Mixed-Mode Oscillations of El Niño–Southern Oscillation

Andrew Roberts Cornell University, Ithaca, New York

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John Guckenheimer Cornell University, Ithaca, New York

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Esther Widiasih University of Hawai‘i at West O‘ahu, Kapolei, Hawaii

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Axel Timmermann University of Hawai‘i at Mānoa, Honolulu, Hawaii

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Christopher K. R. T. Jones University of North Carolina at Chapel Hill, Chapel Hill, North Carolina

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Abstract

Very strong El Niño events occur sporadically every 10–20 yr. The origin of this bursting behavior still remains elusive. Using a simplified three-dimensional dynamical model of the tropical Pacific climate system, which captures El Niño–Southern Oscillation (ENSO) combined with recently developed mathematical tools for fast–slow systems, the authors show that decadal ENSO bursting behavior can be explained as a mixed-mode oscillation (MMO), which also predicts a critical threshold for rapid amplitude growth. It is hypothesized that the MMO dynamics of the low-dimensional climate model can be linked to a saddle-focus equilibrium point, which mimics a tropical Pacific Ocean state without ocean circulation.

Current affiliation: Department of Mathematics, Cornell University, Ithaca, New York.

Corresponding author address: Andrew Roberts, Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853. E-mail: andrew.roberts@cornell.edu

Abstract

Very strong El Niño events occur sporadically every 10–20 yr. The origin of this bursting behavior still remains elusive. Using a simplified three-dimensional dynamical model of the tropical Pacific climate system, which captures El Niño–Southern Oscillation (ENSO) combined with recently developed mathematical tools for fast–slow systems, the authors show that decadal ENSO bursting behavior can be explained as a mixed-mode oscillation (MMO), which also predicts a critical threshold for rapid amplitude growth. It is hypothesized that the MMO dynamics of the low-dimensional climate model can be linked to a saddle-focus equilibrium point, which mimics a tropical Pacific Ocean state without ocean circulation.

Current affiliation: Department of Mathematics, Cornell University, Ithaca, New York.

Corresponding author address: Andrew Roberts, Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY 14853. E-mail: andrew.roberts@cornell.edu

1. Introduction

The ENSO phenomenon is the dominant source of interannual climate variability. El Niño events are characterized by positive sea surface temperature anomalies (SSTA) in the eastern equatorial Pacific that cause anomalous diabatic heating of the atmosphere. This in turn drives global atmospheric planetary wave adjustments impacting weather in regions far away from the tropical Pacific (Bjerknes 1969; Deser and Wallace 1990; Karoly 1989).

ENSO can be regarded as a coupled atmosphere–ocean instability, subject to atmospheric noise, external periodic forcing through the annual cycle, and atmosphere and ocean nonlinearities. From a dynamical systems perspective the oscillatory ENSO mode emerges from a Hopf bifurcation. To capture the irregular nature of ENSO, its occasional amplitude bursting (Timmermann et al. 2003) (Fig. 1), and the presence of extreme El Niño events, nonlinear extensions to the standard ENSO recharge oscillator theory (Jin 1997, 1998) have been suggested, including noise-induced instabilities (Levine and Jin 2010), frequency entrainment, chaotic dynamics originating from annual cycle–ENSO interactions (Chang et al. 1995, 1994; Jin et al. 1994; Liu 2002; Stein et al. 2014; Timmermann and Jin 2002; Timmermann 2003; Tziperman et al. 1994), and homoclinic–heteroclinic dynamics (Timmermann et al. 2003).

Fig. 1.
Fig. 1.

(top) Observed sea surface temperature anomalies averaged over Niño-3 region. High temperatures correspond to strong El Niño events; red lines indicate a repeated El Niño bursting pattern. (bottom) Simulated eastern tropical Pacific SSTA in build-up phase and termination phase of strong El Niño events using (1) and the parameters in Table 1.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0191.1

In view of the socioeconomic impacts of very extreme El Niño events, it is paramount to further refine existing theories for the decadal-scale emergence of these large events. Here we hypothesize that ENSO’s bursting behavior can be interpreted in terms of mixed-mode oscillations. To demonstrate the applicability of this powerful mathematical framework for ENSO, we adopt the same three-dimensional nonlinear ENSO recharge oscillator equations used by Timmermann et al. (2003).

Mixed-mode oscillations (MMOs) comprise large-amplitude oscillations (LAOs) followed by small-amplitude oscillations (SAOs) , then large oscillations , small oscillations , and so on. The sequence is called the MMO signature (Desroches et al. 2012). There are a number of geometric explanations for MMOs, and the Shilnikov mechanism mentioned by Timmermann et al. (2003) is the poster child for MMOs in a system with a single time scale. In systems with multiple time scales, MMO mechanisms can be more robust, and these mechanisms are surveyed by Desroches et al. (2012). The bursting behavior observed in Fig. 1 is often an indicator that the underlying model has a multiple time-scale structure.

Here we set out to analyze a dimensionless version of the model from Timmermann et al. (2003) to show that there is a parameter regime in which the model exhibits MMOs. We isolate two distinct types of MMO: one has an SAO that is monotonically increasing in amplitude while the other has SAOs that first decrease in amplitude and then increase again. They are each formed by a distinct mathematical mechanism that will be explained in the analysis below. We shall refer to them as monotone and nonmonotone SAOs, respectively. We also relate the observed MMOs to the Shilnikov mechanism discussed by Timmermann et al. (2003).

We stress that the model under consideration is idealized and omits many processes certainly relevant to the dynamics of ENSO. The goal of the paper is to show that the basic mechanisms captured by the recharge oscillator can account for the bursting seen in ENSO.

The outline of the paper is as follows: In section 2 we describe the physical model. Then we prepare the model for MMO analysis through a coordinate transformation and nondimensionalization. In particular, we identify the dimensionless model as a multiple time-scale problem and identify the time-scale parameters. In section 3 we analyze the model with a focus on demonstrating the existence of MMOs. This analysis includes an in-depth discussion of the mathematical theory behind both mechanisms leading to MMOs in the ENSO model. Finally, we conclude with a discussion of the physical implications in section 4.

2. A nonlinear model for ENSO

a. The physical model

The fundamental dynamics of ENSO can be described in terms of the recharge oscillator paradigm (Jin 1997). The key regions for ENSO physics are the western and eastern equatorial Pacific. The important dynamical variables are the temperatures in these regions , eastern tropical Pacific subsurface temperature , and its linkage to western tropical Pacific thermocline changes . Following the original model (Jin 1998; Timmermann et al. 2003), the underlying ordinary differential equations describing ENSO and its linkage to the mean steady state can be written as
e1
where , , and represent the equatorial temperature of the western Pacific, equatorial temperature of the eastern Pacific, and thermocline depth of the western Pacific, respectively. The first terms of the temperature equations represent a relaxation of the system back toward a climatological mean state , which represents the radiative–convective equilibrium state (the temperature that the tropical Pacific would attain in the absence of ocean dynamics). The second quadratic term in the equation captures the anomaly wind-driven zonal advection of temperature. The wind anomalies themselves are determined by the east–west temperature gradient , thus leading to a quadratic dependence on temperature.
The eastern tropical Pacific temperature tendency is determined by relaxation toward and the vertical temperature advection term, which depends on and the subsurface temperature . The prognostic equation for the western tropical Pacific thermocline variation captures a relaxation back to mean climatological conditions with a damping time scale r and the effect of wind stress curl changes on the Sverdrup transport. The latter describes the corresponding discharging (recharging) process for El Niño (La Niña) events, which is assumed to be proportional to the east–west temperature gradient. The system of ordinary differential equations is closed by a nonlinear parameterization of eastern tropical Pacific subsurface temperature variations and eastern thermocline depth :
eq1
The difference controls the nonlinear scaling between thermocline anomalies and subsurface temperature anomalies relative to a mean subsurface temperature of ; corresponds to a mean eastern equatorial temperature attained at a depth of about 75 m, H denotes an eastern thermocline reference depth, and indicates the sharpness of the thermocline.

The parameters used by Timmermann et al. (2003) are listed in Table 1. To illustrate the dynamical behavior of this simplified ENSO system, we conduct a numerical simulation of the system (1). The results are shown in the bottom panel of Fig. 1. The large simulated amplitude modulation of eastern tropical Pacific temperatures provides evidence for a separation of time scales. In the paper by Timmermann et al. (2003), the simulated El Niño bursting behavior is attributed to a Shilnikov saddle-focus mechanism. In this paper we seek a more robust mechanism for MMOs, and the analysis relies on a global time-scale separation. Such a global separation can only be seen in a dimensionless model, because nondimensionalizing the system gives insights into the relative sizes of the time-scale parameters that are relevant to the MMO analysis.

Table 1.

Parameters used by Timmermann et al. (2003).

Table 1.

b. The mathematical model

Before taking on the scaling, we introduce a change of variables as follows:
eq2
With this change of variables, system (1) takes the form
e2

In system (2), S is the temperature difference between the eastern and western Pacific surface water, T is the departure of the western Pacific surface ocean temperature from some reference temperature, and h represents the western Pacific thermocline depth anomaly.

Remark 1

Despite the change of coordinates, the new system (2) captures the same dynamics as the original system (1) because the two systems are equivalent. Indeed, they exhibit the same behavior as the dimensionless system (3) that we are preparing to introduce. For example, Kim and An (2011) show that the original system (1) is sensitive to the radiative–convective equilibrium temperature . By translating and , the value is no longer explicitly present in the new equations (2); however, it appears implicitly in the new parameter C (see Table 2). We make this relationship explicit when we introduce the dimensionless parameter c in (4) that corresponds to C. As the analysis will show, the parameter c determines the relative strength of the super El Niño. Here we see a benefit of the nondimensionalization: it is not alone that is important, but rather the difference , which characterizes the maximum possible temperature range in the model. Neither nor alone appear in any of the other dimensionless parameters, implying that their effect on the system is entirely dependent on each other.

Table 2.

Rescaled parameters of the system from Timmermann et al. (2003).

Table 2.

Now seek the dimensionless version of the system. We define the following dimensionless variables:
eq3
If we indicate the new time derivative by a prime (i.e., ), the dimensionless system then takes the form
e3
where the key parameters can be expressed as
e4
The zonal advection feedback is represented by the term and the vertical advection and thermocline feedback are captured through . The term is a representation of the recharging mechanism of the thermocline through wind stress curl-induced Sverdrup transport. Thermal relaxation of sea surface temperatures in this system is expressed in terms of and . The bursting corresponds to the development of extreme El Niño events with eastern sea surface temperatures approximating those in the western Pacific , similar to the bursting in (2). The model simulates a rapid termination of these events into a La Niña state and a slow recharge of the system through growing small-amplitude ENSO oscillations. Upon reaching a threshold state, the system rapidly develops into an extreme El Niño event and the cycle repeats itself.

The system (3) has two global time-scale parameters and . The parameter relates the efficiency of the advective process to the efficiency of the upwelling term, and δ is the relative time scale associated with the thermocline adjustment. Motivated by the bursting behavior seen in Fig. 1, we will assume so that the system has one fast variable (x) and two slow variables (y, z). Note that the value of the standard model setup (Table 2) is , but smaller values can be physically justified, if for instance the recharging time scale increases beyond 400 days, or if the upwelling efficiency ζ increases. In the case where both and , then (3) will be a three-time-scale system with x fast, z slow, and y superslow.

The foundation of the approach we use for analyzing systems with multiple time scales is called geometric singular perturbation theory (GSPT) (Fenichel 1979; Jones 1995). The combination of GSPT with blowup techniques provides results on global behavior, including complicated patterns such as mixed-mode oscillations (Brøns et al. 2006; Desroches et al. 2012; Krupa and Szmolyan 2001a,b). Figure 2 depicts mixed-mode oscillations observed in (3). Similar bursting behavior has been observed in neurophysiological experiments (Amir et al. 2002; Del Negro et al. 2002; Dickson et al. 1998; Gutfreund et al. 1995; Khosrovani et al. 2007). Recent mathematical developments have enhanced our ability to explain such phenomena by exploiting a time-scale separation of the underlying model. The method combines the theory of canards (Benoît 1983; Benoît et al. 1981; Szmolyan and Wechselberger 2001; Wechselberger 2007) with a suitable global return mechanism (Brøns et al. 2006; Guckenheimer 2008; Kuehn 2011; Milik et al. 1998; Roberts et al. 2015; Wechselberger 2005, 2012), and it is now widely accepted as a robust explanation for dynamical behavior that is qualitatively similar to that observed in Fig. 2 (Brøns et al. 2008; see special issue of Discrete and Continuous Dynamical Systems, Series S, 2009, Vol. 2, No. 4; Desroches et al. 2012).

Fig. 2.
Fig. 2.

Examples of MMO patterns in (3) when (a) , , and ; (c) and ; and (d) and . (b) A closer look at the small oscillations in (a). Notice that the size of the small-amplitude oscillations scale with δ, making them difficult to see in (a). El Niño events occur when , whereas strong La Niña events are characterized by .

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0191.1

3. Model analysis

We approach the problem as a singular perturbation problem with respect to the parameter δ and then relate the dynamics for physically relevant parameters to the geometry of the singular limit. Thus, we treat the system as a problem with one fast variable and two slow variables.

a. GSPT

Recall that τ is the dimensionless time variable in (3). We can rescale time by the small parameter δ by setting to obtain the system
e5
where the dot indicates differentiation with respect to the new variable s (i.e., d/ds). This scaling does not change the trajectories of the system—that is, they have identical phase portraits. It does reparameterize the trajectories, changing the speed at which the curves are traced. In that sense, the two systems are equivalent as long as . The initial formulation (3) is referred to as the “fast system” and (5) as the “slow system.” However, in the limit as , the systems are different and the case where is referred to as the singular limit.
When , the fast system (3) becomes the layer problem:
e6
Physically, this corresponds to a state of constant and h but varying . The dynamical evolution of is governed by vertical advective processes in the eastern tropical Pacific. Notice that the dynamics in the y and z directions are trivial in the layer problem, while the x dynamics are not. The y and z dynamics are described in the reduced problem obtained by taking the limit as of the slow system (5):
e7
This set of equations characterizes a balance between advection and thermocline processes for some value of . Physically, this corresponds to a situation in which the western tropical Pacific SST and thermocline depth h essentially decay toward and 0, respectively.

b. The layer problem: The critical manifold and its stability

The set
eq4
is important in both components of the singular limit. In the layer problem [see (6)], M is the set of equilibrium points. As such, the stability of these points for fixed values can be computed by the linearization of the layer equation. We view y and z as parameters that affect both the location and stability of the equilibria. In the reduced problem [see (7)], we observe that M defines the set on which the slow dynamics are defined. In contrast to a “standard” parameter drift approach where y and/or z vary based on an explicit function of time, here y and z vary depending on the fast variable x and, hence, implicitly as a function of time. In the singular limit, the fast system equilibrates to a point in M and then the slow system evolves along M. The importance of M cannot be overstated—it forms the backbone of complicated dynamical behavior—and it is referred to as the critical manifold.
Since the equation in (6) contains a product, the critical manifold has two components:
e8
e9
The first condition characterizes an extreme El Niño event , depicted, for example, in Fig. 2.
GSPT guarantees that for δ small enough, our intuitive approach will work unless the Jacobian of the layer problem is 0. Computing the Jacobian, we get
eq5
Here, M is attracting wherever and repelling wherever . First, looking at the stability of , we see that
eq6
Therefore, is attracting where and repelling when . This condition is crucial in determining the dynamical behavior near an extreme El Niño state . Along the curve , and, geometrically, this curve represents the intersection of with . Thus, is attracting below and repelling above .
Turning to the stability of we see
e10
which has three zeroes for fixed . There exist such that . Generically, there are three possible orders of these zeros: 1) , 2) , and 3) . In each of these cases, let be the roots of . Then the leftmost branch of (i.e., ) is attracting, with stability alternating at each subsequent root. Thus, is attracting for and and is repelling for as well as for The lines
eq7
are called fold curves. As seen in Fig. 3, fold curves separate the stable and unstable regions of . Along the folds, the standard GSPT approach for normally hyperbolic critical manifolds breaks down. However, it is this degeneracy that allows for more complicated interaction between the fast and slow dynamics that lie at the heart of the interesting mathematical and dynamical behavior of the conceptual ENSO model.
Fig. 3.
Fig. 3.

The critical manifold when , , , and . (a) The manifold in 3D. (b) The component . (c) Projection of the component onto the xz plane. Stable regions are shaded in red, while unstable regions are shaded in blue [(b) and (c) only].

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0191.1

c. The reduced problem: Dynamics on the critical manifold

Next, we consider the slow dynamics of the reduced problem (7). In general, a reduced problem describes a flow of reduced dimension restricted to a manifold given by an algebraic condition. Since M is the union of the manifolds and , we need to solve two separate reduced problems—one on each of the two manifolds.

The reduced problem on , described by
e11
is simple because the global coordinate chart describes the manifold . Moreover, the dynamics are already formulated in terms of the coordinate variables y and z. Since (11) is a linear system, it is easy to see that is the unique globally attracting fixed point on (extreme El Niño case). It is worth noting that, while trajectories on will be drawn to this fixed point, it lies in the region where itself is unstable. This observation will play a major role in the global dynamics of (3) when
Next we turn to the reduced problem on , given by
e12
In this case we are not as lucky as we were for the problem on ; the fold lines indicate that we will be unable to formulate a global coordinate chart as . However, using the algebraic condition
eq8
we obtain a single coordinate chart—namely,
e13
where we can study the whole reduced flow on . This is done by differentiating the algebraic condition (13) and substituting it for the equation in (12) to obtain
eq9
Substitution and some minor rearranging yields
e14
Note that the system (14) is singular along the fold curves (denoted by green lines in Fig. 4) since it is precisely the set .
Fig. 4.
Fig. 4.

An MMO orbit and approximating singular cycle for parameters , , , and . (left) Projection of the singular cycle onto the xz plane. The five segments of the singular cycle are drawn as black curves and labeled S1, S2, F1, S3, and F2. Projections of the fold curves are green and labeled and . Projection of the segment S3 onto along the fast x direction is drawn as a red curve and labeled P. The folded-node point is drawn as a large red dot and labeled FN. Its strong stable manifold is drawn blue and labeled SC. The funnel of FN is the region between SC and . The equilibrium point (EQ) is a green ×. (right) A three-dimensional plot of an MMO orbit (blue) and the approximating singular cycle (black). The gray surface is and the light blue surface is .

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0191.1

The system (14) can be desingularized by rescaling the time variable s by a factor of . The new system, called the desingularized system, is
e15
The rescaling is quite powerful in that we are now able to define dynamics on the entirety of , including the folds. Recalling condition (10), we see that the scaling factor changes signs across the fold curves . The time orientation of trajectories is reversed in the region where . As indicated in Fig. 4, we are primarily interested in the region of phase space where . In fact, is invariant, even for in the full system (3). Therefore, trajectories starting in the region will remain there for all time. Physically, this means that no El Niño event can grow beyond the warm pool temperature . When , is attracting wherever Thus, the rescaling reverses trajectories on the repelling branch of where the time variable is scaled by a negative factor.

In the desingularized system (15), it is easy to classify three types of special points of the reduced problem:

  • Equilibria occur where and . At these points, the scaling factor

  • Regular fold points are fold points that are not equilibria of (15). That is, but

  • Folded singularities are equilibria of (15) where owing to the rescaling. That is, folded singularities occur where and , but . The point labeled “FN” in Fig. 4 is an example of a folded singularity.

Some folded singularities lead to the formation of the small-amplitude oscillations in an MMO. Similar to equilibria, folded singularities are classified by the eigenvalues of the linearization of the flow at the equilibrium. A folded singularity with real, negative eigenvalues, for example, is a stable folded node, while a folded singularity with complex conjugate eigenvalues with positive real part is an unstable folded focus. FN in Fig. 4 labels a stable folded node.

d. Geometry of MMOs

1) Nonmonotone SAOs

Folded nodes can produce MMOs with a suitable global return mechanism (Brøns et al. 2006; Desroches et al. 2012; Guckenheimer 2008; Krupa and Wechselberger 2010; Kuehn 2011). In a 2D system such as (15), a node is an equilibrium point with two real eigenvalues of the same sign: a weak eigenvalue and a strong eigenvalue such that . For the folded node denoted FN in Fig. 4, we have
e16
so FN is a stable node. Trajectories approach the node tangent to one of the eigenvectors. The strong stable manifold, corresponding to , is a single trajectory denoted by , while all other trajectories approach the folded node in the direction of the eigenvector of . Trajectories that approach the folded node from the left without hitting the fold line fill a region called the funnel of the folded node. The strong stable manifold (denoted “SC” in the left panel of Fig. 4) bounds one side of the funnel.

The folded node is key to one mechanism for generating SAOs in the ENSO system when . As δ is increased from 0, the attracting and repelling sheets of the critical manifold perturb to invariant slow manifolds that no longer share the same boundaries along the fold lines. The slow manifolds twist in a region surrounding the vanished folded-node point so that they continue to intersect along a finite set of trajectories. The number of intersections is related to the eigenvalues of the folded node (Benoît 1990; Wechselberger 2005). The intersections of the slow manifolds partition the attracting slow manifold into rotational sectors consisting of trajectories that make different numbers of rotations as they pass through the folded-node region. A key characteristic of these nonmonotone SAOs is that the number of small-amplitude oscillations has an upper bound determined by the geometry of the intersecting slow manifolds. Figure 5 depicts a few trajectories lying in different rotational sectors. Note that as the number of rotations grows, their amplitude becomes very small and the oscillations can hardly be seen in the figure (Desroches et al. 2012).

Fig. 5.
Fig. 5.

Local rotation (SAOs) of the full system when , , , , and The black trajectory approaches the stable manifold of the saddle-focus equilibrium along its stable manifold and departs along its unstable manifold. The green, blue, and red trajectories approach the stable manifold to the equilibrium in the region where the reduced system has a folded node. Their SAOs arise from the twisting of the attracting and repelling slow manifolds in this region. The green trajectory that approaches the stable manifold first has more small oscillations and these oscillations have smaller amplitude than those of the blue and red trajectories.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0191.1

2) Monotone SAOs

The reduced system (14) has an equilibrium point that lies close to the folded node. As parameters are varied in the system, this equilibrium crosses the fold curve in a bifurcation that is called a folded saddle-node type II. Figure 6 shows the relative location of the equilibrium of the singular limit to the folded node as the parameter a varies. The folded saddle-node type-II bifurcation is located at the intersection of these two curves. The equilibrium itself changes from a sink to a saddle as it crosses . In the full system with , the equilibrium undergoes a singular Hopf bifurcation at parameter values that are close to where the equilibrium crosses . The adjective singular here refers to the fact that the slow and fast time scales both play a role in the bifurcation: the imaginary eigenvalues of the equilibrium have magnitude intermediate between the two time scales. For parameters to one side of this Hopf bifurcation, the equilibrium point is a saddle focus with a real negative eigenvalue and a pair of complex eigenvalues with positive real parts. Because of the large relative magnitude of the imaginary parts of these eigenvalues, trajectories in the unstable manifold of the equilibrium spiral away with a large number of oscillations as the amplitude of the oscillations grows slowly. These growing oscillations constitute a second mechanism for generating SAOs. All trajectories that come close to the equilibrium, leave along its unstable manifold, and acquire SAOs as they do so.

Fig. 6.
Fig. 6.

Bifurcation diagram showing how close the system is to a folded saddle-node type II when , , and . A folded saddle-node type II occurs when (3) in its singular limit undergoes a transcritical bifurcation where one equilibrium is an ordinary singularity and the other is a folded singularity.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0191.1

This behavior connects the multiple time-scale analysis presented here to the Shilnikov analysis of chaotic behavior associated with homoclinic orbits to saddle-focus equilibria in three-dimensional vector fields (Guckenheimer and Holmes 1983). Shilnikov bifurcation was a mechanism proposed earlier to occur in ENSO models of the type studied here (Timmermann et al. 2003).

Folded nodes and singular Hopf bifurcation are two mechanisms for producing small-amplitude oscillations in systems with two slow variables and one fast variable. For these SAOs to be embedded in MMOs, there must be a global return of trajectories that leave the SAOs. These global returns may have a singular limit as consisting of a singular cycle that concatenates trajectories of the reduced system with trajectories of the layer system. Below we find a singular cycle that approximates MMO orbits of the ENSO model.

3) MMOs in ENSO

Figure 7a shows a three-dimensional phase portrait of an MMO in the ENSO model represented by both systems (3) and (5). Setting produces the singular limit in which trajectories are concatenations of solutions to the reduced and layer equations in (7) and (6). Here, we construct a singular cycle that approximates this MMO orbit and suggests its decomposition into slow and fast segments.

Fig. 7.
Fig. 7.

Attracting MMO when , , , , and (a) The MMO orbit in phase space. (b) The time series for x (corresponding to the east–west temperature gradient). (c) The time series for x + y (corresponding to temperature in the eastern Pacific).

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0191.1

The singular cycle projected onto the xz plane is shown in the left panel of Fig. 4. The cycle is composed of five segments. Beginning at the lower-left corner, which represents the La Niña state of the system following discharge of a large El Niño event, the first segment (S1) is a trajectory of the reduced system on that lies in the funnel flowing to FN. The system recharges during this event; as the western thermocline depth increases, the temperature in the western Pacific increases and the zonal temperature gradient decreases. We anticipate that there may be SAOs with decreasing amplitude in this region when . From the folded node, there is a continuum of possible choices for the second segment of the singular cycle. (Recall that the desingularization process reversed the orientation of the reduced flow on between and , so that trajectories flow away from the folded node in this strip.) We choose the second segment (S2) to be the stable manifold of the equilibrium point (EQ) in the desingularized system. This segment is short, and not much appears to happen in the phase space except that the trajectory approaches the equilibrium, which is a saddle in the reduced system. In the full system, the equilibrium is a saddle focus and SAOs of growing amplitude begin in this region.

Upon reaching the saddle (EQ), the third segment (F1) of the singular cycle is selected to be a jump along a trajectory of the layer equation to the sheet of the critical manifold. This represents a strong El Niño event. Trajectories do not cross since it is invariant in both the full and reduced ENSO models. The fourth segment (S3) of the singular cycle is then a trajectory of the reduced system for . Here the thermocline and eastern Pacific temperatures evolve on the slow time scale while the zonal temperature gradient remains small throughout the strong El Niño event. Where the singular cycle departs from is not clear: theoretical analysis of what happens when two sheets of a slow manifold intersect along a curve is incomplete and a subject for future work (Cortez 2011). The choice we make is to continue the trajectory along until the value of z attains the minimum observed on the MMO trajectory. From this point, the fifth segment (F2) of the singular cycle is a fast jump back to the left-hand sheet of , discharging the strong El Niño event and completing a return to the initial point of the cycle.

The right panel of Fig. 4 superimposes the MMO computed for parameter values , , , , and on the singular cycle displayed in the left panel of Fig. 4. The resemblance of the two is evident. Using the singular cycle, we can decompose the MMO into slow and fast segments. The small-amplitude oscillations occur in the region of the reduced model phase plane close to EQ. The nonmonotone SAOs related to the folded-node mechanism are associated with oscillations in the folded-node region, whereas the monotone SAOs due to the singular Hopf mechanism are associated with oscillations that grow along the unstable manifold of the equilibrium. The two mechanisms are not mutually exclusive, and determining which is prevalent in ENSO will involve comparison with signals in either real-world observation or output from coupled ocean–atmosphere general circulation models.

4. Summary and discussion

Even though the decadal-scale bursting of strong El Niño events has been the focus of several recent studies (Kim and An 2011; Timmermann and Jin 2002; Timmermann 2003), the underlying dynamics still remain elusive. Our paper provides the first insight into possible connections between MMO dynamics and El Niño bursting. We study the dynamical behavior of a conceptual model of the tropical Pacific climate system, which is based on three nonlinear ordinary differential equations. For realistic physical parameters, the system exhibits small-amplitude oscillations (SAOs), which occasionally burst into large events. More specifically, the amplitude of the El Niño/La Niña events slowly increases, until reaching a threshold in phase space where a large El Niño event emerges. The event terminates rapidly as a result of the discharging process (Fig. 1, bottom panel). The discharge is so strong that the system overshoots into a maximum-amplitude La Niña event. Its subsequent evolution follows again the growth of SAOs, until the threshold is reached and the cycle repeats itself.

The underlying equations can be written as a “one-fast, two-slow system.” The fast variable is associated with the east–west temperature gradient and its rapid adjustment to changes in vertical advection in the eastern tropical Pacific. The slow system captures the adjustment of the western tropical Pacific temperature via thermal relaxation and zonal advection and of the western tropical Pacific thermocline by changes in Sverdrup transport. To better understand the complex dynamical behavior, we apply geometric singular perturbation theory to the set of conceptual ENSO equations.

In the specific limit of , which represents either a very slow thermocline discharging process or a very efficient vertical advection, we find that the dynamics of a typical periodic orbit can be spliced together by concatenating trajectories of the decoupled fast and slow dynamics. It is shown that the existence of a folded node, which is characterized by a small El Niño amplitude , plays a key role in organizing the dynamics. This point essentially becomes a point of no return for the rapid development into a strong El Niño event that follows the fast dynamics. The system can remain on the extreme El Niño manifold for quite some time (Figs. 2 and 5), during which the western tropical Pacific temperature slowly decreases along with the western tropical thermocline depth. Upon reaching another threshold jump point at western tropical Pacific temperature , the system quickly discharges which leads to a rapid transition from onto . From there it moves back slowly toward the folded node to resume its trajectory.

We have shown that a global return mechanism is present in this ENSO recharge oscillator model. Associated with this mechanism are two types of MMOs: those with monotone SAOs and those with nonmonotone SAOs. The distinction between them can be seen clearly in Fig. 5. Both are potential explanations of the interspersion of El Niño bursting with small-amplitude oscillations. The key characteristic of the former is the bound on the number of small-amplitude oscillations and, for the latter, it is the monotonically increasing amplitude of the small oscillations. Whether these can be distinguished in an ENSO signal from observations or large-scale computation is an exciting challenge.

Acknowledgments

A.R., E.W., and C.J. were supported by NSF Grant DMS-0940363. A.T. was supported by U.S. NSF Grant 10-49219 and U.S. Department of Energy Grant DE-SC0005110. C.J. was supported by the Office of Naval Research under Grant N00014-15-1-2112. J.G. was supported by U.S. NSF Grant 10-06272. Additionally, A. R. and E. W. would like to thank UH–Manoa for their hospitality when visiting.

REFERENCES

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
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    • Search Google Scholar
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    • Search Google Scholar
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    • Search Google Scholar
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    • Search Google Scholar
    • Export Citation
  • Fenichel, N., 1979: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equations, 31, 5398, doi:10.1016/0022-0396(79)90152-9.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., 1997: An equatorial ocean recharge paradigm for ENSO. Part I: Conceptual model. J. Atmos. Sci., 54, 811829, doi:10.1175/1520-0469(1997)054<0811:AEORPF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., 1998: A simple model for the Pacific cold tongue and ENSO. J. Atmos. Sci., 55, 24582469, doi:10.1175/1520-0469(1998)055<2458:ASMFTP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., J. D. Neelin, and M. Ghil, 1994: El Niño on the devil’s staircase: Annual subharmonic steps to chaos. Science, 264, 7072, doi:10.1126/science.264.5155.70.

    • Search Google Scholar
    • Export Citation
  • Jones, C. K. R. T., 1995: Geometric singular perturbation theory. Dynamical Systems, R. Johnson, Ed., Lecture Notes in Mathematics, Vol. 1609, Springer, 44–118.

  • Karoly, D. J., 1989: Southern Hemisphere circulation features associated with El Niño-Southern Oscillation events. J. Climate, 2, 12391252, doi:10.1175/1520-0442(1989)002<1239:SHCFAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Khosrovani, S., R. S. Van Der Giessen, C. De Zeeuw, and M. De Jeu, 2007: In vivo mouse inferior olive neurons exhibit heterogeneous subthreshold oscillations and spiking patterns. Proc. Natl. Acad. Sci. USA, 104, 15 91115 916, doi:10.1073/pnas.0702727104.

    • Search Google Scholar
    • Export Citation
  • Kim, B.-M., and S.-I. An, 2011: Understanding ENSO regime behavior upon an increase in the warm-pool temperature using a simple ENSO model. J. Climate, 24, 14381450, doi:10.1175/2010JCLI3635.1.

    • Search Google Scholar
    • Export Citation
  • Krupa, M., and P. Szmolyan, 2001a: Extending geometric singular perturbation theory to nonhyperbolic points—Fold and canard points in two dimensions. SIAM J. Math. Anal., 33, 286314, doi:10.1137/S0036141099360919.

    • Search Google Scholar
    • Export Citation
  • Krupa, M., and P. Szmolyan, 2001b: Relaxation oscillation and canard explosion. J. Differ. Equations, 174, 312368, doi:10.1006/jdeq.2000.3929.

    • Search Google Scholar
    • Export Citation
  • Krupa, M., and M. Wechselberger, 2010: Local analysis near a folded saddle-node singularity. J. Differ. Equations, 248, 28412888, doi:10.1016/j.jde.2010.02.006.

    • Search Google Scholar
    • Export Citation
  • Kuehn, C., 2011: On decomposing mixed-mode oscillations and their return maps. Chaos, 21, 033107, doi:10.1063/1.3615231.

  • Levine, A., and F.-F. Jin, 2010: Noise-induced instability in the ENSO recharge oscillator. J. Atmos. Sci., 67, 529542, doi:10.1175/2009JAS3213.1.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., 2002: A simple model study of the forced response of ENSO to an external periodic forcing. J. Climate, 15, 10881098, doi:10.1175/1520-0442(2002)015<1088:ASMSOE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Milik, A., P. Szmolyan, H. Löffelmann, and E. Gröller, 1998: Geometry of mixed-mode oscillations in the 3-D autocatalator. Int. J. Bifurcation Chaos, 8, 505519, doi:10.1142/S0218127498000322.

    • Search Google Scholar
    • Export Citation
  • Roberts, A., E. Widiasih, M. Wechselberger, and C. K. Jones, 2015: Mixed mode oscillations in a conceptual climate model. Physica D, 292–293, 7083, doi:10.1016/j.physd.2014.11.003.

    • Search Google Scholar
    • Export Citation
  • Stein, K., A. Timmermann, N. Schneider, F.-F. Jin, and M. Stuecker, 2014: ENSO seasonal synchronization theory. J. Climate, 27, 52855310, doi:10.1175/JCLI-D-13-00525.1.

    • Search Google Scholar
    • Export Citation
  • Szmolyan, P., and M. Wechselberger, 2001: Canards in . J. Differ. Equations, 177, 419453, doi:10.1006/jdeq.2001.4001.

  • Timmermann, A., 2003: Decadal ENSO amplitude modulations: A nonlinear mechanism. Global Planet. Change, 37, 135156, doi:10.1016/S0921-8181(02)00194-7.

    • Search Google Scholar
    • Export Citation
  • Timmermann, A., and F.-F. Jin, 2002: A nonlinear mechanism for decadal El Niño amplitude changes. Geophys. Res. Lett., 29, doi:10.1029/2001GL013369.

    • Search Google Scholar
    • Export Citation
  • Timmermann, A., F.-F. Jin, and J. Abshagen, 2003: A nonlinear theory for El Niño bursting. J. Atmos. Sci., 60, 152165, doi:10.1175/1520-0469(2003)060<0152:ANTFEN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tziperman, E., L. Stone, M. Cane, and H. Jarosh, 1994: El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator. Science, 264, 72, doi:10.1126/science.264.5155.72.

    • Search Google Scholar
    • Export Citation
  • Wechselberger, M., 2005: Existence and bifurcation of canards in in the case of a folded node. SIAM J. Appl. Dyn. Syst., 4, 101139, doi:10.1137/030601995.

    • Search Google Scholar
    • Export Citation
  • Wechselberger, M., 2007: Canards. Scholarpedia, 2, 1356, doi:10.4249/scholarpedia.1356.

  • Wechselberger, M., 2012: A propos de canards (apropos canards). Trans. Amer. Math. Soc., 364, 32893309, doi:10.1090/S0002-9947-2012-05575-9.

    • Search Google Scholar
    • Export Citation
Save
  • Amir, R., M. Michaelis, and M. Devor, 2002: Burst discharge in primary sensory neurons: Triggered by subthreshold oscillations, maintained by depolarizing afterpotentials. J. Neurosci., 22, 11871198.

    • Search Google Scholar
    • Export Citation
  • Benoît, É., 1983: Systemes lents-rapides dans et leurs canards. Asterisque, 109110, 159191.

  • Benoît, É., 1990: Canards et enlacements. Publ. Math. l'IHÉS, 72, 63–91.

  • Benoît, É., J. L. Callot, F. Diener, and M. Diener, 1981: Chasse au canard (première partie). Collect. Math., 32, 3776.

  • Bjerknes, J., 1969: Atmospheric teleconnections from the equatorial Pacific. Mon. Wea. Rev., 97, 163172, doi:10.1175/1520-0493(1969)097<0163:ATFTEP>2.3.CO;2.

    • Search Google Scholar
    • Export Citation
  • Brøns, M., M. Krupa, and M. Wechselberger, 2006: Mixed mode oscillations due to the generalized canard phenomenon. Bifurcation Theory and Spatio-Temporal Pattern Formation, W. Nagata and N. S. Namachchivaya, Eds., Fields Institute Communications, Vol. 49, American Mathematical Society, 39–63.

    • Search Google Scholar
    • Export Citation
  • Brøns, M., T. J. Kaper, and H. G. Rotstein, 2008: Introduction to focus issue: Mixed mode oscillations: Experiment, computation, and analysis. Chaos, 18, 015101, doi:10.1063/1.2903177.

  • Chang, P., B. Wang, T. Li, and L. Ji, 1994: Interactions between the seasonal cycle and the southern oscillation—Frequency entrainment and chaos in a coupled ocean-atmosphere model. Geophys. Res. Lett., 21, 2817–2820, doi:10.1029/94GL02759.

    • Search Google Scholar
    • Export Citation
  • Chang, P., L. Ji, B. Wang, and T. Li, 1995: Interactions between the seasonal cycle and El Niño-Southern Oscillation in an intermediate coupled ocean-atmosphere model. J. Atmos. Sci., 52, 2353–2372, doi:10.1175/1520-0469(1995)052<2353:IBTSCA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Cortez, M., 2011: Understanding the effects of rapid adaptation on predator-prey interactions using the theory of fast-slow dynamical systems. Ph.D. dissertation, Cornell University, 186 pp.

  • Del Negro, C. A., C. G. Wilson, R. J. Butera, H. Rigatto, and J. C. Smith, 2002: Periodicity, mixed-mode oscillations, and quasiperiodicity in a rhythm-generating neural network. Biophys. J., 82, 206214, doi:10.1016/S0006-3495(02)75387-3.

    • Search Google Scholar
    • Export Citation
  • Deser, C., and J. Wallace, 1990: Large-scale atmospheric circulation features of warm and cold episodes in the tropical Pacific. J. Climate, 3, 12541281, doi:10.1175/1520-0442(1990)003<1254:LSACFO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Desroches, M., J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga, and M. Wechselberger, 2012: Mixed-mode oscillations with multiple time scales. SIAM Rev., 54, 211288, doi:10.1137/100791233.

    • Search Google Scholar
    • Export Citation
  • Dickson, C. T., J. Magistretti, M. H. Shalinsky, E. Fransén, M. E. Hasselmo, and A. Alonso, 1998: Properties and role of Ih in the pacing of subthreshold oscillations in entorhinal cortex layer II neurons. J. Neurophysiol., 83, 2562–2579.

    • Search Google Scholar
    • Export Citation
  • Fenichel, N., 1979: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equations, 31, 5398, doi:10.1016/0022-0396(79)90152-9.

    • Search Google Scholar
    • Export Citation
  • Guckenheimer, J., 2008: Singular Hopf bifurcation in systems with two slow variables. SIAM J. Appl. Dyn. Syst., 7, 13551377, doi:10.1137/080718528.

    • Search Google Scholar
    • Export Citation
  • Guckenheimer, J., and P. Holmes, 1983: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, Vol. 42, Springer-Verlag, 453 pp., doi:10.1007/978-1-4612-1140-2.

  • Gutfreund, Y., Y. Yarom, and I. Segev, 1995: Subthreshold oscillations and resonant frequency in guinea-pig cortical neurons: Physiology and modelling. J. Physiol., 483, 621640, doi:10.1113/jphysiol.1995.sp020611.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., 1997: An equatorial ocean recharge paradigm for ENSO. Part I: Conceptual model. J. Atmos. Sci., 54, 811829, doi:10.1175/1520-0469(1997)054<0811:AEORPF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., 1998: A simple model for the Pacific cold tongue and ENSO. J. Atmos. Sci., 55, 24582469, doi:10.1175/1520-0469(1998)055<2458:ASMFTP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Jin, F.-F., J. D. Neelin, and M. Ghil, 1994: El Niño on the devil’s staircase: Annual subharmonic steps to chaos. Science, 264, 7072, doi:10.1126/science.264.5155.70.

    • Search Google Scholar
    • Export Citation
  • Jones, C. K. R. T., 1995: Geometric singular perturbation theory. Dynamical Systems, R. Johnson, Ed., Lecture Notes in Mathematics, Vol. 1609, Springer, 44–118.

  • Karoly, D. J., 1989: Southern Hemisphere circulation features associated with El Niño-Southern Oscillation events. J. Climate, 2, 12391252, doi:10.1175/1520-0442(1989)002<1239:SHCFAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Khosrovani, S., R. S. Van Der Giessen, C. De Zeeuw, and M. De Jeu, 2007: In vivo mouse inferior olive neurons exhibit heterogeneous subthreshold oscillations and spiking patterns. Proc. Natl. Acad. Sci. USA, 104, 15 91115 916, doi:10.1073/pnas.0702727104.

    • Search Google Scholar
    • Export Citation
  • Kim, B.-M., and S.-I. An, 2011: Understanding ENSO regime behavior upon an increase in the warm-pool temperature using a simple ENSO model. J. Climate, 24, 14381450, doi:10.1175/2010JCLI3635.1.

    • Search Google Scholar
    • Export Citation
  • Krupa, M., and P. Szmolyan, 2001a: Extending geometric singular perturbation theory to nonhyperbolic points—Fold and canard points in two dimensions. SIAM J. Math. Anal., 33, 286314, doi:10.1137/S0036141099360919.

    • Search Google Scholar
    • Export Citation
  • Krupa, M., and P. Szmolyan, 2001b: Relaxation oscillation and canard explosion. J. Differ. Equations, 174, 312368, doi:10.1006/jdeq.2000.3929.

    • Search Google Scholar
    • Export Citation
  • Krupa, M., and M. Wechselberger, 2010: Local analysis near a folded saddle-node singularity. J. Differ. Equations, 248, 28412888, doi:10.1016/j.jde.2010.02.006.

    • Search Google Scholar
    • Export Citation
  • Kuehn, C., 2011: On decomposing mixed-mode oscillations and their return maps. Chaos, 21, 033107, doi:10.1063/1.3615231.

  • Levine, A., and F.-F. Jin, 2010: Noise-induced instability in the ENSO recharge oscillator. J. Atmos. Sci., 67, 529542, doi:10.1175/2009JAS3213.1.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., 2002: A simple model study of the forced response of ENSO to an external periodic forcing. J. Climate, 15, 10881098, doi:10.1175/1520-0442(2002)015<1088:ASMSOE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Milik, A., P. Szmolyan, H. Löffelmann, and E. Gröller, 1998: Geometry of mixed-mode oscillations in the 3-D autocatalator. Int. J. Bifurcation Chaos, 8, 505519, doi:10.1142/S0218127498000322.

    • Search Google Scholar
    • Export Citation
  • Roberts, A., E. Widiasih, M. Wechselberger, and C. K. Jones, 2015: Mixed mode oscillations in a conceptual climate model. Physica D, 292–293, 7083, doi:10.1016/j.physd.2014.11.003.

    • Search Google Scholar
    • Export Citation
  • Stein, K., A. Timmermann, N. Schneider, F.-F. Jin, and M. Stuecker, 2014: ENSO seasonal synchronization theory. J. Climate, 27, 52855310, doi:10.1175/JCLI-D-13-00525.1.

    • Search Google Scholar
    • Export Citation
  • Szmolyan, P., and M. Wechselberger, 2001: Canards in . J. Differ. Equations, 177, 419453, doi:10.1006/jdeq.2001.4001.

  • Timmermann, A., 2003: Decadal ENSO amplitude modulations: A nonlinear mechanism. Global Planet. Change, 37, 135156, doi:10.1016/S0921-8181(02)00194-7.

    • Search Google Scholar
    • Export Citation
  • Timmermann, A., and F.-F. Jin, 2002: A nonlinear mechanism for decadal El Niño amplitude changes. Geophys. Res. Lett., 29, doi:10.1029/2001GL013369.

    • Search Google Scholar
    • Export Citation
  • Timmermann, A., F.-F. Jin, and J. Abshagen, 2003: A nonlinear theory for El Niño bursting. J. Atmos. Sci., 60, 152165, doi:10.1175/1520-0469(2003)060<0152:ANTFEN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Tziperman, E., L. Stone, M. Cane, and H. Jarosh, 1994: El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator. Science, 264, 72, doi:10.1126/science.264.5155.72.

    • Search Google Scholar
    • Export Citation
  • Wechselberger, M., 2005: Existence and bifurcation of canards in in the case of a folded node. SIAM J. Appl. Dyn. Syst., 4, 101139, doi:10.1137/030601995.

    • Search Google Scholar
    • Export Citation
  • Wechselberger, M., 2007: Canards. Scholarpedia, 2, 1356, doi:10.4249/scholarpedia.1356.

  • Wechselberger, M., 2012: A propos de canards (apropos canards). Trans. Amer. Math. Soc., 364, 32893309, doi:10.1090/S0002-9947-2012-05575-9.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    (top) Observed sea surface temperature anomalies averaged over Niño-3 region. High temperatures correspond to strong El Niño events; red lines indicate a repeated El Niño bursting pattern. (bottom) Simulated eastern tropical Pacific SSTA in build-up phase and termination phase of strong El Niño events using (1) and the parameters in Table 1.

  • Fig. 2.

    Examples of MMO patterns in (3) when (a) , , and ; (c) and ; and (d) and . (b) A closer look at the small oscillations in (a). Notice that the size of the small-amplitude oscillations scale with δ, making them difficult to see in (a). El Niño events occur when , whereas strong La Niña events are characterized by .

  • Fig. 3.

    The critical manifold when , , , and . (a) The manifold in 3D. (b) The component . (c) Projection of the component onto the xz plane. Stable regions are shaded in red, while unstable regions are shaded in blue [(b) and (c) only].

  • Fig. 4.

    An MMO orbit and approximating singular cycle for parameters , , , and . (left) Projection of the singular cycle onto the xz plane. The five segments of the singular cycle are drawn as black curves and labeled S1, S2, F1, S3, and F2. Projections of the fold curves are green and labeled and . Projection of the segment S3 onto along the fast x direction is drawn as a red curve and labeled P. The folded-node point is drawn as a large red dot and labeled FN. Its strong stable manifold is drawn blue and labeled SC. The funnel of FN is the region between SC and . The equilibrium point (EQ) is a green ×. (right) A three-dimensional plot of an MMO orbit (blue) and the approximating singular cycle (black). The gray surface is and the light blue surface is .

  • Fig. 5.

    Local rotation (SAOs) of the full system when , , , , and The black trajectory approaches the stable manifold of the saddle-focus equilibrium along its stable manifold and departs along its unstable manifold. The green, blue, and red trajectories approach the stable manifold to the equilibrium in the region where the reduced system has a folded node. Their SAOs arise from the twisting of the attracting and repelling slow manifolds in this region. The green trajectory that approaches the stable manifold first has more small oscillations and these oscillations have smaller amplitude than those of the blue and red trajectories.

  • Fig. 6.

    Bifurcation diagram showing how close the system is to a folded saddle-node type II when , , and . A folded saddle-node type II occurs when (3) in its singular limit undergoes a transcritical bifurcation where one equilibrium is an ordinary singularity and the other is a folded singularity.

  • Fig. 7.

    Attracting MMO when , , , , and (a) The MMO orbit in phase space. (b) The time series for x (corresponding to the east–west temperature gradient). (c) The time series for x + y (corresponding to temperature in the eastern Pacific).

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