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Monte Carlo Basin Bifurcation Analysis

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Abstract and Figures

Many high-dimensional complex systems exhibit an enormously complex landscape of possible asymptotic states. Here, we present a numerical approach geared towards analyzing such systems. It is situated between the classical analysis with macroscopic order parameters and a more thorough, detailed bifurcation analysis. With our machine learning method, based on random sampling and clustering methods, we are able to characterize the different asymptotic states or classes thereof and even their basins of attraction. In order to do this, suitable, easy to compute, statistics of trajectories with randomly generated initial conditions and parameters are clustered by an algorithm such as DBSCAN. Due to its modular and flexible nature, our method has a wide range of possible applications. Typical applications are oscillator networks, but it is not limited only to ordinary differential equation systems, every complex system yielding trajectories, such as maps or agent-based models, can be analyzed, as we show by applying it the Dodds-Watts model, a generalized SIRS-model. A second order Kuramoto model and a Stuart-Landau oscillator network, each exhibiting a complex multistable regime, are shown as well. The method is available to use as a package for the Julia language.
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Monte Carlo Basin Bifurcation Analysis
Maximilian Gelbrechtand J¨urgen Kurths
Potsdam Institute for Climate Impact Research
Institute of Physics, Humboldt University Berlin
Frank Hellmann
Potsdam Institute for Climate Impact Research
(Dated: October 16, 2019)
Many high-dimensional complex systems exhibit an enormously complex landscape of possible
asymptotic states. Here, we present a numerical approach geared towards analyzing such systems.
It is situated between the classical analysis with macroscopic order parameters and a more thorough,
detailed bifurcation analysis. With our machine learning method, based on random sampling and
clustering methods, we are able to characterize the different asymptotic states or classes thereof and
even their basins of attraction. In order to do this, suitable, easy to compute, statistics of trajecto-
ries with randomly generated initial conditions and parameters are clustered by an algorithm such as
DBSCAN. Due to its modular and flexible nature, our method has a wide range of possible applica-
tions. Typical applications are oscillator networks, but it is not limited only to ordinary differential
equation systems, every complex system yielding trajectories, such as maps or agent-based models,
can be analyzed, as we show by applying it the Dodds-Watts model, a generalized SIRS-model. A
second order Kuramoto model and a Stuart-Landau oscillator network, each exhibiting a complex
multistable regime, are shown as well. The method is available to use as a package for the Julia
Multistability is a universal phenomenon of complex
systems. Whether it is hysteresis effects in physics, the
human brain [1, 2], gene expression networks [3], in hu-
man perception [4], power grids [5] or the climate system
[6–9], almost every sufficiently complex system has a mul-
titude of stable asymptotic states and bifurcations that
occur when control parameters are changed. Most tra-
ditional methods of bifurcation analysis, such as AUTO
[10] rely on tracking states by continuation of the inte-
gration, and become increasingly challenging for high-
dimensional systems. Further, for high-dimensional sys-
tems, often one is also more broadly interested in classes
of asymptotic states such as synchronized versus unsyn-
chronized states of oscillator network or states that share
a common symmetry. Here, we fill a gap between a
coarse analysis with macroscopic order parameters and
more thorough bifurcation analysis.
Our machine learning approach, Monte Carlo Basin
Bifurcation Analysis (MCBB), based on random sam-
pling and clustering methods, resolves different classes of
asymptotic behaviour into clusters. Rather than study-
ing the existence of states and orbits on the one hand, or
only tracking changes in a single order parameter on the
other, our approach learns which type of attractors are
most dominant in terms of the volume of their basin of
attraction, and quantifies the changing size of the basin
of attraction of each of these classes as a function of a
control parameter. This provides new insights into the
bifurcation structure of multistable high-dimensional sys-
tems. Thus, we can regard MCBB as a way to interpo-
late between detailed studies of asymptotic bifurcations
tracking every change in asymptotic structure on the one
hand, and statistical physics using specialized order pa-
rameters to study the macroscopic behavior at the other
First we will introduce the method and the idea be-
hind it in the following section. Then, the algorithm will
be explained in Sec. II D. A number of paradigmatic ex-
amples, the Dodds-Watts model of social and biological
contagion and networks of second order Kuramoto oscil-
lators and Stuart-Landau oscillators will follow in Sec.
III. Lastly, these results and the performance and appli-
cability of the presented method will be discussed in Sec.
We aim to learn those classes of similar attractors
of a high-dimensional system that collectively have the
largest basin of attraction with respect to a measure of
initial conditions ρ0. Further we intend to understand
how they, and their basin volumes, change as a function
of a parameter pin a range Ip. A class of attractors C
should denote an equivalence class of attractors, includ-
ing at different p, that have similar invariant measures.
To do so we will interpret ρ0as a probability distribu-
tion. We can then draw initial conditions from ρ0and
parameters from Ipand simulating the system to gen-
erate trajectories. Assuming ergodicity, the tail of the
trajectories then sample the invariant measures on the
attractors. We then use these tail samples to estimate
arXiv:1910.06322v1 [nlin.AO] 14 Oct 2019
whether the invariant measures they were drawn from
are similar in the sense of the defining equivalence of our
classification. This way we identify clusters among the
tail samples that are drawn from the same class. By
then computing the number of samples in each cluster
drawn at a particular p(or a small interval around it),
we provide an estimate for the relative size of the basin
of attraction of a class at p. Further we can use the sam-
ples to study how the members of the class change as p
A key step here is the definition of similarity of invari-
ant measures. Comparing all tail samples to each other
is a potentially prohibitively expensive step. Further, in
high dimensional systems with a large number of asymp-
totic states we might be interested in coarser classes of
behaviour. Therefore we typically define the similarity
between clusters in terms of statistics of the invariant
measures that can easily be estimated using the tail sam-
To make this idea more precise we need to define how
to determine that two asymptotic measures are similar.
We begin by outlining the formal quantities under inves-
A. Classes of attractors and their basin volumes
We investigate a complex system with system parame-
ter pyielding a trajectory x(t;x0, p) for initial conditions
x0. This can be an ordinary differential equations sys-
tem ˙
x=F(x, t;p) or a map xn+1 =F(xn, xn1, ...;p). If
this is a sufficiently well behaved dynamical system, the
measure ρ0will asymptotically evolve into ρ, a linear
combination of invariant measures on the attractors Aof
the system,
As we vary the parameter p, the set of attractors and
invariant measures of the system will change as well.
Given a notion of similarity of invariant measures we de-
fine equivalence classes of asymptotic states C. Denoting
Cpthose elements of the equivalence class that occur for
the system parameter p, we have a parameterized space
of measures for each class. Assuming that there are only
finitely many at each p, we write
ρC(p) = X
for asymptotic measures with support only in class C.
We assume ρC(p) = 0 and bC(p) = 0 if the sum is empty
and PcA= 1 otherwise. Under these assumptions ρ
can be decomposed into classes at each p:
ρ(p) = X
bC(p)ρC(p) (3)
When we sample from Ipand ρ0m then run the system,
the resulting trajectories will have probability bC(p) to
asymptotically sample an invariant measure in C.
B. Similarity of asymptotic measures
The key challenge to make this idea operational is to
define a notion of similarity. We will approach this chal-
lenge to define a computable pseudometric in the follow-
ing. Let us first consider an extremal case: A linear re-
sponse of asymptotic measures suggests to identify ρA(p)
and ρA(p+p) as belonging to the same class if they are
connected by a smooth continuum of measures. That
is, the difference between them vanishes smoothly in an
appropriate sense as ∆pgoes to zero, e.g. in the sense
of [11, 12]. When sampling trajectories, we can build
clusters of samples by requiring some discrete notion of
this continuity, ensuring that it converges to the right
continuum condition in the appropriate limit.
Taking classes built up in this way puts us firmly in the
realm of bifurcation analysis. We would resolve every po-
tential difference in asymptotic states. As noted above
this might not be desirable when the number of asymp-
totic states is large, and designing a discrete similarity
measure on the high dimensional space that is not pro-
hibitively expensive to evaluate is not straightforward.
Going into the other extreme are order parameters.
We could consider ρA(p) and ρA(p+ ∆p) as similar if
they lead to the same order parameter up to some finite
bound. This would place us directly into the realm of
statistical physics, but requires us to know already what
meaningful order parameters for our system are.
Generally speaking we build the classes by making use
of some pseudometric on the space of measures built from
a weighted sum of differences of statistics Sk(ρ) of the
measures. The sampled trajectories then provide us with
a way to estimate these statistics, and thus the pseudo-
metric distance between the underlying invariant mea-
D(ρi, ρj) = X
Specifically we will show that for the examples consid-
ered in this paper it is sufficient to track the mean and
the variance of the measures, encoding the position and
size of the attractor in phase space:
The position of the attractor:
The size of the attractor:
where ρkdenotes the marginal distribution on system
dimension k.
FIG. 1. Schematic illustration of an example dynamic
with stable asymptotic states (solid blue lines) and unstable
asymptotic states (dashed blue lines)
We further consider the histograms of these statistics
over the dimensions of the system. This is particularly
useful when the system consists of many identical el-
ements, and it allows us to identify asymptotic states
related by permutation symmetry. This is critical for
the application to networked systems, for example a dy-
namical system on a fully connected network will have a
symmetry group Sn. A more detailed discussion of the
technical aspects are given in the next section.
Dependent of the investigated systems, other statistics,
such as higher moments or entropy measures can be used
as well. Our implementation of the algorithm provides a
flexible framework for this purpose (see Appendix A).
C. Clustering
Finally to construct a cluster of samples from the esti-
mates of the distance of measures there are two options.
Again following linear response theory, we can require
that the observed distance is (up to a factor) a finite
scaling of the linear response of the asymptotic state. For
every sample with a parameter piwe continue the inte-
gration with pi±δp where δp <minj(||p(i)p(j)||)>i
should be a typical parameter spacing, leading to sam-
ples from the measure ρi±. Then, we compare the dif-
ference D(ρi, ρj) between trial iand jwith the differ-
ence to the results of the continuation of the integration
i=D(ρi, ρi±). If the former is much larger we assume
that there is no direct continuation between the states.
Two states are then in the same cluster if there is a chain
of states connecting them.
Instead of this computationally intensive continuation
study we can also try to extract sensible values for the
distance between samples directly from the data. This
leads then to a constant response size parameter DB for
all trials that is ideally a specific percentile Qk(p(δ±
i)) of
the distribution of actual responses δ±
i. When we incor-
porate the parameter proximity constraint with a weight
wpin the distance calculation as well, the new condition
then reads
iand jare connected if
wk|Sk(ρi)Sk(ρj)|+wp|p(i)p(j)|< DB .(5)
Such a criterion is part of density based clustering algo-
rithms such as DBSCAN [13] which we can thus use to
distinguish the different classes of asymptotic states given
a certain set of suitable statistics. If a single, constant
threshold like DB is used, it also allows us to vary this
threshold in order to resolve different classes of asymp-
totic finer or coarser: if we choose a large DB many simi-
lar asymptotic states will be grouped into a single cluster
that corresponds to a broad class of asymptotic states.
Contrary, a smaller DB will result in more different clus-
ters, hence resolving the asymptotic states finer. Fig. 1
schematically illustrates that: As long as this constant
threshold is smaller than the minimal distance between
trajectories of the two asymptotic states in question, they
will be resolved into different clusters.
Crucially, all steps described here can be performed
in a time that scales at most quadratic in the system
dimension. This means that high dimensional systems
are amenable to being studied in this way.
D. Algorithm
We now describe the algorithm that implements the
ideas described above in more detail.
MCBB is a modular algorithm: most steps can be
modified to suit the dynamical system in question. Algo-
rithm 1 summarizes this procedure and in the following
a detailed description of every step is given.
a. Setup We aim to distinguish different classes of
asymptotic states by using clustering algorithms on sets
of measures that each evaluate one of the NMonte Carlo
trials. Given a dynamical system such as an ordinary
differential equation system ˙x =F(x, t;p) or a map
xn+1 =F(xn,xn1, ...;p) with xRNd, we draw N
initial conditions x(i)
0from the distribution UIC and N
parameter values p(i)from the distribution Up. In what
follows, we will will use uniform distributions for UIC
and Up. While we will mostly focus on systems with
one parameter dimension, it is in principal also possi-
ble to investigate systems with more than one parameter
dimension. In particular setups with two varying param-
eters can provide useful insights into the dynamics of the
investigated systems. However, results for systems with
three or more parameter dimensions are harder to visual-
ize and will need exponentially more trial runs to create
sufficient density in the parameter space. In contrast,
just as for basin stability, the number of necessary sam-
ples does not scale with the dimension of the space of
initial conditions.
b. Integration Subsequently, the system is solved for
all of the Ndrawn configurations (x(i)
0, p(i)). The inte-
gration time has to be set appropriately to the system, so
that the asymptotic states are reached. After discarding
the transient, the system is integrated for a sufficiently
long time. While in theory, this choice of a suitable inte-
gration time and transient time is highly non-trivial, in
practice, one should have prior knowledge about the time
scales of the system. In most situations choosing these
times at reasonably large values and checking them for
individual trajectories is sufficient. A more sophisticated
approach will be discussed in future work.
The Julia package provided with this paper (see Ap-
pendix A) uses DifferentialEquations.jl [14] to solve ODE
systems. The library automatically chooses appropriate
solvers, such as Tsit5 or Verner methods [15, 16]. Even
though these methods feature an adaptive step width
during integration, we save the trajectories at a constant
step width, so that the results of all Ntrials are saved
at the same time steps. We then consider the sample
provided by a set final fraction of the trajectory.
c. Evaluation of the Integration On each of the tail
samples generated this way we evaluate a set of statis-
tics, typically we consider some number Nsof statistics
per system dimension Nd. These include per default the
position and size of the attractor as the mean and stan-
dard deviation of the tail sample. Other statistics are
possible as well, though. Thus, we obtain Nmatrices of
statistics Sieach (Nd×Ns) sized with elements Si,kl.
d. Clustering For most clustering algorithms a dis-
tance matrix between all samples is needed. This (N×N)
distance matrix can computed from the Sis with two dif-
ferent approaches. First, by calculating
Dij =
|Si,kl Sj,kl |
where each measure can be weighted with a weight wi.
The parameter values can be included in the distance
metric with weight wNm+1 to ensure that similar asymp-
totic states with strongly different parameter values are
distinguished from each other. The other possibility is to
first fit a one dimensional histogram Hi,k to each statis-
tic kacross all system dimensions. This can be advanta-
geous when symmetric configuration of asymptotic states
should not be distinguished which is often the case for
networks of identical units. The distance matrix then
follows with a suitable histogram distance DH(Hi,k, Hj,k )
such as the 1-Wasserstein metric with
Dij =
wkDH(Hi,k, Hj,k ) + wNm+1 |p(i)p(j)|.(7)
When all Hi,k for one specific statistic kshare the same
binning and norm, the 1-Wasserstein metric can be com-
puted very efficiently from the empirical CDF of each
histogram. The choice of the weights wdepends on the
statistics used and the expected asymptotic states. Gen-
erally, a good first guess is to give higher moments such as
variance and non-normality measures lower weight than
the mean. Given the distance matrix, a clustering al-
gorithm such as DBSCAN [13], is used. DBSCAN clas-
sifies all points that can be reached through a common
DB -area as one cluster. Estimating an appropriate DB
parameter is a non-trivial task and there are different
possibilities. In [13] the authors recommend to use the
k-Nearest Neighbour (kNN) distance, more specifically
the 4NN distance and use the value of the 4NN distance
at the first visual knee in the ordered 4NN distance graph
of all data points as DB . Another, yet similar possibil-
ity is to use the median of the cumulative kNN distance,
where k is a certain percentage of all points, e.g. 0.5%.
As explained in Sec. II, the DB can also be estimated
by continuing the integration and tracking the response
of D. In the examples we have studied, this yields sim-
ilar values like the more empirical kNN-based methods,
but is computationally more expensive. This is why the
kNN-based methods are preferred for the estimation of
the parameter. Fundamentally there is no ”right” choice
of DB , in combination with the choice of distance mea-
sures it determines how finely we want to distinguish tail
samples. While the choice of statistics and weights deter-
mines what aspects we look at, DB provides us with an
overall resolution that we can vary. As the clustering step
at this point is very quick, it is easy to scan a variety of
values. We will see an example further down where two
clusters that are somewhat similar are no longer resolved
as we increase DB . Density-based clustering algorithms
such as DBSCAN are sensible to outliers. Input that is
strongly dissimilar to all other data is classified as an out-
lier. For our purpose, this will typically happen when an
explosion of multistability, many different, yet dissimilar,
asymptotic states occur.
e. Evaluation of the Clustering The clustering algo-
rithm Cthus returns the cluster assignments
C=C({Si}) (8)
which map each of of the Ntrials to one of the NCclus-
ters with Ci[1, NC] being the number of this cluster
for trial i. The cluster assignments Cenable us to further
analyse the system in question. First of all we can track
the size of the basin of each class of asympotic states for
changing parameters and thus quantify bifurcations and
multistability within the system. This is done by com-
puting the amount of trials within a parameter window
[pmin;pmax ] and sliding this window over the complete
parameter range. For each cluster Ci, thus our estimator
for the relative basin volume at parameter p,ˆ
bCi(p) is
bCi(p) = ||CL(p)
j|| (9)
i=nj|(Cj=i)p(j)[pmin;pmax ]o.(10)
In order to further assess the dynamics of each class of
asymptotic sets, the statistics are subdivided into the
sets belonging to each of the clusters as well. This way
we can track, for example, how the position or size of
samples in a cluster change as a function of p. Investi-
gating solutions of typical trajectories within each cluster
can provide insights as well. In Section III examples of
such analysis are shown.
All in all, the two main parameters of the method are
the weights wof the distance calculation and the clus-
tering parameter, in case of DBSCAN DB . As a default
for w, we take wE= 1, wV ar = 0.5, wp= 1. In the Sec-
tion III we will explain in more detail for every system
why we chose the weights presented. For the cluster-
ing parameter, an estimate with the kNN distance or a
response analysis is made and if needed this value is in-
creased (decreased) if one wants to resolve more (fewer)
clusters. As for most Monte Carlo methods, the num-
ber of trials Nshould be chosen sufficiently large so that
the results are independent from it. A reasonable test is
therefore to run the experiment twice: if the results differ
qualitatively, one has to increase N.
Algorithm 1 Monte Carlo Basin Bifurcation Analysis
1: Given: A system ˙
x=F(x, t;p) with system dimension
Nd(can be an ODE system but also a map)
2: Given: A set of Nsstatistics {S} on the components of
trajectories RNtR(e.g. mean and variance)
3: Given: A distribution UIC of the initial conditions and
parameters Up
4: for i1, N do
5: Sample Ninitial conditions x0from ρ0and Nparam-
eter values p
6: Solve system for a long trajectory x(t;p)
7: for dim 1, Nddo
8: for meas 1, Nsdo
9: compute matrix of statistics Si,dim,s =
Si(xdim(t)) on the tail of the trajectories.
10: end for
11: end for
12: end for
13: Obtained: N(Nd×Ns)-matrices Si
14: Compute (N×N)-sized distance matrix Dof all Sito
each other.
15: Density-based clustering (e.g. DBSCAN) of D
16: Analyze cluster memberships and statistics Sfor each
cluster dependent on the parameter values p
MCBB is a method that can be applied to a wide range
of dynamical systems. Both, systems with discrete and
with continuous state spaces are possible to investigate,
as are systems with discrete and continuous time evolu-
tion. Typical applications are networks of oscillators as
will be shown in the following, but also discrete agent-
based models such as the Dodds-Watts model. Every
system that returns a trajectory given an initial condi-
tion and parameter can in principal be analyzed with
MCBB. In the following the Dodds-Watts model, Ku-
ramoto oscillator networks and Stuart-Landau oscillator
networks will be investigated with MCBB. The source
code of all these results is available as Jupyter notebooks
in the supplementary material.
A. Dodds-Watts model
The Dodds-Watts model of social and biological conta-
gion [17, 18] is a generalization of contagion models such
as the SIS and SIR model [19, e.g.]. Given is a popu-
lation of NIindividuals that are connected to all other
individuals. Each of the individuals is either in the sus-
ceptible (S), infected (I) or recovered (R) state and has
a memory of doses they received within the last Ttime
steps Dt,i. At each time steps each individual icomes
into contact with another individual jthat is randomly
selected from all other individuals. If jis infected, ire-
ceive a dose dwith exposure probability p. The amount
of the dose dis drawn from a distribution f(d). The
dose adds to the dose memory Dt,i of iat time step t
so that Dt,i =Pt
tT+1 dt0,i. If the dose memory of an
individual exceeds the dose threshold d
i, it becomes in-
fected. Latter dose threshold d
iis drawn from a distri-
bution g(d). As soon as Dt,i drops below the threshold,
the individual recovers with probability rat each time
step. A recovered individual becomes susceptible again
with probability s. One gets the classic SIS model for
example for the configuration s= 1, g(d) = δ(d1),
f(d) = δ(d1), T= 1 with pand ras free parame-
ters. For more details on the model, see [18]. For this
NIdimensional model with discrete states si,t [S, I , R]
and discrete time t[1,2, .., tN] we directly evaluate
the count of susceptible N(S) and infected states N(I)
within the time evolution of each individual as measures
for the algorithm. As shown by [18], there are several con-
figuration which possess multistable regimes where also
a mixed population with N(I) unequal 0 or NIcan be
In particular we are investigating the two configura-
tions: (A) with NI= 1000, T= 12, r= 1, g(d) =
δ(d3), s= 1 and (B) with NI= 1000 g(d) =
0.075δ(d1) + 0.4δ(d2) + 0.525δ(d12), T= 20,
r= 1 and s= 1. The number of initially infected indi-
viduals is drawn from a uniform distributed between 0
and NI. We evolve the system for 1000 time steps from
which we regard the first 800 time steps as the transient.
Configuration (B) is roughly similar to the SIS model but
with a dosage memory of 20 steps and a dosage thresh-
old distribution so that roughly half of the population
is quite resilient against becoming infected. For both
configurations N= 5000 trajectories with random ini-
tial conditions and parameter values were computed. As
both of the measures are equally important, we choose
wI=wS= 1 and wp= 0, so that we do not use the pa-
rameter value in the distance calculation. The distance
Dwas constructed using histograms of the statistics as
described in Sec. II D.
Based on a visual inspection of a 4NN-distance graph,
the clustering parameter dDB = 0.15 was chosen for con-
figuration (A). Fig. 2 shows the results of the analysis.
Similar to the results reported in [18], we see for such a
configuration a bifurcation occur around p0.4. For
values larger than this the fully infected state becomes
stable. Its basin of attraction quickly grows, but the fully
healthy state remains stable as well with a very small
basin of attraction for large pvalues.
Configuration (B) exhibits a slightly more complex
structure as Fig. 3 reveals in accordance with the results
in [18]. Additionally, Fig. 3 features sliding histograms
as well. These can be helpful to identify the dynamics of
the clusters. For each sliding parameter window a his-
togram is fitted to all measure results within this window.
These histograms are then plotted directly next to each
so that we can visualize changes of the measures within
each cluster for changing parameter values. In the case of
the Dodds-Watts model where we measure the fraction
of time an individual agent was infected and suscepti-
ble, these are predominantly either 1 or 0 as most agents
are either infected or susceptible the whole time. Fig.
3A shows the behaviour of the system. For small val-
ues ponly the fully healthy state is stable (see also Fig.
3B). The first bifurcation occurs around p= 0.3 when
a mixed state, for which susceptible and infected indi-
viduals coexist, becomes stable. Its basin of attraction
quickly grows, while the healthy state remains stable but
with a very small basin of attraction. Fig. 3 shows that
for growing pthe amount of infected individuals rises.
Eventually, around p= 0.7 a fully (or almost fully) in-
fected state becomes stables. As Fig. 3D shows directly
at the bifurcation point not all individuals of the fully
infected state are infected which is the case for larger p.
Comparing the results to these reported in [18] we see
that the fully infected and the mixed state are indeed
two distinctive stable branches of the system and thus
rightfully classified by MCBB into two separate clusters.
B. Kuramoto Networks
The Kuramoto Model is one of the fundamental exam-
ples of synchronization theory and network science. The
version with inertia has been used in a variety of con-
texts, most importantly to model nodes in power grids
[20–25]. In the transition towards globally stable syn-
chronization, the Kuramoto model with inertia exhibits
an extreme form of multistability, with a large number of
attractors. Studying the dominant patterns of synchro-
nization in the transition region was one of the motivat-
ing questions for the development of MCBB.
FIG. 2. Approximate Relative Basin Volume of the two dif-
ferent classes of asymptotic states, fully infected (blue) and
fully healthy (red), for configuration (A) of the Dodds-Watts
model. The colored areas in the plot represents the basin
volume of the respective state. Computed by using a sliding
parameter window over the clustering results (see Sec. II D),
a window length of 0.05 and an offset of 0.01 were used.
FIG. 3. (A) Approximate Relative Basin Volume of the differ-
ent asymptotic states of configuration (B) of the Dodds-Watts
model. It exhibits a fully infected (blue), fully healthy (red)
and mixed state (green). Computed by using a sliding pa-
rameter window over the clustering results (see Sec. II D), a
window length of 0.05 and an offset of 0.01 were used.
The system is given by the equations
Aij sin(φiφj),(11)
with equally many +1 and 1. For K= 0 the oscillators
rotate freely with ω=±10. As Kincreases synchro-
nization starts to occur in the network. At K= 10 the
system typically synchronizes completely with ωi= 0.
While a large number of works have studied the stability
of this synchronous state as a function of the local net-
Part. S 2
Part. S 1
FIG. 4. (A) Network structure of the investigated Kuramoto
system. Red nodes have a negative drive (minus sign in Eq.
11) and blue nodes a positive one. (B) The basin bifurca-
tion diagram for the system with the synchronized, partially
synchronized and synchronized regimes.
work topology[23, 25–35], comparatively little is known
about the intermediate regime.
As the main dynamics is in the frequency, we will only
consider the frequency dimensions in the analysis here.
Figure 4a shows the network on which the oscillators are
coupled. It is a random regular graph for which every
node has degree k= 3. The statistic we will use on the
asymtptotic state are the positions of the frequency of
all the nodes and the distance is Dcomputed accord-
ing to Eq. 6. The results shown are for N= 25,000
The basin bifurcation structure, with distances calcu-
lated from the per-dimension mean of the frequency, is
given in Figure 4b. We see that for K= 0 the oscilla-
tors rotate freely, the frequencies are located at ω=±10.
This state persists, until its basin starts to shrink from
K= 1 onward. In the intermediate regime most of the
asymptotic states occur. These are classed together in
the outlier cluster here, meaning that they occur so in-
frequently that not enough samples can be obtained for a
statistical treatment. This shows that the basin structure
isn’t dominated by one transitional state but an explosion
of multi-stability occurs. However, the basin bifurcation
diagram also shows two states that achieve a higher basin
in the transition region. Each of these clusters occurred
in more than 0.5% of the total runs, and peaks at taking
up more than 10% of the basin volume at some p.
If we look a bit deeper into these clusters, we find that
they represent partial synchronization, in which a region
of the network is synchronized, while all other oscillators
still rotate at their natural frequency Figure 5.
To understand how these intermediate clusters lose sta-
bility as Kincreases, we can consider the size of the
asymptotic states considered in Figure 6. Here we see
that the size of the attractor increases as Kincreases. In
other words, the frequency itself starts to oscilate around
a stable average frequency. This suggests an interesting
insight into the behaviour for the transition regime. As K
increases some neighbouring oscillators couple and syn-
chronize. As the attractor of the partially synchronized
state grows, the oscillators at non-synchronized nodes
coupling coupling
Cluster 2 - Unsync. Cluster 3 - Partial Sync.
mean frequency
FIG. 5. Analysis of the Clusters exhibiting no synchronization
(A) and partial synchronization (B). (A) and (B) are sliding
histogram plots, similar to Fig. 3 and show the means of
the frequencies over all nodes as histograms depending on the
coupling parameter. (C) and (D) show the mean frequency of
each individual oscillators over all samples in the cluster for
cluster 2 (left) and cluster 3 (right).
FIG. 6. Sliding histogram plot similar to Fig. 5. Here, the
standard deviation of all frequency time series is shown de-
pending on the coupling parameter.
spend more and more time far from their natural fre-
quency. Eventually they would have to spend consider-
able time close to the frequency of a synchronized com-
ponent that they couple to and get entrained.
To verify that these are the mechanisms that drive the
transition, and to understand which network properties
enable early partial synchronized states, is beyond the
scope of MCBB and this paper. However, the basin map
of the bifurcation transition that is revealed by this ap-
proach provides immediate and crucial insights into how
the basin structure and the structure of the attractors
themselves change in the transition. In particular it re-
veals that the attractors do not move, but grow until they
lose stability.
C. Stuart-Landau Oscillator Networks
Another paradigmatic type of oscillator is the Stuart-
Landau oscillator which can be written as
˙z= (λ+− |z|2)z(12)
where zC,λis the bifurcation parameter and ωis
its eigenfrequency. Originally found by Lev Landau and
later derived by Stuart and Watson [36–38] to describe
the transition to disturbance in hydrodynamics, it is also
a normal form of the Andronov-Hopf bifurcation and
hence widely applicable and of great importance in many
fields [39]. Coupling Stuart Landau Oscillator can lead to
several interesting phenomena. Most importantly oscil-
lator quenching in the form of Amplitude Death (AD)
and Oscillator Death (OD) [40, e.g]. An other inter-
esting phenomena are Chimera states [41, e.g.]. These
are states of systems of coupled identical oscillators that
exhibit a inhomogeneous pattern in which phase-locked
states coexist with drifting states. To apply MCBB for
Stuart-Landau systems, we use the configuration of [42]
as it prominently features a multistable regime with trav-
elling wave (TW), oscillation death (OD) and what the
authors refer to as stable amplitude chimera (SAC) dy-
namics. In this setup NNStuart-Landau oscillator with
identical eigenfrequency ωare coupled by attractive cou-
pling to its P1nearest neighbours and repulsive coupling
to its P2nearest neighbours with the following equations:
˙zi= (1 + − |zi|2)zi+K
where <(x) is the real part and =(x) the imaginary part
of x. We can also investigate this setup with the cou-
pling mediated on two Watts-Strogatz random graphs
[43], one for the repulsive and one for the attractive cou-
pling. With the rewiring probability pr= 0, we get the
same equation as above, for pr6= 0 we expect changes in
the dynamic.
1. Parameter Configuration
We choose the same parameter configuration as in [42]:
ω= 2, NN= 100, P1= 1 and P2= 22. In our experi-
ments we vary K,r2=P2/NNand pr. We use random
initial conditions with real and imaginary part uniformly
distributed between 1 and 1 (in contrast to the clus-
ter initial conditions used for some calculations in [42])
and vary Kfrom 1.8 to 2.5. As per dimension measures
we use mean and standard deviation. Since the Stuart-
Landau oscillators are complex valued, all measures are
applied separately to the real and imaginary part. From
our a priori knowledge about Stuart-Landau Oscillators,
we know that their asymptotic states will exhibit differ-
ent kinds of oscillatory behaviour, thus it is a good choice
to put the largest weight on the standard deviation. We
choose wE= 0.25, wSD = 1, wp= 1 and run N= 15,000
trials that are integrated from t0= 0 to tf= 200. The
first 70% of this time span are regarded as the transient
and are not used for the evaluation. The first experiment
is performed with pr= 0 and r2= 0.22 and the distance
Dis calculated using histograms according to Eq. 7.
2. Varying the coupling
After running the experiment and calculating the dis-
tance matrix D, the associated 4-dist graph exhibits the
knee point at around 0.01. We slightly decreased this
value to 0.009 and 0.008 in the reported results. Figs.
7 (A) and (B) show these results for the approximate
relative basin volume. Similar to the results reported
in [42] we see a multistable regime, in which TW dy-
namics are prevalent for K < 1.95 and OD dynamics
are for K > 2.2. In between there are various states in
which some oscillators show OD-like behaviour and oth-
ers exhibit a synchronized oscillation. We thus prefer to
refer to these kinds of states as partially synchronized
(PS) states. Importantly, the PS states are a mixture of
many similarly partially synchronized states and not just
a single asymptotic state. If we choose a larger DB like
in Fig. 7A, the states with full OD and the PS states
with only few partially synchronized oscillators and oth-
erwise mostly OD dynamics are merged into one cluster
(OD+PS). For smaller DB they are separated into two
distinct clusters (Fig. 7A). One particular structured and
more common kind of partially synchronized states can
be found for 1.9< K < 2.0. As Fig. 8 shows, these states
are highly regular stationary waves, interrupted by oscil-
lators exhibiting OD, we thus refer to these states as reg-
ularly clustered stationary wave states (RCSW). Aside
from these more regular dynamics, there are all kinds of
different mixed states between wave-like dynamics and
oscillation death. Many are so dissimilar to each other
that they fall into the outlier cluster. The outlier clus-
ter has the most members during the transitions from
TW to PS via RCSW at K2.0 and at the transition
between OD and PS at K2.2. A handful of smaller
clusters with less than 60 members (or 0.4% of all tri-
als) were neglected. They contain partially synchronized
states with more similarities to each other than to those
in the outlier cluster. We identified these dynamics by
further analyzing the statistics within each cluster. Fig.
8 shows example plots and sliding histogram plots for
two of these clusters. The RCSW states mostly oscil-
late and thus almost all oscillators have a mean of zero
and a constant standard deviation different from zero.
We see that these histograms change little for different
coupling values. The cluster is very homogeneous with
almost all members looking like the example shown in
Fig. 8 C. The PS cluster, on the other hand, is much
more inhomogenous. Its members have in common that
most of the oscillators exhibit OD, thus as Fig. 8 con-
firms, they exhibit nonzero means, with both positive and
negative values while having a vanishing standard devia-
tion which corresponds to the typical stable fixpoints of
OD dynamics. Fig.8 D shows one example, the amount
of oscillators still exhibiting a synchronized oscillators is
different within the cluster, though. Additional results
for the other clusters can be found in the appendix.
3. Varying the coupling and amount of coupled neighbours
Similarly to the additional setup in [42], we can also
investigate this system with two varying parameters with
MCBB. First, we choose to vary K, the coupling, and
r2, the relative amount of neighbours the oscillators are
coupled to repulsively. Fig. 9 shows similar clusters of
similar asymptotic behaviour as in the one-dimensional
setup. We see that TW dynamics are present only for
small Kand large r2values, while OD+PS dynamics are
present even for small Kvalues when r2is small. For
very small r2there is also a desynchronized (DS) cluster.
Most notably the distinctive RCSW type dynamics are
only present for r2>0.1 and its basin becomes larger for
larger r2values.
4. Rewiring of the network
14 When we start to randomize the coupling by
rewiring it according to the scheme of Watts-Strogatz
random graphs, we get the results presented in Fig. 10.
Here, we added the outlier cluster together with several
smaller clusters that all exhibit mixed, partially synchro-
nized, partially OD dynamics to the mixed states (MS)
cluster. The range of Kfor which these kinds of dynam-
ics appear gets wider when the rewiring princreases. TW
dynamics appear less for larger prvalues. RSCW type
dynamics do not appear when we rewire the network.
Given a complex system, such as a ODE system,
like the Kuramoto and Stuart-Landau networks demon-
strated in Sec. III B and III C, or a map like the Dodds-
Watts model presented in Sec. III A, MCBB is able to
analyze and quantify which classes of asymptotic states
are occuring. As demonstrated with the paradigmatic
example systems MCBB is a widely applicable approach.
It is suitable to analyze the behaviour of every high-
dimensional system that returns a trajectory, be it agent-
based models such as the Dodds-Watts model or Differ-
ential Equations like the Kuramoto and Stuart-Landau
networks. The known bifurcations of these systems were
reproduced by MCBB as shown for example with the
Dodds-Watts model. Additionally, it enables us to re-
veal clusters of qualitatively similar asymptotic states
for all these systems as the results investigated in Sec.
III C show. It does successfully identify the sizes of the
basins of the most important asymptotic states even in
transition regimes, what a traditional bifurcation analy-
sis can not reveal. For the Kuramoto system we see how
and when the basins of the unsynchronized states shrinks
and how the basins of the completely synchronized states
emerges. We also get an insight into the transition be-
tween these states, as we can see how the size of the
states increases before they destabilize. Hence, for the
Kuramoto model it provides an intuitive way of visualiz-
ing the synchronization process. When applying MCBB
to a Stuart-Landau system the different asymptotic be-
haviours, travelling wave states, oscillator quenching phe-
nomena such as oscillator death and mixed stated, are
classified in different clusters and interesting dynamics
such as regularly clustered stationary wave states are re-
vealed and their basins quantified.
The analysis can always be fine tuned by changing the
clustering parameters to resolve the asymptotic states
finer or coarser. Additionally, the weights of the distance
calculation provide another mean of adjustment. The
flexible nature of the method also allows for experimen-
tation with the statistics used to evaluate the trajectories
and the exact clustering algorithm. In particular various
entropy-based seem promising to use. While designing
the method we already used the per dimension Kullback-
Leibler divergence of the time series to the Gaussian
measure KLGi= DKL ρG(Ei,Vari)
ρias a statistic
to track structural changes of investigated systems. This
was especially useful for relatively low-dimensional sys-
tems. The curve entropy [44] of the complete trajectory
was tested as well. Additionally, we also experimented
with a distance between histograms of the covariance ma-
trices as a statistic. This expands variance-based size
measure to also take cross-correlations between the di-
mensions into consideration which could be useful for
systems that exhibit multiple possible cross-correlations
structures in the asymptotic states that otherwise behave
similar, e.g. different kinds of collective oscillations. For
the example systems presented here, it was however suf-
ficient to only use the position and size of the attractors
as measures. Additional measures were not necessary to
resolve the different classes of asymptotic states. This
should not stop experimentation with additional mea-
sures though, as some of them are already implemented
in the accompanying software as well with further addi-
tional ones easy to add.
Aside from the approximate basin volume and the slid-
ing histograms shown in this paper, it is also possible to
further investigate the clusters found by the clustering
algorithm, e.g. by analyzing which kind of initial condi-
tions lead to certain class of asymptotic states or by an-
alyzing how each dimension is changing with the control
parameters separately and not in histogram form. These
options are already implemented in the Julia package (see
FIG. 7. Cluster diagram of the Stuart-Landau Oscillator network with pr= 0 for two different values of the clustering
parameter DB . For (A) DB = 0.009 and for (C)&(D) DB = 0.008. MCBB resolves the different classes of asymptotic
states: travelling wave (TW), regular clustered stationary waves (RCSW), (full) oscillation death (OD) and mixed partial
synchronized / oscillation death (PS) states. When increasing D B states in which most (but not all) oscillators exhibit OD,
while the remaining few oscillators are synchronized (PS) and the states in which all oscillators exhibit OD (OD) are merged
to one cluster (OD+PS). The window size used is 0.025 and the offset is 0.01.
1.9 2.0 2.1 2.2 2.3 2.4
Coupling K
Value of Mean
1.9 2.0 2.1 2.2 2.3 2.4
Coupling K
1.0 B
rel. Magnitude
FIG. 8. Introspective analysis of the two of the clusters also
shown in Fig. 7. Plots (A),(B),(E),(F) are sliding histogram
plots. For each sliding window of coupling values K, the re-
spective measures of trajectories within the said cluster are
plotted as a histogram in y-Direction. (A-C) inspect the
RCSW cluster. (A),(B) show the mean and the standard
deviation of the RSCW cluster. (C) and (D) are example tra-
jectories from the respective clusters. (E), (F) show the mean
and standard deviation of the PS cluster.
Appendix A) and more could be envisioned in the future.
It is further possible to extend the method to systems
with unknown background parameters that adhere to cer-
tain distribution and additional control parameters or
forcings, such as some climate models which will be fur-
FIG. 9. Results from the setup with two parameters, vary-
ing the amount of coupled neighbours r2and the coupling
strength K.
ther discussed in future work.
While this work focused on introducing the method
and testing it with fairly theoretical models, we believe
that this opens the door to studying a wide variety of
systems in novel ways. We expect that the method will be
FIG. 10. Results from the setup with two parameters, varying
rewiring prof the Watts-Strogatz random graph that medi-
ates the repulsive coupling and the coupling strength K.
fruitful in diverse contexts where a mix of multistability
and high dimensional behaviour are important. Most
notable among those would be biological networks and
climate systems.
A distinct limit of the approach is that it is only able to
detect and track stable solutions of the investigated sys-
tems. Unstable solutions are not accessible with MCBB.
A further important avenue of investigation is to study
the mathematical properties of the algorithm described
here in much more detail. In particular it would be highly
desirable to understand the convergence properties of the
algorithm. We also suspect that there is considerable
scope for improving the clustering by making use of in-
formation from the continuation, rather than reverting to
a standard density based algorithm. One other avenue
of investigation where we will improve the method fur-
ther is to use the statistics of the tail sample we record
in order to track when the integration has reached the
asymptotic regime in a suitable sense.
MCBB provides an excellent way to visualize the com-
plex behaviour of systems where a traditional bifurcation
analysis is often not useful or difficult to implement. It
resolves the most important classes of asymptotic states
and enables the user to track the size of its basins along
changing parameters.
Appendix A: Julia Package
The algorithm is implemented in Julia. It can be in-
stalled directly from the GitHub repository https:// This
library makes heavy use of Julia’s DifferentialEquations.jl
library [14]. There is an extensive documentation avail-
able that explains the package with many examples that
is linked in the page of the repository.
Appendix B: Logistic Map
While designed for high-dimensional systems, MCBB
will also still work in the fringe case of a one dimensional
system such as the logistic map xn+1 =rxn(1 xn).
FIG. 11. Basin Volume and Bifurcation diagram of a logistic
Fig. 11 shows the approximate relative basin volume
computed with MCBB compared to the bifurcation dia-
gram. It was computed using the mean, standard devi-
ation and Kullbach-Leibler divergence as measures with
the weights 1, 0.5 and 0.5. The major bifurcation points
are reproduced as do the stable regions inside the chaotic
regime form seperate clusters, while most of the chaotic
regime is grouped into to distinct clusters, one before and
one after the larger stable region around r3.8.
Appendix C: More Results
Additionally to the results presented in Sec. III, one
can also further inspect the other clusters found by
MCBB for the Stuart-Landau systems. This is done in
Fig. 12 and 13. The Julia package (see Sec. A also
allows for further other visualizations and inspections of
the measures and the clusters. The documentation of the
package explains these in more detail.
The authors thank Jobst Heitzig, Marc Wiedermann
and Valerio Lucarini for fruitful discussions about the
presented approach. This paper was developed within
the scope of the IRTG 1740/TRP 2015/50122-0, funded
by the DFG/FAPESP, the Condynet2 project by BmBF
FK. 03EK3055A and the DFG project CoCo-Hype KU
837/39-1 / RA 516/13-1. The authors thank the Ger-
man Federal Ministry of Education and Research and the
Land Brandenburg for supporting this project by provid-
ing resources on the high performance computer system
at the Potsdam Institute for Climate Impact Research.
mean mean mean
FIG. 12. Further analysis on the clusters also shown in Fig.
7. (A),(B),(E),(F) are sliding window histograms fits of the
denoted measures for trials with parameters within the re-
spective window. (C) and (D) are example trajectories of
trials within these clusters.
FIG. 13. Further analysis on the clusters also shown in Fig.
7. (A),(B),(E),(F) are sliding window histograms fits of the
denoted measures for trials with parameters within the re-
spective window. (C) and (D) are example trajectories of
trials within these clusters.
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In many natural systems, attractive coupling together with repulsive coupling plays a vital role in determining their evolutionary dynamics. We investigate the stabilization of amplitude chimera through repulsive coupling in the presence of attractive coupling in a system of nonlocally coupled oscillators. The nonlocal repulsive coupling can facilitate the emergence of stable amplitude chimera even for random initial conditions contrasting with the earlier investigations, where the amplitude chimera was observed just as a transient state and that too for a specific cluster initial conditions. The stability of the observed amplitude chimera is analyzed using Floquet theory. To elucidate the transition among the distinct dynamical states, we find the average number of inhomogeneous oscillators as a function of the coupling strength and show that the transition among the dynamical states exhibits hysteresis. Further, we deduce analytically the critical stability curve that separates the oscillatory (amplitude chimera and traveling wave) states from the death (multi-incoherent oscillation death, cluster chimera death, cluster oscillation death) states. We also analyze the influence of the nonisochronicity parameter and noise on the stable amplitude chimera. We report that the nonisochronicity parameter favors the traveling wave state from incoherent death through the stable amplitude chimera state.
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In this paper the concept of thermodynamics of curves is employed for the analysis of dynamic systems. More specifically, a new indicator based on an entropy measure allows to recover some information about the degree of irregularity of a curve. The proposed indicator was used to compare the evolution of trajectories in the state space and some of its properties seem very interesting for a study and a classification of nonlinear systems. Among its properties, the proposed indicator provides a finite value which does not depend on stability issues and it always provides a constant value when applied to linear systems. The proposed indicator was then tested in several benchmark problems, also for chaotic systems, to emphasize the theoretical expectations.
This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number. It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.