We begin by analyzing some basic properties of the autonomous system that will be useful in the subsequent investigation. The bifurcation diagram of Fig. a shows the range of variability of Ψ1 vs. the forcing parameter γ. The value γ=1 corresponds to a global bifurcation that manifests itself by a sudden transition from a small-amplitude limit cycle to a relaxation oscillation with a much higher amplitude. The previous results in this respect will be further bolstered by those in Sect.  herein (Figs.  and ), which are based on the diagnostic tool proposed in Sect. .

Figure b shows the limit cycle in Γ arising from arbitrary initial data for γ=1.1, which corresponds to the red vertical line in the bifurcation diagram of panel (a). For γ=1.35 the attractor is chaotic (green line in panel a; see Sect.  further below). In this case, the map of the suitably normalized decimal logarithm of the probability density function (PDF) of the trajectories in Γ is plotted in Fig. c; it is defined by qkt=log⁡101000×nkt. Here nk is the number of trajectories contained at time t in the kth cell belonging to the same regular grid of N square cells of width ΔΨ that is used in our ensemble simulations, with N=15000 and ΔΨ=2, and it is plotted at t=T∗=400 years, i.e., after spinup.

In an autonomous dynamical system, the attractors do, by definition, not depend on time, i.e., an attractor is a geometric object in phase space that is fixed in time. However, any attractor that is not a fixed point – whether a limit cycle, torus or strange attractor – can contain time-dependent trajectories. Such ensembles of trajectories arising from specific sets of initial states will be plotted to illustrate the attractors of the autonomous system studied herein.

Following , in Fig. a, b the attractors that correspond to the two cases in Fig. b, c are represented by the time evolution of PΨ3(t)=log⁡10(1000×pΨ3), where pΨ3(t) is the PDF of localization of the Ψ3 variable; see for technical details. The dense distribution of PΨ3(t) for γ=1.35, as seen in Fig. b, is clearly associated with the chaotic character of the flow, while the periodic distribution of PΨ3(t) that corresponds to γ=1.1 in Fig. a is due to the different phases that each trajectory attains on the limit cycle, depending on the initial point.

In Fig. c, d, the same attractors are characterized through the metric σ that was introduced by ; this metric measures the mean divergence of trajectories over the total integration time T∗ and is defined as follows. The instantaneous Euclidean distance between two initially close trajectories is δ(t), and its normalized value is given by δn(t)=δ(t)/δ(0). Then σ is simply the average of δn over T∗,

σ(X,Y)=1T∗∫0T∗δn(t)dt,5with(X,Y)≡(Ψ1(0),Ψ3(0))∈Γ. found the quantity σ to be a good indicator of the degree of sensitivity of the system's evolution with respect to the initial state during the phase of convergence to the attractor.

The determination of the PBAs of the periodically forced system and the application of the new qualitative and quantitative diagnostic methods proposed in Sect.  need an analysis of the behavior of trajectories that lie at t=0 on a given subset Ω of phase space, as is the case when calculating σ(X,Y) above. Thus, investigating the behavior of model trajectories as they emerge from Ω is the most unifying and distinctive feature of the present model study.

The map of σ in Fig. c reveals, in the autonomous case at hand, the same striking features found by for the nonautonomous, aperiodic-forcing case, namely the coexistence of extended regions of Γ with σ⩽1, shown by cold colors, and with σ>1, appearing as warm colors. In the first case, two trajectories that are initially close remain close at all times, as seen in Fig. a. In the second case, though, two trajectories that are initially close may attain a large phase difference once they have converged to the attractor (cf. Fig. b), while still remaining perfectly coherent.

In the chaotic case with γ=1.35, the warm-color regions, in which σ>1, overwhelm the cold-color regions, in which σ≤1, cf. Fig. d. To illustrate the two types of behavior, Fig. c, d show the evolution of Ψ3(t) of two initially nearby trajectories. If σ<1, as is the case near P1, the two trajectories are virtually coincident (Fig. c). If, on the contrary, σ>1, as is the case near P2, the two aperiodic signals lose their coherence (Fig. d). Finally, it is worth noting that, for simulations with sufficiently small γ (not shown), σ<1 everywhere.

A different and useful way of looking at these two types of behavior is to analyze the corresponding mixing properties of the flow in the model's phase space. To do so, one can make use of the system's entropy : 6Sϑt=-∑k=1Npkln⁡pk. Here Γ is decomposed into a regular grid of N square cells of width ΔΨ (with N=15000 and ΔΨ=2, so that the grid corresponds to that of the initial data used in our ensemble simulations) and pk(t) is the probability of localization in the kth cell at time t of the trajectories emanating at time t=0 from a given subset ϑ⊂Γ.

Figure a shows the intersection with the (Ψ1,Ψ3) plane at t=300 years of 15 000 trajectories originating from the box ϑ1 that coincides with the ΔΨ×ΔΨ grid cell centered at P1; the red dots correspond to the case γ=1.1 and the green dots to γ=1.35. Figure b shows Sϑ1 for the two cases; note that Sϑ1(0)=0 since all the initial states lie in the single cell ϑ1, and thus p1=1. The entropy of the periodic case γ=1.1, characterized by σ<1, oscillates between 0 and 1, with the final evolution limited to virtually a single cell over the limit cycle; the latter cell is enclosed in the red circle of Fig. a.

In the chaotic case γ=1.35, σ≤1 for 43 % of the points contained in ϑ1, while σ>1 for the remaining points. The evolution of the former leads to the localized blue dots in Fig. a while the evolution of the latter leads to the green dots scattered over the strange attractor. The green line of Fig. b, giving Sϑ1 computed with all the trajectories, shows the gradual spreading of the initial points with σ>1.

Figure c, d show the same quantities for the initial ΔΨ×ΔΨ box ϑ2 centered at P2. The chaotic case is similar to that for ϑ1, but with a greater entropy; however, the periodic case differs in that now σ>1 – cf. Fig. c. Figure c shows that the asymptotic evolution of the very small ϑ2 covers a limited but significant part of the limit cycle, as seen by comparing this figure with Fig. b; the corresponding entropy in Fig. d eventually oscillates periodically between the values Sϑ2∼2.2–3.7. Figure  thus demonstrates clearly the usefulness of the metric σ in characterizing subsets of Γ and the effect of the control parameter γ.

Finally, it is worth stressing that, since the forcing is constant, the range of variability of the entropy in the chaotic case with σ>1 must tend to zero as the number of points tends to infinity. This tendency is clearly illustrated by the green line of Fig. d. However, the range of variability of Sϑ1 in the chaotic case (Fig. b) is still quite large after 400 years because, as pointed out above, the number of points with σ>1 contained in ϑ1 is relatively small.

We conclude the analysis of the autonomous system by discussing an apparent paradox. We have just seen that, in regions of Γ where σ>1, the trajectories for γ=1.1 exhibit sensitive phase dependence on initial data, as shown, for instance, by Fig. b, by the red dots in Fig. c and by the red curve in Fig. d. Sensitive dependence on initial data is usually associated with chaotic dynamics, but in this case the dynamics are periodic.

This paradox is resolved by noting that such sensitivity concerns only the phase of the periodic trajectories, as already noticed in the previous subsection and, in addition, it occurs only if the initial data lie outside the attractor, e.g, elsewhere on Γ; on the attractor, this phase sensitivity disappears, as we will show below. On the contrary, in the chaotic case γ=1.35, the sensitivity to initial data for trajectories with σ>1 always holds, off the attractor as well as on it. This is in excellent agreement with the chaotic character of the dynamics in the latter case.

We must show, therefore, that the trajectories are stable for γ=1.1 and unstable for γ=1.35, once they have settled onto the attractor. This distinction between the two cases can already be inferred from Figs.  and but it is worth investigating the issue in greater detail. The usual quantitative approach relies on the computation of the leading finite-time Lyapunov exponent λ of each trajectory e.g.,.

The results (not shown) are consistent with the assumption above, but the exponents are highly dependent on the time Tλ over which the finite-time exponents are computed, and on the amplitude of the perturbation superimposed on the reference trajectory at each time step Tλ. Moreover, the assumption of exponential divergence of chaotic trajectories is not fully met in our highly nonlinear framework, so that the transition between periodic and chaotic dynamics may actually occur at a value of λ that is not exactly equal to 0. A qualitative diagnostic method is instead illustrated in Fig. , and furthermore we propose an alternative quantitative method in Sect. .

Let us consider the points lying in the black rectangle shown in Fig. c at t=300 years: the corresponding evolution at four subsequent time instants, TΔ=25 years, is shown in Fig. a for γ=1.1 (red dots). Note that the period of the orbits on the attractor is T∗p=14.08 years, i.e., TΔ>T∗p.

The stability of the trajectories under consideration is clearly demonstrated in Fig. a by the compact form and limited extent of the cluster: indeed, these points that start from t=300 years evolve anticlockwise around the attractor, covering it roughly 6 times during the interval 4TΔ=100 years that separates the first snapshot from the last one. On the contrary, Fig. b shows that for γ=1.35 (green dots) the compact form of the initial cluster is lost already after a single 25-year lapse of time. The trajectories will thus soon be scattered over the strange attractor, due to their divergence.

In conclusion, our autonomous system becomes chaotic for sufficiently large values of γ, e.g., for γ=1.35. The system's periodic regime spans a range of γ that includes the bifurcation at γ=1, which is apparent in Fig. a.

Moreover, we have shown that, when γ=1.1, regions of Γ exist within which the mean normalized distance σ between two initially nearby trajectories is larger than unity; see again Fig. c. In this case, despite the attractor's being a limit cycle, the trajectories leaving from such regions of phase space experience sensitive phase dependence on the initial data. Although sensitive dependence is typically associated with chaotic systems, this sensitivity is not in contradiction with the periodic character of the solutions: as a matter of fact, the trajectories under discussion are stable and the sensitive dependence disappears once the trajectories have converged onto the attractor.

The existence of regions with σ>1 for an autonomous periodic system is an important feature for the transition to chaos when the system is subjected to time-dependent forcing: this issue will be discussed in the next section. Besides, the new diagnostic method introduced in Sect.  to help analyze transition to chaos in the nonautonomous case will be applied in Sect.  to the autonomous case.

For all the above reasons, we now present the PBAs of our periodically forced ocean model. As already just mentioned, the PBAs of a periodically forced dissipative system are always periodic, but the system can be either chaotic or nonchaotic, depending on its parameter values. For the sake of simplicity, we will refer below to the PBAs of a chaotic and nonchaotic system, abbreviated as CPBAs and NPBAs, respectively.

For the autonomous case, the time evolution of PΨ3(t) and the map of σ(X,Y) – already shown in Fig. a and c, respectively – are again plotted in Fig. a and e for the sake of comparison. Figure d, h correspond to the reference CPBA studied by . Note also that one of the two reference cases studied by has the same values of γ and ε, but the latter parameter was multiplied by the aperiodic forcing shown in Fig.  below. Two intermediate cases that correspond to ϵ=0.05 and ϵ=0.1 are shown in Fig. b, f and c, g, respectively. In Sect. , we will show in greater detail that the transition to chaotic behavior occurs abruptly, when crossing a critical value ϵc that lies between the two intermediate values of 0.05 and 0.1; here we merely provide some qualitative arguments showing that, in fact, the case ϵ=0.05 is still periodic, while the case ϵ=0.1 is chaotic.

Before proceeding with the analysis of the results in Fig. , we recall that chaotic systems subjected to periodic forcing can be studied either by ensembles of trajectories – as done herein – or by stroboscopic averages using a single long trajectory, provided the assumption of cycloergodicity holds e.g.,: the latter result extends the classical ergodicity property valid for strange attractors of autonomous systems e.g., to chaotic, periodically forced systems. An example of this equivalence is shown in Fig. 3c, d of for the present model and for the parameter values corresponding to the PΨ3(t) and σ(X,Y) plotted in Fig. d, h herein (obviously, the trajectory used in that example was derived from a region with σ>1). However, the existence of regions with σ⩽1, as well as with σ>1, in the two chaotic cases (Fig. g, h) shows that the cycloergodicity assumption fails to hold for our idealized ocean model. Our system must, therefore, be investigated using the ensemble approach, which we pursue throughout this paper. In fact, had we only used the stroboscopic map method, we would never have discovered the existence of two types of local attractors, namely CPBAs and NPBAs, in our model.

Figure  provides further information on the four cases illustrated in Fig. . In Fig. a–d, the magenta dots represent the intersection with the (Ψ1,Ψ3) plane at t=400 years of 15 000 trajectories, whose initial points (in blue) are evenly distributed in Γ at t=0; in addition, in Fig. e–h, the corresponding entropy Sϑ=Γ is plotted as a function of time, along with the number no of cells that are occupied by at least one point. Clearly, the structure of the PBA snapshot in Fig. b, for ε=0.05, is very similar to that of the autonomous case in Figs. b and a, while the PBA snapshots plotted at t=400 years – for ε=0.1 and 0.2 in Fig. c and d, respectively – are quite different.

To understand this difference better, we focused in Fig.  on the subdomain ϑ2 of Γ that was defined in Fig. c and for which σ>1. The autonomous case has already been analyzed in Sect. : in fact, Fig. a, e are equivalent to Fig. c, d. In the case of ε=0.05, the same behavior is found, i.e., the sensitivity to initial data leads to only a compact subset of the attractor being covered; this implies the periodicity of the trajectories.

On the contrary, for ε=0.1 the intersection of the trajectories at t=400 years with the (Ψ1,Ψ3) plane, shown by the magenta dots in Fig. c, is virtually indistinguishable from the one that appears in Fig. c, when the initial data are selected in the whole of Γ: this excellent match is an unequivocal sign of the mixing property of chaotic dynamics, as already discussed for the autonomous case γ=1.35 in connection with Figs.  and . That the same property holds for ε=0.2 in Fig. d is not surprising, since already recognized the model's chaotic behavior for this parameter value.

Finally, it is instructive to visualize Ψ3(t) for a couple of trajectories that are very close at t=0, as done in Fig.  for the autonomous case. Figure  shows Ψ3(t) for the four cases of Figs. – and for trajectories that emerge from ϑ2. In Fig. a, b the two trajectories are periodic, but with a phase difference. In the two chaotic cases of ε=0.1, 0.2 in Fig. c, d, both trajectories are clearly aperiodic. In Fig. c, though, i.e., in the case that is closer to the transition, this aperiodicity is merely associated with a temporary shift in phase of an otherwise periodic signal within the intervals t≃40–120 years and t≃360–400 years.

The intermittent behavior seen in Fig. c appears – from many simulations that are not shown here – to be typical of chaotic solutions near the transition point εc and suggests a possible mechanism through which chaos is induced by an external periodic forcing. For values of ε just past εc, the model still tends to behave periodically, but the external forcing is sufficiently strong to entrain a trajectory occasionally into a nearby region, where the periodicity is preserved but the phase differs by a finite amount. Since these shifts are very sensitive to the initial data, the result is a chaotic trajectory characterized by separate intervals of periodic oscillations with a different phase. This mechanism also explains why the transition to chaos leads to a notable increase in the measure of the regions in Γ where sensitive dependence to initial data occurs; such an increase is visually obvious when comparing Fig. e, f with Fig. g, h. As ϵ increases further, the duration of the intervals of constant phase decreases, and the oscillations tend to become more genuinely aperiodic, as seen in Fig. d.

This behavior is similar to the intermittency found in autonomous dissipative systems e.g.,, in which a trajectory switches back and forth from periodic to aperiodic oscillations provided a certain control parameter of the system – e.g., the amplitude of the steady, time-independent forcing – crosses a given threshold. In our nonautonomous system, the amplitude of the periodic forcing ε plays a similar role. This transition to chaos induced by time-dependent forcing appears, therefore, to be directly linked to the existence of regions in phase space in which sensitive dependence to initial data occurs in the limit of periodic solutions. Thus, the chaotic behavior merely due to the time-dependent nature of the forcing can be traced back to the apparently paradoxical property of the autonomous system that was emphasized in Sect. . This striking observation deserves to be analyzed in greater depth in future studies.

Recognizing qualitatively whether a PBA is chaotic or not is relatively simple; e.g., this can be done through the heuristic arguments illustrated in Figs.  and and through those outlined in the previous subsection and illustrated in Figs. –. But how does one characterize the transition from periodic to chaotic dynamics as a control parameter, such as the amplitude ε of the periodic forcing, changes?

We have already discussed in Sect.  the limitations of using the mean finite-time Lyapunov exponents. Here we propose a new, simple and robust method that is particularly useful in our periodic-forcing case, but can be applied also to any autonomous system and even to aperiodically forced systems; the latter situations will be addressed in Sect. . In Sects.  and , we have relied on the mixing properties of chaotic dynamics, as measured by the system's entropy Sϑ, to recognize the occurrence of chaotic behavior. Now we rely on the emergence of aperiodic signals from a subset of Γ; this subset will necessarily be contained in the region where sensitive dependence on initial data occurs, i.e., where σ>1.

The most obvious approach would be to compute the power spectrum of each trajectory. Periodic signals can, however, be quite complex, as seen, for instance, in Fig. a, b; this complexity makes it quite difficult to identify a parameter whose value will distinguish, accurately and reliably, between periodic and chaotic dynamics, based solely on the Fourier spectra of a finite number of finite-length trajectories.

We propose a simpler alternative method that takes advantage of the ensemble simulations carried out to obtain the PBAs numerically. Let Ψ‾3 and ζ3 be the mean and root-mean square values of Ψ3(X,Y,t), and consider the centered and normalized anomaly time series Ψ̃3(X,Y,t) and Ψ̃3X′,Y′,t, of Ψ3, with 7Ψ̃3=Ψ3-Ψ‾3ζ3, where X,Y and X′,Y′ are two points in Γ that are near to each other, and from which these two time series emerge at t=0. We can then compute the cross-correlation between the two signals, after removing the initial transient, as c(X,Y,τ)=81T∗-2T∫TT∗-TΨ̃3X,Y,tΨ̃3(X′,Y′,t+τ)dt; here T∗=400 years is again the maximum integration time, and -T≤τ≤T, while T=50 years; once more, the following results are independent of T, provided it is sufficiently larger than the typical timescale of the phenomenon. Note also that, in the above definition, we have dropped the dependence of c on (X′,Y′) for the sake of conciseness.

Now, if σ<1, the two signals are periodic and virtually coincident, as seen, for instance, in Fig. a, c. Hence, defining the maximal cross-correlation by 9Θ(X,Y)=max⁡cX,Y,τ:τ∈-T,T, one will have Θ(X,Y)≅1, with Θ being attained at τ=0.

However, if σ>1, there are two possibilities:

either the PBA is not chaotic, in which case all couples (X,Y) and (X′,Y′) yield two periodic and virtually equal signals, apart from a finite phase difference, as seen, for instance, in Figs. b and a, b; in this case, again, Θ≅1, which will now occur at some lag τ≠0 that depends on the phase difference;

This alternative is illustrated in Fig.  for the four couples of trajectories plotted in Fig. a–d, all of which were initialized in ϑ2, where σ>1.

It is then useful to analyze the maps of the parameter Θ. Figure a–d show Θ for the four cases of Figs. –. In the two cases that we have already identified as nonchaotic, Θ varies within a range of values [0.95,1.1] that lies very close to unity, as expected; see Fig. a, b. There is only a small neighborhood of P0=(-16,-83) in which Θ is very small: this is because P0 in the autonomous case is a fixed point; see Fig. .

The two cases that we have identified as chaotic appear here as Fig. c, d, and in them Θ exhibits in fact smaller values. These values lie in the range [0.2,0.8] for a large subset of the domain, where σ>1, as shown in Fig. g, h. Regions in which σ>1 but Θ≃1 are present as well, but the corresponding trajectories are nonetheless unstable once they have converged onto the PBA, because they will always pass sufficiently near trajectories that are chaotic, thanks to the mixing properties of the latter.

In summary, if Θ∼1 everywhere (yellow colors) we have an NPBA, whereas if Θ yields values that are sufficiently smaller than unity (grey colors) then we have a CPBA.

To summarize in a clear and simple way the information provided by the values of both σ and Θ, we introduce the integer-valued parameter Φ, defined as follows: 10ΦX,Y=1 if σ⩽1(yellow),2 if σ>1andΘ>Θ0(green),3 if σ>1andΘ⩽Θ0(red). Here Θ0 is a threshold value and ΦX,Y is plotted in Fig. e–h for Θ0=0.8.

As an example of the usefulness of Φ, let us note that the σ maps in Fig. f and g are fairly similar, except for the more extended warm-color region, where σ>1, in the second map. Recall, however, that the meaning of σ>1 is profoundly different if the system is chaotic, in which case mixing is present, as opposed to when it is not, in which case sensitive dependence to initial data concerns only the phase of the signal and is not accompanied by mixing. This ambiguity is resolved by the use of the step function Φ: if ϵ=0.05 (Fig. f), sensitive dependence to initial data, i.e., σ>1, yields the value of Φ=2 (green regions), since Θ≃1, which tells us that the system is not chaotic. On the contrary, if ϵ=0.1 (Fig. g), regions with Φ=2 (in red) appear within the green regions: this implies low Θ values and therefore chaos.

We conclude by analyzing the transition from NPBAs to CPBAs via a suitable function of the control parameter ε: this metric is provided by the average value ΘΓ of Θ over Γ. The graph of ΘΓ(ε) in Fig.  is obtained by performing many ensemble simulations of system trajectories with many distinct values of ε; the latter values are chosen to lie closer to each other, where the variation in ΘΓ(ε) is stronger. An abrupt transition from NPBAs, with ΘΓ≃1, to CPBAs occurs at εc≅0.09. Many additional analyses (not shown) for values just below and above εc confirm that this is in fact the critical value beyond which chaos sets in.

The diagnostic method proposed in Sect.  to monitor the transition from NPBAs to CPBAs in periodically forced systems relies on two properties: (i) in an NPBA all trajectories are periodic, and (ii) in a CPBA diverging aperiodic trajectories emerge from a subset of Γ, in which necessarily σ>1. Thus, the same cross-correlation-based method can obviously be applied to an autonomous system as well. The method's application to the autonomous model studied in Sect.  will shed new light on the periodic vs. chaotic character of its solutions.

The graph in Fig.  shows ΘΓ(γ) and is obtained, like that of Fig. , by performing many ensemble simulations, each with a different value of γ, rather than ε, which equals zero in the present case. The first thing to notice is the sudden drop of ΘΓ at γ=γc=1, where a global bifurcation separates small-amplitude limit cycles from large-amplitude relaxation oscillations, as shown in and in Sect.  here. In addition, investigated the stochastic version of this deterministic tipping point in the case of random forcing.

The chaotic nature of the attractor for γc is illustrated in Fig. . For γ=0.99, the Ψ3 time series exhibits the typical small-amplitude, purely periodic behavior studied in Sect. , while for γ=1 both small- and large-amplitude oscillations occur irregularly in the same time series. The behavior at γ=1.01 illustrates the return to more regular behavior.

Thus, chaotic dynamics occurring in an extremely restricted γ range separates two different types of limit cycles. Figure  shows this dramatic transition in terms of σ: the chaotic nature of the flow for γc=1 is such that the warm-colored regions in which σ>1 overwhelm the cold-colored regions, as in Fig. d, where γ=1.35.

For γ>1, the system is not chaotic – except for limited γ intervals centered at γ≃1.25 and γ≃1.335 – until a new abrupt drop of ΘΓ at γ=γ0=1.3475, shown by a dashed black line in Fig. . This drop signals the presence of chaotic attractors beyond γ0; in particular, the chaotic case γ=1.35, shown by the solid green line and discussed in Sect. , lies just after this transition. It is worth noting that large fluctuations dominate the chaotic regime.

Our cross-correlation diagnostics have been shown to apply to both periodically forced and autonomous systems. Its validity, however, is even more general, since it extends to a large class of aperiodically forced systems as well. We choose the model setup of to illustrate the latter possibility. A thorough analysis of this application is beyond the scope of the present study: we will therefore limit ourselves to analyzing the basic aspects of the problem and leave the details for a future investigation.

considered the same system – governed by Eq. (), and within the same parameter regime adopted here and in . The forcing, though, was aperiodic and given by the following: 11Gt=γ1+ε′f(t), where ε′>0 is a dimensionless coefficient and f(t) is a normalized, fixed realization of an Ornstein–Uhlenbeck process that has been smoothed to resemble multi-annual wind-stress forcing of the midlatitude oceans' double-gyre circulation. Figure  shows G(t) for γ=1 and ε′=0.2.

Figure  shows the evolution of two initially nearby trajectories emerging from P2, along with the corresponding cross-correlation, for γ=1.1, with ε′=0.05 in panels (a–b) and ε′=0.20 in panels (c–d); both cases have σ>1 and the corresponding time series of PΨ3 are plotted in Fig. 4h, j of . The case ε′=0.2 corresponds to the CPBA analyzed in detail by . The chaotic character of the solution is clearly visible from Fig. c; the cross-correlation between the two signals is plotted in Fig. d and it is accordingly small.

On the contrary, the case ε′=0.05 corresponds to an NPBA: the two signals in Fig. a develop a large phase difference after the initial transient, but are virtually identical and remain coherent at all times. Now, unlike in Figs. b and a, b, the two signals are not periodic because they are modulated by the aperiodic forcing, but this is irrelevant; in fact, the nonchaotic character of the solution can still be highlighted by the corresponding cross-correlation in Fig. b, whose maximum value is Θ≃1, like in the autonomous and periodically forced case. Obviously, this is possible because the period of the modulated relaxation oscillation is much smaller than the timescale of the forcing; this is in fact the only condition required for the applicability of this diagnostic method to aperiodically forced systems.

We can therefore conclude that the parameter ΘΓ can be a valuable tool for monitoring the onset of chaos in aperiodically forced systems as well. For example, this diagnostic method can be applied to study the onset of chaos in systems that possess a drift mimicking global warming and other climate change scenarios as done, for instance, in.