ESDEarth System DynamicsESDEarth Syst. Dynam.2190-4987Copernicus PublicationsGöttingen, Germany10.5194/esd-7-313-2016Early warning signals of tipping points in periodically forced systemsWilliamsonMark S.m.s.williamson@exeter.ac.ukhttps://orcid.org/0000-0002-4548-8922BathianySebastianhttps://orcid.org/0000-0001-9904-1619LentonTimothy M.Earth System Science group, College of Life and Environmental Sciences, University of Exeter, Laver Building, North Park Road, Exeter EX4 4QE, UKAquatic Ecology and Water Quality Management, Wageningen University, PO Box 47, Wageningen, the NetherlandsMark S. Williamson (m.s.williamson@exeter.ac.uk)13April20167231332627October20156November20154February201626March2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://esd.copernicus.org/articles/7/313/2016/esd-7-313-2016.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/articles/7/313/2016/esd-7-313-2016.pdf
The prospect of finding generic early warning signals of an approaching
tipping point in a complex system has generated much interest recently.
Existing methods are predicated on a separation of timescales between the
system studied and its forcing. However, many systems, including several
candidate tipping elements in the climate system, are forced periodically at
a timescale comparable to their internal dynamics. Here we use alternative
early warning signals of tipping points due to local bifurcations in systems
subjected to periodic forcing whose timescale is similar to the period of
the forcing. These systems are not in, or close to, a fixed point. Instead
their steady state is described by a periodic attractor. For these systems,
phase lag and amplification of the system response can provide early warning
signals, based on a linear dynamics approximation. Furthermore, the Fourier
spectrum of the system's time series reveals harmonics of the forcing period
in the system response whose amplitude is related to how nonlinear the
system's response is becoming with nonlinear effects becoming more prominent
closer to a bifurcation. We apply these indicators as well as a return map
analysis to a simple conceptual system and satellite observations of Arctic
sea ice area, the latter conjectured to have a bifurcation type tipping
point. We find no detectable signal of the Arctic sea ice approaching a local
bifurcation.
Introduction
The potential for early warning of an approaching abrupt change or “tipping
point” in a complex, dynamical system has been the focus of much research,
see for example , ,
and . Abrupt change in a
system can occur due to a bifurcation – that is, a small smooth change in
parameter values can result in a sudden or topological change in the system's
attractors. This extreme sensitivity of systems close to criticality is
familiar from studies of critical phenomena in statistical mechanics
and stability analysis in nonlinear
dynamical systems . Much work on the anticipation
of bifurcations from time series data, e.g. in ecosystems
, or the climate system , is based on methods that infer the time taken for the
system to recover to its steady state (the system's timescale) from
perturbations away from that steady state. The methods are usually based on a
clear separation of three timescales: (i) The timescale of the dynamics of
the system one wants to study, (ii) much faster processes than the timescale
of the system, usually thought of as the perturbations the system recovers
from and (iii) much slower processes than the timescale of the system which
govern the present system steady state. In addition, the system dynamics are
modelled as overdamped, the fast dynamics as a noisy, normally distributed
random variable of small variance and the slow dynamics as a constant,
control parameter. Provided these are good working approximations, critical
slowing down – the increase of the system's timescale, is expected prior to
a local bifurcation and can be detected by computing the lag 1
autocorrelation in a sliding window of a system's time series. An increasing
trend in autocorrelation shows the stability of the system is weakening or
equivalently, the system's timescale is increasing – which is a generic
feature of a system approaching a local bifurcation. Provided the variance of
the fast noisy process is constant, increasing variance of the system's time
series is also a good indicator of critical slowing down, although it is less
robust than lag 1 autocorrelation due to its dependence on the noisy process.
For many systems of interest one or more of the above assumptions may be
invalid . In particular, when the forcing of
a system has a comparable period to the timescale of the system, the forcing
cannot be modelled as a slow, constant control parameter or a fast, random
process; however they can still be thought of as a perturbation away from the
system steady state that one can measure the recovery time from, an
observation we exploit in this manuscript. These systems are particularly
relevant in the climate system where periodic forcing is a consequence of the
motion of the Earth relative to the Sun; for example, solar insolation
variation from the diurnal, annual or Milankovich cycles. These systems have
steady states described by periodic attractors rather than the simpler, fixed
point type attractors required for lag 1 autocorrelation and variance to be
used as early warning indicators.
In an elegant study computed the Fourier spectra of
noisy perturbations in systems with periodic attractors. Very close to a
local bifurcation, the dominant system timescale asymptotes towards infinity
causing the dynamics of the noisy perturbations away from the attractor to be
dependent only on the type of bifurcation and not on the details of the
system's specific equations. This observation allowed the author to classify
all codimension 1 bifurcations in an arbitrary periodic system by the
harmonics in the spectra of residuals. He called these early warning signals
noisy precursors.
A common method to study stability changes in periodic attractors is the
return or Poincaré map, see . Here, one converts
the continuous-time periodic orbit into the fixed point of a return map by
sampling the orbit once every period. One can then compute the usual fixed
point indicators for the resulting return map time series such as lag 1
autocorrelation and variance. Advantages of this approach include no
linearity requirement on the dynamics of the periodic attractor although
perturbations away from the attractor must be small enough to be treated
linearly. One can also handle systems with internally generated cycles rather
than those generated by external periodic forcing that we look at in this
manuscript. To detect any change in stability from the return map, the timescale of the system must also be greater than the period of the forcing. To
see this, imagine one samples the cycle to create a point in the return map
and then immediately after perturbs the system away from the stable cycle. If
the system timescale is shorter than the cycle period, which is determined
by the forcing period, the system will have recovered back to the stable
cycle before the system is sampled again for the next point in the return
map. The perturbation and its recovery will therefore be invisible to
stability analysis on the return map time series. It should be noted that
close to a local bifurcation the system timescale approaches infinity so
satisfying this requirement. However, if this requirement is met only over a
few cycles or less it will be very hard to detect.
With this limitation in mind we suggest alternative early warning signals of
approaching local bifurcations when the period of the forcing is similar to
the timescale of the system. We look particularly at sinusoidal forcing
since this approximates the variation of solar insolation well. However, the
method works for any periodic forcing and we give the derivation of the
general case in the Appendix. We demonstrate that increasing the system timescale as it approaches a local bifurcation shows up as an increasing phase
lag in the system response relative to the forcing. In addition, the
amplitude of the system response increases as well. These indicators, like
lag 1 autocorrelation and variance in fixed point attractor methods, assume
the linearised dynamics approximate the true nonlinear dynamics well. One
might ask how well the linear approximation works, especially near the
bifurcation, since bifurcations are strictly nonlinear phenomena. A
quantitative answer to this question can be provided by computing the Fourier
spectrum of the system's time series. In particular, as the system's
behaviour becomes more nonlinear, harmonics of the forcing period are
generated in the system response and their amplitudes may be obtained from
the system's Fourier spectra. Since the system response becomes more
nonlinear as one approaches the bifurcation, one can view the increasing
amplitude of harmonics as another early warning signal.
The paper is organised as follows: in Sect. the
early warning indicators used in the manuscript are introduced, namely the
system response phase lag and amplification as well as harmonic amplitudes.
We also review a common approach to periodic attractors, the return map,
which is complementary to phase lag and response amplification. In Sect. a periodically forced overdamped system in a double
well potential is used to illustrate the timescale separation problem and
the properties of the early warning indicators when a local bifurcation is
approached. In Sect. we apply the early warning
indicators to satellite observations of Arctic sea ice area, a system
conjectured to be approaching a local bifurcation. We conclude in Sect. .
Early warning indicators of local bifurcations in periodic systems
As previously mentioned in the introduction, the Arctic sea-ice has been
conjectured to be approaching a local bifurcation. Treating this system
approximately, one can think crudely of the slow control parameter as the
decrease of outgoing long-wave radiation, giving a warming trend in air
temperature as the Earth's atmospheric CO2 concentration increases. This
is a system that is forced periodically and deterministically by the annual
cycle of short-wave solar insolation. The system response is dominated by
this periodic forcing rather than small amplitude; random noisy forcing is also
present and system timescale is roughly the same order of the forcing
period. In this section, motivated by detection of local bifurcations in
systems like the sea-ice type from time series, we look for suitable methods.
Although this system has no clear separation of timescales with which to use
fixed point methods directly, the fact that this system has a large and
predictable perturbation one can measure the response to reduces the need for
the statistical methods (and therefore large numbers of data) required for
noisy perturbations and so the number of data in a time series becomes less of an
issue.
Students of physics or engineering will likely have solved the equation for
the forced damped harmonic oscillator and observed in the overdamped limit
that the phase and amplitude depend on the damping parameter (see for example
). In the following subsections we propose to
use this fact and phase lag and response amplification as simple
non-statistical indicators of system timescale. We demonstrate their
properties and their functional dependence on system timescale. These early
warnings are based on a linear dynamics approximation but by taking the
Fourier transform of the system response, one can also look at the magnitude of
the nonlinear response. This has two purposes, first one can check the linear
approximation is good and second, because bifurcations are strictly nonlinear
phenomena, the system response will become more nonlinear as one approaches
the bifurcation giving another early warning indicator that can be monitored.
The systems we concentrate on in this manuscript, relevant to externally
forced climate problems, have cycle periods determined by the period of
forcing and a one-way coupling from the forcing to the system and so are
special cases of periodic attractors. For these special cases, when forcing
period and system timescale are similar, phase lag and response
amplification are useful indicators. However, return maps are generally more
useful when treating more general periodic attractors. At the end of the
section we briefly review the method of return maps.
Phase lag and response amplification
We consider systems that can be described by
x˙=f(x)+D(t)
where f(x) is, generally, a nonlinear function of the system state scalar
variable x with forcing D(t) given by
D(t)=Dm+Dacos(ωt).Dm and Da are constants, ω=2πT is the angular
frequency and T is the period of the forcing. We have assumed any other
random, noisy external forcing to be very small and can be neglected. The
solution for a general form for D(t) is given in the Appendix, however here
we use sinusoidal forcing as this is most relevant for many climate systems
and we wish not to obscure the simplicity of the main result. x˙
describes the dynamics of a forced overdamped system. This is a nonautonomous
system whose state can be completely described by t and x. After some
time ts≫τ, where τ is the system timescale, the system will
settle into some sort of steady state, either an orbit or a fixed point whose
mean state x¯ is
x¯=1T∫tsT+tsx(t)dt.
We now Taylor expand f(x) to first order around x¯ so that
x˙≈a-xτ+Dacos(ωt),
where
a=f(x¯)-∂f∂x|x=x¯x¯+Dmτ=-1/∂f∂x|x=x¯
are the linearisation constants. We have assumed higher order terms such as
1n!∂nf∂xn(x-x¯)n, n≥2
are small relative to zeroth and first order terms so that the linearised
dynamics approximates the full nonlinear dynamics well. We show how to check
this approximation in Sect. . Assuming the
approximation is good, one can solve Eq. (). As
t≫τ the system settles into the orbit
limt≫τx(t)=aτ+Daτ1+ω2τ2cos(ωt+ϕ),
where the system response lags the forcing by phase ϕlag=ωtlag=-ϕ given by
ϕlag=arctan(ωτ),
that is, the phase lag is a function of the forcing frequency and the system
response timescale. One also notices that the system response, relative to
the forcing amplitude, Da, is amplified by a factor
τ1+ω2τ2,
which is also a direct function of ω and τ. The more general
derivation when D(t) can be any periodic function is given in the Appendix Sect. .
System nonlinearity and harmonic amplitude from Fourier analysis
By simply looking at the time series of the system response and the forcing
one can determine what the amplitude and phase lag are when the driving is of
the form Eq. () and the system response is
approximately linear without the need for statistics. However, the system is
essentially nonlinear and these nonlinear effects may become large near a
bifurcation or when the system is driven hard. By taking the Fourier
transform of the time series of the system response one can quantify how
large these nonlinear effects are. With a similar motivation
and calculated the
Fourier spectra of the perturbations, rather than the response, away from
periodic attractors very close to local bifurcations with noisy and weak
periodic modulation respectively.
Once the system has settled into an orbit of period T, the full nonlinear
response of an arbitrary system can be written as a Fourier series, a sum of
N sinusoidal functions with angular frequencies ωn=2πnT, amplitudes An and phases ϕn, i.e.
x(t)=∑n=0NAncos(ωnt+ϕn).
The n=0 component is a constant, the long-term mean of the response; the
n=1 component is the linear response of the system and the n≥2
components are the nth order harmonics and come about from the nonlinear
response of the system. Since the system has settled into a periodic orbit
the system must repeat itself every cycle. The only way the system can do
this is by adding harmonics to linear response. By looking at the ratios
An/A1 for n≥2 the nonlinear effects relative to the linear
approximation can be quantified. In practice the largest harmonics will
generally be the 2nd (n=2) and 3rd order (n=3) harmonics and provided
they are an order of magnitude (10 times, An/A1<10-1) less
than the fundamental harmonic, the linear analysis in the last section works
well. Calculation of the amplitudes, An, can be made via a Fourier
transform of the time series.
One may also expect subharmonics, components that have periods that are
integer multiples of the forcing period, to be observed in the system
response. Subharmonics are not possible in the systems we consider here due
to the dimensionality of the phase space.
Systems described by
Eq. () are completely described by the two-dimensional space of variables x and t. Recasting the nonautonomous
system in Eq. () as a two dimensional autonomous
system by identifying a new angular variable ϕ=ωt, the system is
then described by x˙=f(x)+D(ϕ) and ϕ˙=ω. The
resulting phase space (x,ϕ) is then cylindrical as ϕ is 2π
modular. If subharmonics are possible in the periodic system response the
trajectory must wind around the cylinder at least twice before repeating
itself. Such a trajectory implies it crosses itself which is not allowed due
to the existence and uniqueness theorem. Therefore subharmonics cannot exist
in the two-dimensional systems. This is of course not true for three- and
higher-dimensional systems.
Since the ratios An/A1 measure how nonlinear the system is one
expects these to increase as the system approaches a bifurcation. These
ratios can be plotted for a time series by taking the Fourier transform for a
data window consisting of an integer number of cycles and sliding this window
forward by one cycle recursively through the time series. The number of
cycles in the window must be large enough that the harmonics can be
satisfactorily resolved in the Fourier spectra. In addition, each cycle must
be sampled at a time interval Δt≤TNyquist/2 where
TNyquist is the minimum harmonic period you want to resolve.
Lag 1 autocorrelation of a return map
Provided the system timescale is larger than its period, τ/T>1, one can
use return maps to assess the stability of a periodic attractor. The return
map time series, generated by sampling the system response once every cycle,
allows one to apply fixed point statistical early warning indicators such as
lag 1 autocorrelation. It is the random forcing away from the periodic
attractor that this method infers timescale from, rather than response to
deterministic, periodic forcing. This is usually done by calculating the lag
1 autocorrelation for a sliding data window of the return map time series at
least as long as the system timescale but not so long that any increasing
trend in system timescale skews the autocorrelation estimate. It is also
desirable to have many points within this window as the standard error of the
estimate scales as 1/m, where m is the number of cycles (points)
within the window (see for a discussion). For
time series consisting of a small number of cycles this can be a limiting
factor.
Examples
We now demonstrate the early warning indicators in Sect. for different ratios of forcing period, T, (or
equivalently angular frequency ω) to system timescale τ. In
particular, we use a periodically forced double well potential as our main
system. This system has been extensively studied in the context of stochastic
resonance (see and
for reviews) as the simplest model of the phenomena when noise is also added.
Phase and amplitude have been investigated in this setting by
and . This literature is
largely concerned with resonance effects in transition probabilities between
the wells (finite barrier height between the wells) rather than the
anticipation of local bifurcations (barrier height tends to zero).
Our system which has one dynamical variable, x, and evolves according to
x˙=x-x3+D(t),
where overdots denote differentiation with respect to time, t and the
periodic forcing function D(t) is given by Eq. (). Equation () models a
nonautonomous nonlinear system, the overdamped limit of a Duffing oscillator
. When forcing is constant (ω=0) the
familiar, well-studied autonomous fold bifurcation is recovered (for example
see ). For ω=0, the solutions of x˙=0,
give the system's fixed points, x* (the nullclines) and number either one
or three depending on the value of Dm. One can evaluate the stability of
these fixed points by looking at the linearised dynamics close to the fixed
points, J(x*):
J(x*)=∂x˙∂x|x=x*=1-3x*2.
If J(x*) is negative, the fixed point is stable, if it is positive it is
unstable. In the region where three fixed points exist one finds a bistable
region, i.e. two points are stable while the third is unstable. The bistable
region has boundaries marked by the local bifurcations and these can be found
by solving J(x*)=0 for x* when the fixed point becomes neutrally
stable. One can also calculate the e-folding timescale of the system in
state x, τ from the Jacobian τ=-1/J(x). We will refer to the
e-folding time as the system timescale. As the system gets closer to the
bifurcation the system timescale will increase and tend to infinity at the
bifurcation. Early warning indicators are simply functions of J(x*) or
equivalently τ.
The dynamics of the system described by Eq. () in three different timescale regimes. Forcing
parameters are set to Dm=0, Da=1/2. In the upper panel system state x
is plotted against D(t). The black lines are the nullclines and the
coloured lines are the system responses for different periods of forcing. In
the lower panel x is plotted against the number of cycles, t/T, once the
system has reached a steady state. The dotted line is the forcing, D(t)
while the coloured lines are the system responses. The red line is for the
slow forcing limit, τ≪T, T=100π so ωτ≈1/100.
As the system timescale is much faster than the change in the forcing, the
system essentially “sticks” to the fixed points until they become unstable at
the bifurcations and jump to a different attractor. One can regard the system
response in two different ways: (i) a single periodic attractor giving
relaxation oscillations in a monostable region. (ii) Tipping between point
attractors by crossing local bifurcations in a bistable region. This tipping
causes the dynamics to be very nonlinear. The green line is the fast forcing
limit, T≪τ, T=π/100 so ωτ≈100. There are two
possible stable attractors for this set of values. As the system timescale
is much slower than the change in the forcing, the system essentially remains
static and all the dynamics come from the forcing itself. Although it is hard
to see in the figure due to the small amplitude system response, the lag
relative to the forcing is 1/4 of a cycle and the dynamics are approximately
linear. The blue line is the intermediate regime, τ∼T, T=π so
ωτ≈1 and there are two possible stable attractors for this
set of values. As the system timescale is approximately the same as the
period of the forcing, the system response is a competition between the
system's tendency to decay towards the nullcline and the forcing pushing it
away setting up a stable orbit. Notice that there is some phase lag and the
dynamics look approximately linear.
Period of forcing similar to system timescale, ωτ∼1
This regime, T∼τ, is the main focus study in this manuscript, when
the system responds on approximately the same timescale as the period of the
forcing. In this regime the dynamics are a balance between the system's
tendency to want to decay towards the fixed point and the forcing trying to
push it away. After some time, t≫τ, the system will settle into an
orbit rather than a fixed point due to the similarity of the timescales.
Just as there was a bistable region where multiple stable fixed points
existed for a single value of Dm when ω=0, analogously in the case
T∼τ multiple stable periodic attractors are possible given a fixed
set of values for Dm, Da and ω. The system state x is plotted
against D and against t as the blue line in Fig. . Which state the system settles in depends only
on the system's initial condition x(t=0). Local bifurcations are present in
this intermediate region, however they are local bifurcations between orbits
rather than fixed points. In this intermediate regime, one can neither place
the Dacos(ωt) part of D(t) in either the slow or fast processes
and therefore the assumptions of the usual fixed point early warning methods
are not strictly valid. This is, however, where phase lag and response
amplification are useful early warning indicators.
To illustrate the early warning indicators we fix the forcing amplitude Da
and the period T∼O(τ) and take Dm as a control parameter, slowly
varying from negative values towards the local bifurcation in the system
described by Eq. (). We expect to see the system
response become more phase lagged and amplified as we approach the local
bifurcation at Dm≈0.33 when approaching from the lower nullcline
solutions. We also expect the amplitude of the harmonics of the system
response to increase.
We choose to tip the system from one state to another by slowly altering the
mean of the driving Dm. Alternatively, the system could have tipped by
changing one of the other driving parameters such as amplitude Da or
frequency ω. Since the system response amplitude depends on Da and
ω and phase lag depends on ω, one must take this into account
when inferring system timescales from the indicators.
In Fig. the system is run forward in time,
linearly varying Dm from -2 to 2 across the bifurcation over about 25
cycles of the forcing period (for the values of the parameters see the figure
caption). Plotted in Fig. are phase lag and
amplitude of the system response prior to the bifurcation at around t/T=15.
Both are increasing as the bifurcation is approached due to the increase in
τ. Phase lag is calculated from the difference between the times of the
maxima in the forcing and the system response in each cycle. Response
amplitude is calculated by taking half the difference between the maximal and
minimal values in the system response in each cycle. Also plotted are the
ratios of the second and third harmonic amplitudes to the amplitude of the
fundamental harmonic with time using a sliding window of length 5 complete
cycles against the time at the end of the sliding window. The window needs to
be long enough to resolve the harmonics in the spectrum but short enough to
keep Dm approximately constant. For this example, where the harmonic
amplitudes (and the nonlinearity of the response) are quite small, 5 cycles
is the minimum to resolve the peaks. The sliding window is then advanced one
cycle in the time series and the harmonic amplitudes are calculated for this
new window. This process is iterated until the local bifurcation is reached
to produce the lower panel in Fig. which
shows both harmonic amplitudes increasing.
The dynamics of the system are described by Eq. () with varying Dm. Parameters are set to Da=1/2,
T=π (the same order as the system timescale ωτ∼1) and
Dm is varied linearly with time between -2 and 2 over about 25 cycles. In
the upper panel the black lines are the nullclines while the system response
is the blue line plotted against D(t). The orbit loses stability around a
mean value of D≈0.5 and jumps to a new orbit. In the lower panel we
have plotted the system response (blue) against the forcing D against
t/T. One can see the loss of stability of the orbit around t/T≈15
and the prior increase in system response
amplitude.
The early warning indicators, response amplification (upper panel),
A=Daτ1+ω2τ2, and phase lag (middle panel),
ϕlag2π=12πarctan(ωτ) calculated for
the time series in Fig. . We have plotted these
indicators prior to the bifurcation at t/T≈15. The 2nd (blue) and
3rd (red) harmonic amplitudes An/A1 are also plotted in the lower panel
using a sliding window of 5 complete cycles. All indicators are increasing as
expected.
We also plot the complete spectrum of the ratios An/A1 against Tn/T
derived from a Fourier transform of the system response in Fig. . In the upper panel all parameters are the same
as Fig. except we have fixed Dm in each of
the two runs. In the first run Dm=-2, this is far from the bifurcation and
one expects the system to behave more linearly (blue line). One sees a second
harmonic around 2 orders of magnitude smaller than the linear response. In
the second run Dm=0.25 and the orbit is much closer to the bifurcation
(red line). The second harmonic has increased to about an order of magnitude
smaller than the fundamental harmonic and a third harmonic is now also
visible indicating the system has become more nonlinear.
Ratio of the nth order harmonic amplitude to the fundamental
harmonic amplitude An/A1 against the ratio of the nth harmonic period
to the fundamental harmonic period Tn/T. The dynamics of the system are
described by equation . In the upper panel parameters
are fixed to Da=1/2, T=π (the same order as the system timescale
τ). The blue line is for Dm=-2 (far away from the bifurcation), the
nonlinear response is dominated by the second harmonic at Tn/T=1/2
although small, about two orders of magnitude less than the linear response.
The red line is Dm=1/4, close to the bifurcation the system response has
become more nonlinear. The second harmonic (Tn/T=1/2) is now almost 1
order of magnitude less and the third order harmonic (Tn/T=1/3) is also
prominent. In the bottom panel, we show the spectrum when the dynamics are
very nonlinear. Parameters are set to Dm=0, Da=1/2, T=100π so
ωτ≈1/100. This is the slow forcing limit shown in Fig. (red line) which has a very nonlinear relaxation
oscillation type response. Note only odd harmonics (Tn/T=1/3,1/5,1/7,…
etc.) are present due to the system experiencing a symmetric potential
requiring the solution, x(t), to also have this
symmetry.
To give an example of a climate system operating in this regime consider the
annual variation in sea temperatures in Northern Hemisphere temperate
regions. A rough estimate of the ocean surface mixed layer timescale gives
τ∼10 months and this surface layer is heated by the annual cycle of
solar insolation to varying degrees throughout the year. Calculation of the
phase lag for this τ and T yields a lag of about 2.6 months, i.e.
roughly the maximal and minimal sea temperatures are in September and March.
Arctic sea ice extent also falls into this regime and we analyse this system
in Sect. .
A note about return maps
Towards the upper range of ωτ∼1, specifically ωτ>2π, return map analysis via statistical fixed point indicators becomes
useful with the added caveat that the time series of the system must have
enough cycles to produce statistically significant results. Return map
analysis is complementary to phase lag and response amplification since these
quantities start to asymptote when ωτ>2π. This complementarity
is illustrated in the following figures. The blue line in Fig. is essentially the same as
Fig. (ωτ∼1) except Dm is varied over
100 cycles instead of 25. This is because extra data points are needed to
calculate the lag 1 autocorrelation of the return maps with any reliability.
We have also added Gaussian white noise to Eq. ()
of standard deviation 0.01 as the return map method needs small
perturbations with which to infer return times to the cycle. In Fig. we have plotted the early warning indicators for this
time series including the return map calculated with a sliding window of 25
cycles. The black lines are the theoretical curves and the blue lines are the
estimated curves. The key point is the theory and estimated autocorrelations
do not show anything in this regime (ωτ∼1) however the phase
lag and response amplification are clearly increasing.
Conversely, the red lines in Figs. and are the same quantities but with a decreased period of
forcing (T=1/4 so ωτ∼4π). This is a regime in which phase
lag and response amplitude start to asymptote and are therefore not so useful
to infer changing system timescale. However, lag 1 autocorrelation of the
return map now becomes useful as can be seen in Fig. .
Same figure as Fig. in the manuscript
except the variation of Dm is over more cycles to generate more points for
a reliable return map analysis. Weak Gaussian white noise of standard
deviation 0.01 is added to the system. Parameters are set to Da=1/2 and
Dm is varied linearly with time between -2 and 2 over about 100 cycles. In
the upper panel the black lines are the nullclines while the system response
is the blue line for T=π giving ωτ∼1 whereas the red line
has a shorter period of T=1/4 to give ωτ∼4π. These are
plotted against D(t). In the lower panel we have plotted these system
responses as time series against the forcing (black
line).
ωτ∼1: The early warning indicators, response
amplification (upper panel), A=Daτ1+ω2τ2,
and phase lag (middle panel),
ϕlag2π=12πarctan(ωτ) calculated for
the blue time series in Fig. . In the
lower panel, lag 1 autocorrelation of a sliding window of 25 points of the
return map is plotted with standard errors (dashed lines) on the estimate.
Black lines are theoretical curves of all the quantities. The key point is
that
phase lag and amplitude response are useful quantities in this regime however
the return map is not.
Period of forcing much slower than system timescale, ωτ≪1
When Eq. () is operating in this regime (period of
forcing much greater than system timescale T≫τ) the system can
adjust to changing D(t) relatively quickly and effectively remains at a
fixed point. D(t) can therefore be modelled as a slow constant, control
parameter and all the usual timescale separation assumptions apply. Fixed
point indicators such as lag 1 autocorrelation are then good early warning
indicators of local bifurcations. In contrast, phase lag and response
amplitude are not useful as these quantities asymptote to ϕlag→0 and →τ respectively. The system state x is
plotted against D and against t as the red line in Fig. .
An example of a system that has the correct timescale separation and
periodic forcing are the glacial and/or interglacial cycles that have the slow
build, fast collapse type behaviour of relaxation oscillations. Ice sheets
have timescales in the order of thousands of years forced by the solar
insolation variation of Milankovitch cycles. The forcing is a superposition
of many different sinusoidal frequencies, the dominant ones having periods of
41 kyr (related to the obliquity of Earth's orbit), 19 and 23 kyr (related to
the precession). Current thinking, however, favours more complex, two- and
higher-dimensional dynamics to model these cycles than the single variable
models we consider in this paper
.
ωτ∼4π: The early warning indicators, response
amplification (upper panel), A=Daτ1+ω2τ2,
and phase lag (middle panel),
ϕlag2π=12πarctan(ωτ) calculated for
the red time series in Fig. . In the lower
panel, lag 1 autocorrelation of a sliding window of 25 points of the return
map is plotted with standard errors (dashed lines) on the estimate. Black
lines are theoretical curves of all the quantities. Phase lag and amplitude
response have now asymptoted and are not useful quantities however the return
map now becomes useful.
The spectrum of a very nonlinear, relaxation oscillation type, dynamics is
illustrated in the lower panel of Fig. . This
is the spectrum of the slow forcing run (red line) in Fig. . Only odd harmonics appear in its spectrum
because the static potential V=-∫x˙dx is symmetric about x for
this value of Dm=0, i.e. V(x)=V(-x) and therefore any solution of
x˙ must also have this symmetry, x(t+T/2)=-x(t). Only odd harmonics
have this property.
This is not sufficient though as there are other
parameter settings that feature the second harmonic and also have the same
symmetric potential, i.e. Dm=0 and T=π in Fig. (blue line). The difference is that the runs
featuring second harmonic responses only experience a limited part of the
potential, not the full symmetric potential. Even though the potential is the
same, the forcing is quick enough to trap the system in an orbit in just one
of the two potential wells. This local potential well is asymmetric and what
the system sees is effectively described by a Taylor expansion around the
centre of that well. In contrast the relaxation oscillation type run travels
across both wells equally and therefore sees the global symmetric potential
requiring an odd harmonic solution. This is not a generic case however.
Period of forcing much faster than system timescale, ωτ≫1
The system state x is plotted against D and against t as the green line
in Fig. . The forcing is changing much faster
than the system can respond so the system effectively looks static and all
the dynamics come from the forcing directly. In this case we can place D(t)
in the fast dynamics. However, not all of the other assumptions for use of
lag 1 autocorrelation in a fixed point analysis are satisfied. It is true
that D(t) is independent of x, however it is not uncorrelated with itself
at different times and therefore cannot strictly be modelled as a normally
distributed random variable, although at first glance it looks as though it
is again possible to use usual fixed point early warning techniques so one
must be careful. In this regime, phase lag and response amplification
asymptote and again are not very useful to detect a trend in increasing timescale. Phase lag, ϕlag→π/2 and response amplification
→1ω so one may only infer τ≫T.
An example of a system approximately modelled by this limit is the global
terrestrial vegetation carbon which has a dominant timescale on the order of
decades, much larger than its periodic forcing, the annual cycle of solar
insolation. This dominant timescale comes from the large long term carbon
storage, e.g. the timescale taken for a forest to regrow once cut down. One
sees this phase lag of quarter of a cycle in the annual minimum of the Mauna
Loa CO2 record
Pieter Tans, NOAA/ESRL
(www.esrl.noaa.gov/gmd/ccgg/trends/) and Ralph Keeling, Scripps
Institution of Oceanography (www.scrippsco2.ucsd.edu/)
relative to the Northern
Hemisphere solar insolation maximum. This lagged annual minimum in the
integrated response of the total atmospheric carbon results from the
dominance of the Northern Hemisphere's mid latitude terrestrial vegetation
carbon in the global carbon flux. We have plotted the Mauna Loa CO2 record
and the time of year of the minimum concentration in Fig. .
Atmospheric CO2 concentration recorded at Mauna Loa against time
in the upper panel. In the lower panel we have plotted the minimum annual
CO2 concentration against year. One notices the minimum CO2
concentration occurs roughly 3/4 of the way through the year. This is because
maximal carbon uptake occurs during the Northern Hemisphere summer from the
terrestrial vegetation and it is maximally lagged behind the maximum in the
Northern hemisphere solar insolation (best growing conditions) by 1/4 of a
cycle because of the timescale difference between the response of the system
and the period of the forcing.
Looking for a tipping point in Arctic sea ice satellite observations
There has been much research on a possible local bifurcation and tipping
point in the Arctic sea ice, see for example ,
, ,
, and . This
possible bifurcation in the sea ice cover may be due to the well-known ice
albedo feedback first studied by and
. When ice is present it reflects a high proportion of
the incoming solar radiation due to its higher albedo yet when it starts
receding the darker ocean absorbs more radiation increasing heating and
promoting more sea ice retreat. This feedback can result in instability and
multiple steady states.
We calculate all the previously mentioned early warning indicators for a time
series of Arctic sea ice area satellite observations from 1979 to present
day. That is we calculate phase lag, response amplitude, relative size of the
2nd and 3rd harmonics and the lag 1 autocorrelation of the return map with
time to look for signs of critical slowing down that might indicate the
approach of a local bifurcation or “tipping point” in the Arctic sea-ice. We
also calculate the complete Fourier spectra for the entire time series as a
linearity check. In Fig. satellite observations
of Arctic sea ice area are plotted against year. Sea ice area data were
obtained from The Cryosphere Today project of the University of Illinois.
This data set
uses SSM/I and SMMR series satellite products and spans 1979 to present at
daily resolution.
Arctic sea ice area satellite observations from 1979 to present day
(2015) obtained from The Cryosphere Today project of the University of
Illinois.
In Fig. we plot the amplitude of the sea ice area
annual cycle and the phase lag between the sea ice area minimum and maximum
during each cycle. We assume the maximal and minimal driving occurs at the
same time as maximal and minimal of the solar insolation, that is, the
midpoint and end point of the year respectively to obtain phase lags. To
limit the impact of high-frequency variability on the location of the
extrema, we have smoothed the daily data with a sliding window of 30 days.
From Fig. we see the cycle amplitude is increasing with
time although the phase lag does not appreciably change. We first make some
rough calculations to see if these plots are consistent with each other: from
the phase lag figure, a timescale of τ∼[0.33, 0.5] years from the lag
of [0.18, 0.2] of a cycle can be inferred. If we assume for the moment the
amplitude of the forcing Da is not changing throughout the time period of
the observations (this may not be true) and take the smallest value in the
range for τ1978=0.33 years occurring in 1978 and the largest value in
2015, τ2015=0.5 years we can make a rough calculation of how much the
sea ice amplitude would have increased, i.e.
A2015A1978=τ2015τ19781+ω2τ197821+ω2τ2015≈1.06. From Fig. we take the amplitude at 1978 to be
A1978∼4.5 and at 2015 to be A2015∼5 we find
A2015A1978=1.11. These values could therefore be consistent
with a constant Da and a changing timescale. However, the timescales
inferred from either the phase lag or amplitude are not changing appreciably
and therefore it seems unlikely the system is approaching a local
bifurcation.
In the upper panel amplitude of sea ice area within each cycle is
plotted against year. In the middle panel phase lag is plotted between the
sea ice area minimum (red line) and maximum (blue line) and the solar
insolation minimum and maximum respectively against the year. In the lower
panel, the 2nd (blue) and 3rd (red) harmonic amplitudes An/A1 are plotted
against year end using a sliding window of 10 years. The amplitude is
increasing however the phase lag is not. Harmonic amplitudes also show no
convincing trend.
We note that the phase lag is a more robust indicator. This is because the
phase lag depends only on the product of the frequency of the forcing and the
system timescale whereas the amplitude depends additionally on the amplitude
of the driving, Da, which may well be changing throughout the
observational period and could account for some or all of the increase seen
in the amplitude in Fig. . Although the solar insolation
will be a large component of the forcing amplitude and is essentially fixed,
other factors such as clouds as well as air and sea temperatures will also
factor into the driving amplitude. Geometrical constraints imposed by land
masses affecting the maximal extent of the sea ice will also influence the
amplitude of the sea ice oscillation when ice extent is large
. In contrast, we can take the frequency of the
driving to be essentially fixed by the annual solar insolation cycle making
the phase lag more robust.
In the lower panel of Fig. we have plotted the ratio of
the second and third harmonic amplitudes to the amplitude of the fundamental
harmonic with time using a sliding window of 10 complete cycles against the
year at the end of the window. We have used the minimal window length needed
to resolve both harmonics reliably. This indicator also shows no clear trend
with time.
Ratio of the nth order harmonic amplitude to the fundamental
harmonic amplitude An/A1 found from the Fourier transform of the Arctic
sea ice area time series against the ratio of the nth harmonic period to
the fundamental harmonic period Tn/T. One can see the Arctic sea ice
response features prominent second, third, forth, fifth and sixth harmonics
in its spectrum.
We have also calculated the lag 1 autocorrelation of the return map. From
phase lag, the estimate timescale of the sea ice is 0.5 years (ωτ∼π) which is less than τ/T>1 needed for a reliable estimate.
However, the estimated timescale is uncertain and it is conceivable the
return map analysis might work. As there are only 37 complete years of data,
any return map time series has a maximum of 37 data points. To discern any
trend in the autocorrelation one needs as many windows as possible, however
this results in a decreasing number of data points per time series and an
increasing error in the estimate. We have therefore chosen a sliding window
of 20 cycles although the results are invariant to this choice, always being
very uncertain. We linearly detrend the cycle in each sliding window and then
create the return map time series from that detrended window. One can also
choose at which point in the cycle one wants to take the return map from and
this additional freedom is utilised in the right hand panel in Fig. . Lag 1 autocorrelation is plotted against sliding window end
year (x axis) and day of the year in each cycle the return map generated on
(the y axis). We create new return maps every 10 days giving 36 different
points within each cycle. As seen in the figure, autocorrelation depends very
heavily on where in the cycle one chooses to generate the map, a sign that
the return map is not a good approach for this system. A good return map
should be largely invariant to where in the cycle it is taken provided the
cycle is stable and not changing. In the left hand panel of Fig. we plot the standard error divided by the autocorrelation.
Note that most estimates of lag 1 autocorrelation have standard errors larger
than half their value giving very uncertain estimates. In an effort to reduce
the uncertainty in the estimate we have also taken the mean autocorrelation
over all points in the cycle the return map is taken in Fig. . The mean lag 1 autocorrelation is 0.16±0.26 which
corresponds to a (very uncertain) timescale of τ≈0.55 years. This
is consistent with the estimates from the phase lag. This also suggests that
the sampling interval T>τ and therefore determining the timescale
using the return map approach is difficult. We have increased the sliding
window to 37 years to minimize the standard error in the estimate, however
one will not be able to then see a trend in autocorrelation. Even so, the
standard errors are still greater than half the estimate.
We have also plotted the full spectrum of the ratios An/A1 for the entire
time series in Fig. . We note the nonlinear effects are
quite prominent in this system, second and third harmonics are around an
order of magnitude smaller than the linear response, although we can still
probably get away with the linear analysis. Forth, fifth and sixth harmonics
are also visible. These nonlinearities may be due to albedo effect or to the
geometrical effects of the Arctic ocean basin .
Left panel: Lag 1 autocorrelation of the return map against sliding
window end year using a sliding window of 20 years (x axis) and point
within the cycle the return map is created on the y axis (we create return
maps every 10 days). One sees that autocorrelation depends very heavily on where
in the cycle one chooses to generate the return map. Right panel:
standard error of the autocorrelation return map estimate divided by the
estimate against sliding window end year. The estimate is very uncertain
almost everywhere with standard errors generally being at least half as big
as the estimate.
Mean lag 1 autocorrelation of the return map across all starting
points within the cycle using a sliding window of 20 years. This is the same
as Fig. with the mean taken along the y axis. Estimated
autocorrelation is still very uncertain. The mean is the solid line with the
dotted lines being the mean plus or minus the standard error. The mean value
across all years 0.16±0.26 which corresponds to a (very uncertain) timescale of τ≈0.55 years.
To conclude, from this simple analysis it seems that the system's timescale
and therefore stability is not changing appreciably if at all and it is
unlikely to be approaching a local bifurcation. However, simple theoretical
models, such as , and
who also used a return map approach suggest that the
sea ice timescale does not change very much approaching the bifurcation,
even decreasing slightly before rapidly changing over a very small interval
and therefore would be very hard to detect if present.
Conclusions
Much previous work on detecting local bifurcations from time series required
one to be able to partition the universe into widely separated timescales
and model the system dynamics as overdamped. When this is the case one can
use the usual, statistical fixed point early warning indicators of increasing
lag 1 autocorrelation and variance since these indicators measure the
system's response to small perturbations away from its fixed point by the
fast, noisy processes. It is the response to this small, noisy forcing that
allows one to measure the system's timescale. The systems we have been
looking at in this paper do not have fast or random forcing. The systems
considered here have deterministic forcing with a period roughly that of its
timescale although the dynamics are still overdamped. Deterministic forcing
again allows one to infer the system's timescale simply by measuring the
response to the forcing without the need for large amounts of data required
by statistical quantities for robust estimates. We used two analogous early
warning indicators to lag 1 autocorrelation and variance in these systems;
these were phase lag and response amplification respectively. Just as
autocorrelation is more robust as an indicator (it is a function of fewer
parameters), the same is true of phase lag, only depending on the frequency
of the forcing and the timescale of the system. The system response
amplification also depends on the amplitude of forcing, which in many
circumstances is probably difficult to measure.
We also used a Fourier transform of the time series to quantify how nonlinear
the system is behaving and whether the linear approximations usually made are
good. Further, by using a sliding window within the time series, one may also
look at the evolution of the harmonic amplitudes as a further early warning
indicator.
We also discussed return map methods that essentially convert a periodic
attractor to a fixed point type so that one may use the usual fixed point
indicators. We also showed there was a complementarity between return map
indicators and phase lag and response amplification, the latter being more
useful for regimes in which ωτ∼1 and the former being more
useful when ωτ>2π.
We applied these indicators to satellite observations of Arctic sea ice area,
a system whose period of forcing, effectively the annual cycle of insolation,
is similar to the timescale of the system. This is also a system that has
been conjectured to have a tipping point due to a local bifurcation. We did
not find any detectable critical slowing down and therefore signs of this
bifurcation. It should be noted, however, that simple models of the sea ice suggest
critical slowing down only occurs very close to the bifurcation, making it
very hard to detect.
Data availability
The data sets of the Mauna Loa CO2 record and the Arctic sea-ice area analyzed in this manuscript are publicly available at
www.esrl.noaa.gov/gmd/ccgg/trends/ and http://arctic.atmos.uiuc.edu/cryosphere/timeseries.anom.1979-2008 respectively.
Phase lag and response amplification with arbitrary periodic forcing
Phase lag and response amplification can be found for the more general case
of any type of periodic forcing D by solving
x˙+xτ=D(t)τ the timescale of the system (the e folding time). For any periodic
forcing, D(t) with period T can be written as the Fourier series
D(t)=∑i=0NBicos(ωit+χi).Bi are the amplitudes of the different component sinusoidal waves,
ωi=2πiT are the frequencies of the components and
χi are the phases of each of the components. As the equation is linear
the superposition principle holds. That is, we assume the solution has the
form
x(t)=∑i=0Nxi(t)
by setting all but the ith term of the driving to zero we can solve the
N+1 equations
x˙i+xiτ=Bicos(ωit+χi)
for each xi(t). These solutions can be superposed to obtain the full
solution to any periodic driving term.
This is
x(t)=∑i=0NτBi1+ωi2τ2[cosωit+χi-arctan(ωiτ)-e-tτcosχi-arctan(ωiτ)]+x0e-tτ
which settles into orbit
x(t)=∑i=0NτBi1+ωi2τ2cosωit+χi-arctan(ωiτ)
when t≫τ, that is, the solution is just the sum of each of the
forcing components i, each with a response amplification of
τ1+ωi2τ2
and a response lagging the forcing with a phase of
ϕilag=arctan(ωiτ).
One can find out what these phase lags and amplitudes are by taking the
Fourier transform of the time series of both the forcing and response.
Acknowledgements
The research leading to these results has received funding from the European
Union Seventh Framework Programme FP7/2007-2013 under grant agreement no.
603864 (HELIX). We are grateful to Peter Ashwin, Peter Cox, Michel Crucifix,
Vasilis Dakos, Henk Dijkstra, Jan Sieber, Marten Scheffer and Appy Sluijs for
the fruitful discussions over beers and balls.
Edited by: J. Annan
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