remarkRemark hypothesisHypothesis claimClaim Noise interference in
anticipating tipping pointsA. Morr, N. Boers and P. Ashwin

# Internal noise interference in anticipation of tipping points
of generic multi-dimensional dynamical systems

Andreas Morr111Corresponding author:  222Earth System Modelling, School of
Engineering and Design, Technical University of Munich, 80333 Munich, Germany
333Complexity Science, Potsdam Institute for Climate Impact Research, 14473
Potsdam, Germany<EMAIL_ADDRESS>Niklas Boers22footnotemark: 2
33footnotemark: 3 444Department of Mathematics and Statistics, University of
Exeter, Exeter EX4 4QF, United Kingdom Peter Ashwin44footnotemark: 4

###### Abstract

A deterministic complex system that slowly passes through a generic fold-type
(saddle-node) bifurcation can be reduced to one-dimensional dynamics close to
the bifurcation because of the centre manifold theorem. It is often tacitly
assumed that the same is true in the presence of stochasticity or noise so
that, for example, critical slowing down (CSD) indicators can be applied as if
the system were one-dimensional. In this work, we show that this may not be
the case; specifically, we demonstrate that noise in other dimensions may
interfere with indicators of CSD, also referred to as early warning signals
(EWS). We point out a generic mechanism by which both variance and AC(1), as
well as other EWS, can fail to signal an approaching bifurcation. This can in
principle occur whenever one noise source drives multiple components of the
system simultaneously. Even under the favourable assumptions of uncoupled
deterministic dynamics and stationary noise, observables of the system can
then exhibit false negative or false positive CSD indications. We isolate this
phenomenon in an example that represents a generic two-dimensional fold-type
bifurcation setting.

###### keywords:

Abrupt transitions, Critical Slowing Down, Early Warning Signals, Multi-
dimensional systems

37G10, 37H05, 37H20

##  1 Introduction

Many physical systems exhibit abrupt transitions or are suspected to harbour
the potential for them [22, 1, 16, 28, 5, 23]. There has been an increased
interest in the analysis and detection of bifurcation-induced transitions [6,
4, 8, 9, 26, 12]. Such events occur if the system’s current equilibrium state
vanishes in response to an external change to the underlying dynamics. The
simplest model for bifurcation-induced transitions is that of a one-
dimensional fold-type bifurcation [29, 31, 15]. This approach is reasonable if
there is one essential component of the system experiencing negative feedback
on its fast time-scale disturbances. Moreover, bifurcation theory tells us
that this is one of only two possible generic one-parameter bifurcations of an
attracting equilibrium, the other being the Hopf bifurcation. Close enough to
such a bifurcation, one can rigorously and generically reduce the
deterministic system to a one-dimensional model on a centre manifold. The
negative feedback on this essential component weakens in advance of the
critical transition brought about by the external change in dynamics. The
resulting observational characteristic of slower and weaker responses to
perturbations is called critical slowing down (CSD). If the model component
undergoing such a bifurcation is a direct observable of the physical system,
one can hope to employ the conventional techniques for detecting CSD, i.e.
searching for positive trends in variance or lag-1 autocorrelation (AC(1)) in
the time-series data. These trends are also referred to as early warning
signals (EWS) in the literature, and a robust theoretical groundwork exists
[19, 20, 22, 7]. Close to a fold-type bifurcation of a one-dimensional system
whose state is denoted by $x\in\mathbb{R}$, one can approximate the dynamics
by the topological normal form:

$\dot{x}(t)=-x^{2}+\alpha$

For $\alpha>0$, this admits one stable equilibrium at
$x^{*}(\alpha)=\sqrt{\alpha}$ and one unstable equilibrium, whereas for
$\alpha<0$, it admits no equilibria. Linearising around the respective
equilibrium point gives

$\dot{x}(t)=-2\sqrt{\alpha}(x(t)-x^{*}(\alpha))+\mathcal{O}\left((x(t)-x^{*}(\alpha))^{2}\right).$

Adding Gaussian white noise to model the omitted fast dynamics of the original
high-dimensional physical system [32, 18], the problem of determining the
statistical characteristics of the process reduces to the analysis of a one-
dimensional Ornstein-Uhlenbeck process:

$\mathrm{d}X_{t}=-2\sqrt{\alpha}(X_{t}-x^{*}(\alpha))\mathrm{d}t+\sigma\mathrm{d}W_{t},$

where $W$ is a Wiener process supported on the filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathbb{R}_{+}},\mathbb{P})$. For
variance and AC(1) of the observable $X$, one finds in the stationary limit

$\mathrm{Var}\left[X\right]=\frac{\sigma^{2}}{4\sqrt{\alpha}},\quad\mathrm{AC}_{X}(1)=\exp(-2\sqrt{\alpha}),$

which both increase as $\alpha>0$ decreases towards the bifurcation at
$\alpha=0$.

Whether the above model simplification is valid for a given multi-dimensional
physical system depends on several factors. For example, if the type of
bifurcation is Hopf rather than fold, the functional expressions of variance
and AC(1) will differ and may not allow for their use as EWS [8]. If the noise
driving the multi-dimensional dynamics is not coupled to the centre manifold,
EWS may stay silent [3]. A further complication that has not been considered
previously is that even if there is a centre manifold corresponding to a fold-
type bifurcation and directly impacted by white noise, the observable
available for analysis need not be this component. Rather, it may be a
function of multiple components of the system. Insofar as the system can be
represented by a multi-dimensional random dynamical equation, we should expect
the noise disturbances within the system to drive multiple components
simultaneously. The alternative to such a general setup is the specific case
of independent noise terms, each driving only one system component, which is
an unwarranted assumption for many physical systems and can essentially be
disregarded as unrealistic. We will investigate the implications of these
generalisations in the context of critical slowing down. Following the
previous considerations of the one-dimensional fold normal form, we proceed by
introducing an uncoupled state variable $y$ with stationary dynamics.

(1) $\dot{x}(t)=-x^{2}+\alpha,\quad\dot{y}(t)=-y$

This system constitutes the most reductive representation of a two-dimensional
fold-type bifurcation exhibiting an abrupt transition. There is no coupling
between the two deterministic equations of motion. Around the equilibrium
point $(x^{*}(\alpha),y^{*}(\alpha))=(\sqrt{\alpha},0)$, the linearised
dynamics are

$\begin{pmatrix}\dot{x}(t)\\\
\dot{y}(t)\end{pmatrix}\approx\begin{pmatrix}-2\sqrt{\alpha}&0\\\
0&-1\end{pmatrix}\begin{pmatrix}x(t)-x^{*}(\alpha)\\\ y(t)\end{pmatrix}.$

When introducing additive white noise to this multi-dimensional setting, we
are, in general, given three degrees of freedom in the form of the respective
noise-couplings $\sigma_{X}$, $\sigma_{Y}$ and $c$:

$\begin{pmatrix}\mathrm{d}X_{t}\\\
\mathrm{d}Y_{t}\end{pmatrix}=\begin{pmatrix}-2\sqrt{\alpha}&0\\\
0&-1\end{pmatrix}\begin{pmatrix}X_{t}-x^{*}(\alpha)\\\
Y_{t}\end{pmatrix}\mathrm{d}t+\begin{pmatrix}\sigma_{X}&c\\\
0&\sigma_{Y}\end{pmatrix}\begin{pmatrix}\mathrm{d}W^{X}_{t}\\\
\mathrm{d}W^{Y}_{t}\end{pmatrix}$

Note that a matrix with an additional coupling of the white noise term
$\mathrm{d}W^{X}$ to $\mathrm{d}Y$ can always be renormalised to an upper-
triangular matrix. Similarly, any third white noise term could always be
represented in the above model equation containing two white noise terms. We
call $c$ the cross-coupling of the noise, which, as discussed above, will be
non-zero in the general case. We will see that introducing only this
additional coupling to the model can have grave implications on the ability to
detect the approach of a bifurcation through conventional CSD methods. This is
the – rather generic – case if the externally observable quantity is not the
destabilising dimension $X$ itself, but rather an observable $\Psi=f(X,Y)$ of
the system. In general, any non-linear function $f$ can occur as an
observable. We choose the subset of linear observables as our object of study
to show that the described phenomenon of failing EWS already occurs for such a
simple functional relationship between the system’s constituents and the
observed quantity.

(2) $\Psi_{t}:=\cos(\beta)X_{t}+\sin(\beta)Y_{t}$

Eq. Eq. 2 covers all possible linear combinations of $X$ and $Y$, and the
phenomenon is impervious to an absolute scaling of the observable.

Section 2 below is dedicated to an analytical derivation of the statistical
properties of $\Psi$ and an assessment of the potential for deceitful CSD
indications. Having identified possible pitfalls to the EWS in variance and
AC(1) based on a linearised system, we check that those predicted behaviours
indeed manifest in non-linear systems by integrating the dynamical system in
Eq. Eq. 1 with added noise and analysing the resulting time series data in
Section 3.

##  2 Statistical properties of the observable $\Psi$ under linearised
dynamics

To generalise the linear setting to applications beyond the particular
parametrisation of the normal form given in the previous section, we now write
the 2D-Ornstein-Uhlenbeck process in question generally as

(3) $\displaystyle\begin{pmatrix}\mathrm{d}X_{t}\\\
\mathrm{d}Y_{t}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}-\lambda_{X}&0\\\
0&-\lambda_{Y}\end{pmatrix}\begin{pmatrix}X_{t}\\\
Y_{t}\end{pmatrix}\mathrm{d}t+\begin{pmatrix}\sigma_{X}&c\\\
0&\sigma_{Y}\end{pmatrix}\begin{pmatrix}\mathrm{d}W^{X}_{t}\\\
\mathrm{d}W^{Y}_{t}\end{pmatrix}$ $\displaystyle=:A\begin{pmatrix}X_{t}\\\
Y_{t}\end{pmatrix}\mathrm{d}t+\Sigma\,\begin{pmatrix}\mathrm{d}W^{X}_{t}\\\
\mathrm{d}W^{Y}_{t}\end{pmatrix}$

The deterministic linear dynamics are still uncoupled, and we have carried
along the cross-coupling $c$ of the noise terms. Without loss of generality,
we have also centred the dynamics of the $X$ dimension around $0$. In its
stationary limit the zero-mean process $\left(X,Y\right)^{T}$ has the
following covariance matrix

$V:=\mathbb{E}\left[\left(X_{t},Y_{t}\right)^{T}\left(X_{t},Y_{t}\right)\right]=\begin{pmatrix}\frac{\sigma_{X}^{2}+c^{2}}{2\lambda_{X}}&\frac{\sigma_{Y}c}{\lambda_{X}+\lambda_{Y}}\\\
\frac{\sigma_{Y}c}{\lambda_{X}+\lambda_{Y}}&\frac{\sigma_{Y}^{2}}{2\lambda_{Y}}\end{pmatrix}$

This can be obtained by solving the continuous-time Lyapunov equation

$AV+VA^{T}+\Sigma\Sigma^{T}=0.$

Adopting the initial condition $(X_{0},Y_{0})^{T}\sim\mathcal{N}(0,V)$, the
process is stationary, and the time-covariance matrix for all $\tau\geq 0$ is
given by

$R(\tau):=\mathbb{E}\left[\left(X_{t},Y_{t}\right)^{T}\left(X_{t+\tau},Y_{t+\tau}\right)\right]=\exp(\tau
A)V$

Note that $R(\tau)$ for $\tau\neq 0$ will in general not be a symmetric
matrix, since
$\mathbb{E}\left[X_{t}Y_{t+\tau}\right]=\mathbb{E}\left[X_{t+\tau}Y_{t}\right]$
if and only if $c=0$ or $\lambda_{X}=\lambda_{Y}$:

$R(\tau)=\begin{pmatrix}\frac{\sigma_{X}^{2}+c^{2}}{2\lambda_{X}}\exp(-\lambda_{X}\left|\tau\right|)&\frac{\sigma_{Y}c}{\lambda_{X}+\lambda_{Y}}\exp(-\lambda_{X}\left|\tau\right|)\\\
\frac{\sigma_{Y}c}{\lambda_{X}+\lambda_{Y}}\exp(-\lambda_{Y}\left|\tau\right|)&\frac{\sigma_{Y}^{2}}{2\lambda_{Y}}\exp(-\lambda_{Y}\left|\tau\right|)\end{pmatrix}\,.$

The same holds true for the time-correlation matrix

$r(\tau)=\begin{pmatrix}\exp(-\lambda_{X}\left|\tau\right|)&\frac{2c}{\sqrt{\sigma_{X}^{2}+c^{2}}}\frac{\sqrt{\lambda_{X}\lambda_{Y}}}{\lambda_{X}+\lambda_{Y}}\exp(-\lambda_{X}\left|\tau\right|)\\\
\frac{2c}{\sqrt{\sigma_{X}^{2}+c^{2}}}\frac{\sqrt{\lambda_{X}\lambda_{Y}}}{\lambda_{X}+\lambda_{Y}}\exp(-\lambda_{Y}\left|\tau\right|)&\exp(-\lambda_{Y}\left|\tau\right|)\end{pmatrix}$

For any linear observable $\Psi(\beta)=\cos(\beta)X+\sin(\beta)Y$ of the
2D-system, we may now compute the variance and the autocorrelation function in
the stationary limit:

$\mathrm{Var}\left[\Psi\right]=\cos^{2}(\beta)V_{1,1}+\sin^{2}(\beta)V_{2,2}+2\cos(\beta)\sin(\beta)V_{1,2},$

where $V_{i,j}$ is the respective entry of the covariance matrix $V$. For the
autocorrelation in time, we have:

$\displaystyle\mathrm{AC}_{\Psi}(1)$
$\displaystyle:=\mathbb{E}\left[\Psi_{t}\Psi_{t+\tau}\right]/\mathrm{Var}\left[\Psi\right]$
$\displaystyle=(\cos^{2}$
$\displaystyle(\beta)R(\tau)_{1,1}+\sin^{2}(\beta)R(\tau)_{2,2}+\cos(\beta)\sin(\beta)(R(\tau)_{1,2}+R(\tau)_{2,1}))/\mathrm{Var}\left[\Psi\right]$

As expected from the uncoupled nature of the deterministic dynamics, if we
choose $\beta=0$, i.e. $\Psi=X$, we essentially retrieve the conventional
quantities from the one-dimensional case. Also, not surprisingly, if we only
inspect the second dimension $\Psi=Y$, the variance and AC(1) will not depend
on the destabilisation of $X$, i.e. $\lambda_{X}\rightarrow 0$, and can not
function as EWS. See Table 1 for explicit expressions.

Table 1: Edge cases of the linear observable setting. The conventional EWS information is perfectly conserved in the first case and entirely lost in the second. $\beta$ | $\Psi$ | $\mathrm{Var}\left[\Psi\right]$ | $\mathrm{AC}_{\Psi}(1)$
---|---|---|---
0 | $X$ | $(\sigma_{X}^{2}+c^{2})/2\lambda_{X}$ | $\exp(-\lambda_{X})$
$\pi/2$ | $Y$ | $\sigma_{Y}^{2}/2\lambda_{Y}$ | $\exp(-\lambda_{Y})$

In all other cases, the observable $\Psi$ consists of some amplitude of $X$
and $Y$, i.e. $\beta\neq k\pi/2$ for $k\in\mathbb{Z}$. Even here, the EWS seem
to be reliable at first glance. We have

$\displaystyle\mathrm{Var}\left[\Psi\right]\xrightarrow{\lambda_{X}\rightarrow
0}\infty,\quad\mathrm{AC}_{\Psi}(1)\xrightarrow{\lambda_{X}\rightarrow 0}1.$

These increases are monotonic in proximity to the bifurcation, where
$\lambda_{X}=0$. More precisely, for any parameter setting $\lambda_{Y}>0$,
$\sigma_{X}>0$, $\sigma_{Y}>0$ and $c\geq 0$, there exists a
$\lambda_{X}^{*}>0$ with the property that both variance and AC(1) are
monotonically increasing as $\lambda_{X}$ decreases from $\lambda_{X}^{*}$ to
$0$. In that sense, there still is a time span in the advent of the fold-type
bifurcation, in which the linearised dynamics predict a clear increase in both
variance and AC(1), making them applicable as EWS in this case.

However, the practical caveat to this finding is the value of
$\lambda_{X}^{*}$ and the behaviour of the EWS before this threshold. We give
the following example, which illustrates that using EWS has potential pitfalls
and may lead to a misinterpretation of the actual destabilisation in the
underlying dynamics. For this, choose $\lambda_{Y}=1$, $\sigma_{X}=0.1$,
$\sigma_{Y}=2$ and $c=1$, meaning

(4) $\displaystyle\begin{pmatrix}\mathrm{d}X_{t}\\\
\mathrm{d}Y_{t}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}-\lambda_{X}&0\\\
0&-1\end{pmatrix}\begin{pmatrix}X_{t}\\\
Y_{t}\end{pmatrix}\mathrm{d}t+\begin{pmatrix}0.1&1\\\
0&2\end{pmatrix}\begin{pmatrix}\mathrm{d}W^{X}_{t}\\\
\mathrm{d}W^{Y}_{t}\end{pmatrix},$

and examine the observable $\Psi(\beta)$ associated with the mixing angle
$\beta=-\frac{\pi}{4}$, i.e. $\Psi_{t}=2^{-1/2}(X_{t}-Y_{t})$. Fig. 1 shows
the variance and AC(1) of $\Psi$ as a function of the destabilising eigenvalue
$\lambda_{X}$. Although the expected increasing behaviour can be observed when
decreasing $\lambda_{X}$ from $\lambda_{X}^{*}\approx 0.6$ towards $0$, both
CSD indicators, in fact, decrease as long as $\lambda_{X}$ decreases toward
$\lambda_{X}^{*}$. It is this veiling of the critical slowing down of the $X$
dimension that we would like to point out in this work.

The underlying prerequisite of the phenomenon explored above is the following
configuration: The noise component $\mathrm{d}W^{Y}$ is positively coupled to
both dimensions of the deterministic dynamics. At the same time, the
observable $\Psi$ is defined such that the two linear components have opposing
signs. This leads to the disturbances in $X$ and $Y$ interfering with each
other in the summary observable $\Psi$. We show with the following Lemma that
this effect is ubiquitous with respect to the specific parameter choices.

Figure 1: (a) Variance and (b) AC(1) of the observable $\Psi$ for
$\beta=-\frac{\pi}{4}$ in the linearised setting described in the main text.
While these two classical CSD indicators ultimately increase to their expected
limits $\infty$ and $1$ respectively as $\lambda_{X}\rightarrow 0$, the
behaviour up until that increase can be deceitful with respect to critical
slowing down.

######  Lemma 2.1.

Consider the linear stochastic differential equation Eq. 3 with non-zero
constants $\lambda_{Y}$, $\sigma_{X}$, $\sigma_{Y}$ and $c$. There exists an
open interval of observation angles $\beta\in(\beta^{1},\beta^{2})$ for which
both variance and AC(1) of $\Psi$ defined through Eq. Eq. 2 decrease with a
decreasing $\lambda_{X}\in(\lambda_{X}^{1},\lambda_{X}^{2})$.

Proof. We can find such an interval of observation angles
$\beta\in(\beta^{1},\beta^{2})$ close to the choice $\Psi=Y$, i.e.
$\beta=\pi/2$. We have already derived analytical expressions of
$\mathrm{Var}[\Psi]$ and $\mathrm{AC}_{\Psi}(1)$. They are smooth functions of
the system’s parameters. The quantities decrease with a decreasing
$\lambda_{X}$ if their partial derivative with respect to $\lambda_{X}$ is
positive. We find at the trivial choice of $\Psi=Y$ that for all
$\lambda_{X}\in\mathbb{R}$

(5)
$\partial_{\lambda_{X}}\mathrm{Var}[\Psi]\Big{|}_{\beta=\pi/2}=0\quad\mathrm{and}\quad\partial_{\lambda_{X}}\mathrm{AC}_{\Psi}(1)\Big{|}_{\beta=\pi/2}=0$

Furthermore, we find that the derivative of these quantities with respect to
the observation angle $\beta$ is non-zero and equal in sign for both of the
quantities when choosing $\lambda_{X}=\lambda_{Y}$:

$\partial_{\beta}\partial_{\lambda_{X}}\mathrm{Var}[\Psi]\Big{|}_{\beta=\pi/2,\,\lambda_{X}=\lambda_{Y}}=-\frac{c\sigma_{Y}}{2\lambda_{Y}^{2}}\quad\mathrm{and}\quad\partial_{\beta}\partial_{\lambda_{X}}\mathrm{AC}_{\Psi}(1)\Big{|}_{\beta=\pi/2,\,\lambda_{X}=\lambda_{Y}}=-\frac{ce^{-\lambda_{Y}}}{\sigma_{Y}}$

Depending on the signs of $c$ and $\sigma_{Y}$, there will be an interval
$(\pi/2,\beta_{2})$ or $(\beta_{1},\pi/2)$ for which the quantities of Eq. Eq.
5 are positive at $\lambda_{X}=\lambda_{Y}$. Since these two quantities are
continuous in $\lambda_{X}$, they will also be positive for all $\lambda_{X}$
in an interval $(\lambda_{Y}-\delta,\lambda_{Y}+\delta)$. This concludes the
proof. Note that the chosen interval is not the only region for which the
phenomenon discussed in this work is prevalent (see Fig. 2c). $\square$

The Lemma implies that, as long as we have a cross-coupling $c\neq 0$, the
phenomenon persists for all generic choices of $\sigma_{X}$, $\sigma_{Y}$, $c$
and $\lambda_{Y}$. On the other hand, the extent to which it quantitatively
inhibits the use of EWS in an application setting will heavily depend on these
values. Giving a general guideline of which configurations exhibit a lengthy
and pronounced decrease in variance and AC(1), even though CSD would imply an
increase, is not feasible given the number of degrees of freedom. We provide a
Python script with which the prevalence of the effect can be assessed for
different parametrisations. Central to this script is the radial plotting of
the approach to the critical value $\lambda_{X}^{\text{crit}}=0$ from
different angles $\beta$ (see Fig. 2 and its description).

Figure 2: Contour plot of (a) the variance and (b) AC(1) for different values
of $\beta$ and $\lambda_{X}$ in the 2D-Ornstein-Uhlenbeck setting of Eq. 4.
The distance from the centre constitutes the value of $\lambda_{X}$ while the
angle around the circle is $\beta$. The radius of the plots and, therefore,
the maximal value of $\lambda_{X}$ is $4$. The second concentric circle marks
$\lambda_{X}=2$. The colour of the contour plot is red where the radial
derivative of the respective quantity towards the centre is positive and blue
where it is negative. In this sense, all blue regions represent the angles
$\beta$ and values of $\lambda_{X}$ where deceitful CSD indications are
prevalent. Figure (c) is a top-down view of the two contour plots. The blue
and red regions are those where variance and AC(1) agree on the respective
trend. In the purple regions, the trends disagree. The green line shows the
trajectory towards the middle for a fixed $\beta=-\frac{\pi}{4}$, meaning from
the second circle on, it is the same line as plotted in Fig. 1 for variance
and AC(1), respectively.

##  3 Observations in model data

After identifying potential pitfalls to employing variance and AC(1) on linear
observables of the linearised system, we test these predictions on synthetic
data of the true fold bifurcation in Eq. Eq. 1. To this end, we integrate the
following system of stochastic differential equations using the Euler-Mayurama
scheme.

(6) $\displaystyle\begin{split}\begin{pmatrix}\mathrm{d}X_{t}\\\
\mathrm{d}Y_{t}\end{pmatrix}&=\begin{pmatrix}-X_{t}^{2}+\alpha(t)\\\
-Y_{t}\end{pmatrix}\mathrm{d}t+\varepsilon\begin{pmatrix}0.1&1\\\
0&2\end{pmatrix}\begin{pmatrix}\mathrm{d}W^{X}_{t}\\\
\mathrm{d}W^{Y}_{t}\end{pmatrix}\\\
\alpha(t)&=1-\frac{11}{10}\frac{t}{T}\end{split}$

Over the time span of $T=10^{4}$, this integration is performed at a time-step
of $\delta t=1/30$ and subsequently sampled at every 30th value. In order to
ensure that premature noise-induced tipping is restricted to a relatively
short period before the bifurcation, we introduce the noise-scaling parameter
$\varepsilon=0.1$. For better illustration, we artificially introduce a second
stable state for $X$ past the bifurcation point. These dynamics are omitted in
the SDE Eq. 6. The results for the linear observable

$\Psi_{t}=\cos(\beta)X_{t}+\sin(\beta)Y_{t}=\frac{1}{\sqrt{2}}(X_{t}-Y_{t})$

with $\beta=-\frac{\pi}{4}$ are given in Fig. 3. Even though the dynamics are
now non-linear and non-autonomous, the contemporary linearised system still
allows for a good assessment of the expected time-series statistics. The
corresponding estimators of variance and AC(1) indeed give the expected
results. The quantities decrease with time, even though the destabilisation of
the system in the dimension of the $X$ component should conventionally incur
an increase. The interference of the shared driving noise in the linear
observable $\Psi$ constitutes a break with the necessary modelling assumptions
for CSD detection via variance and AC(1) in a single observable. Note that
similarly, a stabilisation of the system, i.e. an increase in $\lambda_{X}$,
would cause the variance and AC(1) to increase. This is a setting in which a
false alarm with respect to CSD would be raised.

Figure 3: (a) \- (c) Sample paths of $X_{t}$, $Y_{t}$ and $\Psi_{t}$ following
the SDE Eq. 6 in the main text with $\beta=-\frac{\pi}{4}$. Up until the
purple dashed line, after which noise-induced tipping may frequently come into
play, the variance and AC(1) of $\Psi$ are each estimated over windows of
length $N=1000$. The averaged results over 10 runs are plotted in (d) and (e)
respectively, along with their theoretically computed values from
considerations of the linearised model. Their decrease would be misinterpreted
as a stabilisation of the system rather than an approach of an abrupt
transition. In (f) \- (i), alternative indicators for CSD are calculated.
These are (f) the maximum of the observed power spectral density (PSD) [8],
(g) the linear restoring rate estimated through a Langevin approach [25], (h)
a generalised least squares estimator for an AR(1) model [4] and (i) the
linearised restoring rate estimated via the PSD [24]. These quantities, too,
exhibit the respective opposite trend of what would be expected of a
destabilising system.

##  4 Discussion

We consider a generic fold-type bifurcation in a class of multi-dimensional
dynamical systems, where we assume that noise drives multiple components of
the system. We demonstrate that such a setup can easily produce observables
that, because of internal noise interference, exhibit deceiving trends in
variance and lag-1 autocorrelation with respect to CSD. Such false or missed
alarms have been shown to be an issue in systems driven by non-stationary or
non-white noise [14, 21, 4, 7, 27, 24, 25] or systems exhibiting deterministic
coupling [8, 3]. It has also been shown that a misrepresentation of the
natural system as a fold-type bifurcation with direct noise influence [17, 11]
or issues with the availability of time series data [13, 2, 30] may lead to
similar problems. This work introduces a previously unknown complication with
the detection of CSD via variance and AC(1), as well as other techniques [8,
25, 4, 24, 10] (see also Fig. 3f-i). Even though the severity of the effect
described here depends heavily on the relative values of the systems restoring
rates and noise couplings, its presence is ubiquitous (see Lemma 2.1).
Furthermore, if present, the effect can only be ruled out by choice of a
different observable $\Psi$, which is often not achievable in the observations
of natural systems.

Whenever possible, a sufficient physical understanding and previous model
validation should confirm that the conventional assumption of a directly
observed noisy dynamical system along its centre manifold is warranted. In
this case, the presented results do not call into question the expressiveness
of the conventional EWS.

## Data Availability

Visit the GitHub repository LinearObservablesCSD to access the code used to
generate all figures in this manuscript.

## Acknowledgments

This work has received funding from the Volkswagen Stiftung, the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No. 820970 and under the Marie Sklodowska-Curie grant agreement No. 956170, as
well as from the Federal Ministry of Education and Research under grant No.
01LS2001A. This is TiPES contribution #X.

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