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

Transcription

1 Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory Peter Ashwin Department of Mathematical Sciences, Laer Building, Uniersity of Exeter, Exeter EX4 4QE, UK Alastair M. Rucklidge and Rob Sturman Department of Applied Mathematics, Uniersity of Leeds, Leeds LS2 9JT, UK September 25, 23 Abstract We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motiated by a low-order model for magnetic actiity in a stellar dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations, where the saddle-node bifurcation is assumed to hae global reinjection of trajectories. The model can produce chaotic behaiour within each of a pair of inariant subspaces, and also it can show attractors that are stuck-on to both of the inariant subspaces. We inestigate the detailed intermittent dynamics for such an attractor, inestigating the effect of breaking the symmetry between the two Hopf bifurcations, and obsering that it can appear ia blowout bifurcations from the inariant subspaces. We gie a simple Marko chain model for the two-state intermittent dynamics that reproduces the time spent close to the inariant subspaces and the switching between the different possible inariant subspaces; this clarifies the obseration that the proportion of time spent near the different subspaces depends on the aerage residence time and also on the probabilities of switching between the possible subspaces. P.Ashwin@ex.ac.uk A.M.Rucklidge@leeds.ac.uk rsturman@maths.leeds.ac.uk

2 ntroduction Modelling chaotic and intermittent changes, for example in the intensity and polarity of magnetic fields caused by stellar dynamos, is a significant challenge, whether ia minimal phenomenological models, mean-field models or detailed numerical simulations. Preious work of Tobias et al. [28] and Knobloch et al. [6] in this direction used the dynamics near an interaction of local bifurcations (saddle-node and Hopf) to suggest low-order phenomenological models for the stellar dynamo problem. The interaction of saddle-node and Hopf bifurcations is one of the simplest bifurcations that gies rise to chaotic attractors local to the bifurcation [2] and hence is useful in proiding a model with chaotic behaiour in a truncated bifurcation normal form. One aim of the present paper is to combine and modify aspects of the models to oercome one of the main problems in [6, 28], namely the fact that the attractors there are only marginally stable and small changes to initial conditions or parameters lead to solutions that depart to infinity. We do this by assuming that the saddle-node bifurcation occurs on a limit cycle [, 4], and so ensure that the dynamics will remain in a compact region in phase space; this allows much more robust simulation of the dynamics than was possible in [6, 28]. We remark that the saddle-node/hopf bifurcation with global reinjection has been subject of recent study by [7, 8] and shows ery rich bifurcation and periodic orbit structure. We will be mostly concerned with the case in which there is a chaotic attractor for two such bifurcations, symmetrically coupled. The model (4) we derie in Section 2 consists of ordinary differential equations (ODEs) in six (real) ariables. Symmetries force the existence of two inariant subspaces of three dimensions; these correspond to pure dipole and pure quadrupole magnetic fields in the analogous model of [6]. The intersection of the two subspaces is of one dimension and this is the ariable that undergoes the saddle-node bifurcation. The dynamics within each of the 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 in arbitrarily small neighbourhoods of the inariant subspace. This leads to the appearance of on off and other types of intermittency; see for example [, 3, 9, 2, 26]; for reiews, see for example [5, 24]. This leads on to the second aim of the paper; to inestigate in detail some examples of intermittent dynamics of some attractors for the model (4) where the dynamics is intermittent to more than one inariant subspace. Although intermittent dynamics has been found preiously in examples of mean field dynamos, as far as we are aware this is the first example of two-state intermittency in such a model that inoles chaotic saddles within the inariant subspaces. To this end, we look at numerical simulations typical for the system in Section 3 and consider their intermittent dynamics in detail. Preious examples of two-state intermittency include, for example, [3] and [, 29], where attractors are on off intermittent to more than one inariant subspace such that the same chaotic saddle goerns approach towards and departure from the inariant subspace. Here we find in out 2

3 intermittency (see [5]) where different saddles are responsible for approach towards and departure from the subspace. This in out intermittency has preiously been found in PDE and ODE dynamo models [9] and in other situations [9] but not as a two-state intermittency. n particular, the dynamics of the intermittent attractor explores a neighbourhood of the inariant subspace that includes the attractor within the inariant subspace, but which also includes unstable dynamics within the subspace. We briefly inestigate the appearance of two-state intermittency ia blowout bifurcations from the inariant subspaces (cf. for example [2, 3,, 22, 23]) and find eidence that subcritical blowout bifurcations from the inariant subspaces is succeeded by a crisis that sets up two-state in out intermittency. Section 4 models two-state intermittency in the model (4) ia the probability density of a Marko model for the distance from each of the inariant subspaces. We use this to obtain estimates for the proportion of time spent near each of the inariant subspaces, and fit the parameters in the model to the numerical examples in preious sections. Finally in Section 5 we 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 inariant subspaces each of three dimensions; these represent pure dipole and pure quadrupole magnetic fields in the motiating models [6, 28] for this work. The intersection of these inariant subspaces is a one dimensional ODE corresponding to zero magnetic field. We aim in this paper to improe the models of [6] and [28] to (a) allow chaotic behaiour within the symmetric subspaces, (b) make the appearance of two-state intermittent attractors between these subspaces more robust and (c) aoid problems with sensitiity to parameters and initial condition that may lead to blowing up in [6], by making the zero magnetic field ariable compact. We defer a detailed discussion of the motiation of the model, and the differences from the models of [6] and [28] to section Saddle-node/Hopf bifurcation We consider a codimension 2 bifurcation. The interaction of a saddle-node bifurcation and a Hopf bifurcation is well understood [2]. This occurs when the Jacobian matrix of a flow has a pure imaginary pair and a simple zero eigenalue. This can be written in normal form, truncated to quadratic order, as ż = (µ + iω)z + ɛz = κ 2 + e z 2 () where z C, R, and µ, ω, ɛ, κ, e R are parameters. A normal form symmetry is present in the equations, namely z ze iφ, for any fixed angle φ. (This includes the symmetry z z.) 3

4 κ H κ = µ 2 /ɛ 2 H2 PSfrag replacements SN µ Figure : Bifurcation diagram for the degenerate case () of the interaction of a saddle-node bifurcation (marked SN, unfolding parameter κ) and a Hopf bifurcation (marked H, unfolding parameter µ). Obsere that there is also a heteroclinic connection on the line marked H2, and this is degenerate in that an open set in the phase space is foliated with tori for these parameter alues. See the text and [2] for more details. The parameter e is frequently rescaled to e = (the negatie sign ensures the Hopf bifurcation is supercritical) [2] and we will take e = throughout. A saddle-node bifurcation occurs in the -direction as κ passes through, giing fixed points, + and, at (z, ) = (, ± κ). The system is axisymmetric because of the normal form symmetry, and so can be transformed into cylindrical polars (r, θ, ) = ( z, arg(z), ) with the angular ariable decoupled. A limit cycle L at (r, ) = ( κ µ 2 /ɛ 2, µ/ɛ) for µ 2 ɛ 2 κ is created as the complex ariable z undergoes a Hopf bifurcation on crossing the line κ = µ 2 /ɛ 2. A heteroclinic connection H from + to exists at κ = (µ > ). This indicates a degenericity in the model, since also on this half-line there is a secondary Hopf bifurcation, in which the limit cycle bifurcates into a nested family of tori. The behaiour of the system is summarized in figure, for the case ɛ > (see [2] for details). This system models a dynamo in which hydrodynamic behaiour (with magnetic field equal to zero) is represented by the -axis. The properties of a dynamo require that this axis remain inariant. Two different conecting states (one hydrodynamically stable but magnetically unstable, 4

5 and one hydrodynamically unstable but magnetically stable) are represented by the two fixed points on the -axis. The complex coordinate z represents the magnetic field. When separated into real and imaginary parts we hae a natural correspondence of real part with the toroidal field, and imaginary part with the poloidal field [28]. The (primary) supercritical Hopf bifurcation represents the onset of magnetic instability. Also in the ż equation is a term giing the contribution of the elocity field to the magnetic flux. (This is the only permissible quadratic term, since 2 would break the inariance of the -axis, and z 2 would break the inariance z z, needed to allow a reersal of the field.) Finally, the 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 to =. n particular, trajectories that begin outside the heteroclinic connection H escape in this way. This can be preented by introducing another attracting fixed point on the -axis, near [28]. This also results in the degeneracy of the secondary Hopf being broken. 2.2 Breaking the degeneracy of the secondary Hopf Adding a cubic term c 3 to the equation retains the axisymmetry, breaks the secondary Hopf degeneracy, and introduces a new fixed point at /c. Thus the system becomes ż = (µ + iω)z + ɛz = κ 2 + c 3 z 2. The constant c is chosen to be negatie to make the new fixed point stable in the -direction. The two original fixed points remain near + and for small c. The bifurcation diagram remains similar to the case aboe, but now the secondary Hopf bifurcation no longer occurs at the same parameter alues as the existence of H. We also hae regions of parameter space created in which stable two-tori are possible (see [2, 5] 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 ariables. Thus no chaotic dynamics is possible. Breaking the axisymmetry allows more complicated trajectories to occur. The system as a dynamo model also demands that the axisymmetry is broken, as it implies the equialence of the poloidal and toroidal fields, which is not true for the dynamo process. Howeer, changing the sign of the magnetic field is a symmetry of the dynamo problem. We break this symmetry in a way that preseres the inariance of the -axis and retains the symmetry z z, by adding a term proportional to z 3. (The symmetry z z is erroneously absent from the model 5

6 in [28].) ż = (µ + iω)z + ɛz + dz 3 = κ 2 + c 3 z 2 (2) Although this results in a system with a degree of non-genericity (the inariance of the -axis), this constraint is required for the system to be a iable model of a dynamo any purely hydrodynamic states must remain purely hydrodynamic for all time. 2.4 Preenting escape of trajectories to infinity n order to preent solutions from escaping to = or being attracted to a fixed point near, we can render the system periodic in. On the -axis, instead of a saddle-node bifurcation on an infinite line, we hae a saddle-node bifurcation on a limit cycle. Any trajectories threatening to escape close to the lower unstable manifold of return close to the upper stable manifold of + : we call this process the global reinjection. This type of reinjection of solutions has been studied in many models, and its interaction with a Hopf bifurcation is studied in [7, 8], for models of optically drien lasers. We choose to make periodic on the interal [πl/2, πl/2], where L is a new parameter. n order to ensure that the ariable and its powers remain continuous on this interal, we obsere that we need trigonometric functions that are (a) periodic on [ πl/2, πl/2], and (b) proportional to, 2, and 3 for close to. Set u = L sin(/l), so that for sufficiently small, u. Similarly, set u 2 = L 2 sin 2 L : for small, u 2 as well. The new ariable u is not periodic on [πl/2, πl/2], but u 2 and u 2 are, as is u2 u 2. The system can thus be made periodic and continuous by substituting u 2 2 u 2 3 u 2 u 2 and confining to the interal [ πl/2, πl/2]. Hence the full system becomes ż = (µ + iω)z + ɛzu 2 + dz 3 = κ u 2 + cu2 u 2 z 2 (3) with u = L sin(/l) and u 2 = L 2 sin 2 L. This system can produce, for suitable parameter choices, all the behaiour discussed in [28], with the added adantage that nearly all trajectories remain in a compact region of phase space. n particular, the secondary Hopf bifurcation creates quasiperiodic behaiour in the form of an attracting two-torus. ncreasing the linear growth term µ triggers a breaking down of the twotorus and a transition to chaos. During this transition, parameter alues exist at which typical trajectories are attracted to frequency-locked limit cycle. n fact, parameter alues exist at which 6

7 PSfrag replacements (a) (c) PSfrag replacements (d) (b) (c) (d).5 (a) (b) (d) (b) - (a) PSfrag replacements (c) (b) (a) PSfrag replacements (c) 2 (d) Figure 2: Attractors for the system (3) on increasing µ, with other parameters ω =., ɛ =., d = 4.9, κ =., c =., L = 2.. n (a) µ =.26 we hae a stable periodic orbit, with the trajectory winding twele times around the -axis during one full period. n (b) µ =.26 again, but different initial conditions gie a chaotic attractor. This is a Poincaré section through =. Superimposed are circles representing the periodic solution of (a). The equilibria (indicated by crosses) are located at (, ) = (,.966), (,.4). n (c) µ =.27, and the chaotic attractor gets closer to the -axis. Finally in (d), with µ =.28, reinjections are common and the trajectory spends a long time with z ery small. 7

8 chaotic motion is bistable with periodic attractors. Figure 2(a) shows a periodic attractor for parameter alues µ =.26, ω =., ɛ =., d = 4.9, κ =., c =., L = 2.. trajectory winds twele times around the -axis during one full period. Similar limit cycles appear at smaller alues of µ, apparently for most initial conditions. At these parameter alues howeer, the basin of attraction for this limit cycle seems ery small. Most initial conditions lead to chaotic motion illustrated in figure 2(b). This is a Poincaré section formed by plotting (, ) when the trajectory hits the section =. The parameter alues are as aboe, but for different initial conditions. t shows the heteroclinic tangle caused by the unstable manifold of + crossing the stable manifold of. (The fixed points + and are gien by crosses on figures 2(b), (c) and (d).) Superimposed on the chaotic solution are circles plotted where the periodic orbit of figure 2(a) crosses the Poincaré section. ncreasing µ further allows the chaotic attractor to much smaller alues of z, close to the -axis, owing to the heteroclinic connection and the reinjection mechanism. Figure 2(c) shows an attractor for µ =.27. Here ery occasional reinjections are required. When the trajectory reaches = πl/2 = π it is reinjected at = π. Figure 2(d) shows an attractor for µ =.28. At this parameter the trajectory makes a reinjection roughly as often as it traels up the -axis. This is as close as the attractor will get to z =, as increasing µ further results in a moe away from the -axis, as reinjections become more common and eentually ineitable. 2.5 nteraction of saddle-node bifurcation with two Hopf bifurcations We now introduce a second transerse complex direction and assume there are Hopf bifurcations in each of the ariables z and z 2. We include an equation representing an antisymmetric elocity component, w along the lines of [6] giing the 6-dimensional system that we consider for the rest of this paper. z = (µ + σ + iω )z + ɛ z u 2 + d z 3 + b z 2 2 z + (β + γ u 2 )wz 2 z 2 = (µ + iω 2 )z 2 + ɛ 2 z 2 u 2 + d 2 z2 3 + b 2 z 2 z 2 + (β 2 + γ 2 u 2 )wz = κ u 2 + cu2 u 2 ( z 2 + z 2 2 ) ẇ = τw + e(z z 2 + z 2 z ), where µ, σ, ω,2, ɛ,2, d,2, b,2, γ,2, κ, c, τ, e R and β,2 C are all parameters. This system of ODEs has symmetries S : (z, z 2,, w) ( z, z 2,, w) S S 2 : (z, z 2,, w) (z, z 2,, w) S 2 : (z, z 2,, w) ( z, z 2,, w) The (4) and in the case that σ = and the parameters are independent of their subscripts, there is an additional symmetry z z 2. n the general case there are three dimensional inariant subspaces 8

9 gien by D = {(z,,, )} and Q = {(, z 2,, )}, (5) and the intersection of these D Q corresponding to z = z 2 = w =. We hae not attempted to include all possible parameters that will unfold the system to any particular order; merely we hae included parameters such that the obsered dynamics appears to be robust. Note that the presence of the symmetries imply that the global reinjection occurs within the inariant subspace D Q and so is persistent as a global connection. 2.6 nterpretation and motiation The physical motiation behind this model comes from stellar dynamo theory. Mean-field models of stellar dynamos often exhibit oscillatory instabilities to magnetic fields with dipole or quadrupole symmetry: fields that either change sign or that are left inariant by reflections in the equatorial plane of the star. n these models, the instabilities to dipole and quadrupole dynamos occur for similar parameter alues and produce similar nonlinear behaiour (apart from symmetry type). The complex ariables z and z 2 represent dipole and quadrupole magnetic fields, with the real and imaginary parts representing toroidal and poloidal components of the field, and the nearequialence of dipole and quadrupole modes is represented by haing σ µ, ω ω 2, d d 2, b b 2, β β 2 and γ γ 2. The interaction of non-chaotic dipole and quadrupole dynamos was considered in [6] using a model similar to (4), but with d = d 2 =. Real stellar dynamos (and most obiously the Solar dynamo) show behaiour that is not simply oscillatory, but consists of oscillations that are modulated chaotically oer a time-scale many times longer than the basic period of the oscillation. The chaotic modulation of a dipole dynamo was considered in [28] using a model similar to (2). Effectiely, we hae extended the dipole-quadrupole model of [6] to include the chaotic modulation of [28], by including cubic terms to break axisymmetry. We hae also made the model more robust by making periodic, preenting trajectories from escaping to infinity. 3 ntermittent attractors for the model We now turn to an inestigation of the properties of the model (4). n particular, we are interested in how the chaotic modulation of [28] influences the intermittent switching between dipole and quadrupole actiity obsered by [6]. n 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 inariant subspaces D and Q in turn. Throughout this paper we fix some parameters as follows except where explicitly stated: ω = ω 2 =., ɛ = ɛ 2 =, κ =, c =., e =., τ =., L = 2, µ =.26, σ =. and d = d 2 = 4.9. (6) 9

10 PSfrag replacements.. z, z2.. e-5 e-6 e-7 e-8 e Figure 3: Time series for symmetric parameters showing switching between inariant subspaces: µ =.26, σ =. The black line denotes z and grey z 2. The time interals between intermittent isits close to D and Q are apparently random, with a preference for alternating, but hae a welldefined mean. The symmetry in the parameters mean that on aerage an equal time is spent near each inariant subspace. t Recall that L goerns the distance between + and ia the reinjection, and hence L = 2 means that the distance between the fixed points in either direction is comparable. We set the nonlinear coupling parameters as β = β 2 =.5 +.5i, γ = γ 2 = 2.5, b = b 2 =.. (7) 3. A numerical example of two-state in out intermittency With these symmetric parameters (σ = ) we can find trajectories which alternate between isiting regions ery close to D and Q, as shown in figure 3. The dynamics is interpreted as follows, using figure 4 as a schematic guide. As the trajectory nears D (resp. Q), the actie ariable z (resp. z 2 ) approaches some inariant (periodic) dynamics D in (resp. Q in ) within that subspace, whilst the

11 D Q D in Q in out D Q out Switching Figure 4: A schematic diagram showing the two-state in out intermittency obsered in Figure 3. The dynamics switches between laminar phases where it remains close to each of the subspaces Q and D. The approach to the subspace, say Q, is close to the stable manifold of a periodic orbit Q in that is unstable within Q to a chaotic attractor Q out with a positie transerse Lyapuno exponent; trajectories then moe away from Q while remaining close the the unstable set of Q out. When reaching a state of approximately the same distance from Q and D, the nonlinear dynamics can send the trajectory either back towards Q or D in a seemingly random manner.

12 quiescent ariable z 2 (resp. z ) is suppressed. The dynamics in D in (resp. Q in ) within D (resp. Q) is unstable to another (chaotic) inariant set D out (resp. Q out ). When the system is in this state, the quiescent ariable is allowed to grow. When the quiescent ariable has grown to a similar magnitude to the actie ariable, a nonlinear switching mechanism sends the system back towards either D or Q in a seemingly random manner but with a preference for a switch between D and Q. The time interals between switches appear random, but hae a well-defined mean. ndeed, for these symmetric parameters, on aerage an equal amount of time is spent near each inariant subspace. Note that here the dynamics appear to flip almost eery time quiescent ariables grows to a comparable magnitude as the actie ariable, but there are occasions when the ariables fail to switch and an actie dipole (say) phase is followed by another (for example at t = 25). 3.2 Detailed description of the mechanisms of the intermittency The suppression or growth of the quiescent ariable is mainly goerned by the magnitude of the actie ariable. The larger the time-aerage of the actie ariable, the greater the suppression (or smaller the growth) of the quiescent ariable (for this choice of parameters). This is because of (for instance) the z 2 2 z term in the z equation in (4). This suppression and growth, and the sets D in etc., can be seen explicitly in figures 5 and 6. Figure 5 shows a detail from the time series in figure 3. The fie marked segments are displayed in figure 6 as plots of Re(z ) (in black) and Re(z 2 ) (in grey) against m(z ) and m(z 2 ), and z and z 2 against. First in segment we see the system heading into the nonlinear switching region. Here z spirals in and z 2 spirals out to meet in a region in which the two ariables are of similar magnitude. Since the parameters are symmetric it is a delicate issue which ariable is faoured in the switching mechanism. Segment shows the system faouring the dipole z ariable, and the system leads into the region D in. Note that the similar magnitude of z and z 2 causes the damping of the ariable ia the ( z 2 + z 2 2 ) term in the equation. n segment we hae reached D in, which is a periodic torus around the -axis. This torus has a sufficiently large aerage z to suppress z 2 further. D in unstable within D and in segment we see both the torus and growing, leading to D out. The set D out can be seen in segment, and is equialent to the chaotic attractor shown in figure 2(b). This allows the growth of the quiescent ariable because although z reaches larger alues, it also spends a long time much closer to the -axis, and this results in a smaller aerage z. 3.3 Breaking the z z 2 symmetry We can break the symmetry of the parameters in the linear growth term, for example, by setting σ to the equations. One might suppose, setting a greater rate of linear growth in z leads to the z (dipole) subspace being preferred, as when a switch is possible, when z and z 2 are of comparable size, the direction with the greatest eigenalue should dominate. Howeer, it appears that the opposite can be true, at least for the parameter alues we hae inestigated. 2

13 PSfrag replacements z, z2.. e-6 e t Figure 5: Close-up of the time series in figure 3 detailing the different episodes during a laminar phase of the intermittency as the trajectory approaches the subspace D. 3

14 -3 z2, m(z2), m(z2) z z2, m(z2) z Re(z ), Re(z 2 ) z z2, m(z2) z Re(z ), Re(z 2 ) z2, m(z2) Re(z ), Re(z 2 ) z z Re(z ), Re(z 2 ) z2 z z2 rag replacements PSfrag replacements PSfrag replacements PSfrag replacements PSfrag replacements, m(z2), Re(z2) z2, m(z2), Re(z2) z z2, m(z2), Re(z2) z , m(z2), Re(z2) z , m(z2), Re(z2) z rag replacements PSfrag replacements PSfrag replacements PSfrag replacements PSfrag replacements, m(z2), Re(z2) z , m(z2), Re(z2) z z , m(z2), Re(z2) z z , m(z2), Re(z2) z z , m(z2), Re(z2) z z 2 rag replacements PSfrag replacements PSfrag replacements PSfrag replacements PSfrag replacements Re(z ), Re(z 2 ) z Figure 6: Projections of phase portraits of different time segments of figure 5 showing the different episodes of the in out intermittency for the actie ariable z, while the quiescent ariable z 2 decays and then grows. n the plots in the left column black lines represent z and grey lines represent z 2. n the trajectory is approaching the nonlinear switching region; shows the trajectory leading into D in ; in it is near D in ; shows it leading towards D out ; finally in the trajectory is near D out. z 2 4

15 PSfrag replacements. z, z2. e-6 e-8 e- e Figure 7: Time series as for figure 3, but with µ =.26 and σ =.. Here the linear growth rate is greater in the dipole ariable. Howeer, the quadrupole subspace is on aerage faoured. t Figure 7 shows a time series as in figure 3, but now with µ =.26 and σ =., so the growth rate for z is greater than that for z 2. The dynamics for these parameters starting with initial conditions solely in Q are equialent to those gien in figure 2(b) (or figure 2(a) for a more specific, and ery precise, choice of initial conditions), and for initial conditions solely in D we hae the dynamics in figure 2(c). For initial conditions not in either inariant subspace, a typical trajectory switches between these behaiours. Despite the larger linear growth rate in z, the trajectory faours the actiity in z 2. There are many factors inoled in taking precedence oer the linear growth rate. These include the transerse Lyapuno exponents at D in, D out, Q in and Q out ; the tangential Lyapuno exponents at D in, D out, Q in and Q out ; the aerage suppression from the actie ariable on the quiescent ariable; the length of the laminar phases between switches; nonlinear effects within the switching mechanism; the type of attracting dynamics within the inariant subspaces (periodic or chaotic); other inariant or nearly-stable sets within the inariant subspaces. These are all important, but we will concentrate in section 4 on two the length of laminar phases and the switching mechanism. We gie results 5

16 to suggest that large bias towards dipole (resp. quadrupole) stems from an increased tendency in the switching mechanism to follow a dipole (resp. quadrupole) actie phase with another. 3.4 Blowout bifurcation to intermittency Consider a family of dynamical systems smoothly parametrized by some λ R with a proper inariant subspace N for all alues of λ. f the dynamics in N is independent of λ we say λ is a normal parameter relatie to N. As discussed in [2, 5] for normal parameters one can expect to locate blowout bifurcations relatiely easily; for more general parameters this is not the case. An examination of the equations (4) reeals that for the inariant subspace Q the following are normal parameters σ, ω, ɛ, d, b, β, γ, τ, e while for D the following are normal parameters ω 2, ɛ 2, d 2, b 2, β 2, γ 2, τ, e n particular, only τ and e are normal for both subspaces. On arying these from the alues (6) we can stabilise an attractor in the inariant subspace ia a blowout bifurcation whilst leaing the dynamics within the subspace unaffected. The blowout appears to be subcritical and we find no cases where there are attractors that are for example stuck on to only one inariant subspace. Figure 8 illustrates the changes in typical dynamics on increasing τ. For τ <.5 we find stable two-state in out intermittency for typical initial conditions, as in figure 8(a). ncreasing τ beyond.5 creates an attractor close to z 2 = but bounded away from it, as in figures 8(b), (c) and (d) (here the initial conditions are chosen close to D - similar time series close to Q can be easily found with different initial conditions.) n figure 8(b) we hae τ = 3. and the resulting attractor has regular oscillations near D. n 8(c) we hae τ = 7. and an aperiodic attractor, and in 8(d) τ = 8. gies another oscillatory attractor of ery long period. ncreasing τ further causes these bounded attractors to gie way to attracting dynamics within D (again for these initial conditions close to D). Both figures 8(e) and 8(f) hae τ =. and the ariable z 2 decays to zero. The different rates of decay is due to the different dynamics within D in 8(e) the initial conditions lead to the chaotic attractor of figure 2(b), whereas in 8(f) the initial conditions lead to the periodic attractor of figure 2(a). As the parameters admit the permutation symmetry z z 2 this means that the intermittency will be the same to both D and Q subspaces. 3.5 nfluence of noise ncluding additie noise in the model effectiely destroys the inariance of the subspaces D and Q. We introduce the noise by adding to each ariable a random ariable from a normal (Gaussian) distribution scaled with some noise leel ξ, and with the square root of the preious time step. The results of this, for ξ = 6, ξ = 4 and ξ = 3 are shown in figure 9. Small amounts of 6

17 z, z2 z, z2 z, z2 g replacements (a) t (c) t (e) t... (b) t... (d) t (f) t Figure 8: Dynamics for the symmetric parameters (6) and (7) on increasing τ. All figures plot z in black and z 2 in grey. Figure (a) has τ =., and shows stable two-state intermittent dynamics. ncreasing τ beyond.5 creates an attractor close to z 2 = but bounded away from it, as in figure (b), with τ = 3.. ncreasing τ further causes this attractor to moe closer to the inariant subspace D, but remain bounded away from it. Figure (c) has τ = 7.. n figure (d) we hae long period oscillatory behaiour, with τ = 8.. n figures (e) and (f) we hae τ =. and D has become attracting. Different initial conditions in (e) and (f) lead to different dynamics within D. n (e) the initial conditions lead to the chaotic attractor of figure 2(b), whereas in (f) the initial conditions lead to the periodic attractor of figure 2(a). The larger aerage alue of z in this periodic orbit causes a much quicker decay in z 2. 7

18 PSfrag replacements Proportion of time spent in dipole Figure 9: The effect of noise on the proportion of time spent near each inariant subspace. The data are produced by running a typical intermittent trajectory and recording the proportion of time that z > z 2, up to t = 2 6 for each σ. The noise leels are ξ =. (empty circles), ξ = 6 (solid hexagons), ξ = 4 (squares), ξ = 3 (triangles). σ noise hae, as expected, little effect on the proportion of time spent near each inariant subspace, but for larger ξ we find the effect on the proportion of time spent near a specific subspace upon breaking the symmetry of the parameters is een more pronounced. The behaiour of the laminar phases are not changed greatly by the addition of noise, but within the switching mechanism we see a greater tendency for the trajectory to be pushed towards D (say) many time in succession, without isiting Q (see section 4. and figure 2). 4 Modelling two-state intermittency as a Marko process n the section we construct a probabilistic Marko chain model for the two-state intermittency obsered in the the dynamics of the attractor (such as that in figure 3) where trajectories moe between neighbourhoods of pure dipole and pure quadrupole inariant subspaces. 8

19 Switching mechanism r 5 r 4 r 3 r 2 r r r 2 r 3 r 4 r 5 γq γ ε ε ε ε d ε ε ε ε q q q q d d d d s 5 s 4 s 3 s 2 s s s 2 s 3 s 4 s 5 Out n Pure Quadrupole Delay T q Delay T d Pure Dipole Figure : The Marko chain used to model the two-state in out intermittency obsered in the saddle-node/hopf model. The states r n and represent transerse distances ρ n from an inariant subspace near the out dynamics while s n represent transerse distances ρ n from the in dynamics for some < ρ <. The switching mechanism is determined also by a random process with probabilities γ d,q and is subject to an additional delay of T q,d 2 steps, so the minimum time spent in the d, q chain is T d,q. Examining the dynamics of the intermittency, we note for example from figure 3 that the approach towards D in, Q in is approximately uniform, as is the departure from D out,q out. therefore use a chain to model these as shown in figure and assume that the leakage from the in chain to the out chain happens at a uniform rate ɛ d,q. We also assume that the switching mechanism is similarly goerned by a Marko process parametrized by constants γ d,q and that there is a delay of T d,q steps during the switching mechanism. The Marko model as shown in figure has states that correspond to the intermittent model as follows: We model approach to D in by a chain of states {s n }, Q in by a chain of states {s n }, D out by a chain of states {r n }, and Q out by a chain of states {r n }, where we approach the releant inariant set in the limit as n. More precisely, there is a ρ < such that for example p n gies the probability of being approximately distance ρ n from D. f P (a, b) is the probability of a transition from a to b we assume that within the chains, for n P (r n+, r n ) =, P (r n, r n ) = P (s n+, r n+ ) = ɛ d, P (s n, r n ) = ɛ q, P (s n+, s n+2 ) = ɛ d P (s n, s n 2 ) = ɛ q for constants ɛ d,q (, ). All other transitions within the chains hae zero probability, and all transitions take one time step, apart from the transition from s to s 2 which takes T d 2 steps and from s to s 2 which takes T q 2 steps. The transitions in the switching mechanism between the chains are assumed to hae probabilities P (r, s ) = γ d, P (r, s ) = γ d, P (r, s ) = γ q, P (r, s ) = γ q, We 9

20 for constants γ d,q also in (, ). The constants ɛ d,q can be interpreted as the probability per unit time of leaking from the in dynamics to the out dynamics near the subspaces D, Q, and this gies rise to an exponential distribution of the durations that a trajectory is near the inariant subspaces after the constant delay T d,q. The constants γ d,q correspond to the probability that the dynamics near the switching region send a trajectory that enters from D out, Q out back into D in, Q in respectiely. One can easily calculate the inariant probability p of the chain as p(r ) = p(s ) = A p(r n ) = p(s n ) = A( ɛ d ) (n 2), n 2 p(r ) = p(s ) = B p(r n ) = p(s n ) = B( ɛ q ) ( n+2), n 2 implying that the residence time near each of the inariant subspaces is exponentially distributed. We hae p(r n ) = p(s n ) since the trajectories passing through r n are precisely those that hae preiously passed through s n. One can find that aerage residence times A d,q in the chains: A d = T d + 2 n= nɛ d ( ɛ d ) (n ) = T d + 2 ɛ d and similarly A q = T q + 2 ɛ q. Moreoer, the transition between the q and d states at the end of the laminar phases is goerned by γ d and γ q ; obsere that and so A = p(s ) = ( γ q )p(r ) + γ d p(r ) = ( γ q )B + γ d A, = A(T d + 2/ɛ d ) + B(T q + 2/ɛ q ) A = ɛ d ɛ q (γ q ) 2ɛ d (γ d ) + 2ɛ q (γ q ) + ɛ d ɛ q (γ d T q + γ q T d T d T q ) and there is a similar formula for B. The inariant probability density can be used to find the probability of a laminar phase in the dipole and quadrupole chains is P d = γ q 2 γ d γ q, P q = γ d 2 γ d γ q (8) giing the proportion of time spent in the dipole and quadrupole chains: M d = P d A d P d A d + P q A q, M q = P q A q P d A d + P q A q. (9) Note that, as discussed in Section 3.3, the relatie proportion of isits to say the dipole subspace depend on both the aerage residence times A d,q and the probabilities of switching, namely γ d,q. One can erify the alidity of the indiidual components of the Marko model for the original ODE problem. For example in figure we show the distribution of the lengths of the laminar phases 2

21 σ T d T q ɛ d ɛ q γ d γ q Table : Numerical estimates for the Marko chain parameters. Obsere that the greatest effect of the symmetry breaking term σ =.3 is to change the transition probabilities γ d,q within the switching mechanism. for a long timeseries with (a) corresponding to figure 3 (µ =.26, σ = ) and (b) corresponding to µ =.26, σ =.3. Note that the switching mechanism for the ODE takes a certain minimum time after which the frequency of occurrence drops off exponentially. To estimate the proportion of time spent in near D and Q we take a threshold ( z < 2 ) to determine when we are close the inariant subspace and find the transition probabilities γ q,d. These can be estimated by classifying the timeseries into D and Q phases and so associate the timeseries with a string {w i } n i= with w i {D, Q}. We can then estimate γ d = #{ i < n : w i = w i+ = D}, #{ i n : w i = D} that is, γ d is the frequency of obsering D as the next laminar phase gie that the preious one was D. γ q can be estimated similarly. Table gies estimates of the parameters as computed from the timeseries (we take T d,q to the nearest integer for the discrete time Marko chain). Once haing fitted the parameters ɛ d,q, T d,q and γ d,q one can use the model to predict for example using (9), the proportion of time spent near the dipole model. As an example, for σ = we estimate M d from the parameters and M d from a (different) timeseries and find M d =.4997, Md =.523 for σ =, while M d =.5724, Md =.576 for σ = Two-state in out intermittency and noise The details of the dynamics near the inariant subspaces for instance in Figure 3 reeals that the in dynamics within D is periodic while the out dynamics is chaotic, and so one could construct a model where the deterministic flow away from the inariant subspace is replaced by a biased random walk. This will not substantially increase the accuracy for this case, but in cases where the ariance of the out chain propagation is much larger it would be necessary; this would gie a more sophisticated model than here or in [4]. Similarly, one could make the Marko model a lot 2

22 D or Q D or Q fit Frequency Length of phase (a) D Q D fit Q fit Frequency Length of phase (b) Figure : Distributions of the lengths of laminar phase T (near either D or Q) subspaces for (a) the symmetric case shown in Figure 3 (µ =.26, σ = ) and (b) an asymmetric case µ =.26, σ =.3. Obsere that there is a good fit to a constant delay and an exponential probability distribution of phase length T in both cases where for (a) we hae P e.227t for approaches to either, and (b) we hae P e.277t for approaches to D and P e.27t for approaches to Q. Note that the lengths of laminar phases change ery little. 22

23 PSfrag replacements. z, z2... e-5 e Figure 2: Time series showing intermittency in the presence of noise. The parameter alues are as in figure 3, but with σ =.3 and ξ = 4. The dipole subspace is greatly preferred because the transition probabilities in the switching mechanism now faour a dipole dipole switch. t more sophisticated by including a continuum of states and by including noise effects along the lines of [4, 8, 25]. n fact, the introduction of noise to the system accentuates the findings of the aboe that the greatest effect of the symmetry breaking term σ is to change the transition probabilities γ d,q within the switching mechanism. Figure 2 shows a time series showing the intermittency for σ =.3 and ξ = 4, with the other parameters as before. Here it is clear that the dipole subspace is much faoured, and this bias is far more pronounced than with ξ = (recall that for σ =.3, ξ = we had M d =.576 here M d =.8976). t is clear that the tendency within the switching mechanism is now to follow a dipole actie phase with another dipole phase, and in fact we can estimate the transition probabilities as before as γ d = and γ q =

24 5 Discussion and conclusions n summary, we hae examined a system that displays intermittent switching between chaotic dynamics in two inariant subspaces. n detail, the intermittency mechanism inoles four states, two in each of the inariant subspaces. Two of these states (D in and Q in ) are transersely stable but unstable within the inariant subspaces. The other two ( D out and Q out ) are transersely unstable but attractors within the inariant subspaces. Once the system moes away from the inariant subspaces, there is nonlinear switching and reinjection towards D in and Q in. We hae deeloped a Marko model of this intermittency mechanism that is capable of capturing the exponential distribution of times spent near each inariant subspace. This switching mechanism is a new feature of two-state intermittency that is not present in preious examples of in out intermittency. Our work also sheds light on the switching behaiour obsered in other dynamo models, for example the mean-field dynamo model [6] and the model of geodynamo reersals in [2]. n the latter case the switching mechanism seems to be be comparable to the mechanism in the present paper. Although we hae deried the model from perturbations near the codimension-two interaction of a saddle-node bifurcation with two symmetry-related Hopf bifurcations, and we hae not included all possible interaction terms een up to cubic order, we presume (but cannot proe) that the dynamics we hae described is robust or at least prealent. We hae used the degeneracy as an organizing centre to gie dynamics that has two-state intermittency. The model we hae considered is also physically motiated by consideration of hydrodynamic and magnetic instabilities see for example [6]. n addition, the Hopf bifurcations to dipolar and quadrupolar dynamo actiity often occur for similar (or een the same) parameter alues in mean-field dynamo models, so it is reasonable to regard a notional symmetry between these Hopf bifurcations as being only weakly broken. One of the most surprising results of this inestigation has been the recognition that linear growth rates (or aerage residence times) near the inariant subspaces D and Q do not determine the aerage proportion of time spent close to the two subspaces. There are are many other factors that determine which of the two subspaces is preferred. The most important, at least for the Marko model, is the switching mechanism operating in the fully nonlinear regime that determines a preference of one phase to the other, once the equialence of the two inariant subspaces is broken. We obsere that switching can be greatly affected by the addition of noise. As discussed in [6] and [7], numerical simulations of cycling chaos need to be carried out with great care. The simulations presented in this paper were performed to double precision accuracy using the Bulirsch Stoer adaptie step integrator [27] with a relatie tolerance of 8 at each time step. Rounding errors may force the dynamics into inariant subspaces, and care was taken to interpret correctly when this occurred. The dynamics of (4) was also simulated with the inclusion of small amounts of additie isotropic noise, in which case the effects of rounding into an inariant subspace could be easily aoided. 24

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