omega_{\rm cyc} \propto \Omega^n$

the very small value $$n \simeq -0.09$$ for their full data-set. They derive values of 1.15, 0.8 and 0.4 for their groupings 'inactive', 'active' and 'superactive'. With the same data base Böhm-Vitense (2007) finds similar relations with $$n \simeq 1\ .$$ The situation can also be realized in Figure 4 where the data of Saar & Brandenburg (1999) and Olah et al. (2000) are collected. The active stars are given in pink while the less active stars are given in blue and black. The latter really form an own branch with n of order unity while there is no clear rule for the fast-rotating stars. Considering only the shortest cycle for a given rotation rate, one finds the small value n = 0.18 (Olah et al. 2007). For fast rotating stars the angular velocity itself seems to be not the only determinant of the cycle time.

Figure 4: Cycle periods in years vs. rotation times in days, compiled from Saar & Brandenburg (1999) and Olah et al. (2000). The linear relations with $$n \simeq 1$$ by Böhm-Vitense (2007) are given by straight lines with corresponding colors.

## Theory

Schwabe's (1844) 11-year sunspot cycle, Carrington's (1863) differential rotation (of the solar surface) and Spörer's law (of the equatorward migration of sunspots during the cycle, 1894) are the basic properties of the solar activity that must be explained by the dynamo theory. As these three basics are understandable even with a linear theory (which already demonstrates the complexity of the problem) only a linear approach is here presented. There are many more nonlinear extensions of the models but the main principles and problems already appear in the given linear approach.

### Convection zone dynamo

In the solar surface there is turbulence with velocity fluctuations of 1 km/s and correlation times of about 10 min. The magnetic diffusivity $$\eta_{\rm T}$$ formed by such high-speed short-living granular motion is of order 1012 cm2/s. This value allows a skin depth of an electromagnetic oscillator with a timescale of 10 yrs of more than 100 000 km -- close to the depth of the solar convection zone (200 000 km). This is the background for the common believe that the entire solar convection zone should be the site of the solar dynamo. If this is true then also the large ratio of the mean cycle period and the rotation time of the stars shown in Figure 4 becomes understandable. The basic time scale in convection zones is the correlation time of the turbulent cells which at the bottom of the solar convection zone reaches the value of the rotation time (25 days). In a thick convection shell the number $$\tau_{\rm cyc}/\tau_{\rm corr}$$ reflects the square of the ratio of the stellar radius to the correlation length so that numbers exceeding 100 are easily possible.

Most solar dynamo models also operate in a linear way. Rather weak seed fields are amplified in the sense of a self-excitation of large-scale magnetic fields. The global magnetic field can be described by a superposition of a poloidal field (the field projected in a meridional plane) and the toroidal (azimuthal) field component. Both field components would independently decay if there are no extra processes which maintain the magnetic field against the Ohmic decay which would destroy the fields after about 10 yrs.

A dynamo can thus only operate if processes exist which transform poloidal fields into toroidal fields and v.v. One of the processes is well-known: if the angular velocity in the convection zone is not uniform, then it easily transforms poloidal fields into toroidal fields by wiggling them up. The effectivity of this transformation scales with the magnetic Reynolds number of the differential rotation, i.e $$B_\phi/B_r\simeq \delta\Omega\ D^2/\eta_{\rm T}$$ (with D as the depth of the convection zone) which is about 100 for $$\eta_{\rm T}=5\cdot 10^{12}$$ $${\rm cm}^2/{\rm s}$$ and 5000 for $$\eta_{\rm T}= 10^{11}$$ $${\rm cm}^2/{\rm s}\ .$$ It is thus easy to understand that the amplitude of the toroidal field of the Sun exceeds the amplitude of the poloidal field. Obviously, the observed factor of about 1000 needs a rather small eddy diffusivity.

### Electromotive force (EMF) in turbulent plasma

More complicated is the regeneration of the poloidal field from the toroidal one. In principle it can be done by the action of a meridional flow but in this case the resulting magnetic field must be nonaxisymmetric in a large scale (Cowling theorem). One has thus to look for the hydromagnetic properties of the convection/turbulence in the solar convection zone. It is in particular the structure of the turbulent electromotive force $$\mathbf{\mathcal{E}}=\langle \mathbf{u}'\times \mathbf{B}'\rangle$$ which might be important. It appears in the Ohm's law for mean-field quantities

$\tag{2} \langle\mathbf{J}\rangle= \sigma \big(\langle\mathbf{E}\rangle+ \langle\mathbf{u}\rangle\times \langle\mathbf{B}\rangle+\mathbf{\mathcal{E}}\big)$

and plays the central role in the mean-field dynamo theory.

The simplest electrodynamic effect of a turbulent field of electrically conducting matter is its diamagnetism, i.e. the tendency to transport a mean magnetic field into regions of lower turbulence intensity. It can simply be written as

$\tag{3} \mathbf{\mathcal{E}}=\mathbf{\gamma} \times \langle \mathbf{B}\rangle + \dots$

with $$\mathbf{\gamma}=-\frac{1}{2} \nabla \eta_{\rm T}\ .$$ The field is thus transported towards the minimum of the eddy diffusivity $$\eta_{\rm T}\ .$$ The effect is also known as turbulent pumping; it is the reason that in dynamo simulations the flux is mainly transported downwards rather than upwards as the various buoyancy effects would suggest. If the downward pumping is too strong then the dynamo does not operate. For $$\eta_{\rm T}\simeq 10^{12}\ {\rm cm}^2/{\rm s}$$ the amplitude of the downward pumping exceeds the value of 1 m/s.

Another turbulence-induced term runs with the mean-field electric current, i.e.

$\tag{4} \mathbf{\mathcal{E}}= \dots -\eta_{\rm T}\, {\rm curl}\, \langle\mathbf{B}\rangle,$

which describes the strongly enhanced dissipation of the field by the turbulence. The ratio of the eddy diffusivity $$\eta_{\rm T}$$ to the microscopic diffusivity of the gas is extremely high. If the basic stellar rotation is included into the computations of the EMF the magnetic eddy diffusivity forms a complicated tensor with many components and consequences also for the dynamo theory (see Kitchatinov et al. 1994).

### The alpha-effect due to rotation

The transformation of the toroidal field to the poloidal field needs another term. It is the alpha-effect by Steenbeck, Krause & Rädler (1966) which is the basis of almost all of the solar dynamo models developed so far. After the alpha-effect concept in rotating turbulences of conducting matter a relation

$\tag{5} \mathbf{\mathcal{E}}=\alpha\ \langle\mathbf{B}\rangle+...$

exists so that by the action of the turbulence an electric current flows along the magnetic field lines. The current along the toroidal field is always accompanied by a poloidal field, hence the loop can be closed and the magnetic field can be maintained against its Ohmic decay.

An alpha-effect only exists if the star rotates. Additionally there must be a stratification of density and/or turbulence intensity otherwise the alpha-effect is zero. The following argument describes the situation. The quantity alpha is a pseudo-scalar as it connects the polar vector $${\mathbf{\mathcal{E}}}$$ with the axial vector $$\langle\mathbf{B}\rangle\ .$$ The simplest pseudoscalar one can form in turbulent flows is $$\langle \mathbf{u}'\cdot {\rm curl}\, \mathbf{u}'\rangle$$ that describes a helicity in the turbulence field. Helicity means that the turbulence is not mirror-symmetric, i.e. (say) left-handed helical motions are more frequent than right-handed ones or v.v. This is only possible if the star rotates.

The simplest pseudoscalar one can construct with global quantities is the scalar product of the vector of the density stratification $$\mathbf{g}=\nabla {\rm log}\,\rho$$ and the vector of the angular velocity of the basic rotation so that $$\alpha \propto \mathbf{g}\cdot\mathbf{\Omega}$$ should be true. The sign of this quantity differs in the northern and the southern hemisphere and so both the alpha-effect and also the helicity do. At the equator the alpha-effect vanishes, but it exists at the poles. For mesogranulation the helicity can directly be observed at the solar surface. It proves to be negative (positive) at the northern (southern) hemisphere.

There is indeed a theoretical relation of the alpha-effect and the numerical value of the helicity, i.e.

$\tag{6} \alpha \simeq -\tau_{\rm corr} \langle \mathbf{u}' {\rm curl}\, \mathbf{u}'\rangle,$

which has been confirmed by many numerical simulations. Highly nonlinear calculations with forced turbulence also reveal the contributions of magnetic helicity to the EMF which can also be observed at the solar surface (see Brandenburg & Subramanian 2005).

### Dynamo waves

The simplest dynamo models even work in low dimensions producing by the common action of alpha-effect and differential rotation traveling magnetic waves with a frequency of

$\tag{7} \omega_{\rm cyc}\simeq \left(\frac{\alpha^2 }{\eta}(r\frac{{\rm d}\Omega}{{\rm d}r})^2\right)^{\frac{1}{3}}$

(Tuominen et al. 1988). To obtain a cycle time of about 20 yrs an alpha of order 1 m/s is thus necessary. Simulations of the alpha-effect in the solar convection zone yield $$\alpha \simeq 10^{-(1\dots 2)}\ u_{\rm T}$$ with $$u_{\rm T}$$ as the characteristic turbulence velocity so that with $$u_{\rm T}\simeq 100\ {\rm m/s}$$ indeed an alpha-effect of 1 m/s results. The expression (7) also explains the rather weak dependence of the cycle times of stellar dynamos on the rotation time. Theory and observation of the stellar differential rotation lead to the common result that the rotation law in stars hardly depends on their rotation rate (see below).

Also the alpha-effect does not depend strongly on the rotation rate. Ossendrijver et al. (2001) report an increase of the alpha-effect (in horizontal direction) by a factor of only 2.5 if the rotation rate varies by two orders of magnitude. Hence, the result (7) complies with the observed small exponent n in (1).

### The differential solar rotation

Figure 5: The internal rotation law of the Sun from the inversion of SOHO data. The curves are marked with the latitude (Courtesy NSF National Solar Observatory).
Figure 6: The solar rotation law computed for 25 days rotation period . The model includes two parameters only for the anisotropic correlation tensor of the turbulence in the convection zone (Kitchatinov & Rüdiger 2005).

The rotation law within the solar convection zone is known by the inversion procedures of the helioseismology ( Figure 5). Formally, the observed equatorial acceleration at the surface proves to be the result of a strong polar subrotation ($$\partial\Omega/\partial r<0$$) and a weaker equatorial superrotation ($$\partial\Omega/\partial r>0$$) inside the convection zone. The strong polar subrotation forms the main problem for the solar dynamo theory as naturally the strongest fields are there induced which, however, are not observed. The solar core rotates rigidly with approximately the same angular velocity as the surface at midlatitudes. The solution of the Reynolds equation on the basis of the (also turbulence-induced) Lambda-effect for rotating density-stratified turbulence do indeed explain these observations ( Figure 6).

The stellar-surface rotation laws which are empirically known so far lead to the result that the absolute equator-pole differences of the solar-type stars hardly differ from star to star.

Figure 7: The sofar known equator-pole differences (in rad/day) of very young stars, the MOST observations and the present-day Sun. For LQ Hya the equatorial acceleration is reported as highly time-dependent. LQ Lup is too fast for present-day theory. Rotation period in days.

Data from the MOST satellite recently led to the detection of rotation laws for two new young solar-type stars with rotation periods of about 10 days. For $$\epsilon$$ Eri it is $$\delta\Omega=0.061\ {\rm rad/day}$$ and for $$\kappa$$1Ceti it is $$\delta\Omega=0.064\ {\rm rad/day}$$ (Walker et al. 2007). Note the almost perfect coincidence of these values with the solar value (see Figure 7). The two new examples are of extraordinary meaning insofar as their rotational rates start to fill the gap between the fast rotating very young stars and our (old) present-day Sun. A characteristic value of the equator-pole difference of $$\Omega$$ for stars (including the Sun) is $$\delta\Omega= 0.07\ {\rm rad/day}\ .$$ This value excellently complies with the results of the theory of the rotation laws of stars with outer convection zones.

### Meridional flow

All nonuniform rotation laws exist together with meridional flows as the centrifugal force of nonrigid rotation cannot longer be written as a conservative force. The flow which occurs in the simulations at the bottom of the convection zone drifts always equatorwards with an amplitude of $$u^{\rm m}\simeq 5\rm m/s$$ (Küker & Rüdiger 2005). Such a flow needs slightly more than 7 yrs to travel from the pole to the equator. However, with $$\eta_{\rm T}= 10^{12}\ {\rm cm}^2/{\rm s}$$ the magnetic Reynolds number $${\rm Rm}= u^{\rm m} R/\eta_{\rm T}$$ (R ~ 700.000 km, solar radius) does not strongly exceed unity so that the meridional flow will not be important in the dynamo equation. Only if the magnetic eddy diffusivity is reduced by (say) a factor of 10 the meridional circulation dramatically changes the dynamo regime by fixing the cycle time and transporting the azimuthal magnetic field towards the equator.

## Numerical models

The dynamo equation $\tag{8} \frac{\partial \langle\mathbf{B}\rangle}{\partial t}= {\rm curl} \big((\mathbf{u}+\mathbf{\gamma})\times \langle\mathbf{B}\rangle + \alpha \langle\mathbf{B}\rangle - \ \eta_{\rm T}{\rm curl}\,\langle\mathbf{B}\rangle\big)$

(with the pumping term included) must be solved with $$\mathbf{u}$$ as the given large-scale flow pattern in the solar convection zone. Simple models only apply the solar rotation law ( Figure 5). If the fields are weak enough then any magnetic feedback to the terms of the electromotive force can be ignored. Calculations with this assumption are called as 'kinematic' dynamo models. Nonlinear dynamo models are much more complicated.

### Distributed dynamos

Let the magnetic eddy diffusivity $$\eta_{\rm T}$$ be homogeneous within the entire convection zone (thickness $$D\simeq 200.000$$ km) while the solar core is considered as a perfect conductor. The alpha-effect always runs with $$\cos\theta$$ so that it peaks at the poles. If the angular velocity has only a radial gradient, then the cycle time turns out to be of order $$\tau_{\rm cyc}\simeq 0.26 DR/\eta_{\rm T}\ .$$ The value $$\eta_{\rm T} \simeq 10^{12}\ {\rm cm}^2/{\rm s}$$ is thus the reference value leading to a cycle time of 10 yrs.

Figure 8: A solar dynamo model with alpha-effect in the entire convection zone and $$\eta_{\rm T}= 5\cdot 10^{12}\ {\rm cm}^2/{\rm s}\ .$$ Left: The geometry of the induced magnetic field. The colors represent the magnitude of the toroidal field while the arrows form the field lines of the poloidal field. Note the equatorial asymmetry. Right: Phase relation of the field components (BT toroidal, BP poloidal). The cycle time is given in units of the diffusion time (here 33 yrs).

With the observed solar rotation law and positive (in the northern hemisphere) depth-independent alpha-effect Eq. (8) yields magnetic fields which are represented in Figure 8. One finds a radial drift of the toroidal field belts but not an equatorial migration,i.e. the butterfly diagram ( Figure 1) can not be explained. The parity of the solution is dipolar, the toroidal magnetic amplitudes are much too weak. Also the cycle time is too short.

The model can be improved if the alpha-effect is concentrated to a thin layer at the bottom of the convection zone. Then the amplitude ratio exceeds values of 200 but one always gets more magnetic belts than needed. The butterfly diagram appears at too high latitudes and will never reach the low latitudes which are necessary to reflect the solar observations correctly.

Too simple kinematic $$\alpha\Omega$$-dynamos working in the convection zone with positive alpha-effect and the solar rotation law are thus not able to fulfill the main observational constraints.

### Boundary-layer dynamos

The idea of the boundary-layer (BL) dynamo bases on the findings that i) the alpha-effect in the bottom of the solar convection zone, where the convection overshoots into the stably stratified radiative zone, becomes negative and ii) the radial gradients of the angular velocity peak there (see Belvedere et al. 1991). One could thus indeed realize an equatorwards drift of the toroidal field belts in case the alpha-effect vanishes at the poles. If the boundary layer, however, is too thin then the cycle time becomes too short for $$\eta_{\rm T}$$ = 1012 cm2/s and also the number of magnetic belts becomes too high. Indeed, after current convection zone simulations the overshoot phenomenon restricts to a layer with less than 10.000 km depth. It is hard to believe that in such a thin layer the averaging procedure for the turbulence can be done leading to a mean-field formulation of the EMF. Current dynamo models thus prefer to work with the positive alpha-effect dominating the entire convection zone. In these cases the butterfly diagram can only result from the action of the meridional circulation flowing towards the equator at a radius where the toroidal magnetic field has its maximum.

The inclusion of the meridional flow has strong impacts to the mean-field dynamo models only when the eddy diffusivity is so low that the magnetic Reynolds number of the flow reaches values of the order of 103. The consequences for the dynamo regime are drastic. Models with rather small values of the eddy diffusivity have been a subject of intense numerical investigation (Choudhuri et al. 1995, Charbonneau & Dikpati 2000, Dikpati & Gilman 2006). Models with a positive alpha-effect and high magnetic Reynolds number provide correct cycle periods, equatorial parity and produce rather good butterfly diagrams.

Figure 9: Advection-dominated solar dynamo with alpha-effect located at the bottom of the convection zone where the magnetic diffusivity has been assumed as very small ( 2*1010 cm2/s) and with a flow amplitude of 5 m/s. Design as in Figure 8.

In the simplest model the alpha-effect exists in the whole convection zone and the drift amplitude lies between 2 m/s and 10 m/s. Almost always solutions with the (wrong) quadrupole solution occur for the lowest eigenvalue. Because of the small value of the eddy diffusivity the cycle time becomes too long compared with the 22 yrs of the Sun. The toroidal field belts are migrating equatorwards but the maximum field amplitude still exists in the polar region.

These results can drastically be improved if the alpha-effect is located mainly at the bottom of the convection zone. The $$\eta_{\rm T}$$ is 1012 cm2/s in the bulk of the convection zone but it is smaller (We choose the more complex situation the rotation is also unstable without the density stratification (2*1010 cm2/s) at its bottom. Now the dipolar parity is indeed preferred. The meridional flow advects the field equatorwards producing a butterfly diagram of the observed type (see Figure 9). For such models one finds that the inclusion of the meridional circulation into models with low values of the magnetic diffusivity

• improves the cycle time problem
• transports the fields towards the equator
• favors the dipolar solutions against the quadrupolar solutions
• produces strong enough toroidal fields
• avoids the high number of field belts which often appears in thin layer solutions.

The needed circulation seems really to exist. The surface counterpart of this flow moving towards the poles with an amplitude of about 15 m/s has been observed (Zhao & Kosovichev 2004). However, the reason for the small value of the eddy diffusivity (i.e. the high magnetic Prandtl number) is still unknown. The dependence of the cycle time on the amplitude of the meridional flow at the bottom of the convection zone is shown in Figure 10 as a summary of various model calculations (see Rüdiger & Hollerbach 2004). One can easily see that the cycle time is much too large for too slow circulation but for drift rates in excess of 6 m/s the resulting cycle times well reproduce the observed cycle time of 22 yrs.

Figure 10: The cycle times for the dipolar modes of advection-dominated dynamo models of Bonanno et al. (2002). There is a clear anticorrelation of the drift velocity and the cycle time. Note the very long cycle times for too slow meridional circulation which are due to the small values of the used turbulent magnetic diffusivity.

### Simulation models

Many of the known mean-field phenomena can also be simulated with high-developed supercomputer codes which numerically solve the first-principle equations of magnetohydrodynamics. By heating a convectively unstable spherical shell from below rotating magnetoconvection results which transports angular momentum and generates large-scale magnetic fields. Basic details of the mean-field calculations are reproduced. The toroidal magnetic fields in the convection zone appears to be weak and highly nonaxisymmetric but the field in the 'tachocline' layer below the bottom of the convection zone becomes strong (~ 3000 G), axisymmetric and antisymmetric with respect to the equator. The latter result can be described as due to the action of an alpha-effect in the convection zone (Browning et al. 2006). Also the rotation law in the convection zone shows the main properties of the observed pattern ( Figure 5). Still some of the results are consequences of special inputs but basic ingredients of the solar dynamo model -- such as the alpha-effect in the convection zone, differential rotation, turbulent pumping (see Dorch & Nordlund 2001) and effective toroidal stretching of the magnetic field lines -- can be obtained by direct numerical simulations. However, as the solar cycle lasts more than 200 rotation times it will be a highly innovative work to reproduce a full solar cycle with the supercomputer.

## References

Internal references

• Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.