Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory

Physica D 194 (2004) 30–48

Two-state intermittency near a symmetric interaction ofsaddle-node and Hopf bifurcations: a case

study from dynamo theory

Peter Ashwina, Alastair M. Rucklidgeb, Rob Sturmanb,∗a Department of Mathematical Sciences, Laver Building, University of Exeter, Exeter EX4 4QE, UK

b Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

Received 25 September 2003; received in revised form 27 January 2004; accepted 3 February 2004Communicated by A. Doelman

Abstract

We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limitcycle, motivated by a low-order model for magnetic activity in a stellar dynamo. This model consists of coupled interactionsbetween a saddle-node and two Hopf bifurcations, where the saddle-node bifurcation is assumed to have global reinjectionof trajectories. The model can produce chaotic behaviour within each of a pair of invariant subspaces, and also it can showattractors that arestuck-onto both of the invariant subspaces. We investigate the detailed intermittent dynamics for such anattractor, investigating the effect of breaking the symmetry between the two Hopf bifurcations, and observing that it can appearvia blowout bifurcations from the invariant subspaces. We give a simple Markov chain model for the two-state intermittentdynamics that reproduces the time spent close to the invariant subspaces and the switching between the different possibleinvariant subspaces; this clarifies the observation that the proportion of time spent near the different subspaces depends onthe average residence time and also on the probabilities of switching between the possible subspaces.© 2004 Elsevier B.V. All rights reserved.

Keywords:Dynamo theory; Bifurcation with symmetry; Intermittency; Cycling chaos

1. Introduction

Modelling chaotic and intermittent changes, for example in the intensity and polarity of magnetic fields caused bystellar dynamos, is a significant challenge, whether via minimal phenomenological models, mean-field models ordetailed numerical simulations. Previous work of Tobias et al.[29] and Knobloch et al.[16] in this direction used thedynamics near an interaction of local bifurcations (saddle-node and Hopf) to suggest low-order phenomenologicalmodels for the stellar dynamo problem. The interaction of saddle-node and Hopf bifurcations is one of the simplestbifurcations that gives rise to chaotic attractors local to the bifurcation[12] and hence is useful in providing a modelwith chaotic behaviour in a truncated bifurcation normal form.

∗ Corresponding author. Tel.:+44-113-343-5184; fax:+44-113-343-5090.E-mail addresses:[email protected] (P. Ashwin), [email protected] (A.M. Rucklidge), [email protected] (R. Sturman).

0167-2789/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2004.02.002

P. Ashwin et al. / Physica D 194 (2004) 30–48 31

One aim of the present paper is to combine and modify aspects of the models to overcome one of the mainproblems in[12,29], namely the fact that the attractors there are only marginally stable and small changes to initialconditions or parameters lead to solutions that depart to infinity. We do this by assuming that the saddle-nodebifurcation occurs on a limit cycle[1,14], and so ensure that the dynamics will remain in a compact region in phasespace; this allows much more robust simulation of the dynamics than was possible in[16,29]. We remark that thesaddle-node/Hopf bifurcation with global reinjection has been subject of recent study by[17,18] and shows veryrich bifurcation and periodic orbit structure. We will be mostly concerned with the case in which there is a chaoticattractor for two such bifurcations, symmetrically coupled.

The model(4) we derive inSection 2consists of ordinary differential equations (ODEs) in six (real) variables.Symmetries force the existence of two invariant subspaces of three dimensions; these correspond to pure dipoleand pure quadrupole magnetic fields in the analogous model of[16]. The intersection of the two subspaces is ofone dimension and this is the variable that undergoes the saddle-node bifurcation. The dynamics within each ofthe three-dimensional subspaces can be chaotic, and attractors may include points within (or be ‘stuck on’ to)these subspaces and hence typical trajectories show intermittency where they remain for arbitrarily long times inarbitrarily small neighbourhoods of the invariant subspace. This leads to the appearance of on–off and other typesof intermittency; see for example[11,13,19,20,27]; for reviews, see for example[5,25].

This leads on to the second aim of the paper; to investigate in detail some examples of intermittent dynamics ofsome attractors for the model(4) where the dynamics is intermittent to more than one invariant subspace. Althoughintermittent dynamics has been found previously in examples of mean field dynamos, as far as we are aware thisis the first example of two-state intermittency in such a model that involves chaotic saddles within the invariantsubspaces. To this end, we look at numerical simulations typical for the system inSection 3and consider theirintermittent dynamics in detail.

The phenomenon oftwo-state intermittency, where attractors are on–off intermittent to more than one invariantsubspace such that the same chaotic saddle governs approach towards and departure from the invariant subspace,was first reported in[21]. For other examples of two-state on–off intermittency see for example[3,10,30]. Here wefind in–out intermittency(see[5]) where different saddles are responsible for approach towards and departure fromthe subspace. This in–out intermittency has previously been found in PDE and ODE dynamo models[9] and inother situations[19] but not as atwo-stateintermittency.

In particular, the dynamics of the intermittent attractor explores a neighbourhood of the invariant subspace thatincludes the attractor within the invariant subspace, but which also includes unstable dynamics within the subspace.We briefly investigate the appearance of two-state intermittency via blowout bifurcations from the invariant subspaces(cf. for example[2,3,11,23,24]) and find evidence that subcritical blowout bifurcations from the invariant subspacesare succeeded by a crisis that sets up two-state in–out intermittency.

Section 4models two-state intermittency in the model(4) via the probability density of a Markov model for thedistance from each of the invariant subspaces. We use this to obtain estimates for the proportion of time spent neareach of the invariant subspaces, and fit the parameters in the model to the numerical examples in previous sections.Finally in Section 5we discuss some generalities of the model and its dynamics, and of the probabilistic model.

2. Formulation of the model

The main model we study in this paper is a system of six coupled ODEs with two invariant subspaces each of threedimensions; these represent pure dipole and pure quadrupole magnetic fields in the motivating models[16,29] forthis work. The intersection of these invariant subspaces is a one dimensional ODE corresponding to zero magneticfield.

32 P. Ashwin et al. / Physica D 194 (2004) 30–48

We aim in this paper to improve the models of[16,29] to (a) allow chaotic behaviour within the symmetricsubspaces, (b) make the appearance of two-state intermittent attractors between these subspaces more robust and (c)avoid problems with sensitivity to parameters and initial condition that may lead to blowing up in[16], by makingthe ‘zero magnetic field’ variable compact. We defer a detailed discussion of the motivation of the model, and thedifferences from the models of[16,29] to Section 2.6.

2.1. Saddle-node/Hopf bifurcation

We consider a codimension 2 bifurcation. The interaction of a saddle-node bifurcation and a Hopf bifurcation iswell understood[12]. This occurs when the Jacobian matrix of a flow has a pure imaginary pair and a simple zeroeigenvalue. This can be written in normal form, truncated to quadratic order, as

z = (µ+ iω)z+ εzv, v = κ − v2 + e1|z|2, (1)

wherez ∈ C, v ∈ R, andµ,ω, ε, κ, e1 ∈ R are parameters. A normal form symmetry is present in the equations,namelyz → zeiφ, for any fixed angleφ. (This includes the symmetryz → −z.) The parametere1 is frequentlyrescaled toe1 = −1 (the negative sign ensures the Hopf bifurcation is supercritical)[12] and we will takee1 = −1throughout. A saddle-node bifurcation occurs in thev-direction asκ passes through 0, giving fixed points,v+ andv−,at(z, v) = (0,±√

κ). The system is axisymmetric because of the normal form symmetry, and so can be transformedinto cylindrical polars(r, θ, v) = (|z|,arg(z), v) with the angular variable decoupled. A limit cycleL at (r, v) =(√κ − µ2/ε2,−µ/ε) forµ2 ≤ ε2κ is created as the complex variablez undergoes a Hopf bifurcation on crossing the

lineκ = µ2/ε2. A heteroclinic connectionH fromv+ tov− exists atκ = 0 (µ > 0). This indicates a degenericity inthe model, since also on this half-line there is a secondary Hopf bifurcation, in which the limit cycle bifurcates intoa nested family of tori. The behaviour of the system is summarised inFig. 1, for the caseε > 0 (see[12] for details).

This system models a dynamo in which hydrodynamic behaviour (with magnetic field equal to zero) is representedby thev-axis. The fact that the absence of a magnetic field must be maintained means that thev-axis must bedynamically invariant. Two different convecting states (one hydrodynamically stable but magnetically unstable, andone hydrodynamically unstable but magnetically stable) are represented by the two fixed points on thev-axis. Thecomplex coordinatez represents the magnetic field. When separated into real and imaginary parts we have a naturalcorrespondence of real part with the toroidal field, and imaginary part with the poloidal field[29]. The (primary)supercritical Hopf bifurcation represents the onset of magnetic instability. Also in thez equation is a term giving thecontribution of the velocity field to the magnetic flux. (This is the only permissible quadratic term, sincev2 wouldbreak the invariance of thev-axis, and|z|2 would break the invariancez → −z, needed to allow a reversal of the field.)Finally, thev equation also contains a term modelling the Lorentz force (reaction of the magnetic field) on the flow.

There is an open set of initial conditions for which trajectories escape tov = −∞. In particular, trajectories thatbegin outside the heteroclinic connectionH escape in this way. This can be prevented by introducing another attract-ing fixed point on thev-axis, near−∞ [29]. This also results in the degeneracy of the secondary Hopf being broken.

2.2. Breaking the degeneracy of the secondary Hopf

Adding a cubic termcv3 to thev equation retains the axisymmetry, breaks the secondary Hopf degeneracy, andintroduces a new fixed point atv ≈ 1/c. Thus the system becomes

z = (µ+ iω)z+ εzv, v = κ − v2 + cv3 − |z|2.The constantc is chosen to be negative to make the new fixed pointv– stable in thev-direction. The two originalfixed points remain nearv+ andv− for smallc. The bifurcation diagram remains similar to the case above, but now


Fig. 1. Bifurcation diagram for the degenerate case(1) of the interaction of a saddle-node bifurcation (marked SN, unfolding parameterκ) anda Hopf bifurcation (marked H1, unfolding parameterµ). Observe that there is also a heteroclinic connection on the line marked H2, and this isdegenerate in that an open set in the phase space is foliated with tori for these parameter values. See the text and[12] for more details.

the secondary Hopf bifurcation no longer occurs at the same parameter values as the existence ofH. We also haveregions of parameter space created in which stable two-tori are possible (see[12,15] for details).

2.3. Breaking the axisymmetry

The axisymmetry of the system, which is a normal form symmetry, allows the angular component to be decoupled,and so the system is two-dimensional in the remaining variables. Thus no chaotic dynamics is possible. Breakingthe axisymmetry allows more complicated trajectories to occur. The system as a dynamo model also demands thatthe axisymmetry is broken, as it implies the equivalence of the poloidal and toroidal fields, which is not true for thedynamo process. However, changing the sign of the magnetic field is a symmetry of the dynamo problem. We breakthis symmetry in a way that preserves the invariance of thev-axis and retains the symmetryz → −z, by adding aterm proportional toz3. (The symmetryz → −z is erroneously absent from the model in[29].)

z = (µ+ iω)z+ εzv+ dz3, v = κ − v2 + cv3 − |z|2. (2)

Although this results in a system with a degree of non-genericity (the invariance of thev-axis), this constraint isrequired for the system to be a viable model of a dynamo—any purely hydrodynamic states must remain purelyhydrodynamic for all time.

2.4. Preventing escape of trajectories to infinity

In order to prevent solutions from escaping tov = −∞ or being attracted to a fixed point near−∞, we canrender the system periodic inv. On thev-axis, instead of a saddle-node bifurcation on an infinite line, we have asaddle-node bifurcation on a limit cycle. Any trajectories threatening to escape close to the lower unstable manifold


of v− return close to the upper stable manifold ofv+: we call this process the global reinjection. This type ofreinjection of solutions has been studied in many models, and its interaction with a Hopf bifurcation is studied in[17,18], for models of optically driven lasers.

We choose to makev periodic on the interval [−πL/2, πL/2], whereL is a new parameter. In order to ensure thatthev variable and its powers remain continuous on this interval, we observe that we need trigonometric functionsthat are (a) periodic on [−πL/2, πL/2], and (b) proportional tov, v2, andv3 for v close to 0. Setu1 = L sin(v/L),so that for sufficiently smallv, u1 ≈ v. Similarly, setu2 = (L/2) sin(2v/L): for smallv, u2 ≈ v as well. The newvariableu1 is not periodic on [−πL/2, πL/2], butu2 andu2

1 are, as isu21u2. The system can thus be made periodic

and continuous by substituting

v → u2, v2 → u21, v3 → u2

1u2,

and confiningv to the interval [−πL/2, πL/2]. Hence the full system becomes

z = (µ+ iω)z+ εzu2 + dz3, v = κ − u21 + cu2

1u2 − |z|2 (3)

with u1 = L sin(v/L) andu2 = (L/2) sin(2v/L).This system can produce, for suitable parameter choices, all the behaviour discussed in[29], with the added

advantage that nearly all trajectories remain in a compact region of phase space. In particular, the secondary Hopfbifurcation creates quasiperiodic behaviour in the form of an attracting two-torus. Increasing the linear growth termµ triggers a breaking down of the two-torus and a transition to achieve chaos. During this transition, parametervalues exist at which typical trajectories are attracted to frequency-locked limit cycle. In fact, parameter valuesexist at which chaotic motion is bistable with periodic attractors.Fig. 2(a) shows a periodic attractor for parametervaluesµ = 0.026, ω = 10.0, ε = 1.0, d = 4.9, κ = 1.0, c = −0.1, L = 2.0. The trajectory winds twelve timesaround thev-axis during one full period. Similar limit cycles appear at smaller values ofµ, apparently for mostinitial conditions. At these parameter values however, the basin of attraction for this limit cycle seems very small.Most initial conditions lead to chaotic motion illustrated inFig. 2(b). This is a Poincaré section formed by plotting(Re(z), v)when the trajectory hits the section Im(z) = 0. The parameter values are as above, but for different initialconditions. It shows the heteroclinic tangle caused by the unstable manifold ofv+ crossing the stable manifold ofv−. (The fixed pointsv+ andv− are given by crosses inFig. 2(b)–(d).) Superimposed on the chaotic solution arecircles plotted where the periodic orbit ofFig. 2(a) crosses the Poincaré section. Increasingµ further allows thechaotic attractor to much smaller values of|z|, close to thev-axis, owing to the heteroclinic connection and thereinjection mechanism.Fig. 2(c) shows an attractor forµ = 0.027. Here very occasional reinjections are required.When the trajectory reachesv = −πL/2 = −π it is reinjected atv = π. Fig. 2(d) shows an attractor forµ = 0.028.At this parameter the trajectory makes a reinjection roughly as often as it travels up thev-axis. This is as close as theattractor will get to|z| = 0, as increasingµ further results in a move away from thev-axis, as reinjections becomemore common and eventually inevitable.

2.5. Interaction of saddle-node bifurcation with two Hopf bifurcations

We now introduce a second transverse complex direction and assume there are Hopf bifurcations in each of thevariablesz1 andz2. We include an equation representing a real antisymmetric velocity component,w along the linesof [16] giving the six-dimensional system that we consider for the rest of this paper.

z1 = (µ+ σ + iω1)z1 + ε1z1u2 + d1z31 + b1|z2|2z1 + (β1 + γ1u

21)wz2,

z2 = (µ+ iω2)z2 + ε2z2u2 + d2z32 + b2|z1|2z2 + (β2 + γ2u

21)wz1,

v = κ − u21 + cu2

1u2 − (|z1|2 + |z2|2), w = −τw+ e(z1z2 + z2z1), (4)


Fig. 2. Attractors for the system(3) on increasingµ, with other parametersω = 10.0, ε = 1.0, d = 4.9, κ = 1.0, c = −0.1, L = 2.0. In (a)µ = 0.026 we have a stable periodic orbit, with the trajectory winding twelve times around thev-axis during one full period. In (b)µ = 0.026again, but different initial conditions give a chaotic attractor. This is a Poincare section through Im(z) = 0. Superimposed are circles representingthe periodic solution of (a). The equilibria (indicated by crosses) are located at(Re(z), v) = (0,0.966), (0,−1.114). In (c)µ = 0.027, and thechaotic attractor gets closer to thev-axis. Finally in (d), withµ = 0.028, reinjections are common and the trajectory spends a long time with|z|very small.

whereµ, σ, ω1,2, ε1,2, d1,2, b1,2, γ1,2, κ, c, τ, e ∈ R andβ1,2 ∈ C are all parameters. This system of ODEs hassymmetries

S1 : (z1, z2, v, w) → (−z1, z2, v,−w), S1S2 : (z1, z2, v, w) → (z1,−z2, v,−w),S2 : (z1, z2, v, w) → (−z1,−z2, v, w),

and in the case thatσ = 0 and the parameters are independent of their subscripts, there is an additional symmetryz1 ↔ z2. In the general case there are three-dimensional invariant subspaces given by

D = {(z1,0, v,0)} and Q = {(0, z2, v,0)}, (5)

and the intersection of theseD ∩Q corresponding toz1 = z2 = w = 0.We have not attempted to include all possible parameters that will unfold the system to any particular order;

merely we have included parameters such that the observed dynamics appears to be robust. Note that the presenceof the symmetries implies that the global reinjection occurs within the invariant subspaceD∩Q and so is persistentas a global connection.


2.6. Interpretation and motivation

The physical motivation behind this model comes from stellar dynamo theory. Mean-field models of stellar dy-namos often exhibit oscillatory instabilities to magnetic fields with dipole or quadrupole symmetry: fields that eitherchange sign or that are left invariant by reflections in the equatorial plane of the star. In these models, the instabili-ties to dipole and quadrupole dynamos occur for similar parameter values and produce similar nonlinear behaviour(apart from symmetry type). The complex variablesz1 andz2 represent dipole and quadrupole magnetic fields, withthe real and imaginary parts representing toroidal and poloidal components of the field, and the near-equivalence ofdipole and quadrupole modes is represented by having|σ| � µ,ω1 ≈ ω2, d1 ≈ d2, b1 ≈ b2, β1 ≈ β2 andγ1 ≈ γ2.The interaction of non-chaotic dipole and quadrupole dynamos was considered in[16] using a model similar to(4),but withd1 = d2 = 0.

Real stellar dynamos (and most obviously the Solar dynamo) show behaviour that is not simply oscillatory, butconsists of oscillations that are modulated chaotically over a time-scale many times longer than the basic period ofthe oscillation. The chaotic modulation of a dipole dynamo was considered in[29] using a model similar to(2).

Effectively, we have extended the dipole–quadrupole model of[16] to include the chaotic modulation of[29],by including cubic terms to break axisymmetry. We have also made the model more robust by makingv periodic,preventing trajectories from escaping to infinity.

3. Intermittent attractors for the model

We now turn to an investigation of the properties of the model(4). In particular, we are interested in how thechaotic modulation of[29] influences the intermittent switching between dipole and quadrupole activity observedby [16]. In this section we show some numerical results and their interpretation in terms of forms of intermittency.Dynamics is possible which spends time close to each of the invariant subspacesD andQ in turn. Throughout thispaper we fix some parameters as follows except where explicitly stated:

ω1 = ω2 = 10.0, ε1 = ε2 = 1, κ = 1, c = −0.1, e = 1.0, τ = 1.0,

L = 2, µ = 0.026, σ = 0.0 and d1 = d2 = 4.9. (6)

Recall thatL governs the distance betweenv+ andv− via the reinjection, and henceL = 2 means that the distancebetween the fixed points in either direction is comparable. We set the nonlinear coupling parameters as

β1 = β2 = −0.5 + 0.5i, γ1 = γ2 = 2.5, b1 = b2 = −0.1. (7)

3.1. A numerical example of two-state in–out intermittency

With these symmetric parameters (σ = 0) we can find trajectories which alternate between visiting regions veryclose toD andQ, as shown inFig. 3. The dynamics is interpreted as follows, usingFig. 4 as a schematic guide.As the trajectory nearsD (resp.Q), the active variablez1 (resp.z2) approaches some invariant (periodic) dynamicsDin (resp.Qin) within that subspace, whilst the quiescent variablez2 (resp.z1) is suppressed. The dynamics inDin

(resp.Qin) withinD (resp.Q) is unstable to another (chaotic) invariant setDout (resp.Qout). When the system is inthis state, the quiescent variable is allowed to grow. When the quiescent variable has grown to a similar magnitudeto the active variable, a nonlinear switching mechanism sends the system back towards eitherD orQ in a seeminglyrandom manner but with a preference for a switch betweenD andQ. The time intervals between switches appearrandom, but have a well-defined mean. Indeed, for these symmetric parameters, on average an equal amount of time


Fig. 3. Time series for symmetric parameters showing switching between invariant subspaces:µ = 0.026, σ = 0. The black line denotes|z1|and grey|z2|. The time intervals between intermittent visits close toD andQ are apparently random, with a preference for alternating, but havea well-defined mean. The symmetry in the parameters means that on average an equal time is spent near each invariant subspace.

Fig. 4. A schematic diagram showing the two-state in–out intermittency observed inFig. 3. The dynamics switches between laminar phaseswhere it remains close to each of the subspacesQ andD. The approach to the subspace, sayQ, is close to the stable manifold of a periodic orbitQin that is unstable withinQ to a chaotic attractorQout with a positive transverse Lyapunov exponent; trajectories then move away fromQwhileremaining close the unstable set ofQout. When reaching a state of approximately the same distance fromQ andD, the nonlinear dynamics cansend the trajectory either back towardsQ orD in a seemingly random manner.


Fig. 5. Close-up of the time series inFig. 3 detailing the different episodes during a ‘laminar phase’ of the intermittency as the trajectoryapproaches the subspaceD.

is spent near each invariant subspace. Note that here the dynamics appear to flip almost every time the quiescentvariable grows to a comparable magnitude as the active variable, but there are occasions when the variables fail toswitch and an active dipole (say) phase is followed by another (for example att = 2500).

3.2. Detailed description of the mechanisms of the intermittency

The suppression or growth of the quiescent variable is mainly governed by the magnitude of the active variable.The larger the time-average of the active variable, the greater the suppression (or smaller the growth) of the quiescentvariable (for this choice of parameters). This is because of (for instance) the−|z2|2z1 term in thez1 equation in(4).This suppression and growth, and the setsDin, etc., can be seen explicitly inFigs. 5 and 6. Fig. 5shows a detail fromthe time series inFig. 3. The five marked segments are displayed inFig. 6as plots of Re(z1) (in black) and Re(z2)(in grey) against Im(z1) and Im(z2), and|z1| and|z2| againstv. First in segment I we see the system heading intothe nonlinear switching region. Herez1 spirals in andz2 spirals out to meet in a region in which the two variablesare of similar magnitude. Since the parameters are symmetric it is a delicate issue which variable is favoured in theswitching mechanism. Segment II shows the system favouring the dipolez1 variable, and the system leads into theregionDin. Note that the similar magnitude ofz1 andz2 causes the damping of the variablev via the−(|z1|2+|z2|2)term in thev equation. In segment III we have reachedDin, which is a periodic torus around thev-axis. This torushas a sufficiently large average〈|z1|〉 to suppressz2 further. The torusDin is unstable withinD and in segment IVwe see both the torus andv growing, leading toDout. The setDout can be seen in segment V, and is equivalent to


Fig. 6. Projections of phase portraits of different time segments ofFig. 5showing the different episodes of the in–out intermittency for the activevariablez1, while the quiescent variablez2 decays and then grows. In the plots in the left column black lines representz1 and grey lines representz2. In I the trajectory is approaching the nonlinear switching region; II shows the trajectory leading intoDin; in III it is nearDin; IV shows itleading towardsDout; finally in V the trajectory is nearDout.


Fig. 7. Time series as forFig. 3, but withµ = 0.026 andσ = 0.001. Here the linear growth rate is greater in the dipole variable. However, thequadrupole subspace is on average favoured.

the chaotic attractor shown inFig. 2(b). This allows the quiescent variablez2 to grow because although|z1| reacheslarger values, it also spends a long time much closer to thev-axis, and this results in a smaller average〈|z1|〉.

3.3. Breaking thez1 ↔ z2 symmetry

We can break the symmetry of the parameters in the linear growth term, for example, by settingσ �= 0 to theequations. One might suppose, setting a greater rate of linear growth inz1 leads to thez1 (dipole) subspace beingpreferred, as when a switch is possible, whenz1 andz2 are of comparable size, the direction with the greatesteigenvalue should dominate. However, it appears that the opposite can be true, at least for the parameter values wehave investigated.

Fig. 7shows a time series as inFig. 3, but now withµ = 0.026 andσ = 0.001, so the growth rate forz1 is greaterthan that forz2. The dynamics for these parameters starting with initial conditions solely inQ are equivalent tothose given inFig. 2(b) (orFig. 2(a) for a more specific, and very precise, choice of initial conditions), and for initialconditions solely inD we have the dynamics inFig. 2(c). For initial conditions not in either invariant subspace, atypical trajectory switches between these behaviours.

Despite the larger linear growth rate inz1, the trajectory favours the activity inz2. There are many factors involvedin taking precedence over the linear growth rate. These include the transverse Lyapunov exponents atDin,Dout,Qin

andQout; the tangential Lyapunov exponents atDin,Dout,Qin andQout; the average suppression from the activevariable on the quiescent variable; the length of the laminar phases between switches; nonlinear effects within theswitching mechanism; the type of attracting dynamics within the invariant subspaces (periodic or chaotic); other


invariant or nearly stable sets within the invariant subspaces. These are all important, but we will concentrate inSection 4on two—the length of laminar phases and the switching mechanism. We give results to suggest that largebias towards dipole (resp. quadrupole) stems from an increased tendency in the switching mechanism to follow adipole (resp. quadrupole) active phase with another.

3.4. Blowout bifurcation to intermittency

Consider a family of dynamical systems smoothly parametrised by someλ ∈ R with a proper invariant subspaceN for all values ofλ. If the dynamics inN is independent ofλ we sayλ is a normal parameter relative toN.As discussed in[2,5] for normal parameters one can expect to locate blowout bifurcations relatively easily; formore general parameters this is not the case. An examination ofEq. (4)reveals that for the invariant subspaceQthe following are normal parametersσ, ω1, ε1, d1, b1, β1, γ1, τ, e while forD the following are normal parametersω2, ε2, d2, b2, β2, γ2, τ, e.

In particular, onlyτ ande are normal for both subspaces. On varying these from the values(6) we can stabilisean attractor in the invariant subspace via a blowout bifurcation whilst leaving the dynamics within the subspaceunaffected. The blowout appears to be subcritical and we find no cases where there are attractors that are for examplestuck on to only one invariant subspace.Fig. 8illustrates the changes in typical dynamics on increasingτ. Forτ < 1.5we find stable two-state in–out intermittency for typical initial conditions, as inFig. 8(a). Increasingτ beyond 1.5

Fig. 8. Dynamics for the symmetric parameters(6) and (7)on increasingτ. All figures plot|z1| in black and|z2| in grey. Part (a) hasτ = 1.0,and shows stable two-state intermittent dynamics. Increasingτ beyond 1.5 creates an attractor close toz2 = 0 but bounded away from it, as in(b), with τ = 3.0. Increasingτ further causes this attractor to move closer to the invariant subspaceD, but remain bounded away from it. Part(c) hasτ = 7.0. In (d) we have long period oscillatory behaviour, withτ = 8.0. In (e) and (f) we haveτ = 10.0 andD has become attracting.Different initial conditions in (e) and (f) lead to different dynamics withinD. In (e) the initial conditions lead to the chaotic attractor ofFig. 2(b),whereas in (f) the initial conditions lead to the periodic attractor ofFig. 2(a). The larger average value of|z1| in this periodic orbit causes a muchquicker decay in|z2|.


creates an attractor close toz2 = 0 but bounded away from it, as inFig. 8(b)–(d) (here the initial conditions arechosen close toD—similar time series close toQ can be easily found with different initial conditions.) InFig. 8(b)we haveτ = 3.0 and the resulting attractor has regular oscillations nearD. In Fig. 8(c) we haveτ = 7.0 and anaperiodic attractor, and inFig. 8(d) τ = 8.0 gives another oscillatory attractor of very long period. Increasingτ

further causes these bounded attractors to give way to attracting dynamics withinD (again for these initial conditionsclose toD). BothFig. 8(e) and (f) haveτ = 10.0 and the variablez2 decays to zero. The different rates of decay aredue to the different dynamics withinD—in Fig. 8(e) the initial conditions lead to the chaotic attractor ofFig. 2(b),whereas inFig. 8(f) the initial conditions lead to the periodic attractor ofFig. 2(a). As the parameters admit thepermutation symmetryz1 ↔ z2 this means that the intermittency will be the same to bothD andQ subspaces.

3.5. Influence of noise

Including additive noise in the model effectively destroys the invariance of the subspacesD andQ. We introducethe noise by adding to each variable a random variable from a normal (Gaussian) distribution scaled with some noiselevelξ, and with the square root of the previous time step. The results of this, forξ = 10−6, ξ = 10−4 andξ = 10−3

are shown inFig. 9. Small amounts of noise have, as expected, little effect on the proportion of time spent neareach invariant subspace, but for largerξ we find the effect on the proportion of time spent near a specific subspaceupon breaking the symmetry of the parameters is even more pronounced. The behaviour of the laminar phases arenot changed greatly by the addition of noise, but within the switching mechanism we see a greater tendency for

Fig. 9. The effect of noise on the proportion of time spent near each invariant subspace. The data are produced by running a typical intermittenttrajectory and recording the proportion of time that|z1| > |z2|, up to t = 2 × 106 for eachσ. The noise levels areξ = 0.0 (empty circles),ξ = 10−6 (solid hexagons),ξ = 10−4 (squares),ξ = 10−3 (triangles).


the trajectory to be pushed towardsD (say) many times in succession, without visitingQ (seeSection 4.1andFig. 12).

4. Modelling two-state intermittency as a Markov process

In this section we construct a probabilistic (Markov chain) model for the two-state intermittency observed in thedynamics of the attractor (such as that inFig. 3) where trajectories move between neighbourhoods of pure dipoleand pure quadrupole invariant subspaces.

Examining the dynamics of the intermittency, we note for example fromFig. 3that the approach towardsDin,Qin

is approximately uniform, as is the departure fromDout,Qout. We therefore use a chain to model these as shown inFig. 10and assume that the leakage from the ‘in’ chain to the ‘out’ chain happens at a uniform rateεd,q. We alsoassume that the switching mechanism is similarly governed by a Markov process parametrised by constantsγd,q

and that there is a delay ofTd,q steps during the switching mechanism.The Markov model as shown inFig. 10has states that correspond to the intermittent model as follows: We model

approach toDin by a chain of states{sn},Qin by a chain of states{s−n},Dout by a chain of states{rn}, andQout bya chain of states{r−n}, where we approach the relevant invariant set in the limit asn → ∞. More precisely, thereis aρ < 1 such that for examplepn gives the probability of being approximately distanceρn fromD. If P(a, b) isthe probability of a transition froma to b we assume that within the chains, forn ≥ 1

P(rn+1, rn) = 1, P(r−n−1, r−n) = 1, P(sn+1, rn+1) = εd, P(sn+1, sn+2) = 1 − εd,P(s−n−1, r−n−1) = εq, P(s−n−1, s−n−2) = 1 − εq

for constantsεd,q ∈ (0,1). All other transitions within the chains have zero probability, and all transitions take onetime step, apart from the transition froms1 to s2 which takesTd − 2 steps and froms−1 to s−2 which takesTq − 2steps. The transitions in the switching mechanism between the chains are assumed to have probabilities

P(r1, s1) = γd, P(r1, s−1) = 1 − γd, P(r−1, s−1) = γq, P(r−1, s1) = 1 − γq

for constantsγd,q also in(0,1). The constantsεd,q can be interpreted as the probability per unit time of leaking fromthe ‘in’ dynamics to the ‘out’ dynamics near the subspacesD,Q, and this gives rise to an exponential distributionof the durations that a trajectory is near the invariant subspaces after the constant delayTd,q. The constantsγd,q

correspond to the probability that the dynamics near the switching region send a trajectory that enters fromDout,Qout

back intoDin,Qin, respectively.

Fig. 10. The Markov chain used to model the two-state in–out intermittency observed in the saddle-node/Hopf model. The statesrn representtransverse distancesρ|n| from an invariant subspace near the ‘out’ dynamics whilesn represent transverse distancesρ|n| from the ‘in’ dynamicsfor some 0< ρ < 1. The switching mechanism is determined also by a random process with probabilitiesγd,q and is subject to an additionaldelay ofTd,q − 2 steps, so the minimum time spent in the ‘d’, ‘q’ chain isTd,q.


One can easily calculate the invariant probabilityp of the chain as

p(r1) = p(s1) = A, p(rn) = p(sn) = A(1 − εd)(n−2), n ≥ 2, p(r−1) = p(s−1) = B,

p(r−n) = p(s−n) = B(1 − εq)(n−2), n ≥ 2

implying that the residence time near each of the invariant subspaces is exponentially distributed inn. We havep(rn) = p(sn) since the trajectories passing throughrn are precisely those that have previously passed throughsn.One can find that average residence timesAd,q in the chains:

Ad = Td + 2∞∑n=1

nεd(1 − εd)(n−1) = Td + 2

εd,

and similarlyAq = Tq + 2/εq. Moreover, the transition between the ‘q’ and ‘d’ states at the end of the laminarphases is governed byγd andγq; observe that

A = p(s1) = (1 − γq)p(r−1)+ γdp(r1) = (1 − γq)B + γdA, 1 = A

(Td + 2

εd

)+ B

(Tq + 2

εq

),

and so

A = εdεq(γq − 1)

2εd(γd − 1)+ 2εq(γq − 1)+ εdεq(γdTq + γqTd − Td − Tq),

and there is a similar formula forB. The invariant probability density can be used to find the probability of a laminarphase in the ‘dipole’ and ‘quadrupole’ chains is

Pd = 1 − γq

2 − γd − γq, Pq = 1 − γd

2 − γd − γq(8)

giving the proportion of time spent in the ‘dipole’ and ‘quadrupole’ chains:

Md = PdAd

PdAd + PqAq, Mq = PqAq

PdAd + PqAq. (9)

Note that, as discussed inSection 3.3, the relative proportion of visits to say the dipole subspace depend on boththe average residence timesAd,q and the probabilities of switching, namelyγd,q.

One can verify the validity of the individual components of the Markov model for the original ODE problem.For example inFig. 11we show the distribution of the lengths of the laminar phases for a long timeseries with(a) corresponding toFig. 3 (µ = 0.026, σ = 0) and (b) corresponding toµ = 0.026, σ = −0.003. Note that theswitching mechanism for the ODE takes a certain minimum time after which the frequency of occurrence drops offexponentially.

To estimate the proportion of time spent in nearD andQ we take a threshold (|z1| < 10−2) to determine whenwe are close the invariant subspace and find the transition probabilitiesγq,d. These can be estimated by classifyingthe timeseries intoD andQ phases and so associate the timeseries with a string{wi}ni=1 with wi ∈ {D,Q}. We canthen estimate

γd = #{1 ≤ i < n : wi = wi+1 = D}#{1 ≤ i ≤ n : wi = D} ,

that is,γd is the frequency of observingD as the next laminar phase give that the previous one wasD. γq can beestimated similarly.Table 1gives estimates of the parameters as computed from the timeseries (we takeTd,q to thenearest integer for the discrete time Markov chain). Once having fitted the parametersεd,q, Td,q andγd,q one can


Fig. 11. Distributions of the lengths of laminar phaseT (near eitherD or Q) subspaces for (a) the symmetric case shown inFig. 3(µ = 0.026, σ = 0) and (b) an asymmetric caseµ = 0.026, σ = −0.003. Observe that there is a good fit to a constant delay and an ex-ponential probability distribution of phase lengthT in both cases where for (a) we haveP ∼ e−0.02127T for approaches to either, and (b) wehaveP ∼ e−0.02177T for approaches toD andP ∼ e−0.02107T for approaches toQ. Note that the lengths of laminar phases change very little.

Table 1Numerical estimates for the Markov chain parameters

σ Td Tq εd εq γd γq

0 130 130 0.04254 0.04254 0.16215 0.16319−0.003 159 176 0.04354 0.04214 0.45547 0.20377

Observe that the greatest effect of the symmetry breaking termσ = −0.003 is to change the transition probabilitiesγd,q within the switchingmechanism.


use the model to predict, for example using(9), the proportion of time spent near the dipole model. As an example,for σ = 0 we estimateMd from the parameters andMd from a (different) timeseries and find

Md = 0.4997, Md = 0.5023

for σ = 0, while

Md = 0.5724, Md = 0.5760

for σ = −0.003.

4.1. Two-state in–out intermittency and noise

The details of the dynamics near the invariant subspaces for instance inFig. 3 reveals that the ‘in’ dynamicswithinD is periodic while the ‘out’ dynamics is chaotic, and so one could construct a model where the deterministicflow away from the invariant subspace is replaced by a biased random walk. This will not substantially increase theaccuracy for this case, but in cases where the variance of the ‘out’ chain propagation is much larger it would be nec-essary; this would give a more sophisticated model than here or in[4]. Similarly, one could make the Markov modela lot more sophisticated by including a continuum of states and by including noise effects along the lines of[4,8,26].

In fact, the introduction of noise to the system accentuates the findings of the above—that the greatest effectof the symmetry breaking termσ is to change the transition probabilitiesγd,q within the switching mechanism.Fig. 12shows a time series showing the intermittency forσ = −0.003 andξ = 10−4, with the other parameters as

Fig. 12. Time series showing intermittency in the presence of noise. The parameter values are as inFig. 3, but withσ = −0.003 andξ = 10−4.The dipole subspace is greatly preferred because the transition probabilities in the switching mechanism now favour a dipole→ dipole switch.


before. Here it is clear that the dipole subspace is much favoured, and this bias is far more pronounced than withξ = 0 (recall that forσ = −0.003, ξ = 0 we hadMd = 0.5760—hereMd = 0.8976). It is clear that the tendencywithin the switching mechanism is now to follow a dipole active phase with another dipole phase, and in fact wecan estimate the transition probabilities as before asγd = 0.93467 andγq = 0.35801.

5. Discussion and conclusions

In summary, we have examined a system with an attractor that displays intermittent switching between chaoticdynamics in two invariant subspaces. In detail, the intermittency mechanism involves four states, two in each ofthe invariant subspaces. Two of these states (Din andQin) are transversely stable but unstable within the invariantsubspaces. The other two (Dout andQout) are transversely unstable but attractors within the invariant subspaces.Once the system moves away from the invariant subspaces, there is nonlinear switching and reinjection towardsDin andQin. We have developed a Markov model of this intermittency mechanism that is capable of capturing theexponential distribution of times spent near each invariant subspace. This switching mechanism is a new feature oftwo-state intermittency that is not present in previous examples of in–out intermittency. Our work also sheds lighton the switching behaviour observed in other dynamo models, for example the mean-field dynamo model[16] andthe model of geodynamo reversals in[22]. In the latter case the switching mechanism seems to be comparable tothe mechanism in the present paper.

Although we have derived the model from perturbations near the codimension 2 interaction of a saddle-node bi-furcation with two symmetry-related Hopf bifurcations, and we have not included all possible interaction terms evenup to cubic order, we presume (but cannot prove) that the dynamics we have described is robust or at least prevalent.We have used the degeneracy as an organising centre to give dynamics that has two-state intermittency. The modelwe have considered is also physically motivated by consideration of hydrodynamic and magnetic instabilities—seefor example[16]. In addition, the Hopf bifurcations to dipolar and quadrupolar dynamo activity often occur forsimilar (or even the same) parameter values in mean-field dynamo models, so it is reasonable to regard a notionalsymmetry between these Hopf bifurcations as being only weakly broken.

One of the most surprising results of this investigation has been the recognition that linear growth rates (or averageresidence times) near the invariant subspacesD andQ do not determine the average proportion of time spent closeto the two subspaces. There are many other factors that determine which of the two subspaces is preferred. The mostimportant, at least for the Markov model, is the “switching mechanism” operating in the fully nonlinear regime thatdetermines a preference of one phase to the other, once the equivalence of the two invariant subspaces is broken.We observe that switching can be greatly affected by the addition of noise.

As discussed in[6,7], numerical simulations of cycling chaos need to be carried out with great care. The simulationspresented in this paper were performed to double precision accuracy using the Bulirsch–Stoer adaptive step integrator[28] with a relative tolerance of 10−8 at each time step. Rounding errors may force the dynamics into invariantsubspaces, and care was taken to interpret correctly when this occurred. The dynamics of(4)was also simulated withthe inclusion of small amounts of additive isotropic noise, in which case the effects of rounding into an invariantsubspace could be easily avoided.

Acknowledgements

We thank Chris Jones, Edgar Knobloch, Reza Tavakol, Steve Tobias and Nigel Weiss for several very interestingconversations in relation to this work. This research was supported by EPSRC grant number GR/N14408.


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Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory