A four-dimensional nonlinear spectral ocean model is used
to study the transition to chaos induced by periodic forcing in systems that
are nonchaotic in the autonomous limit. The analysis relies on the
construction of the system's pullback attractors (PBAs) through ensemble
simulations, based on a large number of initial states in the remote past. A
preliminary analysis of the autonomous system is carried out by investigating
its bifurcation diagram, as well as by calculating a metric that measures the
mean distance between two initially nearby trajectories, along with the
system's entropy. We find that nonchaotic attractors can still exhibit
sensitive dependence on initial data over some time interval; this apparent
paradox is resolved by noting that the dependence only concerns the phase of
the periodic trajectories, and that it disappears once the latter have
converged onto the attractor. The periodically forced system, analyzed by the
same methods, yields periodic or chaotic PBAs depending on the periodic
forcing's amplitude

Understanding the mechanisms that lead to the onset of chaos in dissipative
dynamical systems is of fundamental importance both from a cognitive
viewpoint and for the correct use of the mathematical models on which the
systems are based. Chaos arises in such systems as a control parameter in the
governing equations crosses a given threshold. A huge amount of work has been
devoted to analyzing the transition to chaos in the framework of autonomous
dynamical systems, i.e., in systems in which the external forcing and the
coefficients do not depend on time. The various routes to chaos in autonomous
dissipative systems – in the presence of time-independent forcing —
include period-doubling cascades, intermittency and crisis, quasiperiodic
routes, and global bifurcations

Nonautonomous dissipative dynamical systems represent a crucial extension of
autonomous systems for practical applications, since the external forcing in
most real systems – whether deterministic, random or both – depends,
typically, on time. Despite their importance, nonautonomous systems have
received, until recently, less attention than autonomous systems. Transition
to chaos induced by time-dependent forcing has, nonetheless, been studied in
several significant cases. A classical example is the Van der Pol oscillator

The onset of chaos is analyzed here in the framework of nonautonomous
systems, which has been received rapidly increasing attention recently in the
context of climate dynamics

The study makes use of ensemble simulations performed with many initial
states distributed in a given subset of phase space, following the
methodology of

Such an attractor is called a pullback attractor

The paper is organized as follows. In Sect.

The highly idealized model of the oceans' wind-driven, double-gyre
circulation

The dynamics are governed by the evolution equation of potential vorticity in
the quasigeostrophic approximation on the beta plane for a shallow layer of
fluid, superimposed on an infinitely deep quiescent lower layer.

A four-dimensional spectral model is obtained by expanding the streamfunction
in a rectangular domain as follows:

The four nonlinear coupled ordinary differential equations that govern the
evolution of the vector

To construct the system's PBAs, ensembles of forward time integrations are
carried out; each of these starts at

Behavior of the
autonomous ocean model, for which

The ensembles will consist of 15 000 initial data at

We begin by analyzing some basic properties of the
autonomous system that will be useful in the subsequent investigation. The
bifurcation diagram of Fig.

Distinct autonomous regime behavior for

Figure

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

Typical behavior of time evolution of

In Fig.

The determination of the PBAs of the periodically forced system and the
application of the new qualitative and quantitative diagnostic methods
proposed in Sect.

The map of

Chaotic and nonchaotic behavior of the autonomous model, for
time-independent forcing intensity

In the chaotic case with

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

Figure

In the chaotic case

Figure

Finally, it is worth stressing that, since the forcing is constant, the range
of variability of the entropy in the chaotic case with

We conclude the analysis of the autonomous system by discussing an apparent
paradox. We have just seen that, in regions of

Chaotic and nonchaotic behavior of the autonomous model.

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

We must show, therefore, that the trajectories are stable for

The results (not shown) are consistent with the assumption above, but the
exponents are highly dependent on the time

Let us consider the points lying in the black rectangle shown in
Fig.

The stability of the trajectories under consideration is clearly demonstrated
in Fig.

In conclusion, our autonomous system becomes chaotic for sufficiently large
values of

Moreover, we have shown that, when

The existence of regions with

For the idealized double-gyre model governed by Eq. (

We know from the rigorous proof of

Transition from a periodic to a chaotic PBA as the amplitude

It is common, though, in the literature of periodically forced dynamical
systems to study the asymptotic behavior of such a system by an iterated
stroboscopic map

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

Intersection with the

Before proceeding with the analysis of the results in Fig.

Same as Fig.

Figure

To understand this difference better, we focused in Fig.

On the contrary, for

Increasing instability of trajectories as

Finally, it is instructive to visualize

The intermittent behavior seen in Fig.

This behavior is similar to the intermittency found in autonomous dissipative
systems

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.

We have already discussed in Sect.

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.

We propose a simpler alternative method that takes advantage of the ensemble
simulations carried out to obtain the PBAs numerically. Let

Now, if

However, if

either the PBA is not chaotic, in which case all couples

or the PBA is chaotic, in which case all couples yield two aperiodic and
significantly different signals, as seen, for instance, in Figs.

This alternative is illustrated in Fig.

It is then useful to analyze the maps of the parameter

The two cases that we have identified as chaotic appear here as
Fig.

In summary, if

To summarize in a clear and simple way the information provided by the values
of both

As an example of the usefulness of

Cross-correlation

PBA diagnostics for

Fixed point

We conclude by analyzing the transition from NPBAs to CPBAs via a suitable
function of the control parameter

The diagnostic method proposed in Sect.

Transition from periodic to chaotic behavior, illustrated by the
metric

The graph in Fig.

The chaotic nature of the attractor for

Same as Fig.

Critical transition in the autonomous system at

Critical transition in the autonomous system at

Thus, chaotic dynamics occurring in an extremely restricted

For

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

Figure

Aperiodic forcing

Role of the cross-correlation diagnostics in characterizing chaotic
behavior for an aperiodically forced system, given by Eqs. (

On the contrary, the case

We can therefore conclude that the parameter

In this paper, we studied the transition from nonchaotic to chaotic PBAs in a
nonautonomous system whose autonomous limit is nonchaotic, and in which,
therefore, chaos is induced by the periodic forcing. The illustrative example
chosen for this general problem was a low-order quasigeostrophic model of the
midlatitude wind-driven ocean circulation, subject to periodic forcing. The
model was described and connected with previous work in
Sect.

We first investigated, in Sect.

To do so, we used the metric

Next we investigated, still in the autonomous case, the apparent paradox of
regions with

We conclude that sensitive phase dependence on initial data in a periodic regime may be present if the trajectories are initialized off the attractor, but that it disappears once the trajectories have converged onto the attractor. Clearly, generic sensitive dependence on initial data is, therefore, a necessary but not sufficient condition for chaotic behavior.

In Sect.

We found that the first two cases, namely

An analysis of the trajectories, as shown in Fig.

In Sect.

A simple and robust way to do this is to compute the cross-correlation

This cross-correlation-based method has also been applied to the autonomous
system, in which the conditions required for the periodically forced case do
apply as well. The diagram of

We have then applied the cross-correlation-based method to the aperiodically
forced system studied by

Finally, the coexistence of local PBAs with chaotic vs. nonchaotic behavior
within a global PBA – as first described by

Overall, this paper provides additional insights into the complex and varied behavior that arises even in highly idealized atmospheric, oceanic and climate models from the interaction of nonlinear intrinsic dynamics with various types of external forcing. In addition, it stresses the importance of using the framework of nonautonomous dynamical systems and of their PBAs for a deeper understanding of this complexity and variety.

No data sets were used in this article.

The purpose of this Appendix is to provide further insight into the
coexistence of local PBAs with quite different stability properties, as
illustrated in the main text by Figs.

Our goal here is to illustrate how multistability of nonautonomous systems
manifests itself unambiguously through the existence of disjoint local PBAs.
In the case of periodically forced systems, such as those considered in this
article, similar results can be inferred, of course, from the analysis of
Poincaré maps. Nevertheless, the presence of other frequencies in the
internal dynamics, the external forcing or the noise may render the analysis
of Poincaré maps difficult, whereas the framework of PBAs naturally includes
such additional levels of complexity

Furthermore, the purpose of this Appendix is also to illustrate that multistability of local PBAs arises not only for our quasigeostrophic model, as discussed in the paper's main text. More generally, PBA coexistence occurs fairly often for externally forced systems, although a careful analysis of the flow's dependence on initial data may be required in practice in order to conclude on multistability.

A paradigm of multistability is provided by dissipative nonlinear systems
that become Hamiltonian in the limit of vanishing dissipation, as is the
case, for instance, in celestial mechanics

To illustrate this multistability phenomenon, we consider the following
periodically forced Van der Pol–Duffing oscillator, given by the
second-order nonlinear (ordinary differential equation) ODE:

Multistability was already numerically documented for
Eq. (

Coexistence of local forward attractors.

Coexistence of local PBAs. The initial data leading to quasiperiodic
(and chaotic) orbits are taken from the small domain

Return maps of the minima of

The numerical protocol followed to analyze multistability for
Eq. (

The majority of the initial data taken in the smaller domain

By allowing the quasiperiodic trajectories that emanate from

To illustrate further the distinction between quasiperiodic and chaotic
orbits, the return maps for the minima of the

The initial data taken in

The features that we find here to be exhibited by the local chaotic PBA are
highly reminiscent of those that were obtained, in this deterministic case,
by applying a standard Poincaré section analysis;
cf.

Finally, we emphasize that it is not the disjointness of the two domains,

Other such quasiperiodic PBAs exist (not shown), whereas only one chaotic PBA seems to exist for the parameter regime analyzed herein.

Indeed, as mentioned earlier, even though the area ofSP devised the study, carried out the computations and wrote the first draft of the paper. MG helped integrate the work into the broader picture of applying the theory of nonautonomous and random dynamical systems to the study of climate variability and climate change. MDC devised the appendix to further illustrate the coexistence of local PBAs that are chaotic and nonchaotic, and carried out the corresponding computations. All three authors contributed to the writing of the paper.

Stefano Pierini is a member of the editorial board of the journal. The authors declare no conflict of interest.

This article is part of the special issue “Numerical modeling, predictability and data assimilation in weather, ocean and climate: A special issue honoring the legacy of Anna Trevisan (1946–2016)”. It is a result of a Symposium Honoring the Legacy of Anna Trevisan – Bologna, Italy, 17–20 October 2017.

Stefano Pierini would like to thank the “Università di Napoli Parthenope” for having supported his visit to the University of California at Los Angeles in August 2017 through grants D.R. 539 (29-6-2016) and D.R. 953 (28-11-2016, DSTE315) and for the partial support provided by the grant DSTE315B. Stefano Pierini and Michael Ghil gratefully acknowledge partial support from the MOMA project (PNRA16-00196-B) of the Italian P.N.R.A. This work has been partially supported by the Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) grants N00014-12-1-0911 and N00014-16-1-2073 (Mickaël D. Chekroun and Michael Ghil). Mickaël D. Chekroun also gratefully acknowledges support from the National Science Foundation (NSF) grants OCE-1243175, OCE-1658357 and DMS-1616981.

This paper is dedicated to the memory of Anna Trevisan and to her contributions to the applications of dynamical systems theory to the climate sciences. Edited by: Juan Manuel Lopez Reviewed by: two anonymous referees