Delayed feedback versus seasonal forcing: Resonance ... › ~berndk › transfer › kkp14_enso_p… · Delayed feedback versus seasonal forcing 3 In contrast to an ordinary di erential

Delayed feedback versus seasonal forcing: Resonance phenomena in

an El Niño Southern Oscillation model

Andrew Keane∗, Bernd Krauskopf and Claire M. PostlethwaiteDepartment of Mathematics, The University of Auckland,

Private Bag 92019, Auckland 1142, New Zealand

December 2014

Abstract

Climate models can take on many different forms, from very detailed highly computational models withhundreds of thousands of variables, to more phenomenological models of only a few variables that aredesigned to investigate fundamental relationships in the climate system. Important ingredients in thesemodels are the periodic forcing by the seasons, as well as global transport phenomena of quantities such asair or ocean temperature and salinity.

We consider a phenomenological model for the El Niño Southern Oscillation system, where the delayedeffects of oceanic waves are incorporated explicitly into the model. This gives a description by a delaydifferential equation, which models underlying fundamental processes of the interaction between internaldelay-induced oscillations and the external forcing. The combination of delay and forcing in differentialequations has also found application in other fields, such as ecology and gene networks.

Specifically, we present exemplary stable solutions of the model and illustrate bistability in the form ofone-parameter bifurcation diagrams for the seasonal forcing strength parameter. So-called maximum mapsare calculated to show regions of bistability in a two-parameter plane for the seasonal forcing strength andoceanic wave delay time. To explain the observed solutions and their multistabilities, we conduct a bifur-cation analysis of the model by means of dedicated continuation software. Knowing for which parametervalues certain bifurcations take place allows us to explain and expand on some features of the model foundin previous publications concerning the existence of unstable solutions, multistability and chaos. We un-cover surprisingly complicated behaviour involving the interplay between seasonal forcing and delay-induceddynamics. Resonance tongues are found to be a prominent feature in the bifurcation diagrams and they areresponsible for a high degree of multistability in the model. We find bistability within certain resonancetongues as a result of a symmetry property of the governing delay differential equation. We investigatethe co-existence of stable tori, how they relate to each other and bifurcate, which involves bifurcations ofinvariant tori.

1 Introduction

El Niño is a climate phenomenon that is characterised by the warming of oceanic waters off the western coast ofequatorial South America. It has been well-known to humans for hundreds of years, as the water temperatureshave a direct impact on the quality of the fishing in those areas. Furthermore, El Niño is of relevance on a globalscale: for example, it has been shown to trigger droughts in Australia and South-East Asia [2, 17, 48] and italso seems to be coupled with weather behaviour across the Indian Ocean [4] and even the Atlantic Ocean [21].

The Southern Oscillation is an oscillation in the surface air pressure between the eastern and western tropicalPacific. A common measure of the strength of the oscillation is the Southern Oscillation Index, defined as thesurface air pressure difference between Tahiti and Darwin, Australia. In 1969, Jacob Bjerknes proposed that ElNiño and the Southern Oscillation were part of the same phenomenon, which could be described as a coupled

∗corresponding author: [email protected]

1

Delayed feedback versus seasonal forcing 2

system [10]. This gives the El Niño Southern Oscillation (ENSO) system with an ocean component (El Niño)and an atmosphere component (Southern Oscillation). As well as the ENSO warm phase, referred to as ElNiño, there is also a cool phase called La Niña.

The ENSO system has been modelled in the past in various mathematical forms and at varying degreesof sophistication. For the purpose of forecasting there exist intermediate dynamical models (for example, see[13, 16]), high-dimensional coupled ocean-land-atmosphere models (see [50]), statistical models (for example, see[9, 20]) and hybrid models (see [32, 38]). Because of the shear complexity of these models used for forecastingthey are not suitable or practical for a rigorous investigation of the underlying mechanisms and their interactions.Low-dimensional coupled ocean-atmosphere models (for example, see [25, 29, 30, 52, 56, 60]) on the other hand,have proven beneficial for furthering the qualitative understanding of ENSO. For an overview of modellingENSO as a dynamical system, see [33].

In this paper, we consider a basic low-dimensional model for ENSO in the form of a scalar delay differentialequation (DDE); it was introduced in [60] and then simplified in [24] to a form that focuses only on the interplaybetween the negative feedback via ocean-atmosphere coupling and the seasonal forcing. The DDE model takesthe form

ḣ(t) = −a tanh [κh(t− τ)] + b cos (2πt). (1)It describes the evolution of the thermocline depth h (see Section 2 for more detail) at the eastern boundaryof the Pacific Ocean (more specifically, its deviation from the annual mean) as a function of time measuredin years. The first term of Eq. (1) is a nonlinear delayed negative feedback of a form justified in [44], whichreflects the saturation in the ocean-atmosphere coupling at large deviations from the mean thermocline depth.Parameters a and κ represent the negative feedback amplification factor and the ocean-atmosphere couplingstrength, respectively, and τ is the delay time in years needed for the propagation of oceanic waves across thePacific Ocean that form the negative feedback mechanism (see Section 2 for details). The second term of Eq. (1)is periodic seasonal forcing with a period of one year to reflect the annual cycle of the seasons, where b is theforcing amplitude.

In [24] it was shown that these two mechanisms are both essential for ENSO variability and sufficient forcreating rich behaviour and mimicking important features seen in real-world observations and more sophisticatedhigh-end models; see Section 2 for details on the role of these mechanisms in the ENSO system. For the remainderof the paper, we set the parameters a = 1 and κ = 11, which are the values that were used and justified inprevious investigations [24, 64].

The authors of [24] highlight the possible complexity of the dynamics of Eq. (1). Their numerical explorationincluded the calculation of maximum maps, where the maxima of simulated solutions for a single fixed initialcondition are illustrated as a function of the two parameters b and τ . These maximum maps revealed largedomains in the (b, τ)-plane where clear jumps (discontinuities) in the observed maxima, max(h(t)), occur withchanges in parameters. Since the solutions depend continuously on the parameter values, these jumps in maximasuggest the existence of unstable solutions that separate the stable ones in the phase space. In their follow-uppaper [64], the authors present results about phase locking to the seasonal forcing, which is an important featureof the ENSO system, and they show the co-existence of distinct solutions (i.e. bistability) obtained by numericalintegration (simulation) from different fixed initial conditions.

Here, we take a dynamical systems point of view and conduct a bifurcation analysis of Eq. (1), where thetechnique of investigation includes the use of the state-of-the-art continuation software DDE-BIFTOOL [22, 53].We focus our investigation on the (b, τ)-plane, in order to explain features of the model observed in [24, 64].Due to uncertainties and ambiguities in how these parameters relate to the observable world, we are interestedin the model sensitivity to changes in these parameters.

Delay and periodic forcing are key features in the model. They both influence and/or induce bifurcations(for example, see [3, 34, 46]). Feedback mechanisms that involve delays may be present in many forms, not justin climate systems. For example, DDEs have been used to describe the dynamics of epidemics [40], networksof neurons [51], coupled chemical oscillators [11] and lasers [36, 41]. Delayed feedback is also successfully andwidely-used as a form of control [47]. The specific combination of both delayed effects and periodic forcing hasalso been the subject of investigation in different fields, for example, in ecology [35], gene networks [62] andlaser dynamics [5]. As such, the results presented here are of relevance in a broader sense than simply withinthe context of phenomenological climate modelling.


In contrast to an ordinary differential equation (ODE), a DDE such as Eq. (1) with a single fixed delayrequires a whole initial history segment over the time interval [−τ, 0] as an initial condition to define the initialvalue problem, i.e. the future evolution of the system. This means that the DDE in its autonomous form hasthe infinite-dimensional phase space C([−τ, 0];Rn) × R, where Rn is the physical space of the variables of theDDE, C([−τ, 0];Rn) is the infinite-dimensional space of continuous functions and R represents time.

For b = 0 in Eq. (1) (i.e. without periodic forcing), we are left with a scalar DDE that has been studiedanalytically in the past. Of specific interest here is the result that for τ above the critical delay time τc = π/(2κ)the zero solution h ≡ 0 loses stability in a Hopf bifurcation. For τ ≥ τc there exists a set of stable periodicsolutions of period T = 4τ [14, 18, 45]. When a strong ocean-atmosphere coupling is used (such as κ = 11), thefirst term in Eq. (1) acts approximately as a delayed switching function, resulting in solutions with time seriesof a zigzag form (for an example, see Fig. 2(a1)).

Self-sustained oscillations exist due to the delayed feedback alone, and the addition of periodic forcingintroduces a second frequency. This implies the possibility of dynamics on an invariant torus, which may belocked or unlocked depending on the frequencies involved. If the two frequencies have an irrational ratio, anytrajectory will not close and the solution is quasi-periodic. If the frequencies have a rational ratio, then thereis a pair of periodic orbits (one stable and one unstable) on the torus with a finite period; one speaks of lockeddynamics. The regions in the parameter space where the dynamics on the torus are locked are known asresonance tongues, which are a well-studied phenomenon (for example, see [26, 39, 42, 54]); they are discussedin the context of Eq. (1) in Sections 4–5.

The periodic forcing depends explicitly on time t, so Eq. (1) is a non-autonomous DDE. An autonomousdescription can be given by increasing the dimension of the system by one by letting t ≡ z ∈ S1 and dz/dt = 1.Hence, there are no equilibrium solutions. Because the continuation software DDE-BIFTOOL is not designedto allow for non-autonomous DDEs, we introduce two additional dependent variables to mimic the periodicforcing by including the Hopf normal form:

ẏ1(t) = λy1(t)− ωy2(t)− y1(t)(y21(t) + y22(t)) (2)ẏ2(t) = ωy1(t) + λy2(t)− y2(t)(y21(t) + y22(t)), (3)

where λ and ω are constant parameters. For our purposes, we choose λ = 1 and ω = 2π, for which the unitcircle is a stable periodic orbit in the (y1, y2)-plane with a period of one. Therefore, rewriting Eq. (1) as

ḣ(t) = −a tanh [κh(t− τ)] + by1(t) (4)

provides us with the required seasonal forcing in an autonomous form that embeds S1 into R2, which we canimplement in DDE-BIFTOOL.

Once a periodic solution to Eq. (1) is found for certain parameters, we can use DDE-BIFTOOL to con-tinue (or track) it numerically while varying parameters. DDE-BIFTOOL can also determine the stability ofperiodic solutions by calculating their Floquet multipliers, which in turn can be used to identify bifurcations.The bifurcation theory of DDEs with a single fixed delay is analogous to the theory of ODEs (relevant bifur-cation theory for ODEs can be found in, for example, [39, 55]). Since there are no equilibrium solutions ofEq. (1), in later sections the bifurcation types of interest will be those of periodic orbits: we find saddle-nodebifurcations of periodic orbits, period-doubling bifurcations and torus (or Neimark-Sacker) bifurcations. Allof these bifurcations can be continued numerically in two-dimensional parameter space with the latest versionof DDE-BIFTOOL by fixing constraints on the Floquet multipliers. For details about the numerical methodsimplemented in DDE-BIFTOOL, see [22, 43, 49].

We begin this work by presenting some examples of stable periodic solutions that show evidence of multi-stability in model (1). We then illustrate bistability clearly in the form of one-parameter bifurcation diagrams,obtained by tracking solutions for both increasing and decreasing parameter b, while using the previous so-lution as an initial condition history. This allows us to calculate maximum maps in the (b, τ)-plane for bothincreasing and decreasing b to map out regions of bistability. The maximum maps presented in [24] used asingle fixed initial condition and, as such, did not show the parameter regions where bistability is present. Themaximum maps that we show here reveal (as noted in [24]) sharp interfaces that represent rapid transitions (orjumps) in the observed maxima for varying parameters. We then overlay the maximum maps with bifurcation


curves calculated with DDE-BIFTOOL. Overall, our analysis and general theory [39] describe a parameterplane divided by curves of torus bifurcations, which are bridged by an infinite number of resonance tongues andsmooth curves of quasi-periodic solutions. The bifurcation curves agree well with the sharp interfaces seen in themaximum maps and allow for a detailed interpretation of numerical simulations. We compare our bifurcationcurves with simulation results from [64] to show that the dynamics for a parameter set that was believed to bechaotic actually consists of quasi-periodic (or high-period) solutions. The bifurcation analysis reveals resonancetongues as a prominent feature in the parameter plane. We discuss and demonstrate the role they play formultistability; in particular, we identify a symmetry property in Eq. (1) to be a source of bistability withinp :q resonance tongues of even p or q. Some of the sharp interfaces in the maximum maps cannot be explainedby the bifurcation curves calculated with DDE-BIFTOOL, which leads us to a discussion about the changingcriticality along a curve of torus bifurcations and then a detailed study of bifurcations of invariant tori in thesystem. We provide evidence for the presence of fold bifurcations of tori and associated resonance tongues. Thisincludes bifurcations of quasi-periodic solutions that differ from bifurcations of periodic orbits, since the twoinvariant tori involved mutually destroy each other before they reach the bifurcation point. We show how twosuch bifurcations can be connected by a branch of solutions on an invariant torus of saddle-type by locatingresonant solutions along the branch.

The paper is organised as follows. In Section 2 we explain in more detail the physical processes of ENSO thatlead to the model. Section 3 contains results obtained by numerical integration, including time series of sampleperiodic solutions, one-parameter bifurcation diagrams and maximum maps of Eq. (1). The overall bifurcationset in the (b, τ)-plane is presented in Section 4. In Section 4.1 we focus on the role of resonance tongues for themultistability of the system. Further results concerning bistability within resonance tongues are presented inSection 4.2. Section 4.3 addresses the changing criticality of the torus bifurcation and how it affects numericalobservations. The properties of tori and their bifurcations are discussed in Section 5. Finally, in Section 6 wedraw some conclusions and point to future work.

2 Background on the ENSO

We now give some further details of properties of ENSO and a brief description of where the terms of the modelEq. (1) have their origin. For further details on the associated climate processes we refer to [19].

The thermocline is a relatively thin oceanic layer between the deep cold waters and the warmer well-mixedlayer above. The depth of this layer is different in different regions of the ocean; in the eastern Pacific Oceanit has an average depth of about 50m. The variable h(t) of Eq. (1) represents the deviation from the meanthermocline depth at the eastern boundary. The quantity h is measured downwards from the surface, so anincrease in h means an increase in the depth of the thermocline away from the ocean surface. The thermoclinedepth is often used as a proxy for the regional sea-surface temperature (SST), since a deeper thermocline meansless upwelling (i.e. vertical transport of colder waters towards the surface) and, hence, a higher SST. However,the exact relationship itself between the two is non-trivial and includes delays [27].

Figure 1 illustrates the interactions via the ocean-atmosphere coupling that influence the thermocline depthat the eastern Pacific. In equilibrium, the thermocline is typically deeper in the west of the Pacific Ocean andshallower in the east. An atmospheric convection loop exists above the equatorial ocean as a result of thisdifference, where hot air rises in the west and cold air sinks in the east, giving rise to the easterly trade winds(easterly meaning blowing from east to west). As shown by the arrows in the atmosphere component of Fig. 1,a positive perturbation in h slows down these winds, creating westerly wind anomalies (i.e. deviations fromthe mean) over the equatorial Pacific Ocean. These anomalies together with the effect of the so-called Ekmantransport phenomenon1 cause surface water in the central part of the Pacific basin (where the ocean-atmospherecoupling is strongest) to move towards the equator, shown by the arrows of the ocean component in Fig. 1.This, in turn, induces two sets of equatorial waves, which are waves trapped near the equator by the Coriolis

1Because of the Coriolis force the surface flow generated by the wind is at 45◦ to the wind direction (to the left/right in thesouthern/northern hemisphere). However, dividing the body of water into thin layers, this angle shifts further for each deeper layersince the drag force is not from the wind itself but the layer above. The spiral form of flow shifting in direction and graduallybecoming weaker for deeper layers is known as the Ekman spiral. Integrating over the Ekman spiral gives a net water transportation90◦ to the left/right of the surface wind in the southern/northern hemisphere — this is known as Ekman transport.


h

+

Atmosphere

Rossby wave

Kelvin wave

Equator

Figure 1: The variable h represents deviations from the mean thermocline depth at the eastern boundary of theequatorial Pacific Ocean. The coupling between ocean and atmosphere allows for the creation of negative andpositive feedback mechanisms as indicated by the arrows (see text for details).

force. The deficit of warmer surface water in the off-equatorial central Pacific Ocean decreases the depth ofthe thermocline, which decreases the SST due to more upwelling. This negative signal propagates westwardand towards the equator as a so-called Rossby wave; see Fig. 1. At the western boundary of the Pacific Oceanthe Rossby wave is reflected and travels back eastward as a negative Kelvin wave (it is again the Coriolis forcethat allows Kelvin waves to only travel in an eastward direction). After a certain time delay, represented bythe parameter τ in Eq. (1) and assumed to be constant, the negative signal finally arrives back at the easternboundary of the Pacific Ocean, where it leads to a decrease of h. This process provides the negative feedbackmechanism. The time needed for the disturbance in the thermocline to propagate westward as a Rossby andeastward as a Kelvin wave back to the eastern boundary of the ocean is about 6 months.

Besides feedback mechanisms caused by delayed oceanic waves, the literature [15, 23, 31, 58] indicates thatthe seasonal forcing and subsequent resonances play an important role in describing the dynamics of ENSO. Infact, the very name ‘El Niño’, referring to the timing of the warming events around Christmas, suggests thatresonance effects with the seasons are present. Indeed, the negative delayed feedback and the seasonal forcingare the only two mechanisms incorporated into the DDE model (1).

There also exists a positive feedback mechanism, which was included in the model in [60] and is also shownin Fig. 1 for completeness. Just as the above mentioned wind anomalies cause a deficit in the off-equatorialwaters, there is a surplus of warm surface water at the equator, which increases the depth of the thermocline.This positive perturbation of the thermocline travels eastward in the form of an equatorial Kelvin waves. Aftera time delay of about a month the Kelvin wave arrives back at the eastern boundary to provide a positivefeedback mechanism that increases h further.

An ENSO model that includes the positive feedback mechanism, as well as multiplicative parametric forcing,rather than additive forcing, was recently investigated in [37], where the authors demonstrated some of theabilities of DDE-BIFTOOL. As mentioned above, this positive feedback mechanism is not included in Eq. (1),but will be revisited in future work.

3 Stable solutions and maximum maps

Numerical integration of Eq. (1) offers some initial insight into the behaviour of the system. A time series is themost intuitive way of representing solutions in the context of the observable ENSO system: maxima representwarming El Niño events and minima cooling La Niña events. The strength of the event is indicated by themagnitude of the maximum or minimum. Concerning the dynamics in a more abstract sense, phase spaceprojections can give us an idea of the shape of the attractor and are useful for identifying attractor types, as we


0 5 10

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Figure 2: Stable solutions of Eq. (1), shown as time series in panels (a1)–(e1) and as projections onto the(h(t− τ), h(t))-plane in panels (a2)–(e2); throughout a = 1, κ = 11 and τ = 1.2, b = 0 for (a), τ = 1.2, b = 3 for(b) and (c), and τ = 0.62, b = 3 for (d) and (e).

will see in the examples that follow. We follow the common choice of projection onto the (h(t− τ), h(t))-planeor the (h(t− τ), h(t− τ/2), h(t))-space.

3.1 Time series and phase space diagrams

Figure 2 shows five examples of stable solutions. They are obtained by numerical integration of Eq. (1) withthe Euler method. Fixed initial conditions of either h ≡ 0 (for rows (a), (b) and (d)) or h ≡ 1 (for rows (c) and(e)) are used. All solutions, including those in later sections, are excluding transients, i.e. the solutions shownare the trajectories after they have had time (up to hundreds of years, although mostly 30-40 years is adequate)to approach and reach a stable attractor. Panels (a1)–(e1) are examples of solutions shown as time series; theyare represented in two-dimensional projections of the phase space in panels (a2)–(e2).


Row (a) displays the solution for b = 0 and τ = 1.2. As seen in panel (a1), the solution is periodic with analmost zigzag form and a period T = 4τ = 4.8 years. The phase space projection onto the (h(t− τ), h(t))-planein panel (a2) shows the solution as a closed loop. An interpretation of this solution in row (a) in the context ofthe El Niño phenomenon yields a case where there is no seasonal forcing and the oceanic waves that producethe negative feedback mechanism take 1.2 years to reach the eastern boundary of the Pacific (cf. Section 2).The El Niño event then occurs every 4.8 years.

By contrast, row (b) displays a solution for τ = 1.2, but for b = 3, where we see a periodic solution of periodT = 1 with what appears to be a sinusoidal form. A closed loop is seen in panel (b2). In this case, wherethe dynamics is influenced by both the internal feedback mechanism and the seasonal forcing, the solution isdominated by the seasonal forcing, which is why it has a period of one.

Row (c) is for the same parameter values as row (b), but with a different initial history. Panel (c1) revealsa different solution from that of panel (b1): a periodic solution of period T = 5. This solution has a more com-plicated trajectory in the phase space projection in panel (c2) compared to the previous example in panel (b2).Note that the self-intersections seen in panels (c2)–(e2) are a result of the projection of the trajectory ontotwo dimensions. An interpretation of the example shown in row (c) would indicate multiple El Niño eventsof varying strength, with the largest occurring every 5 years. This time series is a case where two frequencies(from the delayed feedback and the seasonal forcing) have a clear influence.

Row (d) for b = 3 and τ = 0.62 gives an example where the stable solution is quasi-periodic (or of a very highperiod). The quasi-periodic behaviour can be seen particularly well in the phase space projection in panel (d2),where the trajectory over 50 years traces out a torus. When interpreting the parameters compared to the lastexample, the strength of the seasonal forcing is the same, but the oceanic waves now travel across the Pacificin just above half the time.

The final example in row (e) is calculated for the same parameters as row (d), but with a different initialhistory, which results in a periodic solution of period T = 3. As in the last two examples, both time series andphase space projection shows the influence of two frequencies on the dynamics.

Comparing rows (d) and (e), we find a case of bistability between a quasi-periodic and a periodic solutionfor the same values of the parameters. Hence, the solution that the system converges towards depends on theinitial history. Similarly, bistability is seen when comparing rows (b) and (c). This clearly shows that there areregions in the (b, τ)-plane where bistability (possibly multistability) exists.

3.2 One-parameter bifurcation diagrams

To further investigate the bistabilities observed in Figs. 2(b)–(c) and Figs. 2(d)–(e), we calculate one-parameterbifurcation diagrams. Figure 3 shows the overall maxima of solutions for a range of b values for τ = 1.2 inpanel (a) and for τ = 0.62 in panel (b). A maximum is simply taken as the largest value from the time series;because the solutions may not necessarily be periodic, the length of time from which the maximum is obtainedmust be sufficiently long (it was typically about 100 years). The diagrams in Fig. 3 show maxima of solutionscalculated while sweeping both up and down in the parameter b. This is done by setting the initial history usedto calculate a solution as the previous solution (i.e. that of a slightly lower or higher value of b, depending onthe direction that b is being changed). The black arrows indicate the direction in which b is changed and wherethere are jumps (or rapid transitions) in the maxima obtained.

In both Figs. 3(a) and (b) there exist an upper and a lower branch of maxima when increasing and decreasingb, respectively. In both cases we see an overlapping range of b values for which stable solutions from bothbranches exist, yielding the hysteresis loops indicated by the black vertical arrows in Fig. 3.

In Figs. 3(a) and (b) the upper branches originate from solutions dominated by the internal feedback mech-anism for low values of b. On the lower branches one sees maxima of solutions that are initially dominated bythe seasonal forcing for large values of b. Where these branches overlap there is bistability between them. Notethat the solution seen for (b, τ) = (3, 1.2) in Fig. 2(b) with period T = 1 lies on the lower branch of Fig. 3(a),while the solution seen in Fig. 2(c) with period T = 5 lies on the upper branch. Similarly, the quasi-periodicand periodic of T = 3 solutions seen in Figs. 2(d) and (e) are found on the lower and upper branch of Fig. 3(b),respectively.

A number of small kinks are visible in the graph of max(h(t)) in Fig. 3. They correspond to parameter


0 2 4 60.5

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b

6

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Figure 3: One-parameter bifurcation diagrams showing maximum values of h(t) for a = 1, κ = 11 and τ = 1.2(a) and τ = 0.62 (b). Each plot shows two sets of global maxima: one for increasing b and the other fordecreasing b. The black arrows indicate the direction of changing b, as well as rapid transitions in the maximumvalue of h(t).

values where the solution enters, exits or briefly passes through a resonance tongue containing only lockedsolutions. Namely, in contrast to a quasi-periodic solution, a locked periodic solution does not cover the entiretorus and, therefore, will not necessarily attain the overall maximum on the torus. For example, small kinksseen in Fig. 3(b) represent the solution as it passes through a small resonance tongue at b ≈ 0.3, enters a largeresonance tongue at b ≈ 1 and exits the same resonance tongue as the solution becomes unstable at b ≈ 3. Atb ≈ 3.1 there is a torus bifurcation, as will be detailed in Section 4.

3.3 Maximum maps

A maximum map plots the maximum of attractors as a function of two parameters, where the maximum of eachsolution max(h(t)) is displayed according to a colour scheme. This provides a quick overview of some featuresof the dynamics. As mentioned in Section 1, maximum maps in the (b, τ)-plane were calculated in [24] for asingle fixed initial history. In Figs. 4(a) and (b), we instead show two maximum maps where, for each row offixed delay τ , the parameter b is scanned up and down (using previous solutions as initial histories in the samefashion as for Fig. 3), as is indicated by the arrows.

In both panels of Fig. 4 one can identify two regimes — one in the upper-left and one in the lower-right ofthe (b, τ)-plane. They are divided by a sharp interface that runs from the bottom-left corner to the upper-rightcorner. There are also elongated shapes, particularly in the upper-left corner of the plane. The sharp interfacesthat form these structures and the dividing curve represent where there are rapid transitions in max(h(t)) (forexample, those seen in Fig. 3). Sharp interfaces in max(h(t)) were also noted in maximum maps by the authorsof [24].

For sufficiently large values of b, the solutions represented in Fig. 3 are dominated by the seasonal forcingand have a period of T = 1. In Figs. 4 (a) and (b), these forcing-dominated solutions can be found in thelower-right half of the plane.

Comparing panels (a) and (b) of Fig. 4 one notices clear differences in the interface dividing the two mainregions, which are the result of bistabilities, or perhaps even multistabilities. The solution that the systemconverges to depends on the direction in which b is varied, that is, it depends on the initial history used. Thetwo maximum maps in Fig. 4 hence reflect the bistabilites seen in the one-parameter bifurcation diagrams ofFig. 3. For example, the upper and lower branches in Fig. 3(a) coincide with the maxima at τ = 1.2 in panels(a) and (b), respectively, of Fig. 4.


0 2 4 6 80

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b

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�

max(h(t))

Figure 4: Maximum maps displaying the maximum value of h(t) according to the colour scheme as b is increased(a) and decreased (b); here a = 1 and κ = 11.

4 The bifurcation set in the (b, τ)-plane

We now investigate the dynamics causing the sharp interfaces in max(h(t)) and the associated structures inthe maximum maps. Figure 5 shows the maximum maps from Fig. 4 in gray-scale together with bifurcationcurves found with DDE-BIFTOOL, namely: saddle-node bifurcations of periodic orbits (blue), period-doublingbifurcations (black) and torus bifurcations (red). As indicated by the arrows, b is increased in panel (a) anddecreased in panel (b).

The bifurcation curves in Fig. 5 divide the (b, τ)-plane into regions of qualitatively different solution types,which allows us to explain the features seen in the maximum maps. In both panels (a) and (b) closed curves ofsaddle-node bifurcations agree well with the elongated shapes; also compare with Fig. 4. Furthermore, closedcurves of period-doubling bifurcations are found within some of the closed curves of saddle-node bifurcations ofperiodic orbits.

It is in Fig. 5(a), where b is being increased, that the curves of saddle-node bifurcations of periodic orbitsagree to a larger extent with some of the sharp interfaces that form the elongated shapes. Except for smallvalues of τ , the curve T of torus bifurcations (in red) does not agree well with the sharp interfaces seen inpanel (a), suggesting that the solution undergoing the torus bifurcation is not the one being followed whileincreasing b.

On the other hand, in Fig. 5(b), where b is being decreased, the curve T agrees well with the sharp interfacein max(h(t)) that divides the parameter plane. Regarding this large sharp interface, we know that the smallermaxima seen for larger b values (i.e. to the right of the large sharp interface) represent the solutions dominatedby the seasonal forcing. This implies that, as b decreases, these solutions undergo a torus bifurcation at thecurve T and become unstable. This is the reason why the curve T agrees well with the sharp interface in thecase of decreasing b seen in Fig. 5(b). There are, however, some ranges of τ where the sharp transitions do notoccur exactly at the curve T; the reason for this is discussed in Section 4.3.

In both panels (a) and (b) there remain some sharp interfaces that do not coincide with any bifurcationcurve. This is because these sharp transitions in max(h(t)) are due to bifurcations that cannot be readilycontinued numerically; this is discussed in Section 5.

The elongated shapes bounded by curves of saddle-node bifurcations of periodic orbits are in fact resonancetongues. Numerical simulation confirms that they contain stable frequency locked solutions, meaning that allsolutions inside each resonance tongue have the same fixed frequency ratio. The resonance tongues shownhere are a selection of those present in the system: there are actually infinitely many resonance tongues. Theresonance tongues are rooted on the line of zero forcing (where b = 0) and/or on the curve of torus bifurcationsat points of p : q resonance. They become very thin in the parameter plane for larger q. General theory [39]


0 2 4 6 80

0.5

1

1.5

2

τ

-

(a) SN

PD

T

4:3HHY1:1

1:2 3:7PPi 3:8P

Pi

1:3

1:4

1:1 1:5

1:6

1:7 1:7

0 2 4 6 80

0.5

1

1.5

2

0

0.5

1

1.5

2

τ

b

max

(h(t))

�

(b) SN

PD

T

4:3HHY1:1

1:2 3:7PPi 3:8P

Pi

1:3

1:4

1:1 1:5

1:6

1:7 1:7

Figure 5: Maximum maps of Fig. 4 overlaid with curves of saddle-node bifurcations of periodic orbits (SN),period-doubling (PD) and torus bifurcations (T), which are drawn in blue, black and red, respectively. Severalfrequency ratios of resonance tongues are indicated; here a = 1 and κ = 11.

tells us that along the torus bifurcation curve, the rotation number of the emerging invariant tori is changingcontinuously with the parameters (b, τ). If the rotation number is a rational number, the bifurcating solution islocked on the torus and a resonance tongue will branch off at this point. So for every rational rotation number


p/q, the resonance tongue will contain a family of p : q resonant periodic orbits that are locked to the forcing.Such periodic orbits form p :q torus knots as they wind around the torus.

The zero-forcing line (b = 0) is a straight curve of torus bifurcations for τ > π/(2κ), where delay-inducedoscillations exist: once the seasonal forcing is switched on (i.e. b > 0), a second frequency is introduced into thedynamics and an invariant torus is formed. Because T = 4τ as mentioned in Section 1, we see in the bifurcationset that p : q resonance tongues are rooted along the zero-forcing line at τ = q/4p. Examples of these, a 4 : 3,3 :7 and a 3 :8 resonance tongue, are included in Fig. 5 branching off at τ = 3/16, 7/12 and 8/12, respectively,from the zero-forcing line. General theory also tells us that for an irrational value of τ along the zero-forcingline, or for an irrational rotation number along the curve of torus bifurcations, the location of the solution willbe the starting point of a smooth curve of unlocked quasi-periodic solutions that exist on a torus [39].

An example of another source of resonance is shown in Fig. 5: the smaller resonance tongue that branches offat the point (b, τ) = (0, 1.25) has the same shape as the resonance tongue that branches off at (b, τ) = (0, 0.25)with each tongue containing solutions of period T = 1. This is due to the repeating nature of periodic solutionsof DDEs. The idea is detailed in [63], where the basic concept is that, given a periodic solution of period Tto a certain DDE with delay time τ , another solution with an identical time series will exist for delay timeτ ′ = τ + T . Because the solution is periodic, when the feedback term of the DDE calls on h(t − τ − T ) itis receiving exactly the same input as for h(t − τ). Particularly at larger τ values this will contribute to anincreasing number of resonance tongues.

Note that, besides periodic (locked) and quasi-periodic (unlocked) behaviour, there may be small domainsin the parameter space where chaos exists. However, chaotic behaviour does not seem to be a significant featurein the model for the parameter range investigated here.

4.1 Transition through resonance tongues

Figures 6 (a1) and (a2) are reproduced from [64] and display the local maxima and minima in h(t) of stablesolutions found by numerical integration of Eq. (1) for b = 2 and corresponding values of τ with a fixed initialcondition h ≡ 1. Panel (a1) shows alternating regions of small finite numbers of local maxima and minima withregions of very large (possibly representing infinite) numbers of local maxima and minima. The differences inthe numbers of local maxima and minima indicate different solution types: a periodic solution will have a finitenumber of local maxima and minima, while an aperiodic solution could have an infinite number of local maximaand minima over time.

Panel (a2) is an enlargement of (a1) showing only local maxima for a smaller range of τ ∈ [0.50, 0.59] valuesand b = 2. Here, most of the range of τ corresponds to solutions with a very large (possibly infinite) number oflocal maxima with windows of small finite numbers of local maxima. The authors of [64] suggested that chaosis present here between windows of periodic solutions.

The different numbers of local maxima and minima seen in panels (a1)–(a2) can be understood from thebifurcation analysis. Panel (b1) is a transposed version of Fig. 5 (b). The green line at b = 2 indicatesthe position of the parameter section represented in Fig. 6(a1). Along the green line the solution types in thebifurcation analysis coincides very well with the local minima and maxima shown above in panel (a1). Beginningwith small values of τ in panel (b1), the solution is dominated by the seasonal forcing until a torus bifurcationat τ ≈ 0.51 (red curve). For the same values of τ in panel (a1), there is just one set of minima and maxima,reflecting the case of seasonal forcing domination. After the torus bifurcation at curve T, a stable invarianttorus is born. We now see an infinite (or very large) number of local minima and maxima in panel (a1), sincethese solutions are quasi-periodic or of a very high period. As τ increases in panel (b1), the solutions alternatebetween being locked (periodic) and unlocked (quasi-periodic) on the torus, depending on whether the givenτ value lies within a resonance tongue or not. For values of τ for which the solution is within a resonancetongue, there is a small finite number of local minima and maxima. Due to the nature of the seasonal forcing,there is one local maximum every year; for example, the solutions with a period of three years (in the 1 : 3resonance tongue) have three local maxima. Inside some resonance tongues are period-doubling bifurcations(see Fig. 6(b1)), which is why the local minima and maxima sometimes split into two, for example, for valuesclose to τ = 1 in panel (a1).

Fig. 6(b2) is an enlargement of panel (b1), where some example resonance tongues are shown, which are


0 0.5 1 1.5 20

2

4

6

8

b

τ

(b1)

(a1)

T

0.5 0.52 0.54 0.56 0.581.8

1.9

2

2.1

2.2

b

τ

(b2)

(a2)

4:93:7 2:5

3:8

CCCW

T

Figure 6: Panel (a1) displays the local maxima and minima of simulated solutions for b = 2 and τ ∈ [0, 2] andpanel (a2) shows the same maxima for τ ∈ [0.5, 0.59]; these two figures are reproduced from Ref. [64]. Panel (b1)is a transposed version of Fig. 5(b) with a green line indicating b = 2. Panel (b2) shows the maximum mapwith bifurcation curves over the same τ -range as (a2) with some resonances indicated. Here a = 1 and κ = 11.

bounded by the blue curves of saddle-node bifurcations of periodic orbits. Again, the green line indicates theposition of the parameter section shown in panel (a2). By comparison with panel (b2), we see that panel (a2)shows in finer detail the formation of the stable invariant torus from the torus bifurcation at τ ≈ 0.51, afterwhich both locked and unlocked solutions exist. Therefore, as seen in the context of the bifurcation investigationcarried out above, the behaviour observed throughout the τ -range considered in panel (a2) is not chaos butquasi-periodic behaviour. At some τ values there are windows of smaller finite numbers of local maxima. Theserepresent solutions that lie within thin resonance tongues, some of which can be seen in panel (b2), includinga 4:9, 3 :7, 2 :5 and 3:8 resonance tongue. The agreement between Fig. 6(a2) and (b2) is very good, but theremay be small discrepancies that arise because the set of local maxima from [64] were calculated by numericalintegration from the same fixed initial condition for each value of τ , whereas the set of solutions represented inpanel (b2) were calculated by scanning the parameter plane.

4.2 Bistability within resonance tongues

We observe that some resonance tongues in the maximum maps appear to be striped with alternating horizontallines; an example is the 2 : 5 resonance tongue in Fig. 6(b2). A clear example is also the 1 : 2 resonance tonguein Fig. 7, which is an enlarged version of part of Fig. 5(b) with a different colour scheme; note that increasingb produces a qualitatively similar map. The resonance tongue in Fig. 7 is bounded by curves of saddle-nodebifurcations of periodic orbits; it is rooted on the zero-forcing line at one end and on the (red) curve of torusbifurcations at the other end. Inside the tongue there are stripes representing solutions of both larger andsmaller maxima.

Normally, within a resonance tongue there is one stable and one unstable solution that approach each other


0.5 1 1.5 20.48

0.49

0.5

0.51

0.52

0.53

0.54

0.3

0.4

0.5

0.6

0.7

τ

b

max(h(t))

TSN

Figure 7: Maximum map for decreasing b showing the 1 : 2 resonance tongue bounded by the blue curves ofsaddle-node bifurcations (SN) of periodic orbits. Also shown is the red curve of torus bifurcations (T); herea = 1 and κ = 11.

as parameters are varied, then coincide and disappear at the boundary of the resonance tongue in a saddle-nodebifurcation of periodic orbits. However, Fig. 7 suggests that there are two sets of stable periodic solutions withinthe tongue. More specifically, for a given τ , as b is increased or decreased, the solution reaches one of the twosolutions depending on the initial condition when the tongue is entered, leading to the visible horizontal stripesas a result of sweeping in b. Note that producing this figure by varying τ for fixed b would result in verticalstripes.

To explain this phenomenon, Figs. 8(a)–(b) show two stable (blue) solutions and two unstable (red) solutions,respectively, for the same parameter set (b, τ) = (1, 0.5). These solutions are shown as a projection onto the(h(t), h(t− τ), h(t− 12τ))-space in panel (c). Note that viewing this projection from different angles (not shown)reveals that trajectories do not intersect. Panel (d) shows a one-parameter bifurcation diagram of these solutionswhen they are continued in τ for b = 1. The gray line at τ = 0.5 intersects the solutions shown in panels (a)–(c).Small blue-filled circles represent saddle-node bifurcations of periodic orbits at the boundary of the resonancetongue.

Comparing the two solutions in each panel (a)–(b), one can see that the symmetry

h2(t) = −h1(t+1

2) (5)

gives two distinct solutions that are symmetric counterparts of each other. In general, this symmetry is aninherent property of Eq. (1), resulting from both the periodic nature of the forcing term and the fact that thedelay term is an odd function. The symmetry 5 does not depend on the parameter values; however, for p : qlocked solutions with odd p and q integers, h2 ≡ h1. In this case, there is only one distinct solution with thesymmetry h1(t) = −h1(t+ 12 ). This explains why only some of the resonance tongues (i.e. those with even p orq) appear striped in the maximum maps.

The symmetry (5) appears in the phase space projection in panel (c) as a rotational invariance of 180 degrees.The two symmetric counterpart solutions can also be seen in the bifurcation diagram in panel (d). Continuingthe solutions shown in panels (a)–(b) of Fig. 8 across the resonance tongue for varying τ reveals that either sideof the tongue is bound by, not just one, but two symmetric saddle-node of periodic orbits bifurcations. This wasnot visible in Fig. 5 because both sets of saddle-node bifurcation curves that bound either side of the resonancetongue lie on top of each other as they relate to symmetrically related periodic solutions.


−0.5

0

0.5

h(t)

(a)

0 1 2 3 4

−0.5

0

0.5

h(t)

t

(b)

−0.5

0

0.5

−0.5

0

0.5

−0.5

0

0.5

h(t−τ)

h(t− 12τ)

h(t)

(c)

0.49 0.5 0.51

0.2

0.4

0.6

0.8

max(h(t))

τ

(d)

Figure 8: Two stable (blue) and two unstable (red) periodic orbits within the 1:2 resonance tongue at (b, τ) =(1, 0.5) are shown in panels (a) and (b), respectively, as time series and in panel (c) as a projection onto the(h(t), h(t − τ), h(t − 12τ))-space. Panel (d) is the one-parameter bifurcation diagram in τ for b = 1, where theblue and red curves correspond to stable and unstable solutions, respectively. Saddle-node bifurcations of theperiodic orbits are indicated by blue-filled circles; intersection points with the gray line at τ = 0.5 yield thesolutions observed in panels (a)–(c). Here a = 1 and κ = 11.

4.3 Criticality of torus bifurcation

For some parameter values in Fig. 5(b) there are discrepancies between the curve of torus bifurcations and thesharp interface seen in the maximum map. This can be seen more clearly in Fig. 9, which is an enlargementof part of Fig. 5. The curves of saddle-node bifurcations of periodic orbits are not shown here; nonetheless,the resonance tongues are easy to recognise. Figure 9(a) shows the maximum map for increasing b, where thecurve T does not seem to affect the solutions being followed. Instead, there are other sharp interfaces that willbe discussed in Section 5. Figure 9(b) shows a region where the curve T agrees only partially with the sharpinterface. For τ . 1.5 or τ & 1.6 the curves agree, where the maximum of the solutions change rapidly at curveT (from dark blue to red) as b is decreased. However, for 1.5 . τ . 1.6, there is a gradual change (to lightblue) after the torus bifurcation curve as b is decreased, before a rapid change in maximum at b values beyondthe torus bifurcation curve.

The reason for the discrepancies between the curve of torus bifurcations and the sharp interface in panel (b)is that the torus bifurcation changes criticality along the curve. For τ . 1.5 or τ & 1.6 in panel (b), the (darkblue) solution, which is dominated by the seasonal forcing, simply becomes unstable at the torus bifurcationcurve as b is decreased and the next solution jumps to a larger (red) maximum. The solution with a larger (red)maximum is one that lies on a different, larger torus that co-exists for these parameters, that is all parametersfor which the maximum appears red in panel (a). This implies that the torus bifurcation at these values of


3 4 5 61.45

1.5

1.55

1.6

1.65

-

τ

b

(a)

T

3 4 5 6

0.6 0.8 1 1.2 1.4 1.6 1.8

�

b

max(h(t))

(b)

T

Figure 9: Maximum maps for increasing b (a) and decreasing b (b) with a (red) curve of torus bifurcations (T);here a = 1 and κ = 11.

τ is subcritical (resulting in an unstable invariant torus of saddle-type to the right of the torus bifurcationcurve). As b is decreased for 1.5 . τ . 1.6, the (dark blue) solution also becomes unstable at the curve oftorus bifurcations. However, for these τ values, a stable torus emerges with a stable periodic or quasi-periodicsolution. It grows in maximum (light blue) and, at some value of b after the torus bifurcation curve, this torusloses stability, at which point the maximum jumps to another solution with a larger (red) maximum. Thisimplies that the torus bifurcation for those values of τ is supercritical (resulting in a stable invariant torus).

With the knowledge gained from the bifurcation analysis in Section 4, the examples shown in Fig. 2 canbe understood in terms of their position on the (b,τ)-plane relative to the bifurcation curves. The solutions inFigs. 2(c) and (e) belong to the resonance tongues of T = 5 and T = 3, respectively; see Fig. 5(a). The solutionin Fig. 2(b) is dominated by the seasonal forcing and has a b value larger than the torus bifurcation curve; seeFig. 5(b). The solution in Fig. 2(d) is a high-period or quasi-periodic solution for a value of b slightly belowthat of a supercritical torus bifurcation; see Fig. 5(b).

The bifurcation curves shown in Fig. 5 explain most, but not all, of the results obtained by numericalintegration in Sections 3–4. For example, one might ask the question: why does the stable invariant torus seento emerge from the curve of torus bifurcations in Fig. 9(b) disappear at certain combinations of parameters?What causes the sharp interfaces seen in Fig. 9(a)? These questions are discussed in the next section.

5 Bifurcations of tori

We now consider the sharp interfaces seen in the maximum maps of Figs. 5 and 9 that still remain unexplained.For example, as b is increased in Fig. 9(a) for each value of τ , the tori being followed have a relatively largemaximum values (appearing red on the maximum map). However, these tori seem to suddenly lose their stabilityand disappear, after which the next stable solution has a considerably smaller maximum (appearing blue on themaximum map). As b is being decreased in Fig. 9(b) for τ values where the torus bifurcation is supercritical,small (light blue on the maximum map) stable tori emerge from the torus bifurcation curve. They then soonlose their stability and disappear, after which the next stable solution has a larger (red) maximum value.

Figure 10(a) is a one-parameter bifurcation diagram for the parameter range b ∈ [2.9, 3.2] and τ = 0.94— which crosses a region in the (b, τ)-plane where such unexplained sharp interfaces can be seen in Fig. 5.For larger values of b, the solutions in Fig. 10(a) are periodic and dominated by the seasonal forcing and areannotated 1 :1. These periodic solutions can be continued with DDE-BIFTOOL through the torus bifurcation(T). The stable torus can be followed by numerical integration while decreasing b in small steps, until there is arapid transition in max((h(t)) at b ≈ 2.95. Notice the kink at b ≈ 2.97, where the torus passes through the 3:10


2.9 2.95 3 3.05 3.1 3.15 3.20.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

2.9695 2.97 2.97050.815

0.82

0.825

3.00753 3.00754 3.00755

0.95

0.955

0.96

(a)(b)

(c)

5 :17

7:24

6

?

��

�

-

max(h(t))

b

1:1

torus

3:10

torus2:7

T

SNT

SNTSN

8:2713:44

5:17

12:417:24

Figure 10: Panel (a) is a one-parameter bifurcation diagram in b for τ = 0.94, where blue and red lines indicatestable and unstable solutions, respectively. The branches of stable tori were found by parameter sweepingwith numerical simulation, and they end at the points denoted SNT. The black arrows indicate the directionof change of b and a hysterisis loop. The red circles represent unstable locked periodic solutions. Panels (b)and (c) are one-parameter bifurcation diagrams in b for τ = 0.94 of the periodic orbits in the 5 : 17 and 7 : 24resonance tongues, respectively. Here a = 1 and κ = 11.

resonance tongue and its rotation number is 3/10. Enlarging the blue curve would reveal further smaller kinksrepresenting thinner resonance tongues. For smaller b and larger max((h(t)) values, there is a 2 : 7 resonancetongue whose periodic orbits can be continued with DDE-BIFTOOL; it terminates in a saddle-node bifurcationof periodic orbits (SN). While increasing b, the solutions after the 2 : 7 resonance tongue can be followed bynumerical integration and we find an upper stable torus until b ≈ 3.01.

Notice how the upper and lower blue curves representing solutions on tori bend downwards and upwards,respectively, and become vertical before their rapid transitions. This is very reminiscent of a saddle-nodebifurcation of periodic orbits. This comparison suggests that there is a fold or saddle-node bifurcation of tori,denoted SNT in Fig. 10(a). We discuss the intricacies of this phenomenon in more detail below. On the levelof Fig. 10(a), the suggestion is that there is a branch of unstable tori between the two points labelled SNT. Itis not possible with existing techniques to readily find and follow unstable tori in a DDE by continuation orsimulation. We can, however, use DDE-BIFTOOL to locate locked periodic orbits along the unstable branch.

The small red circles in Fig. 10(a) represent periodic solutions in narrow resonance tongues. To find theseunstable locked solutions, we make use of the fact that resonance tongues are ordered in the Farey sequence:the largest resonance tongue that exists between a p :q and a r :s resonance tongue is a p+r :q+s resonance (forexample, see [28]). Therefore, we know that between the 2 : 7 and 3 : 10 resonance tongues seen in Fig. 10(a)the torus must pass through a 5 : 17 resonance tongue. To find a periodic solution in this resonance tongue,we construct an initial guess and then use DDE-BIFTOOL to correct it. To achieve an approximation of a5 : 17 solution, we take a 17 year section from the time series of a nearby periodic solution, in this case the3 : 10 periodic orbit. Based on an estimate of where the 5 : 17 solution would lie on the plot in Fig. 10(a), wescale the time series such that max(h(t))≈ 0.85 and let b ≈ 2.98. DDE-BIFTOOL is then able to correct thisconstructed initial guess to the true 5 : 17 periodic solution. Other unstable locked periodic orbits were foundsimilarly. Shown as red circles in Fig. 10(a) are the associated very narrow resonance locations of the frequency


ratios indicated, which appear to lie on a curve between the two points labelled SNT.Figures 10(b) and (c) show the unstable 5 : 17 and 7 : 24 periodic solutions, respectively, continued for

changing b. In each case, this gives us a slice of the resonance tongue to which the periodic solution belongs.Note that the respective b-ranges are very small. Also notice the double set of saddle-node bifurcations ofperiodic orbits in panel (c), as is expected for an even period (see Section 4.2). As indicated by their red colour,all points in these slices of resonance tongues are unstable solutions. We find that these solutions always haveat least one unstable Floquet multiplier, which implies that the torus along this part of the branch is indeed ofsaddle-type.

5.1 Resonance tongues and Chenciner bubbles

Figure 10(a) presents a convincing bifurcations diagram where branches of stable and saddle tori exist and comevery close to each other near the points labelled SNT. Nevertheless, it is important to realise that the tori losesmoothness and cease to exist once they become sufficiently close to each other near SNT [6, 12]. In otherwords, the precise bifurcation diagram is not so simple and involves the break-up of tori. A good approach forinvestigating the sharp interface boundary formed by these bifurcations of tori is to consider resonance tonguesof locked tori in the nearby (b, τ)-plane.

We begin by continuing all the resonances identified in Fig. 10 in the (b, τ)-plane. Figure 11 shows maximummaps with b increasing in panel (a) and decreasing in panel (b), upon which we overlay curves of saddle-nodebifurcations of periodic orbits (SN) that form the boundaries of these resonance tongues; they are labelledp : q at the points where they bifurcate from the curve T. Near the curve T, where the resonance tonguesbecome extremely narrow, the continuation of the saddle-node bifurcations of periodic orbits eventually becomesimpractical. To represent the extremely narrow segments of the tongues rooted on the curve T, we thereforecompute and plot a curve of a single periodic orbit in each p : q resonance tongue. Most resonance tonguesappear as single curves because they are very thin. Although this is not visible, except for the 5 : 17 tongue,one boundary is drawn in a lighter blue. The points where the resonance tongues intersect the line shown atτ = 0.94 coincide with the red circles and the 3:10 kink seen in Fig. 10(a).

By calculating the Floquet multipliers of the p :q periodic solutions with DDE-BIFTOOL, we establish that,as they bifurcate from curve T in Fig. 11, the invariant tori are stable, meaning that all of the resonance tonguescontain a set of stable and unstable solutions. Yet, it was shown in Fig. 10 that these resonance tongues, exceptthe 3 : 10 tongue, contain only unstable periodic solutions. To explain how this happens, let us consider, forexample, the 7 : 24 resonance tongue. Following the 7 : 24 resonance tongue from the curve T, it contains a setof stable and unstable solutions. At b ≈ 2.92 the boundary curves of this resonance tongue have local minimain b. Since these curves are so close together, this can be interpreted as a fold of the resonance tongue. Thereis another fold of the 7 : 24 resonance tongue at b ≈ 3.01, where it has a local maximum in b. Calculationsreveal that in-between the two folds with respect to b all solutions lie on a torus of saddle-type and have atleast one unstable Floquet multiplier. This is why the 7 :24 resonance tongue contains only unstable solutionsas it passes τ = 0.94 (cf. Fig. 10(c)). On the other hand, past the local maximum there is again a set of stableand unstable periodic solutions in the resonance tongue. The other resonance tongues in Fig. 11 have the samefolding and stability properties. At τ = 0.94 only the 3 : 10 resonance tongue has not undergone any fold and,hence, is seen to be on the stable branch in Fig. 10(a). Overall, the folding of resonance tongues explains whycertain locked solutions seen in Fig. 10(a) lie on either a stable or saddle torus at τ = 0.94.

As can be seen in Fig. 11, the folds of resonance tongues coincide with the saddle-node bifurcations of tori,represented by the sharp interface in max(h(t)). However, there is actually no smooth curve of saddle-nodebifurcations of tori. Near their folds the resonance tongues form so-called Chenciner bubbles: the invarianttorus loses normal hyperbolicity and breaks up as it enters the region of Chenciner bubbles in the transitionfrom a stable torus to a torus of saddle-type. In Chenciner bubbles the dynamics are generally very complicated[8, 61]. In Fig. 11(b) the sharp interface in maximum values might appear as a smooth curve; however, lookingcloser would reveal further smaller Chenciner bubbles. In this case the resonance tongues are simply thinner,so the Chenciner bubbles are smaller and not visible on the scale of Fig. 11.

Since we found that the resonance tongues bifurcate from the torus bifurcation curve T, we can identifymany more in an extended region of the (b, τ)-plane. Figure 12 shows (dark and light blue) curves of saddle-


0.93

0.94

0.95

0.96

-

τ

(a)

T

SN 3:10

8:2713:445:1712:417:24

2.9 2.95 3 3.05 3.1

0.93

0.94

0.95

0.96

0.4

0.6

0.8

1

max(h(t))

�

τ

b

(b)

T

SN 3:10

8:2713:445:1712:417:24

Figure 11: Maximum maps with increasing b (a) and decreasing b (b). The red and blue curves are torusbifurcations (T) and saddle-node bifurcations of p : q periodic orbits (SN), respectively. One boundary of eachresonance tongue is drawn in a lighter blue. The green line at τ = 0.94 intersects the solutions seen in Fig. 10.Here a = 1 and κ = 11.

node bifurcations of the p : q periodic orbits for p < q ≤ 30 that bifurcate from a segment of the curve T (for0.92 . τ . 1). Again, the resonance tongues fold near the two sharp transitions of the maximum maps forincreasing and decreasing b, respectively. More specifically, the envelopes of these folds form the two boundaries.Notice also that in Fig. 12 the two (dark and light blue) boundary curves of the shown resonance tongues start toseparate considerably near the second fold from T (their local maxima in b). Indeed, the two large indentationsin the boundary of the maximum map for increasing b are formed by light blue boundary curves of the lowresonances 2 : 7 and 3 : 11; compare with Fig. 11(a). Not only do the light and dark blue boundary curves ofeach tongue separate but these two sets of boundary curves converge to different limits in Fig. 12. This leadsto parameter regions where many resonance tongues overlap. We find that, once this overlapping occurs, the


0.88

0.92

0.96

1

-

τ

(a)

T

6:19�

3:10�7:24�

7:27�

2.6 2.7 2.8 2.9 3 3.1 3.2

0.88

0.92

0.96

1

0.4

0.6

0.8

1

1.2

max(h(t))

�

τ

b

(b)

T

6:19�

3:10�7:24�

7:27�

Figure 12: Maximum maps with increasing b (a) and decreasing b (b). The red and blue curves are torusbifurcations (T) and saddle-node bifurcations of p : q periodic orbits, respectively; shown are all p : q resonancetongues with p < q ≤ 30 bifurcating from a segment of the torus bifurcation curve. The upper/lower boundaryof each resonance tongue is drawn in dark/light blue. Here a = 1 and κ = 11.

resonance tongues contain cascades of period-doubling bifurcations and chaotic behaviour may occur.Figure 13 shows two simultaneously stable solutions for τ = 0.91 and b = 2.6, obtained by numerical

integration of Eq. (1) with the Euler method. As seen in Fig. 12, this is a point in the (b, τ)-plane where manyresonance tongues are overlapping.

Row (a) of Fig. 13 displays a periodic solution that belongs to the 1 : 3 resonance tongue. It has a largemaximum every three years (see panel (a1)) and corresponds to a closed loop in projection onto the (h(t −τ), h(t))-plane in panel (a2). This periodic solution is similar to the one shown in Fig. 2(e1)–(e2), which alsobelongs to the 1 :3 resonance tongue. An interpretation of this solution is that a strong El Niño event appearsevery three years.


0 5 10

−1

0

1

−1 0 1

(a1) (a2)

h(t)

0 10 20 30 40 50

−1

0

1

−1 0 1

(b1) (b2)

h(t)

t h(t− τ)

Figure 13: Stable solutions of Eq. (1), shown as time series in panels (a1) and (b1) and as projections onto the(h(t− τ), h(t))-plane in panels (a2) and (b2); here τ = 0.91, b = 2.6, a = 1 and κ = 11.

Row (b) of Fig. 13 shows a different solution of Eq. (1) for the same parameter values. The time seriesin panel (b1) seems irregular, with the largest maxima occurring every 3-7 years. In projection onto the(h(t − τ), h(t))-plane, 200 years of trajectory traces an attracting object. The solution looks as if it might beperiodic with period T = 37 in panel (b1). Although it is not shown in the previous figures, there exists an11:37 resonance tongue between the 3:10 and 8:27 tongues. However, this resonance tongue, like those nearby,has already undergone a cascade of period-doubling bifurcations at τ = 0.91 and b = 2.6. Upon close inspectionof panel (b1), one can see that the two local maxima at t ≈ 12 differ very slightly from the two local maximaat t ≈ 49. In fact, this trajectory appears to be chaotic: it is actually very sensitive to perturbations, as hasbeen checked with numerical simulations. A chaotic solution, as in Fig. 13(b), reflects the observed irregularityof the time intervals between successive large El Niño events, which occurs every 3-7 years.

6 Discussion

We investigated the interaction of the negative time-delayed feedback mechanism and seasonal forcing in asimplified ENSO model. The bifurcation analysis of the governing DDE with the continuation software DDE-BIFTOOL allowed us to explain certain features seen in numerical simulations in previous works [24, 64].More specifically, we presented maximum maps, calculated by scanning the (b, τ)-plane up and down in theparameter b, on which we overlaid the relevant bifurcation curves. Our bifurcation analysis revealed that whatwas previously thought in [64] to be chaotic dynamics is actually quasi-periodic or high-period locked behaviourresulting from a torus bifurcation. We discussed resonance tongues and their role in multistability, includingbistabilities within p : q resonance tongues with even p or q. Our analysis found that the relevant parameterplane is organised by an infinite number of resonance tongues rooted on curves of torus bifurcations. We alsofocussed on sharp interfaces in the maximum maps that could not be explained by continuing bifurcations ofperiodic orbits. We presented evidence that they are due to the phenomenon of Chenciner bubbles associatedwith the folding of resonance tongues. Following the boundary curves of these resonance tongues also revealedparameter regions where they overlap and more complicated behaviour ensues.

The study we presented may be of interest more generally, because it showcases how state-of-the-art contin-uation methods (in particular, those for periodic solutions) can be utilised for the bifurcation analysis of a DDE.Compared to an investigation solely reliant on numerical simulations, the approach of continuing bifurcationcurves of various types offers a more complete picture of the dynamics in dependence of the model parameters.Given that the scalar DDE (1) contains just two terms, the model shows surprisingly rich behaviour. Thisobservation may be of interest for other application areas where one finds competition between time-delayed


feedback and periodic forcing.The wealth of bifurcations, even within small parameter ranges, highlights the relevance of parameter sen-

sitivity in climate modelling. In particular, it is interesting in the context of climate tipping. Some climatetipping events correspond to certain bifurcations, where the response of a climate system to a slight variationin parameter is a qualitative or drastic change of observed behaviour [7]. Saddle-node bifurcations have beenidentified as potential mechanisms for particular climate tipping events (for example, see [1]). As far as we areaware, the saddle-node bifurcation of tori (characterised by complicated dynamics in the associated Chencinerbubbles) discussed in Section 5 has not yet been considered in the context of tipping. Following the categori-sation of [57], it constitutes a further “dangerous” bifurcation, where there is a sudden disappearance of anattractor and the system jumps to some other “unknown” attracting state. The irreversibility of this form oftipping event is illustrated by the hysteresis loop in Fig. 10(a).

In future work it will be interesting to perform bifurcation studies of models that incorporate additionaleffects; for example, the positive feedback mechanism [60] mentioned in Section 2 or seasonally dependent ocean-atmosphere coupling [59]. The work presented here may serve as a foundation for understanding the dynamics ofsuch ENSO models. Preliminary results (not presented here) reveal that, once the positive feedback mechanismis included, chaos is considerably more prominent over a large range of parameters; this appears to agree betterwith the irregularity seen in real-world El Niño observations.

6.1 Acknowledgements

We thank Jan Sieber for his help with DDE-BIFTOOL, especially regarding the continuation of resonancetongues from their root points. The research of A.K. has been funded by a University of Auckland DoctoralScholarship.

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