1. Introduction and motivation
Weather and climate predictability have been investigated for several decades, starting from the pioneering work of Lorenz (1963), by relying on the concepts and methods of autonomous dissipative nonlinear dynamical systems (Ghil and Childress 1987, and references therein). For such systems, any initial volume in phase space contracts in time, on the average, thus eventually reducing to a time-invariant set of zero volume called an attractor. In the autonomous case, an attractor can be a fixed point, a limit cycle, an invariant torus, or a strange attractor (Eckmann and Ruelle 1985; Ott 2002; Tél and Gruiz 2006).
Investigating a system’s attractors, in particular the strange attractors that describe the statistical properties of deterministic chaos, is fundamental in this context (Eckmann and Ruelle 1985) and has helped considerably in the study of multiple weather regimes and of their predictability on time scales of weeks to months (Ghil and Robertson 2002, and references therein). On the other hand, when studying the longer time scales—interannual, interdecadal, and longer—that are associated with climate change, it becomes necessary to take into consideration the time dependence of both anthropogenic and natural forcing (Martinson et al. 1995; Mann et al. 1998; Chang et al. 2015). The proper framework for studying changes in the system’s internal variability, and not just in its mean properties, when subject to variable forcing is that of nonautonomous and random dynamical systems (Drótos et al. 2015; Ghil 2015, 2016).
In the latter framework, the natural generalization of the attractors is provided by pullback attractors (PBAs). A global PBA is defined in the mathematical literature as a time-dependent set 
The concepts and methods of nonautonomous and random dynamical systems, including pullback attraction, were introduced into climate dynamics by Ghil et al. (2008) and Chekroun et al. (2011), and they were also pursued vigorously by T. Tél and his group (Bódai et al. 2011; Bódai and Tél 2012; Bódai et al. 2013). Several recent studies have explored the properties of PBAs for low-order climate models using ensemble simulations (Bódai et al. 2011, 2013; Bódai and Tél 2012; De Saedeleer et al. 2013; Pierini 2014a; Drótos et al. 2015).
Drótos et al. (2015) used the simple conceptual climate model of Lorenz (1984), forced by a periodic seasonal component and a linear decrease in the imposed equator-to-pole temperature difference. These authors argued that the PBAs—or snapshot attractors, using the simpler concept introduced by Romeiras et al. (1990)—and their natural probability distributions are the only tools with which mathematically sound statements can be made at a given point in time within a changing climate. Their claim was based on the finding that the time-dependent, chaotic attractor of the Lorenz (1984) model was independent of the initial states, or initial data (IDs) for short, within the parameter ranges and for the ensembles of IDs they considered. This claim appears to be fully justified for the Lorenz (1984) model, which seems to possess a unique global PBA in those parameter ranges.
The ensemble simulations of a periodically forced quasigeostrophic ocean model by Pierini (2014a) also provided evidence of ID independence in chaotic regimes, but dependence on the IDs did emerge for particular parameter ranges, for which chaotic basins of attraction coexist with nonchaotic basins; trajectories starting in the latter converge onto small subsets of the PBA. Thus, the fundamental problem of the dependence of the natural probability distribution of a changing climate on initial states needs further investigation.
In this paper, we carry out such an investigation. The four-variable quasigeostrophic model of the ocean’s wind-driven, double-gyre circulation formulated by Pierini (2011) and used in the periodically forced simulations of Pierini (2014a) is studied here via ensemble simulations subject to a deterministic aperiodic forcing that mimics time dependence dominated by interdecadal climate variability. Two reference cases separated by a tipping point are considered: above the tipping point, large-amplitude relaxation oscillations shape the behavior of the autonomous system, whereas below it small-amplitude oscillations are dominant. In this second range the system is excitable, because an appropriate time-dependent forcing can excite the relaxation oscillations (e.g., Pikovsky and Kurths 1997; Pierini 2011, 2012). Various properties of these two reference cases are analyzed, such as the time of convergence from arbitrary IDs to PBAs and the sensitivity of the latter to forcing amplitude, as well as the dependence of the local attractors on the choice of the initial ensemble of states.
The paper is organized as follows. In section 2, the model is described and basic aspects of the autonomous system’s solutions and of the two reference ensemble simulations under aperiodic forcing are discussed. In section 3, the time of convergence from arbitrary IDs to the PBAs is estimated. In section 4, the sensitivity of PBAs with respect to the forcing amplitude is investigated. Section 5 is devoted to the dependence of PBAs on the choice of the ensemble of IDs: chaotic and nonchaotic basins of attraction are analyzed in detail. In section 6, the main results are summarized and discussed, and an appendix provides the rigorous mathematical background.
2. Model description
a. Model formulation
The mathematical model used in this study was developed by Pierini (2011) to analyze fundamental dynamical features of the decadal variability in the Kuroshio Extension (Dijkstra and Ghil 2005, and references therein). This model is derived through projection of the governing equations onto an orthonormal basis and subsequent low-order truncation, following the approach of Platzman (1960), Saltzman (1962), Lorenz (1963), and many others since, such as Lorenz (1982), Legras and Ghil (1985), Ghil and Childress (1987), Olbers (2001), and Crucifix (2012). S. Pierini and colleagues had already investigated this decadal variability with a relatively realistic primitive equation ocean model (Pierini 2006; Pierini et al. 2009; Pierini 2010, 2014b).
The model is derived via a severe truncation of the spectral representation of the evolution equation of potential vorticity in the quasigeostrophic, reduced-gravity approximation, in a rectangular domain. The Cartesian orthonormal basis onto which the streamfunction 
The analytical expressions of the Jacobian coefficients 
b. Model behavior
In the autonomous case, there is no t dependence and we let 
Bifurcation diagram of our idealized ocean model (1) in the autonomous case, i.e., with 
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
The value 
The value 
Pierini (2011, 2012, 2014a) did, in fact, study already the present model in the presence of both stochastic and deterministically periodic forcing, and found that such a homoclinic bifurcation does occur in it and that this bifurcation gives rise to what is currently termed a tipping point by the climate community (e.g., Lenton et al. 2008). Such a dynamical mechanism is a typical paradigm of abrupt climate change associated with an intrinsically nonlinear mode of variability (Lorenz 1963; Ghil and Childress 1987; Crucifix 2012), and it is sometimes called coherence resonance (Pikovsky and Kurths 1997). When coherence resonance does occur, the system displays strong dependence on forcing amplitude, as we will see in section 4.
In the nonautonomous case, 
Ensemble behavior of forced solutions of the double-gyre ocean model. (a) Time dependence of the total forcing 
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
Apart from this oceanographic interpretation, the forcing in Eq. (3) extends the analysis of Pierini (2014a) from purely periodic to deterministically aperiodic. This choice disrupts the cyclostationarity and cycloergodicity properties of PBAs and makes the forced model’s behavior more realistic.
The evolution of the model solutions subject to the forcing of Fig. 2a is shown in Figs. 2b and 2c for two reference cases, 
Figures 2b and 2c provide a first representation of the sets that approximate the corresponding PBAs; the small number of trajectories is in fact chosen for the sake of graphical clarity. The correct identification and characterization of the PBAs requires, however, an analysis of the PDFs that evolve along the trajectories and of the convergence to the appropriate invariant time-dependent sets. This convergence is investigated in the next section.
3. Convergence to PBAs
As mentioned in the introduction, PBAs are the extension of the attractors of dissipative autonomous dynamical systems to the nonautonomous case. When time-dependent forcing is present, the trajectories originating from the set 
The limit 
Evolution in time of 
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
In Figs. 2d and 2e, we plot 
Evolution in time of 
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
To estimate the rate of converge to the PBA, five cases are considered in Fig. 3, for both 
4. Sensitivity of PBAs to the amplitude of the forcing
Model robustness is important in making any inferences about the role of time-dependent forcing (Ghil 2015, and references therein). We thus proceed to analyze in this section the sensitivity of the present model’s PBAs with respect to the amplitude ε of the time-dependent component of the forcing, while in the next section dependence on IDs will be analyzed.
Figure 4 shows the dependence of 
Inspecting the panels of the left column, which correspond to 
This result agrees with Pierini (2012), who noticed that model (1), when subjected to stochastic forcing, behaves approximately like an adiabatically changing autonomous dynamical system, as far as coherence resonance is concerned. Likewise, when 
The same behavior is found for 
5. PBA dependence on the choice of initial ensemble
In section 3, we have investigated the evolution of the same set of IDs starting from different initial times τ; here we study the evolution of different initial states starting from the same initial time 
In the periodic-forcing case, Pierini (2014a) provided numerical evidence that the present ocean model’s PBA can be obtained from any set of IDs, provided its evolution is chaotic; see Figs. 3e and 3f therein. Drótos et al. (2015) have recently stressed that the chaotic PBAs of the Lorenz (1984) atmospheric model are independent of IDs and have suggested that this is likely to be the typical situation in climate models.
PBA dependence on IDs may, however, be more complex. In his systematic study of the present model’s PBAs as a function of the forcing period, Pierini (2014a) identified periods for which a chaotic basin of attraction—which converges onto the global PBA—coexists with nonchaotic basins. Thus, for example, a chaotic basin can coexist with another basin whose phase-space flow converges onto a single periodic trajectory lying on the global PBA; see, for example Figs. 6a–c of Pierini (2014a). It is therefore natural to investigate whether a similar behavior may be present also in the aperiodic-forcing case studied in the present paper. In the following, we use two complementary mathematical tools to explore this possibility.
a. Finite-time Lyapunov exponents
The typical approach to the investigation of chaotic versus nonchaotic behavior is to evaluate the finite-time Lyapunov exponents (FTLEs) of the system (Nese 1989). FTLEs have been widely used in physical oceanography to help identify localized coherent structures and material surfaces (Haller 2001; D’Ovidio et al. 2009).
To test whether chaotic behavior may coexist with nonchaotic trajectories, we computed maps of 
Maps of FTLEs for the present model. Maps of 
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
The first thing to note in all six panels is the existence of extensive regions in which 
The remarkable decrease of the range of 
b. Mean normalized distance
Mean normalized distance 
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
To check the character of the two types of evolution, we plot in Fig. 7 many trajectories that leave two small subsets of 
Evolution of ensembles of 420 trajectories each that emanate from small subsets of 
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
Figures 7a and 7c confirm the extreme sensitivity to IDs that lie in boxes A and C: a comparison of these two panels with Figs. 2b and 2c, respectively, shows that the attracting sets to which trajectories started in A and B converge are dense in the PBAs obtained by starting from the whole 
c. Discussion
It is worth exploring in greater detail the evolution of some of the trajectories plotted in Fig. 7. We display in the top graphs of Figs. 8a–d the 
Superposition of two time series of 
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
In the chaotic cases, represented by both graphs in Figs. 8a, and 8c, the values of 
Inspection of the bottom graphs in Figs. 8a–d confirms the validity of the mean normalized distance σ of Eq. (6) as a sharper diagnostic tool than the FTLEs. Longer integrations (not shown) simply extend in time the ensemble of trajectories shown in Figs. 7a and 7b, but the chaotic versus nonchaotic character of the trajectories—as diagnosed by σ—is preserved.
To further analyze the structure of the model’s PBAs, the intersection of many trajectories with the (
(a) Intersection with the (
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
The red dots arise from the small boxes A and C that are located in the chaotic regions of Figs. 6a and 6b, respectively. These red dots, too, are distributed all over the corresponding attractors in the two panels, as expected from the previous analysis; in fact, only the 420 red dots that correspond to the trajectories of Figs. 7a and 7c have been plotted here, to avoid complete overlapping with the gray dots.
On the contrary, the cyan dots that arise from the large nonchaotic boxes 1, 2, 3, and 4 of Figs. 6a and 6b converge onto small subsets of the PBAs. This convergence onto a smaller-dimensional set is very similar to what Pierini (2014a) found for some forcing periods that yielded periodic PBAs; for example, see Figs. 6a–c therein.
For 
Graphs like those of Fig. 9 cannot provide quantitative information on the probability of localization of trajectories on the model’s global PBA. Such information can be obtained, though, from the three-dimensional PDFs plotted in Fig. 10.
PDF of localization of the 15 000 trajectories of Figs. 6 and 9 projected onto the (
Citation: Journal of Climate 29, 11; 10.1175/JCLI-D-15-0848.1
For the subcritical case 
In summary, in the small regions of phase space in which the invariant sample measure on our model’s global PBA peaks in a filamentary pattern, both chaotic and nonchaotic trajectories coexist. The PDF peaks in Figs. 10a and 10b are clearly an indication of enhanced overall predictability, and recall those observed in Figs. 6 and 7 of Chekroun et al. (2011) for a highly idealized model of El Niño–Southern Oscillation with seasonal forcing.
6. Summary and conclusions
In this study, a low-order quasigeostrophic model of the wind-driven ocean circulation has been used as a prototype of an unstable, nonlinear, nonautonomous dynamical system of climatic relevance. We have studied this double-gyre model’s PBAs (i.e., those time-dependent invariant sets that attract all trajectories initialized in the remote past) along with the invariant sample measures that live on these sets. PBAs (Arnold 1998; Ghil et al. 2008; Chekroun et al. 2011) and their close cousins, snapshot attractors (Romeiras et al. 1990; Bódai et al. 2011, 2013; Bódai and Tél 2012; Drótos et al. 2015), have only recently been recognized as the natural tools for investigating basic features of a changing nonequilibrium climate, and the theory of nonautonomous dynamical systems as the proper mathematical framework for such investigations.
We chose here an aperiodic forcing dominated by interdecadal variability to mimic the effects of multidecadal climate changes (Chang et al. 2015) on the midlatitude double-gyre circulation; see section 2. The present study generalizes therewith a set of results previously obtained in the presence of periodic forcing (Pierini 2014a). We recall that, in the absence of time-dependent forcing [i.e., when 
In section 3, the convergence of ensembles of trajectories to PBAs was assessed in order to estimate the time required to identify the PBAs. To do so, we considered—in addition to projections of the trajectories from the full, four-dimensional phase space onto a plane spanned by the 
The sensitivity of PBAs with respect to the amplitude ε of the time-dependent portion of the forcing was studied in section 4. Below the tipping point 
Finally, the dependence of the attracting sets on the choice of the ensemble of initial states is investigated in section 5. Our investigation relies, on the one hand, on the classical FTLEs to measure the rate of divergence of trajectories, but it introduces also a novel metric, namely the time mean σ of the normalized distance 
In our model, the statistics of an ensemble of trajectories in a chaotic regime does depend on the ensemble of states chosen in the remote past. Figures 6, 7, and 9, in particular, illustrate very well that the model’s global PBA—whose existence is rigorously proven in the appendix—contains both chaotic and nonchaotic trajectories.
Both of these types of trajectories have attractor basins that seem to be separated by fractal boundaries: the former fill the global PBA, while the latter form small and smooth subsets of the PBA. This complex, interwoven pattern of rapidly divergent and nondivergent trajectories shows that a time-independent PDF, as often used in predictability studies, is not sufficient to describe the model’s long-term behavior or to characterize its predictability.
These intriguing results have been obtained with a highly idealized nonlinear model subject to deterministic aperiodic forcing. As usual in this kind of approach, they raise as many questions as they answer about the broader issues of climate modeling and climate predictability. The two most urgent questions for the climate sciences are these:
To which extent is this simple model’s behavior characteristic of a wide class of nonautonomous dynamical systems, whether deterministic or stochastic?
How relevant is this class to climate modeling and climate predictability studies?
Acknowledgments
SP and MG would like to thank the TEMASAV Project of the Regione Campania of Italy (POR Campania FSE 2007/2013) and the coordinator, Giancarlo Spezie, for having supported exchange visits between the Environmental Research and Teaching Institute (CERES-ERTI) in Paris and the Department of Science and Technology (DiST) of the University of Naples Parthenope, thus promoting this collaboration. The research herein was supported by Grants N00014-12-1-0911 and N00014-16-1-2073 from the Multidisciplinary University Research Initiative (MURI) of the Office of Naval Research and by the U.S. National Science Foundation Grant OCE 1243175 (MDC and MG). The authors wish to thank two anonymous reviewers, whose detailed and constructive comments helped improve the manuscript.
APPENDIX
Existence of a Global Pullback Attractor
We present in this appendix rigorous mathematical results on the existence of pullback attractors (PBAs) in ordinary differential equation (ODE) models—like the one given by Eqs. (1) and (2) here—and, in particular, the conditions on the forcing term 
In particular, these existence results are shown to apply to Eqs. (1) and (2) considered in the main text. We mention also that the approach presented below can be adapted to the infinite-dimensional setting and thus to the original two-dimensional quasigeostrophic partial differential equation model, by working in the appropriate function spaces to define the corresponding solutions.
The less mathematically inclined reader can skip appendix section a and proceed directly to section b.
a. Rigorous mathematical background
In what follows X denotes the Euclidean space 
Theorem 1
Remark 2
- We recall that a propagator on X is a two-parameter family of continuous mappings
applies. It is trivial to show that when a nonautonomous ODE system, such as Eq. (A1), possesses a unique global solution for any
- A time-dependent set
where
Proof
We have thus proved that 
b. Interpretation
The result of theorem 1 states thus that, for any forcing 
The identity (A8) means that 
However, the Hausdorff semidistance is not a distance in the classical sense. For instance, 
As shown in these figures, the boundaries between the corresponding local basins of attractions, as measured with the metrics of section 5, are seemingly intricate and strongly suggest that these objects have fractal features. An attractor basin of chaotic solutions being separated by a fractal boundary from the attractor basin of a fixed point has been documented in a quadratic autonomous model governed by four ODEs that arises in population dynamics; see Fig. 5a in Roques and Chekroun (2011).
In that ecological model, however, no abutting of two different attractor basins of chaotic regions on the global attractor is present. A key difference between the ocean model studied herein and the population dynamics model of Roques and Chekroun (2011) lies in the quadratic terms here being energy preserving [cf. Eq. (A2)], while this is not the case in population dynamics.
There is, therefore, a Hamiltonian skeleton here that may help explain the difference between the two phase portraits: The pullback dissipation being very small argues for the asymptotic trajectories—in spite of the dissipative nature of the dynamics—lying quite close to this skeleton. It is thus not surprising to recover interleaved chaotic islands, as shown in Fig. 6. In the autonomous context, the effects of small dissipation have been studied by Ghil and Wolansky (1992), Feudel and Grebogi (1997), and Seoane et al. (2007), among others.
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