In this paper, we extend our previous template analysis of a self-exciting Faraday disc dynamo with a linear series motor to the case of a nonlinear series motor. This introduces two additional nonlinear symmetry-breaking terms into the governing dynamo equations. We investigate the consequences for the identification of a possible template on which the unstable periodic orbits (UPOs) lie. By computing Gauss linking numbers between pairs of UPOs, we show that their values are not incompatible with those for a template for the Lorenz attractor for its classic parameter values.
In a recent pair of papers, Moroz (2008a,b) showed that the chaotic attractor associated with a nonlinear three-dimensional self-exciting Faraday disc homopolar dynamo, introduced in Hide et al. (1996) (hereafter abbreviated as HSA), might be topologically equivalent to that of the Lorenz (1963) attractor (see, Gilmore & Lefranc 2002). This was achieved by using the method of close returns on a Poincaré section to extract the lowest order unstable periodic orbits (UPOs) from chaotic time series (Moroz 2007), and computing the Gauss linking numbers between pairs of orbits (Gilmore 1998). The values of these linking numbers were then compared with those for the same lowest order UPOs that are found on the Lorenz template for its classic parameter values (see below). If no discrepancies arise, then we have a possible template for the chaotic attractor. This is what Moroz (2008a,b) found for the HSA dynamo.
This Faraday disc dynamo comprises a disc and a coil, arranged in series with an electric motor. The axle of the disc is connected to one end of the coil by a sliding contact, while the other end of the coil is connected to one end of the motor. The other end of the motor connects to the rim of the disc via a second sliding contact. A diagram of the configuration is shown in fig. 2 of HSA. The dynamo is governed by the following non-dimensional equations: 1.1a 1.1b and 1.1c where x(t) is the current flowing through the dynamo, the angular rotation rate of the disc and z(t), that of the motor. The key bifurcation parameters are and , where α measures the steady applied couple driving the disc, β measures the inverse moment of inertia of the armature of the motor, κ measures the friction in the disc, and λ, the friction in the motor.
The governing equations for the HSA dynamo possess a reflectional symmetry. If we take two different initial conditions, (x,y,z) and (−x,y,−z), then one solution is simply the rotated image of the other.
Hide (1997) introduced a variation into equation (1.1), by incorporating a nonlinear relationship between the torque on the armature of the motor and the electric current, x(t), generated by the dynamo as parameterized by ϵ: (1−ϵ)x+ϵx2. When ϵ=0, we recover equation (1.1); when ϵ=1, nonlinear quenching of oscillatory solutions occurs. Moroz (2002), investigating the intermediate case of 0<ϵ<1, showed that this quenching was the result of the double-zero bifurcation, associated with the onset of oscillatory solutions, tending to infinity (see below for details).
Introduction of ϵ, as well as increasing the richness of linear stability possibilities, also breaks the reflectional symmetry of the original dynamo equations through the introduction of two new nonlinear terms, one into equation (1.1a) and the other into equation (1.1c). This has the effects of introducing additional codimension-one Hopf and steady-state bifurcations and other codimension-two bifurcations not present in equation (1.1) (Moroz 2002).
Moroz (2002) also illustrated the effects of varying ϵ in a series of linear stability curves, plotted in the -plane. Bifurcation transition diagrams were also plotted as functions of x for fixed β, as α varied. Here, we vary β and fix α at the two values chosen by HSA in their fig. 9. Further details are given below. In the current paper, we return to the model studied by Moroz (2002), and repeat the UPO analyses of Moroz (2007, 2008a,b) to explore the consequences of varying ϵ on the Gauss linking numbers between pairs of UPOs, and so on the possible template of the original HSA dynamo.
This paper is organized as follows. In §2, we introduce the dynamo equations and summarize the relevant linear stability results. We present a selection of bifurcation transition curves in §3, and report on the extraction of UPOs from close returns on the Poincaré section. Section 4 contains the template analysis, including tables of linking numbers. We conclude, in §5, with a discussion of our results.
2. The dynamo equations
In this section, we summarize the main findings from Moroz (2002). Incorporating the nonlinear dependence of the torque on the current, the non-dimensional system of governing equations becomes 2.1a 2.1b and 2.1c where 2.2
(a) Summary of the linear stability analysis
There are three equilibrium solutions to equation (2.1): the trivial equilibrium solution 2.3 and the two non-trivial equilibrium states 2.4 where xe are given by the real roots of the quadratic equation 2.5 Setting ϵ=0, we recover the reflectionally symmetric equilibrium solutions of HSA: 2.6
(i) Bifurcations from the trivial equilibrium
The trivial equilibrium state undergoes a steady bifurcation along the line 2.7 and a Hopf bifurcation along 2.8 with frequency, ω, given by . This Hopf bifurcation originates from a Takens–Bogdanov double-zero bifurcation at the point 2.9 Each of these bifurcation criteria reduce to those for the HSA system when ϵ=0. In addition, we also see that in the limit (, from equation (2.2)), , so that no Hopf bifurcations can occur for finite values of β. This explains the nonlinear quenching reported in Hide (1997).
(ii) Bifurcations from the non-trivial equilibria
When 0<ϵ<1, there are two distinct curves of steady-state bifurcations. We have the curve of pitchfork bifurcations, given by equation (2.7), together with a saddle-node bifurcation, given by the positive real root of 2.10 There are now two distinct double-zero bifurcation points. The first is given by equation (2.9), while the second is given by 2.11 where is given by the positive real root of the quadratic equation 2.12 with 2.13a 2.13b and 2.13c A different codimension-two bifurcation, involving a Hopf bifurcation for the trivial equilibrium and the second (saddle-node) steady bifurcation of the non-trivial state, is possible when 2.14 provided L2(1+λ)>ϵ2λ.
As well as the two double-zero bifurcations, given by (2.11)–(2.13), there are two Hopf bifurcation curves, each of which emanates from one of the two double-zero points. Their derivation is lengthy and is described in detail in Moroz (2002), to which we refer the interested reader. Moroz (2002) also contains a selection of linear stability curves, plotted in the -plane as functions of ϵ for 0.1≤ϵ≤0.6, since codimension-two bifurcation points for the non-trivial equilibria (2.4) were shown to be restricted to ϵ≤0.6.
(b) The Poincaré section
For the HSA dynamo, equation (1.1), Moroz (2007) constructed the Poincaré section by translating the ‘quadratic’ variable y(t) by its non-trivial equilibrium value. When ϵ≠0, there are two nonlinear variables: x(t) and y(t). For consistency with Moroz (2007, 2008a,b), we choose to use y(t), so that we recover the results in those two papers when ϵ=0. In addition, there is a second choice to be made, since equation (2.5) has two roots, which we term xep and xem. Here ep and em refer to equilibria in x>0 and x<0, respectively. This yields yep and yem from equation (2.4). Which non-trivial equilibrium value we translate by is determined by the steady-state solutions of the bifurcation transition diagram (see §3 for details).
Denoting xe and ye as a shorthand for either equilibrium state, following Moroz (2007), we introduce , so that equation (2.1) becomes 2.15a 2.15b and 2.15c The translated equilibria become 2.16 with the Poincaré section given by 2.17
3. Numerical investigations
HSA presented two examples of chaotic behaviour for and Moroz (2007) presented bifurcation transition curves for α=20 and 100 which included these values of β,κ and λ. Other bifurcation transition curves were also produced for (κ,λ)=(1,1.2), and UPOs were extracted.
Moroz (2008a,b) then used topological methods to compute linking numbers between pairs of UPOs, construct tables of these linking numbers, and so identify a possible template. It was shown that, although different UPOs occur for different choices of α and β, the template was the same in each case, namely that for the Lorenz (1963) equations.
Here, we repeat the analysis when ϵ≠0, for the same choices of α, namely α=20 and 100, with (κ,λ)=(1,1.2), and study the effects of the nonlinear symmetry-breaking ϵ terms on the UPOs, their linking numbers and the template. Specifically, we investigated α=20 and 100 for β=2 when ϵ=0.01,0.1; α=20 and β=4.08 for ϵ=0.1; α=20 and 100 for β=3.6 when ϵ=0.2. These choices enable the effects of increasing ϵ to be studied for large and small values of α, when β is both fixed and varied.
(a) Bifurcation transition curves
In all of our integrations, we took (κ,λ)=(1,1.2). Moroz (2002) produced bifurcation transition sequences for as α was increased, for fixed β. Here, we follow Moroz (2007) and display the bifurcation transition sequences for as β is increased. We label the UPOs using symbol sequences of R, if the trajectory cycles around the equilibrium in xe>0, and L, if the trajectory cycles around the negative equilibrium in xe<0. Thus, R2L will be a period-3 orbit, that cycles twice around xe>0, before cycling once around xe<0.
Figure 1 shows the bifurcation transition sequence when ϵ=0.01 for α=20 (figure 1a) and 100 (figure 1b). Despite the relatively small size of ϵ, these sequences already differ from the symmetric ϵ=0 cases, shown in figs 2 and 6 of Moroz (2007). As β increases, there is an abrupt transition from a steady state to chaotic behaviour. For α=20, the chaos persists until there is a loss of stability to a stable period-2 LR orbit at β≈3.08. For α=100, we found a small window which, for β=4.7, is associated with a stable RLR2L3(R3L3)4(R2LRL2)2(R2L2)5 orbit with period ≈34.4 s. When β≈5.36 the chaotic behaviour loses stability to a stable period-4 L2R2 orbit.
The effects of the asymmetries are even more apparent when ϵ=0.1. Figure 2a shows the bifurcation transition plot for α=20, while figure 2b shows the plot for α=100. Note the sequences of periodic windows, interspersed by short intervals of chaos. When α=20 (figure 2a) and β=4.8, we found a stable period-7 (RL)3L orbit of period 7.96 s. As β is increased to 5.8, there is a stable period-9 (RL)4L orbit with period 10.04 s. In the next two larger periodic windows, we found stable period-11 (RL)5L and period-13 (RL)6L orbits for β=6.2 and 6.5, respectively. When β>7.4, a stable period-2 LR orbit prevails.
In figure 2b when α=100, the sequence of larger windows correspond to the stable periodic orbits (R2L3)2R2L2RL2 for β=6.3, (R2L3)R2L2RL2 for β=8, R2L3R2L2RLRL2 for β=10, R2L2RLRL2 for β=16 and R2L2(RL)2RL2 for β=20.
Similar bifurcation transition sequences with periodic windows can be seen in figure 3 for ϵ=0.2. In figure 3a, for α=20, there is a stable period-3 RL2 orbit at β=6, a stable period-11 (RL2)3RL orbit at β=7, a stable period-8 (RL2)2RL orbit at β=7.8 and a stable period-5 RL2RL orbit at β=10. We also found a stable period-7 RL2(RL)2 orbit at β=11.5 and a stable period-9 RL2(RL)3 orbit at β=12.8. This cascade follows the pattern of figure 2a until for β>15.08, a stable period-2 LR orbit again emerges.
In figure 3b, for α=100, the periodic window at β=5.8 is associated with the orbit R2L5, and that at β=7.2 with the orbit R2L5R2L4. For β=7.7, we found the orbit R2L5(R2L4)2RL2RL3, while for β=8.8, the orbit R2L4(RL)2RL3 is found. For β=10, the window is associated with the orbit R2L4, while for β>11.48, there is a stable period-3 RL2 orbit.
We also computed the bifurcation transition sequences for ϵ=0.4 for both α=20 and 100, with β increasing. All we obtained were simple periodic solutions with steadily increasing amplitudes.
(b) Unstable periodic orbits
Following Moroz (2007), we integrated the translated equations (2.15) for 60 000 s with a time step of 0.001 s, and discarded the first 100 s as representing transients. The UPOs were extracted as close returns on the Poincaré section given by equation (2.17), with a close return defined simply as 3.1 where ϵ=0.005 or ϵ=0.01, and i and j denote the ith and jth intersections with .
Following Hénon (1982), we can ensure that trajectories land precisely on the Poincaré section by replacing the independent time variable by Y (t), and dividing each equation in equation (2.15) by dY/dt. This yields a system of equations for (dx/dY,dt/dY,dz/dY), which is then integrated for one step in −dY from Ya to Y =0, where ta is the time just before a change in sign of Y . We then recorded x and z as their values on Y =0.
Figure 4 shows a comparison of histograms of close returns for UPOs for α=20 and 100, when ϵ=0.01 and 0.1. Figure 4 should be read in conjunction with table 1, which lists the labelled UPOs, their symbol sequence and their periods.
When ϵ=0.01 and α=20, the behaviour is dominated by the period-2 LR UPO (labelled A with period 2.598 s) and the period-4 L2R2 UPO (B with period 4.935 s). For α=100, the period-10 L5R5 UPO (P with period 5.465 s) dominates, followed by the period-8 L4R4 UPO (N with period 4.511 s). When ϵ=0.1 and α=20, the period-6 L4R2 UPO (W with period 7.483 s) dominates, whereas when α=100, it is the period-12 L8R4 UPO (h with period 6.673 s).
UPOs L2R2 and L3R3 are found in the ϵ=0.01 investigations for both α=20 and 100. When α=20, L2R2 (B) has period 4.935 s and L3R3 (D) has period 7.391 s, while for α=100, L2R2 (K) and L3R3 (L) have periods 2.502 and 3.541, respectively. One effect of increasing α is therefore to reduce the period of the UPO. Orbits labelled by C and G in table 1 and by B and H in table 2 mean a repetition of the particular UPO in brackets.
When ϵ=0.2 and β=3.6, histograms of UPOs are shown in figure 5 for α=20 and 100, with explanations for the letter labels, given in table 2. For α=20, the period-4 L3R UPO (A with period 4.806 s) dominates. When α=100, it is the period-9 L7R2 UPO (F with period 5.002 s), followed by the period-10 L8R2 UPO (G with period 5.393 s). F appears to be a triple peak. However, closer scrutiny of the data reveals the same period-9 L7R2 UPO but with slightly differing periods.
Figure 6 shows the corresponding x(t) time series for the cases chosen in figures 3–5. The effects of asymmetry, as ϵ increases, are clearly visible, with many more oscillations about the unstable fixed point in x<0 than about the corresponding point in x>0.
4. Template analysis
(a) General procedure
Having extracted the UPOs for a particular choice of ϵ, α and β, we proceed with the template analysis. The template is a branched manifold on which the UPOs are organized in a unique way. There are certain topological invariants that can be calculated from the UPOs, which aid in the identification of the particular template (see, the review article by Gilmore 1998). One such invariant is the Gauss linking number L(A,B) between two distinct closed curves xA and xB: 4.1
L(A,B) takes integer values and is invariant under deformations of the curves, provided they do not cross one another.
To determine the template, we did the following. Using Fortran codes kindly supplied by Bob Gilmore, we computed the linking numbers between the lowest order UPOs, extracted by the method of close returns, described in §3. Identification of the correct template requires information about the number of branches of the template, how the branches are layered and the twisting and crossing of the branches.
Moroz (2008a,b) identified a possible template for the HSA equations (with ϵ=0) by using the Lorenz (1963) equations as a test bed. By computing linking numbers of all the lowest order UPOs up to, and including, the period-6 orbits, for the Lorenz equations at their classic choices of , Moroz (2008a,b) was able to identify which of these orbits also appear in the HSA system, and, moreover, to compare the corresponding values for their linking numbers.
Both the Lorenz and the HSA systems have two branches, since trajectories cycle around two unstable foci, and orbits can be described in terms of combinations of two symbols L and R. The right hand branch overlays the left hand branch, so that the layering information is (1,−1). The torsion matrix T(i,j) gives information about the twisting and layering of the two branches. For both the Lorenz and HSA systems T(i,j)=0 for i,j=1,2 since neither branch twists.
When this additional information is used in a second Fortran code, the output is a table of linking numbers for UPOs which are compatible with that particular template. This table is then compared with the table of linking numbers obtained through the Gauss integral calculations. Should the two tables agree in each of their entries, we conclude that the template is a possible template for the system of equations. Should there be disagreement, then the template is not the correct one for the system, and we must try again.
Moroz (2008a,b) showed that the HSA equations have the same topological structure as the Lorenz (1963) system (for its classic parameter values of ) with the same template, even though we did not find all of the UPOs of the Lorenz system to be present in the HSA system (see table 3 of Moroz (2008b)). The purpose of the present investigation is to determine how breaking the reflectional symmetry of the HSA system, through the ϵ parameter, affects the topological structure and the choice of template.
The effect of a non-zero ϵ is to reduce the number of possible UPOs found from the ϵ=0 case. When ϵ=0, two different initial conditions (x,y,z) and (−x,y,−z) lead to two UPOs, one being the reflectionally symmetric image of the other. When ϵ≠0, this is no longer the case. The two extra nonlinear terms, introduced into the dynamo equations (1.1), one into equation (1.1a), the other into equation (1.1c) break this symmetry.
Figure 6 shows that trajectories cycle around the unstable focus in x<0 many more times than around the corresponding focus in x>0. The effects of this bias become more pronounced as ϵ increases. This lengthening of the number of oscillations about the unstable focus in x<0 affects the extraction of the lowest order UPOs, as well as their reflectionally symmetric partners. For example, in figure 6c, the lowest order UPOs we found included L2R, L3R and L3R2; we found no examples of their reflectionally symmetric counterparts LR2, LR3 and L2R3.
Nevertheless, when we computed the Gauss linking numbers between the various UPOs, their values agreed precisely with values for the Gauss linking numbers when such orbits appeared in the original HSA system with ϵ=0. The symmetry-breaking terms had no effect on the linking number values, in all of the cases we considered. Moreover, when we ran the template verification code, using the same information about the torsion matrix and the layering of branches as for the Lorenz template, we obtained exact agreement between the two sets of linking numbers. The symmetry-breaking terms in the modified HSA dynamo equations (2.15) had no effect on the topological structure of the templates.
Table 3 shows the linking numbers for UPOs for (α,β)=(20,2) for both ϵ=0 and 0.01. As well as producing asymmetries, increasing ϵ lengthens the periods of the UPOs by approximately 1 per cent.
Figure 7 shows the (x,Y) phase portraits of the two UPOs RLR2L2 and LRL2R2 for ϵ=0.01 and α=20. The asymmetry is clear. Table 4 shows the corresponding comparisons for α=100 as were shown in table 3 for α=20. In nearly all cases, except for the UPO L4R3, increasing ϵ increased the period of the UPO by approximately 1 per cent. For the L4R3 UPO, the period was decreased by this amount.
In this paper, we have investigated the consequences of including a variable nonlinear series motor, as parameterized by ϵ, for the identification of a possible template for the HSA Faraday disc dynamo. Such a nonlinear motor introduces symmetry-breaking terms into two of the governing equations (Moroz 2002): (1−ϵ)x+ϵx2, where x(t) is the dimensionless current flowing through the dynamo. When ϵ=0, the equations possess reflectional symmetry: if (x,y,z) is a solution, then so is (−x,y,−z). When ϵ≠0, two new nonlinear terms break this symmetry.
Using the method of close returns on a Poincaré section that passes through one of the two non-trivial fixed points, we extracted UPOs from integrations of the dynamo equations (2.15), for three choices of ϵ: ϵ=0.01,0.1 and 0.2; for two choices of α: α=20 and 100; and for three choices of β: β=2,3.6 and 4.08. This allowed us to study the effects of varying ϵ, α and β on the UPO analysis. These values of α were selected because they featured in the original HSA paper, as well as in the later studies by Moroz (2007, 2008a,b). We were interested in how varying ϵ influenced the values of the Gauss linking numbers between pairs of UPOs.
When ϵ=0, Moroz (2008a,b) showed that the template for the HSA dynamo might be the same as that for the Lorenz equations at their classic parameter values. Moroz (2008a,b) produced a complete table of linking numbers extracted from the Lorenz equations for all the possible UPOs up to, and including, orbits of period 6. This table was used to verify the linking number tables for the HSA dynamo, even though not all UPOs were present in the dynamo equations.
When ϵ≠0, in each of the cases we investigated, we have shown that the Gauss linking numbers between pairs of UPOs are not inconsistent with the Lorenz template as being a possible template for the HSA dynamo with the nonlinear series motor. Usually the template is determined from the lowest order UPOs, such as those of period 2, 3 or 4. In our analysis, we included orbits of period 10 and higher (not included here) with no inconsistencies. Apart from changes in the bifurcation transition diagrams, the only consequence for the UPOs, is to forbid reflectionally symmetric orbits. The extra bias, in which the trajectories cycle around the equilibrium in xe<0 more times than about that in xe>0 does not affect the values in the linking number calculations, and so the possible template.
- Received April 4, 2011.
- Accepted August 15, 2011.
- This journal is © 2011 The Royal Society