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.

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

For many systems of interest one or more of the above assumptions may be
invalid

In an elegant study

A common method to study stability changes in periodic attractors is the
return or Poincaré map, see

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.

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 CO

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

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.

We consider systems that can be described by

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. (

Once the system has settled into an orbit of period

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. (

Since the ratios

Provided the system timescale is larger than its period,

We now demonstrate the early warning indicators in Sect.

Our system which has one dynamical variable,

The dynamics of the system described by Eq. (

This regime,

To illustrate the early warning indicators we fix the forcing amplitude

We choose to tip the system from one state to another by slowly altering the
mean of the driving

In Fig.

The dynamics of the system are described by Eq. (

The early warning indicators, response amplification (upper panel),

We also plot the complete spectrum of the ratios

Ratio of the

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

Towards the upper range of

Conversely, the red lines in Figs.

Same figure as Fig.

When Eq. (

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

The spectrum of a very nonlinear, relaxation oscillation type, dynamics is
illustrated in the lower panel of Fig.

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.

The system state

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 CO

Pieter Tans, NOAA/ESRL
(

Atmospheric CO

There has been much research on a possible local bifurcation and tipping
point in the Arctic sea ice, see for example

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.

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.

From Fig.

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

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,

In the lower panel of Fig.

Ratio of the

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 (

We have also plotted the full spectrum of the ratios

Left panel: Lag 1 autocorrelation of the return map against sliding
window end year using a sliding window of 20 years (

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.

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

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

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.

The data sets of the Mauna Loa CO

Phase lag and response amplification can be found for the more general case
of any type of periodic forcing

which settles into orbit

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