critical slowing

Physica A 544 (2020) 123396
Contents lists available at ScienceDirect
Physica A
journal homepage: www.elsevier.com/locate/physa
Studying the performance of critical slowing down indicators
in a biological system with a period-doubling route to chaos
∗
Nastaran Navid Moghadam a , Fahimeh Nazarimehr a , , Sajad Jafari a ,
Julien C. Sprott b
a
b
Biomedical Engineering Department, Amirkabir University of Technology, Tehran 15875-4413, Iran
Department of Physics, University of Wisconsin - Madison, Madison, WI 53706, USA
article
info
Article history:
Received 30 March 2019
Received in revised form 31 August 2019
Available online 4 November 2019
Keywords:
Bifurcation parameter
Critical slowing down
Period-doubling route to chaos
a b s t r a c t
This paper aims to investigate critical slowing down indicators in different situations
where the system’s parameters change. Variation of the bifurcation parameter is important since it allows finding bifurcation points. A system’s parameters can vary through
different functions. In this paper, five cases of bifurcation parameter variation are considered in a biological model with a period-doubling route to chaos. The first case is a slow
and small stepwise variation of the bifurcation parameter. The second case is a cyclic,
state-dependent variation of the bifurcation parameter. In the third case, a small cyclic
variation is combined with a sizeable stochastic resonance. The fourth case involves
variations by a large noise, and finally, in the fifth case, significant stepwise changes
in the parameter are studied. To identify the conditions under which critical slowing
down occurs, an improved version of four well-known critical slowing down indicators
(autocorrelation at lag-1, variance, kurtosis, and skewness) are used. The results show
that when bifurcations are caused by a sudden change in a parameter or state, critical
slowing down cannot be observed before the bifurcation points. However, in cases with
slowly varying parameters, critical slowing down can be detected before the bifurcation
points. Thus critical slowing down indicators can predict these bifurcation points. In
other words, in three cases, the system approaches bifurcation points slowly. In other
cases, the bifurcations occur suddenly because of a significant shift in the parameter
or state. Thus critical slowing down indicators cannot predict those bifurcation points.
However, critical slowing down indicators can predict the bifurcation points in other
cases.
© 2019 Elsevier B.V. All rights reserved.
1. Introduction
Dynamical systems can show variance dynamics [1–3]. In real-world systems, a small change in the parameters of a
dynamical system may cause a significant change in its behavior [4]. In other words, in a complex dynamical system,
smooth changes in parameters may cause a shift from one type of attractor to another [5–8]. This shift occurs when
parameters are near a critical threshold which has been called a tipping point. Near the critical threshold, systems lose
their resilience [9,10]. Tipping points are bifurcation points in which the system’s state changes from one attractor to
another. Bifurcation analysis of dynamical systems is critical [11,12]. Dynamical networks have many complex dynamics
than their single agent [13–16]. Delay play a significant role in the dynamics of complex networks [17–19].
∗ Corresponding author.
E-mail address: [email protected] (F. Nazarimehr).
https://doi.org/10.1016/j.physa.2019.123396
0378-4371/© 2019 Elsevier B.V. All rights reserved.
2
N. Navid Moghadam, F. Nazarimehr, S. Jafari et al. / Physica A 544 (2020) 123396
Before reaching a tipping point, the system may exhibit only a small change, which makes it challenging to predict
tipping points [9]. However, a system may show some general symptoms as tipping points are approached [9]. Finding
ways to predict when tipping points are imminent is an important task [5,9,20]. Much research has been done on
proposing tipping point indicators [21–24]. Close to the tipping points, systems return slowly to the attractors under
small perturbations to their state [20,25]. This phenomenon is called ‘‘critical slowing down’’ [9]. There are many tipping
point indicators based on critical slowing down. However, indicators of critical slowing down cannot always detect where
the system is about to shift to another attractor [25]. Critical slowing down indicators are categorized as metric-based
or model-based. Metric-based indicators extract a feature from the time series to predict tipping points, while modelbased indicators model the time series and predict tipping points from the model [21]. The four well-known metric-based
tipping point indicators are autocorrelation at lag-1, variance, kurtosis, and skewness [9]. However, they cannot predict
tipping points involving complex dynamical behaviors [20]. In [20], a modified version of these four well-known indicators
was proposed that can predict different bifurcation points in systems with a period-doubling route to chaos. The type of
parameter variation in the vicinity of a tipping point is vital to the appearance of critical slowing down. Dakos et al. in [25]
used different functions of the variation of the bifurcation parameter in a system which can jump from one equilibrium
point to another.
In this paper, we investigate tipping point indicators in five cases of bifurcation parameter variations. To predict tipping
points, we use metric-based indicators including modified autocorrelation at lag-1, variance, kurtosis, and skewness in
a biological model which can show different bifurcations, including a period-doubling route to chaos. We discuss the
sensitivity and limitation of these indicators. We also investigate their efficiency in the presence of noise and external
disturbances.
2. Tipping point indicators with different functions of the parameter variation
Evidence shows that various dynamical behaviors, including chaos, exist in biological systems [26–28]. Some examples
of complex dynamics are the electroretinogram of a salamander with varying flash frequency [29], the pulse of spontaneously cardiac cells as the stimulation frequency changes [30], and temperature gradient in a Benard cell [30]. In this
paper, different functions are investigated for variation of the bifurcation parameter, which has an important effect on
reaching a tipping point. A biological system with different dynamical behaviors is used [31]. In Section 2.1, we explain
the system. The four well-known indicators of tipping points are described in Section 2.2. Variation of the bifurcation
parameter is important since it allows getting close to or far from the tipping point. In Section 2.3, we describe different
methods for changing parameters in the studied system.
2.1. Simulated data
In this paper, a model of attention deficit disorder (ADD) is used to investigate the performance of the modified
version of four well-known tipping point indicators with different variations of the bifurcation parameter. This model was
proposed by Baghdadi et al. in [31]. People with ADD have difficulty keeping attention for a long time. Their attention
swings from one activity to another. The nonlinear neural network models inhibitory and excitatory neurons as given by
xn+1 = B ∗ tanh (w1 ∗ xn ) − A ∗ tanh (w2 ∗ xn )
(1)
where B = 5.821, w1 = 1.487, w2 = 0.2223, and A is the bifurcation parameter [31]. Fig. 1 shows a bifurcation diagram
of the ADD model as a function of the parameter A for two initial values. The system has two coexisting attractors in
some ranges of parameter and shows different bifurcations (e.g period-doubling route to chaos and its inverse route).
2.2. Tipping point indicators
Tipping points are conditions in which a smooth change in the parameters of a system causes a significant change in its
dynamic [4,32]. If one is not aware of tipping points, some unpredicted and unwanted dynamics can occur. When a system
reaches a tipping point, a regime shift occurs [4]. Fortunately, before reaching a tipping point, systems may show a few
warning signs. However, it is not easy to use them successfully. Thus, many researchers try to find ways to predict when a
tipping point will occur [21,22]. To predict tipping points in elementary dynamical behaviors (e.g. going to fixed points or
period-one oscillations), the four well-known tipping point indicators (variance, autocorrelation, skewness, and kurtosis)
are useful. However, those methods cannot predict tipping points in complex dynamics such as a period-doubling route
to chaos [20]. Modified versions of these four indicators have been proposed in [20], and those will be used in this paper.
Variance is a useful tipping point indicator [9] because systems close to a tipping point return slowly to their attractor
when perturbed. Such slowing down increases the variance. Variance is a statistical measure which shows how far the
variables are from their mean [33]. We use it as an early warning sign to predict tipping points by varying parameters.
As discussed in [20], this indicator only can predict regime shifts of a period-one type. Therefore, we use the improved
variance proposed in [20] to predict more complex dynamics. In the modified version, the variance is applied to each
period-component of the system, and then its mean value is reported.
N. Navid Moghadam, F. Nazarimehr, S. Jafari et al. / Physica A 544 (2020) 123396
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Fig. 1. Bifurcation diagram of the ADD model when parameter A changes in the interval [5, 30] with initial condition x0 = 0.1 in black and x0 = −0.1
in cyan. The ADD model is multistable since its dynamic changes by varying initial conditions.
Close to a tipping point, the system returns slowly from perturbations to its original attractor. Critical slowing down
can be detected by autocorrelation [9]. The modified autocorrelation at lag-1 is used in this paper to predict critical
slowing down [20]. It can detect period-doubling bifurcations, the edge of periodic windows and tipping points at pitchfork
bifurcations.
Skewness is another critical slowing down indicator [21]. When the system approaches a tipping point, the system
becomes slower. It causes an increase in the skewness of the time series because of its asymmetry [34]. Classical skewness
can only predict the transition between period-one dynamics. Therefore, we use the improved skewness to predict tipping
points of higher periods [20].
In the vicinity of tipping points, the stability of the attractor decreases. Thus the variance of the system’s state
increases. It widens the tail of the time series distribution. By measuring the kurtosis, the elongation of the distribution is
obtained [35]. Similar to the other indicators for predicting transitions with a period higher than one, we use the improved
kurtosis [20].
2.3. Parameter changes in the route to bifurcations
Dakos et al. [25] have studied different parameter changes in an ecosystem. They showed that the critical slowing
down indicators failed in some of the conditions. In that study, they focused on a bifurcation in which a stable equilibrium
loses its stability, and another stable equilibrium is born when increasing the bifurcation parameter. Also, there are some
parameter ranges where both of the equilibria are stable.
However, there are other types of bifurcations in real dynamical systems [29,30]. In this paper, based on [25], we
test five types of variations of the bifurcation parameter in the ADD model which has a period-doubling route to chaos
for changing the parameter A. The first route is a slow and small stepwise variation of the bifurcation parameter. The
second route is a cyclic, state-dependent variation of the bifurcation parameter. In the third route, a small cyclic variation
is combined with a sizeable stochastic resonance. The fourth route involves variations by a large noise, and finally, in
the fifth route, significant stepwise changes in the parameter are studied. The results of different modified early warning
indicators are presented.
2.3.1. Slow stepwise variations of a parameter near a tipping point
Usually, nature is expected to respond to external factors smoothly and linearly to a gradual change. However, some
systems may reach critical situations in a particular range and give a more severe response. Slow changes of a bifurcation
parameter can show different dynamics of a system. Fig. 2(a) shows the stepwise variations of parameter A for the ADD
model in the interval A ∈ [5, 30], and Fig. 2(b) shows a zoomed region of the parameter’s variation. The parameter A varied
after 200 iterations, and its value increases by 0.005 at each step. The index of the step is named ‘‘t’’ in this paper. The
stepwise variation of the parameter A helps us explore different dynamics of the ADD model in the interval A ∈ [5, 30].
It is the most practical method for the variation of a bifurcation parameter. The ADD model has a bifurcation diagram
for changing the parameter A using the mentioned method (as shown in blue in parts (b)–(e) of Fig. 2). The bifurcation
parameter is varied in a stepwise fashion, and the value of the parameter at each step follows a specified function. In other
words, after every 200 iterations, the parameter changes to a new value, and the new value follows a specific function
of the index of steps (t). Part (c) of the figure shows the improved variance indicator in red. For A = 6.25, where the
dynamics of the system changes from period-one to period-two, the improved variance indicator increases. This happens
because near the tipping points, the stability of the attractor decreases, and causes an increase of the variance. Also, this
phenomenon can be seen in other types of bifurcations such as near A = 7.76 where the dynamics of the system changes
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N. Navid Moghadam, F. Nazarimehr, S. Jafari et al. / Physica A 544 (2020) 123396
Fig. 2. (a) Stepwise variation of parameter A in the interval [5, 30]. (b) Zoomed view of the first five steps of the parameter A, (c) bifurcation
diagram of the ADD model in blue and the improved variance method in red, (d) bifurcation diagram of the ADD model in blue and the improved
autocorrelation method in red, (e) bifurcation diagram of the ADD model in blue and the improved skewness method in red, (f) bifurcation diagram
of the ADD model in blue and the improved kurtosis method in red. The results show that bifurcation points, in this case, can be predicted using
the indicators.
from period-two to period-four, near A = 27.24 where the dynamics changes from period-four to period-two, and near
A = 28.99 where the dynamics changes from period-two to period-one. Parts (d), (e) and (f) of Fig. 2 show the results
of the improved autocorrelation at lag-1, skewness, and kurtosis, respectively. They can predict different bifurcations like
the variance method.
2.3.2. State-dependent cyclic transition
Bifurcation parameters can vary with a cyclic increasing and decreasing pattern. To apply this case to the ADD model,
the bifurcation parameter is varied by changing the system’s state at the end of each 200 iterations. The parameter is
changed stepwise, but its variation in each step does not follow a linear function as in the previous case. At the beginning
of each step, the bifurcation parameter A is varied according to the following algorithm,
Algorithm 1
■
■
A0 = 5, b = 0.02
If A ≥ 5 & A ≤ 30
A = A + b(xend )
elseif A = A0
init = −xend
endif.
Variations of the bifurcation parameter versus the index of steps (t) are shown in Fig. 3(a). The bifurcation parameter at
each step depends on the end value of the state in the previous step. When the end of the state is positive, the parameter
N. Navid Moghadam, F. Nazarimehr, S. Jafari et al. / Physica A 544 (2020) 123396
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Fig. 3. (a) State-dependent cyclic change of the parameter A, (b) bifurcation diagram of the ADD model in a state-dependent cyclic change of the
parameter A in blue and the improved variance method in red, (c) bifurcation diagram of the ADD model in a state-dependent cyclic change of
parameter A in blue and the improved autocorrelation method in red, (d) bifurcation diagram of the ADD model in a state-dependent cyclic change
of the parameter A in blue and the improved skewness method in red, (e) bifurcation diagram of the ADD model in a state-dependent cyclic change
of the parameter A in blue and the improved kurtosis method in red. The results show that bifurcation points, in this case, can be predicted using
the indicators.
increases for the next step, and when it is negative, the parameter decreases for the next step. Considering the bifurcation
diagram of the ADD model (Fig. 1) and the variations of the bifurcation parameter A (part (a) of Fig. 3), parameter A
increases until A = 9.5. Then the attractor of the ADD model expands in a crisis, and its attractor crosses into negative
values. Thus the parameter A starts to decrease. The decreasing trend ends when the parameter A reaches its upper bound
of A = 5 and changes the sign of the initial condition and throws the system to another coexisting attractor.
Fig. 3 shows four well-known improved indicators of tipping points for state-dependent cyclic variation of parameter A.
Part (b) of the figure shows the variance indicator. Variance increases before the tipping points at t = 23, t = 194, t = 364
where the dynamics of the system changes from period-one to period-two and t = 44, t = 214, t = 385 where the
dynamics changes from period-two to period-four. It also increases at t = 127, t = 297, t = 468 where the dynamics
changes from period-four to period-two and at t = 148, t = 318, t = 488 where the dynamics changes from period-two
to period-one. When the bifurcation of the system is spontaneous and is caused by a jump in the initial value, the variance
does not show a proper increasing trend like t = 170 and t = 340. This result was expected since those variations are
spontaneous and do not have any history which can cause a slowing down. Fig. 3(c) shows the autocorrelation indicator
in this case, with similar results to variance. The skewness indicator is shown in part Fig. 3(d). The skewness decreases
before tipping points at t = 23, t = 194, t = 364 where the dynamics of the system varies from period-one to period-two,
and at t = 44, t = 214, t = 385 where the dynamics changes from period-two to period-four. It also increases before
t = 127, t = 297, t = 468 where the dynamics changes from period-four to period-two and at t = 148, t = 318, t = 488
where the dynamics changes from period-two to period-one. However, the variations at t = 170 and t = 340 do not have
any history, and they cannot be predicted using tipping point indicators. Fig. 3(c) shows the improved kurtosis indicator
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N. Navid Moghadam, F. Nazarimehr, S. Jafari et al. / Physica A 544 (2020) 123396
Fig. 4. (a) Stochastic resonance variation of the parameter A, (b) bifurcation diagram of the ADD model in stochastic resonance change of the
parameter A and the improved variance method in red, (c) bifurcation diagram of the ADD model in stochastic resonance change of the parameter
A and the improved autocorrelation method in red, (d) bifurcation diagram of the ADD model in stochastic resonance change of parameter A and
the improved skewness method in red, (e) bifurcation diagram of the ADD model in stochastic resonance change of parameter A and the improved
kurtosis method in red. The results show that bifurcation points, in this case, can be predicted using the indicators. However, there are fluctuations
in the trend of these indicators caused by the stochastic perturbation.
in this case. The kurtosis shows tipping points similar to the skewness but has almost the same value at different tipping
points.
2.3.3. Oscillation of a parameter in combination with stochastic resonance
In the real world systems, environmental conditions can swing independent of the system’s dynamic, and it can cause
different bifurcations in combination with stochastic perturbations. To simulate this case in the ADD model, we change
the bifurcation parameter A, as shown in Algorithm 2.
Algorithm 2
■
■
■
b = 0.06
xnew (0) = xold((end) +
) 0.05 (rand − 0.5)
t
A = A + b sin 2π 200
As before, the bifurcation parameter A is changed stepwise. In this case, A is changed at the beginning of each step
(which is shown with t) with a sinusoidal function and remains unchanged for the 200 iterations, after which the
parameter changes for the next step. Also, a stochastic perturbation is added to the state of the system at the beginning
of each step. Variations of the bifurcation parameter with the step index are shown in Fig. 4(a).
Four tipping point indicators are calculated for changing the parameter A in this case. The variance indicator is shown
in Fig. 4(b). The variance increases before the tipping points at t = 40, t = 240, t = 438 where the dynamics of the system
N. Navid Moghadam, F. Nazarimehr, S. Jafari et al. / Physica A 544 (2020) 123396
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Fig. 5. (a) Bifurcation diagram of the ADD model in the case of noise-induced in blue and the improved variance method in red with A = 5, (b)
bifurcation diagram of the ADD model with noise-induced in blue and the improved autocorrelation method in red, (c) bifurcation diagram of the
ADD model with noise-induced in blue and the improved skewness method in red, (d) bifurcation diagram of the ADD model with noise-induced
in blue and the improved kurtosis method in red. The results show that the bifurcations for this case cannot be predicted using early warning
indicators.
changes from period-one to period-two and at t = 64, t = 264, t = 464 where the dynamics changes from period-two to
period-four. It also increases at t = 137, t = 337, t = 537, where the dynamics changes from period-four to period-two.
At t = 164, t = 364, t = 564, the dynamics changes from period-two to period-one, and the variance increases before
that. However, there are fluctuations in the trend of this indicator caused by the stochastic perturbation. Fig. 4(c) shows
the autocorrelation indicator. The autocorrelation shows tipping points similar to the variance, but with almost the same
value for different tipping points. The skewness (part (d) of Fig. 4) decreases before t = 40, t = 240, t = 438 where
the dynamics of the system changes from period-one to period-two and at t = 64, t = 264, t = 464 where dynamics
changes from period-two to period-four. This indicator increases before t = 137, t = 337, t = 537 where the dynamics
vary from period-four to period-two and before t = 164, t = 364, t = 564 where the dynamics changes from period-two
to period-one. The kurtosis indicator reveals tipping points similar to the skewness, but it has the same value for different
tipping points. The perturbation of this case affects all the indicators and causes them to fluctuate.
2.3.4. Noise-induced transition
In this case, the bifurcation parameter is unchanged, and a huge change can perturb the system state for each block
of 200 iterations. The system state changes at the beginning of each epoch of 200 iterations, as shown in Algorithm 3.
Algorithm 3
■
xnew (0) = xold (end) + 5 randn
Fig. 5 shows the four tipping point indicators in this case. The system is run for 20 000 iterations which are divided into
100 epochs of 200 iterations each. At the beginning of each epoch, a massive perturbation is applied to the system state.
The bifurcations of the system, in this case, result from these perturbations. Thus it is expected that the early warning
indicators cannot show these bifurcations. Parts (a)–(d) of Fig. 5 show the improved variance, autocorrelation, skewness,
and kurtosis indications. They show that the bifurcations for this case cannot be predicted using early warning indicators.
As the figure shows, the system state jumps between 3 and −3 (at A = 5), and none of the indicators can predict tipping
points.
2.3.5. Large stepwise change in the bifurcation parameter
A large sudden and unexpected change may occur in the environmental conditions, or the system encounters a
considerable perturbation. If the change in the parameter is permanent, it will take the system to a new attractor. Like the
previous case, it is not expected that the system exhibits critical slowing down before the regime shift. For the simulation
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N. Navid Moghadam, F. Nazarimehr, S. Jafari et al. / Physica A 544 (2020) 123396
Fig. 6. (a) Bifurcation diagram of the ADD model in blue and the improved variance method in red, (b) bifurcation diagram of the ADD model in
blue and the improved autocorrelation method in red, (c) bifurcation diagram of the ADD model in blue and the improved skewness method in red,
(d) bifurcation diagram of the ADD model in blue and the improved kurtosis method in red. The tipping point indicators cannot show any evidence
before the occurrence of bifurcation points.
of this case, the system is run for 200 epochs, and each epoch has 200 iterations. At the beginning of each epoch, a large
perturbation is applied to the system state that can cause the system to jump from one attractor to another. After 100
epochs, a significant shift occurs in the bifurcation parameter where it changes from A = 5 to A = 6.25. For A = 5, the
ADD model has a period-one dynamic. However, a sizeable stochastic noise has been added to the system state at the
beginning of each epoch, which can cause a regime shift. Fig. 6 shows four tipping point indicators in this case. As in the
previous case, the changes do not have any history, and critical slowing down is not seen. Thus tipping point indicators
cannot show any evidence before the occurrence of bifurcation points. Fig. 6 shows that at A = 5, the system can jump
between two equilibrium points at 3 and −3. When the parameter A changes to 6.26, the system state jumps between
two period-two coexisting attractors. The figure shows that none of these indicators can predict tipping points.
3. Conclusion
In this paper, we have applied five methods to change the bifurcation parameter of the ADD model that could produce
different bifurcations. Some of those methods have shown critical slowing down, and others have not. In previous research,
different functions of parameter’s variations have been studied from the aspect of critical slowing down in a system that
can jump from one equilibrium point to another. We have used the ADD model because it can show more complex
bifurcations in a period-doubling route to chaos. To predict critical slowing down before bifurcations, the improved
version of four well-known indicators has been used. Then the efficiency of these indicators in each case of parameter
variations has been investigated. The results showed that in the first three cases, the system approaches bifurcation points
slowly. Thus critical slowing down occurs before the tipping points and critical slowing down indicators can predict them.
In the last two cases, the bifurcations happen suddenly because of a significant shift in the parameter or state. Thus
critical slowing down cannot be seen before tipping points, and critical slowing down indicators cannot predict those
bifurcation points. Put it in a nutshell, in some cases the variations happened slowly, and critical slowing down happed
before bifurcation points. So the indicators of critical slowing down were helpful in these cases. In some other cases, the
variations were very harsh without any slowing down before them. In these cases, the critical slowing down indicators
were failed in prediction.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
N. Navid Moghadam, F. Nazarimehr, S. Jafari et al. / Physica A 544 (2020) 123396
9
Acknowledgment
This work is supported by Iran Science Elites Federation Grant No. M-97171.
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