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 ε. A new diagnostic method – based on the
cross-correlation between two initially nearby trajectories – is proposed
to characterize the transition between the two types of behavior. Transition
to chaos is found to occur abruptly at a critical value εc
and begins with the intermittent emergence of periodic oscillations with
distinct phases. The same diagnostic method is finally shown to be a useful
tool for autonomous and aperiodically forced systems as well.
Introduction and motivation
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 e.g.,.
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
, in which chaotic relaxation oscillations
emerge under the effect of an external periodic forcing. A few more recent
examples in the climate sciences include (i) transition to chaos due to
quasiperiodic forcing ; (ii) modification of the
autonomous transition by periodic forcing ; and (iii) the important contributions of Anna Trevisan and colleagues to the data
assimilation problem for chaotic systems, in which the data stream can be
seen essentially as a time-dependent forcing .
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 .
Our study focuses on a four-dimensional nonlinear spectral ocean model
, which is subjected to periodic forcing, chosen as the
simplest form of time dependence. Cases will be considered that are
nonchaotic in the autonomous limit, so that the chaos that emerges in the
system is strictly associated with the nonstationarity of the forcing.
The study makes use of ensemble simulations performed with many initial
states distributed in a given subset of phase space, following the
methodology of and . The overall idea is
that the relevant information in the climate system must be derived from
statistical analyses of an ensemble of different system trajectories, each
corresponding to a different initial state, provided that the corresponding
trajectories have converged to the system's time-dependent attractor.
Such an attractor is called a pullback attractor PBA; e.g., in the mathematical
literature and a snapshot attractor e.g., in the physical literature; it provides the natural extension to
nonautonomous dissipative dynamical systems of the classical concept of an
attractor that is fixed in time for autonomous systems. A global PBA is
defined as a time-dependent set A(t) in the system's phase space
that is invariant under its governing equations, along with the equally
time-dependent, invariant measure μ(t) supported on this set, and to
which all trajectories starting in the remote past converge
. In the deterministic case, it is understood that
A(t) depends also on the particular forcing, say F(t), that is
being applied, but this dependence is usually not kept track of in the
notation. In the random case, the PBA is called a random attractor, and the
dependence on the specific realization ω of the noise process is often
included in the notation, as A(t,ω).
rigourously proved that a weakly dissipative nonlinear model
like the one used there and herein does possess a global PBA, subject to mild
integrability conditions on the forcing. In the present study, the numerical
approach used for the systematic investigation of the system's PBAs follows
and . Further diagnostic tools will be
introduced for the present periodic-forcing setup, and a new diagnostic tool
will also be proposed to monitor the onset of chaos in our nonautonomous
system.
The paper is organized as follows. In Sect. , the
mathematical model is described. In Sect. , the main
properties of the autonomous system are summarized and an apparent paradox
related to the sensitivity to initial states in the periodic regime is
discussed. In Sect. , the results obtained for the
periodically forced system are presented and discussed; the new
cross-correlation-based method specifically formulated to characterize the
onset of chaos is introduced and applied to the specific case at hand. This
method helps characterize the transition to chaos as the amplitude of the
periodic forcing increases, as well as document the coexistence of local PBAs
with chaotic and nonchaotic behavior within the model's global PBA. In
Sect. , the same method is shown to be a useful tool also for
autonomous and aperiodically forced systems. Finally, in
Sect. the results are summarized and conclusions are
drawn. An appendix illustrates in greater detail the coexistence of local
PBAs that are chaotic and nonchaotic in the setting of a periodically forced
Van der Pol–Duffing oscillator.
Model description
The highly idealized model of the oceans' wind-driven, double-gyre
circulation and references therein used in the present study
is governed by the system of four nonlinear, coupled ordinary differential
equations derived by ; ,
and used this model to
represent the ocean component in their low-order climate models. The author
introduced such a low-order model to complement the process studies on the
Kuroshio Extension's low-frequency variability previously carried out with a
much more detailed, primitive equation ocean model
e.g.,. The same low-order
model was later used by and to explore the
PBAs of the system in various cases. Here we merely review the main aspects
of the model; for all the technical details and parameter values, the
interested reader should kindly refer to .
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.
found such a reduced-gravity model to be a good
approximation for process studies of the Kuroshio Extension's low-frequency
variability. The flow is described by the streamfunction
ψx,t: like in the previous studies, ψ, the
horizontal coordinates x=(x,y) and the time t are
dimensionless, but the dimensional time will be plotted in all the time
series presented in this study to emphasize the typical timescales of the
oceanic phenomena under investigation.
A four-dimensional spectral model is obtained by expanding the streamfunction
in a rectangular domain as follows:
ψx,t=∑i=14Ψiti.
The orthonormal basis i is defined as follows:
1=e-αxsinxsiny;2=e-αxsinxsin2y,3=e-αxsin2xsiny;4=e-αxsin2xsin2y,
where α is a real positive constant. This basis satisfies the
free-slip boundary conditions along the borders of the rectangular domain; it
also captures the oceanic flow's westward intensification thanks to the
exponential factor first introduced in the two-dimensional model of
.
The four nonlinear coupled ordinary differential equations that govern the
evolution of the vector Ψt=Ψ1,Ψ2,Ψ3,Ψ4 can be written (see
) as
dΨdt+ΨJΨ+LΨ=Gtw.
The coefficients of the nonlinear and linear terms in the equation are
encapsulated by the rank-3 and rank-2 tensors J and L,
respectively, and the forcing is represented by the vector wsee. The forcing w is obtained from a
suitable double-gyre surface wind stress curl, while G is defined in the
present paper to be periodic,
G(t)=γ1+εsinωt,
with period Tp=2π/ω, while γ and ε are
positive dimensionless parameters.
To construct the system's PBAs, ensembles of forward time integrations are
carried out; each of these starts at t=0 from a different initial point
contained in a given subset Ω of the model's four-dimensional phase
space and ends at t=T∗=400 years; as shown in Figs. and
below, T∗ is much greater than the spinup time in
all cases. Following and , the
four-dimensional hypercube Ω is defined as follows:
Ψ1,Ψ2∈-70,150,Ψ3,Ψ4∈-150,120,
and the initial data are all chosen to satisfy Ψ1=Ψ2 and
Ψ3=Ψ4, i.e., to lie within a plane set embedded in Ω.
Behavior of the
autonomous ocean model, for which ε=0 in Eq. ().
(a) Bifurcation diagram, in which the range of the variable Ψ1
is plotted vs. the wind stress intensity γ; the two cases γ=1.1
and 1.35 discussed in the text are indicated with a red and a green
vertical line, respectively. (b) Limit cycle in the
(Ψ1,Ψ3) plane, plotted after spinup, that arises from Γ for
γ=1.1. (c) Map of the suitably scaled PDF of trajectories
given by qk; see text for details.
The ensembles will consist of 15 000 initial data at t=0 that are regularly
spaced either in Ω or in a small subset thereof. For the sake of
graphical representation, maps of various quantities will be plotted in the
rectangle Γ≡{-70≤Ψ1≤150,-150≤Ψ3≤120}⊂Ω that lies in the Ψ1,Ψ3 plane. In the discussion of the results, we will refer, for the sake of
simplicity and concision, to the model's trajectories as being defined in the
Ψ1,Ψ3 plane but, naturally, the actual
trajectories evolve in the full four-dimensional phase space.
The autonomous systemThe autonomous model's attractors
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. .
Distinct autonomous regime behavior for (a, c)γ=1.1 and (b, d) for γ=1.35. (a, b) Time evolution
of PΨ3 for (a)γ=1.1 and
(b)γ=1.35. (c, d) Maps of the mean normalized
distance σ for (c)γ=1.1 and (d) for
γ=1.35; the points P1=(85,100) and P2=(25,5) appear in the
panels (c) and (d), respectively. Note the different scales
in the two maps; 15 000 trajectories, with regularly spaced initial points
in Γ, were used for both maps.
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=log101000×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)=log10(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.
Typical behavior of time evolution of Ψ3(t) for different
values of the parameter γ and different initial points in Γ.
(a) Two trajectories obtained for γ=1.1 and initialized at
P1 (red line) and at a nearby point (blue line). (b) Same as in
panel (a), but for two trajectories starting from the point P2
(red) and near it (blue). (c, d) Same as in panels (a) and
(b) but for γ=1.35.
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.
Chaotic and nonchaotic behavior of the autonomous model, for
time-independent forcing intensity γ=1.1 (red) and γ=1.35
(green), respectively. Typical behavior of (a, c) the trajectories
in the model's phase plane (Ψ1,Ψ3) and (b, d) of the
model's entropy Sϑ(t). (a) Intersection with the
(Ψ1,Ψ3) plane at t=300 years of 15 000 trajectories emanating
from the small square box ϑ1 of width ΔΨ=2 and centered
at the point P1 (black dot), for γ=1.1 (red dots, enclosed in the
red circle) and for γ=1.35 (green dots); for the blue dots see the
text. (b) Time evolution of the corresponding entropy
Sϑ1 for γ=1.1 (red line) and γ=1.35 (green
line). (c, d) Same as panels (a) and (b), but for
the initial box ϑ2 centered at P2, likewise shown as a black
dot in panel (c). For the evolution of the points contained at
t=300 years in the black rectangle of panel (c), see
Fig. below.
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 :
Sϑt=-∑k=1Npklnpk.
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.
An apparent paradox
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.
Chaotic and nonchaotic behavior of the autonomous model.
(a) Evolution of the 1774 points (red dots) contained at
t=300 years in the black rectangle of Fig. c for
γ=1.1; the attractor is illustrated by the entire set of
15 000 points, shown in light red. (b) Same but for the 135 points
(green dots) that lie within the same rectangle for γ=1.35; in this
case, the attractor composed of the 15 000 points is shown in light green.
Due to the chaotic nature of the dynamics in this case, only one snapshot,
after 25 years, is drawn.
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.
The periodically forced system
For the idealized double-gyre model governed by Eq. (), we have
seen that the autonomous system given by ε=0 in
Eq. () exhibits a limit cycle when γ=1.1; this limit
cycle corresponds to the large-amplitude relaxation oscillation of
Fig. b. However, showed that,
when the same model, with the same value of γ, is subjected to
periodic forcing with ε=0.2 and Tp=30 years, it exhibits
chaotic, cyclostationary and cycloergodic behavior; see Figs. 2 and 3 therein
and the related discussion. To understand the transition to deterministically
chaotic behavior induced by the forcing, we will now apply in
Sect the methodology used in Sect.
to the attractors corresponding to γ=1.1 and Tp=30 years
across the intervening parameter range 0≤ε≤0.2. In
Sect. , the transition to chaos induced by the
periodic forcing will be analyzed in greater detail through an additional
method here developed explicitly for this purpose.
We know from the rigorous proof of Appendix A that our
idealized ocean model possesses a global PBA, in the general case of
time-dependent forcing, whether periodic or aperiodic. PBAs are, in fact,
time-dependent mathematical objects that characterize the asymptotic behavior
of a nonautonomous dissipative dynamical system e.g.,Fig. 2.
Transition from a periodic to a chaotic PBA as the amplitude
ε of the periodic forcing in Eq. () increases.
(a–d) Time evolution of PΨ3 and (e–h) maps of
σ in the (Ψ1,Ψ3) plane for γ=1.1, Tp=30 years and ε=0,0.05,0.1 and 0.2, respectively.
Panels (d) and (h) correspond to the reference case studied
by .
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 {u(t)→u(t+T)}, where u is the variable and T
is the period. In the particular case in which the system's driver is
periodic, so is the PBA; see Sect. 2.3.2 of for a rigorous
proof. In contradistinction, a “normal” – i.e., forward rather than
pullback – attractor visualized in the embedded space (u(t),u(t+T)) and
built by using points along a long trajectory {u(t):t0≤t≤t0+T} is static and therefore contains less information than the
corresponding PBA, no matter how long the interval T may be.
Furthermore, PBAs built from ensembles of initial data allow us to visualize
in one single picture the coexistence of different types of dynamical
behavior in terms of disjoint PBAs; see, for instance, Fig. in
Appendix A herein. Besides, the PBA framework is useful for the visualization
of fractal structures that arise when noise is superimposed to the periodic
forcing. A stroboscopic map analysis may not easily reveal such fractal
features; see , as well as Sect. 3.4 of .
The pullback attractors of the forced system
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.
Intersection with the (Ψ1,Ψ3) plane at t=400 years
(magenta dots) of 15 000 trajectories emanating from Γ at t=0 for
γ=1.1 and Tp=30 years. The complete set of the initial
points covering Γ is in blue. (a)ϵ=0,
(b)0.05, (c)0.10, and (d)0.2.
(e–g) The corresponding entropy Sϑ2 is plotted, along
with the number no of occupied cells.
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.
Same as Fig. , but for 15 000 trajectories emanating
at t=0 from the small rectangle ϑ2 of width ΔΨ=2
centered at P2.
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.
Increasing instability of trajectories as ε increases.
Time evolution of Ψ3(t) for the trajectory initialized at the point
P2 (red line) and at a nearby point (blue line) for γ=1.1,
Tp=30 years, and (a–d)ε=0,0.05,0.1 and
0.2.
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.
Transition to chaos studied by a cross-correlation method
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
Ψ̃3=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
Θ(X,Y)=maxcX,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;
or the PBA is chaotic, in which case all couples yield two aperiodic and
significantly different signals, as seen, for instance, in Figs. d and c, d;
in this case, Θ will be substantially less than unity.
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:
ΦX,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.
Cross-correlation c(τ) between the two initially nearby
trajectories shown in Fig. a–d, computed for the centered
and normalized anomalies Ψ̃3, according to Eq. (), for
ε=0 (red line), 0.05 (orange line), 0.1 (green line) and
0.2 (blue line).
PBA diagnostics for γ=1.1 and Tp=30 years, with
ε=0,0.05,0.1 and 0.2, respectively, for the two sets of four
maps. Upper-row panels (a)–(d) show the field of
Θ(X,Y) in the (Ψ1,Ψ3) plane, as defined in Eq. ();
the color bar for the Θ(X,Y) values is shown to the right of each
panel, and it extends over the range [0.1,1.12]. Lower-row
panels (e)–(h) show the field of Φ(X,Y) in the
(Ψ1,Ψ3) plane, as defined in Eq. , with the threshold
value Θ0=0.8; here Φ=1 is colored yellow, Φ=2 is green
and Φ=3 is red. The corresponding maps of σ(X,Y) appear in
Fig. e–h, respectively.
Fixed point P0 of the autonomous case, with ε=0.
Time evolution of Ψ3 for the trajectory initialized at P0=(-16,-83) (blue line) and at a nearby point (solid red line) for
γ=1.1 and Tp=30 years.
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.
Further applications of cross-correlation diagnosticsApplication to the autonomous system
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.
Transition from periodic to chaotic behavior, illustrated by the
metric ΘΓ plotted vs. the
amplitude ε of the periodic forcing in Eq. ();
γ=1.1 and Tp=30 years.
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.
Same as Fig. , but for the autonomous model with
ε=0 and the amplitude γ of the time-independent forcing on
the abscissa. The vertical red and green lines denote the periodic and the
chaotic cases, respectively, that were analyzed in Sect. ;
see again Fig. . Please see the text for the interpretation of
the dashed lines corresponding to γc=1 and γ0=1.3475.
Critical transition in the autonomous system at γc.
Panels (a) and (d), (b) and (e), and
(c) and (f) correspond to γ=0.99,1.0 and 1.01,
respectively. (a–c) Time evolution of Ψ3 for the trajectory
initialized at P1 (red line) and at a nearby point (blue line);
(d–f) same but for trajectories initialized at P2.
Critical transition in the autonomous system at γc,
illustrated by maps of the mean normalized distance σ in the
(Ψ1,Ψ3) plane for γ=0.99, 1.0 and 1.01.
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.
Application to an aperiodically forced system
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:
Gt=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.
Aperiodic forcing G(t)/γ of the idealized ocean model –
defined by Eq. () herein, and plotted using the value
ε′=0.2, as adopted in .
Role of the cross-correlation diagnostics in characterizing chaotic
behavior for an aperiodically forced system, given by Eqs. () and
(); γ=1.1. (a) Time evolution of Ψ3
for ε′=0.05 in Eq. () and
(b) corresponding cross-correlation. (c, d) Same as
panels (a) and (b) but for ε′=0.2. The
trajectories initialized at P2 are in red and those initialized at a
nearby point are in blue.
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.
Summary and conclusions
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. , the autonomous system,
following up on the work of , who obtained its
governing equations and analyzed their solutions. Here, ensemble simulations
based on a large number of initial data and the calculation of the system's
entropy allowed us to determine novel and interesting features of the system
subject to steady forcing.
To do so, we used the metric σ that was introduced by and
measures the time average of the distance between two trajectories that are
very close at t=0 on a subset Γ of phase space. The analysis based
on this metric yielded regions in Γ with σ values that can be
either larger or less than unity. The spatial structure of these regions
(cf. Fig. c, d here) is very similar to that of the nonautonomous
case investigated by , as seen in Fig. 6 therein. This similarity
suggests that the nonautonomous behavior of a dynamical system is profoundly
influenced by the convergence properties of trajectories initialized off the
attractor in the autonomous case; this finding, in turn, implies that
ensemble simulations are very helpful in studying said properties.
Next we investigated, still in the autonomous case, the apparent paradox of
regions with σ>1 coexisting in the periodic regime of γ=1.1
with the expected regions of σ<1. A large number of trajectories
emanating from the small square box of Fig. c, for which
σ>1, evolves into the extended red line shown in the same figure at a
given time, t=300 years: this line belongs to the periodic attractor shown
in Fig. b and the entropy evolution along it oscillates
periodically (cf. Fig. d); but it is clearly distinct from the
chaotic attractor apparent as the green cloud of points in the same figure
panel. Further evidence for the stability of the trajectories in this
periodic case with σ>1 is provided by Fig. a.
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 studied the onset of chaos in the system
subject to the periodic forcing given by Eq. () in the case of
γ=1.1, in which the system is nonchaotic in the autonomous limit, so
that chaos is induced by the periodic forcing. The PBAs were analyzed at
first for four different sinusoidal-forcing amplitudes, with ϵ=0,0.05,0.10 and 0.2, while using the same γ=1.1 and Tp=30 years.
We found that the first two cases, namely ϵ=0 and 0.05, are nonchaotic, while the other two, namely ϵ=0.10 and 0.20, are chaotic.
A large number of trajectories emanating from the small square box where
σ>1 in Fig. a–d evolves – depending on the value of
ε – into two very different fixed-instant subsets of the
model's PBA, with the snapshot taken at t=400 years, i.e., after convergence
of the trajectories to the PBA. In the first two cases, this snapshot is a
curved-line segment that belongs to the PBA, while in the latter two cases it
covers the whole PBA, due to the typical mixing property of chaos.
An analysis of the trajectories, as shown in Fig. , for
instance, indicates that the transition to chaos occurs via an intermittent
emergence of periodic oscillations with different phases; see again
Fig. c. We have shown that, for values of ϵ in the
chaotic regime just above the transition, the periodic character of the
system is still predominant, but the external forcing is now sufficiently
strong to cause a trajectory's occasionally shifting to a phase space region
in which a different phase prevails: since the shifts are very sensitive to
the initial data, the result is a chaotic trajectory characterized by
irregular jumps of the oscillatory solutions between distinct phases.
In Sect. , we introduced a novel diagnostic method
for the study of the transition between nonchaotic and chaotic behavior as
the amplitude ϵ of the periodic forcing increases. The method's
basic idea is that in a nonchaotic regime any couple of initially nearby
trajectories emerging from Γ remain coherent at all times, while in a
chaotic regime aperiodic diverging trajectories emerge from a subset of
Γ. Hence, a systematic recognition of the character of all the
trajectories allows one to diagnose which of these two types of behavior
occurs or whether the two actually coexist.
A simple and robust way to do this is to compute the cross-correlation c(X,Y,X′,Y′,τ) at lag τ between two initially nearby trajectories
started at (X,Y) and (X′,Y′) in Γ, then compute its maximum value
Θ(X,Y) over τ. If Θ(X,Y)≃1 everywhere in Γ,
then the system is nonchaotic and is therefore periodic under periodic
forcing; on the contrary, if Θ is appreciably smaller than unity in
some subset of Γ, the system is chaotic. The diagram of the average
ΘΓϵ of
Θ(X,Y) over Γ, as plotted in Fig. , reveals an
abrupt transition to chaos at ϵc≃0.09.
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 ΘΓγ in Fig. reveals that
chaotic dynamics occur at first within an extremely restricted range
centered at the global bifurcation point γ=γc=1, which
separates small-amplitude, fairly smooth oscillations below γc
from large-amplitude relaxation oscillations above it. A considerably broader
range of chaotic behavior occurs in the autonomous case for values of
γ greater than a threshold γ0=1.3475.
We have then applied the cross-correlation-based method to the aperiodically
forced system studied by . Our results show that, in fact, this
method can be applied to systems subject to aperiodic forcing when the
system's intrinsic periodicity and the characteristic timescale of the
external forcing are sufficiently well separated from each other, which is
the case in ; cf. Fig. here. Once more, the
cross-correlations in Fig. c, d agree remarkably well with the
character of the model trajectories in Fig. a, b.
Finally, the coexistence of local PBAs with chaotic vs. nonchaotic behavior
within a global PBA – as first described by in the
aperiodic-forcing case – was confirmed here for the periodically forced
case; cf. Figs. and . This
situation was explored in greater depth in Appendix
A for an even simpler, weakly dissipative nonlinear model, namely a
Van der Pol–Duffing oscillator e.g.,, and
additional references were given for this type of PBA bistability.
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.
Data availability
No data sets were used in this article.
Coexistence of pullback attractors in a Van der Pol–Duffing oscillator
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. and . In
complex autonomous systems, several local attractors may coexist for a given
set of the system's parameters; each of these attractors possesses attracting
sets of initial data that are typically separated by fractal boundaries
. Basins of attraction with fractal boundaries have
consequences for predictability: uncertainties in the initial state x0 may
result in different types of dynamical behavior, depending on which basin
x0 lies in; see, for instance, . When the number
of the coexisting attractors is two, one speaks of bistability, and
multistability refers colloquially to more than two coexisting attractors.
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 . ,
for instance, already demonstrated the usefulness of this framework in the
context of bistability.
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 e.g.,. In this
situation, it is expected that the number of coexisting attractors exceeds
any fixed bound in approaching this limit, as documented for various
nearly integrable maps and flows; cf. ,
, ,
and
, and references therein.
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:
x¨=μ(1-x˙2)x-bx3+Fsin(ωt),
where μ, b, F and ω are positive constants that determine the
dynamical behavior of the system. This system is nonautonomous and its PBAs
are analyzed hereafter in the (x,x˙) plane. This nonlinear ODE arises
in various applications such as in engineering, electronics, biology and
neurology
. It
combines the nonlinearity of the dissipation -μx˙2x, which
characterizes the oscillator with that of the internal
force -bx3, which characterizes the
oscillator.
Multistability was already numerically documented for
Eq. () by relying on Poincaré maps
. In our calculations,
we have followed and assumed μ=0.2, F=b=1.0 and ω=0.955. While this parameter regime does not correspond to
the limit of vanishing dissipation that was mentioned above, it still allows
for a coexistence of PBAs, given a careful choice of initial states.
Coexistence of local forward attractors. (a) Quasiperiodic
forward attractor (blue) and chaotic attractor (red). (b) Power
spectrum associated with the quasiperiodic orbit (blue) and the chaotic one
(red).
Coexistence of local PBAs. The initial data leading to quasiperiodic
(and chaotic) orbits are taken from the small domain D2 (and D1)
described by Eq. (). The snapshot of the PBAs shown here is taken
at the fixed time t=2000.
Return maps of the minima of x(t) for (a) the
quasiperiodic orbit shown in blue in Fig. (a), and for
(b) the chaotic orbit shown in red in Fig. a; see text
for details.
The numerical protocol followed to analyze multistability for
Eq. () in terms of PBAs is described next. First, the
initial data have been drawn uniformly in the two disjoint domains D1 and
D2 of the (x,x˙) plane, with
D1=[-2.5,-1.5]×[-2.5,-1.5],A2D2=[-0.34,-0.33]×[-0.34,-0.33].
A total of 6000 initial data from each domain were propagated according to
Eq. (). The ODE was integrated using a Runge–Kutta
fourth-order method with a constant time step Δt=10-2, generating
a total of 12 000 trajectories and keeping 106 data points for each,
after removal of the transient.
The majority of the initial data taken in the smaller domain D2 leads to a
quasiperiodic orbit, while each of the 6000 initial data taken in D1
leads to a chaotic trajectory – as do a few “rare” initial data
from D2. An example of such a quasiperiodic trajectory is shown in blue in
Fig. a, within the (x,x˙) phase plane. This blue
trajectory is superimposed upon a red, chaotic trajectory emanating from an
initial point taken in D1. The corresponding power spectra are shown in
Fig. b, with the same blue and red color coding. The chaotic
trajectory is clearly more diffuse within the phase plane than its
quasiperiodic counterpart, and its power spectrum is quite a bit noisier.
Both of these features are well known to be symptomatic of deterministic
chaos .
By allowing the quasiperiodic trajectories that emanate from D2 to evolve
up to t=2000, one obtains the set of blue points shown in
Fig. . Somewhat surprisingly, this set does not form a closed
curve: each blue dot in Fig. actually corresponds to the state
at t=2000 in the phase plane of a quasiperiodic orbit. One such orbit is
represented in blue in Fig. a, after removal of the transient
dynamics. Each blue dot in Fig. corresponds to a different
quasiperiodic orbit, whose frequency characteristics may change slightly from
one blue dot to another. All these quasiperiodic orbits share, however, a
spectral signature that resembles the one shown by the blue curve in
Fig. b.
To illustrate further the distinction between quasiperiodic and chaotic
orbits, the return maps for the minima of the x(t) variable have been
computed. As is well known e.g.,, if the
return map contains just one point, the solution is periodic in time, with
all minima having the exact same value, and the period of the oscillation can
be estimated by calculating the time interval between two consecutive minima.
If the return map contains continuous-looking curves that fill up with more
and more points as the length of the orbit increases, the solution is
quasiperiodic, while the presence of folds and self-similarity in the return
map provides strong evidence for chaotic solutions. For the blue and red
trajectories of Fig. a, we plot the corresponding return maps
in Fig. a and b, respectively. The two plots clearly
discriminate between the quasiperiodic nature of the former and the chaotic
nature of the latter solution.
The initial data taken in D1, when allowed to flow according to
Eq. (), lead to a totally different local PBA that is
formed by the red points shown in Fig. . Although the
approximation of this local PBA shown here is relatively sparse, one can
clearly discern the fact that its constitutive points are arranged according
to a stretching and folding pattern that is typical of nonlinear, chaotic
dynamics in the autonomous as well as in the nonautonomous setting
.
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. Fig. 17c. In the presence of noise, though,
the fine structure of the PBAs that results from stretching and folding in
phase space is still captured by the PBA framework , whereas a
Poincaré-map approach would lead only to a cloud of points with no
particular geometric structure. This statement was
numerically illustrated by in their Fig. 7, by contrasting the
upper-right panel vs. the six lower panels of that figure.
Finally, we emphasize that it is not the disjointness of the two domains,
D1 and D2, that leads to the two distinct types of PBA, chaotic and
quasiperiodic.
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 of D2 is
small, it still contains initial data whose evolution lands within the local
PBA associated with chaos, i.e., with the other red points shown in
Fig. .
Author contributions
SP 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.
Competing interests
Stefano Pierini is a member of the editorial board of the
journal. The authors declare no conflict of interest.
Special issue statement
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.
Acknowledgements
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
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