# Nonlinear Impulse Pattern Formulation dynamical social and political
prediction algorithm for city planning and public participation

R. Bader<EMAIL_ADDRESS>Institute of Systematic Musicology, University
of Hamburg, Neue Rabenstr. 13, 20354 Hamburg  S. Linke  Hamburg University for
Applied Sciences, Ligeti Center, Veritaskai 1, 21079 Hamburg  S. Gernert
Freie und Hansestadt Hamburg Landesbetrieb Immobilienmanagement und
Grundvermögen Millerntorplatz 1 20359 Hamburg

###### Abstract

A nonlinear-dynamical algorithm for city planning is proposed as an Impulse
Pattern Formulation (IPF) for predicting relevant parameters like health,
artistic freedom, or financial developments of different social or political
stakeholders over the cause of a planning process. The IPF has already shown
high predictive precision at low computational cost in musical instrument
simulations, brain dynamics, and human-human interactions. The social and
political IPF consists of three basic equations of system state developments,
self-adaptation of stakeholders, two adaptive interactions, and external
impact terms suitable for respective planning situations. Typical scenarios of
stakeholder interactions and developments are modeled by adjusting a set of
system parameters. These include stakeholder reaction to external input,
enhanced system stability through self-adaptation, stakeholder convergence due
to mediative interaction adaptation, as well as complex dynamics in terms of
direct stakeholder impacts. A workflow for implementing the algorithm in real
city planning scenarios is outlined. This workflow includes machine learning
of a suitable set of parameters suggesting best-practice planning to aim at
the desired development of the planning process and its output.

††preprint: AIP/123-QED

> Modern city planning, as well as social or political decision-making, depend
> on predictions of the outcome of actions, which are mainly based on
> statistical data or expert opinions. Still, the complexity of modern
> societies is steadily increasing while resources are getting scarce. Also,
> public participation has become increasingly important, as modern city
> planning is often based on the upcycling of materials and increased
> densification of cities rather than the planning of new urban areas. To
> increase predictive robustness, efficient use of resources, and social
> participation and public acceptance of planned actions, a physical modeling
> algorithm is proposed. Such an algorithm needs to be computationally cheap
> to allow large-scale system prediction in a reasonable time. It needs to
> account for typical planning situations with sudden phase changes,
> adaptation, and mutual interactions, and it needs to be able to connect
> stakeholders in a planning process to be modeled with parameters relevant to
> these stakeholders like health, artistic freedom, social peace, or financial
> feasibility. The algorithm proposed is, therefore, based on a Physical
> Culture Theory, taking physics as a common ground. Such an algorithm also
> needs to be able to predict scenarios that have not been present before.
> Therefore, physical modeling is superior to a machine learning algorithm,
> which only knows what it has learned. Such an algorithm could, in the long
> run, be used in public participation processes on a web-based modification
> and suggestion tool that is freely available online.

##  I Introduction

"We are already colourful" A statement that does not seem to fit into the
scientific context and yet has led to the introduction of an algorithm in
urban planning to test, question or confirm this statement, for a review see
Abas2023 . The statement of the citizens who see themselves exposed to the
plans for a future change in their neighbourhood. How can urban planners
advance their projects in a targeted manner in the interests of the community
and expect a "Yes please, let’s examine the parameters for a liveable future
together" instead of a "No thanks, we’ve already set up our lives"? Many
approaches have been proposed mitigating this problem, often involve machine
learning algorithms.

In times of scarcity of space and land, diverging political goals such as the
net zero land strategy and alliances in favour of building even more housing,
the smallest parameters that are not taken into account can have a decisive
influence on the already fragile balance and even prevent planning
altogetherCharturvedi2021 . Active citizens’ initiatives can lead to
referendums and long-term uncertainties in urban planning. On both sides of
the process, resources and lifetimes are spent fighting against their own
system. There is no question that planning in an increasingly dense city is
becoming more complex and planning for the future more uncertain. But building
takes time and what is built is there for the time being: it creates space and
thus shapes society. Both the new neighbourhood and the old neighbourhood need
to be well planned or replanned, taking into account all known parameters, so
that stable, good communities and thus stable societies can form. With
forward-looking planning that takes environmental concerns, demographic
aspects and crises into account, an equal society is conceivable.Wilkinson2024
.

The world is more colourful than a single planning team can grasp. An
algorithm can provide support here and visualise how complex living beings and
things behave. However, it can show citizens just as credibly that this is a
forward-looking and necessary decision by the planners.

One example of a complex planning situation that involved and activated many
stakeholders was the replanning of the "Frappant" shopping centre in Hamburg
Altona, Germany, which was built in 1973. Since the 1980s, the huge "concrete
block" has been in decline and stood completely empty for several years until
the 2005. Then, one by one, artists moved in as temporary tenants and the city
declared the neighbourhood a redevelopment area. By 2009, 150 artists had come
together to form the "frappant.org" initiative and were using the spaces with
different aims. Some appreciate the large and affordable studios, others want
to establish a new cultural centre for the city. In autumn 2009, IKEA buys the
complex to establish a city centre branch. Citizens’ movements for (e.g.
business men) and against (e.g. local residents) IKEA are formed. The old
Frappant building is demolished at the end of 2010Necker2014 .

Considering self-organizing the foundation of social and political systems and
therefore claiming a need for a physics- and nature-based politics has
prominently been advocated for on philosophical groundsLatour2022 ; Latour2017
. These ideas have been formulated in a much broader sense as ecosystems and
life in general to be of self-organizing natureMargulis1998 ; Lovelock1988 ;
Smith2016 ; Watson1983 .

Following this road, nonlinear dynamical models for political and social
issues have extensively been proposed in terms of social complexity measures
(seeFlack2011  for a review), smart city infrastructureRajaan2024 , power
grids energy management stability analysis using machine learningTitz2024 , or
low-carbon transport city planningXinghui2012 , next to many others. The need
to rethink city planning and urbanism is felt strongly in the fields of city
planning and architecture in the direction of using sustainable materials and
ecological urbanismTucci2022 ; Coskun2023 .

Social polarization has often been studied using different nonlinear dynamical
methods. Using a Hamiltonian approach of connected social network coherence
minimization, a phase transition was found when the amount of connectivity in
the society rises above a certain level, which only has happened through
extensive internet connectivity over the last decadesMinhPham2020 . This model
allows only a polar decision regarding friend/enemy.

Using a time-step method of individual opinion development changing according
to the impact of all other network participants, coherent and polarized social
states can be found by alternating the impact strength and the shape of a
hyperbolic tangent functionBaumann2020 . This model enforces polarization due
to positive or negative opinions from the start but allows the opinions to be
continuously moderate to extreme.

In sociology, the structural balance theory initially assumed a set of three
members being friends or enemies and changing relations according to a
balanced or unbalanced stateHeider1946 . It has been enlarged to large
societiesTaylor1970 . In a model of a set of differential equations for
homogeneous or diverging societies, it was found that initial model conditions
determined the output, where initial mean positive relations lead to
conformity. In contrast, initial mean hostile relations lead to
divergenceMarvel2011 .

Conformity simulation in consumer models shows a relation between conformity
network members’ stability depending on the network topologyJedrzejewski2020 .
Conformity stability is much more stable with balanced local/global
connections than with an unbalanced network.

All these models are not perfectly suitable for the task of city planning.
There, different stakeholders participating in a planning process have
different interests like social balance, artistic freedom, financial interest,
etc. These have an initial state at the start and will change throughout the
construction process due to complex interactions among stakeholders, internal
adaptation processes, or external influences. Therefore, a dynamic system must
be formulated based on all stakeholders and their interactions.

Public participation is crucial in deciding if and how a planning process
needs to take place. During this stage, different opinions exist on the impact
of political action on different stakeholders. Some might find the social or
financial balance in a city environment at risk, shop owners might fear
financial downgrading, and artists might fear losing their performance or
exhibition spaces.

Therefore, a robust and persuasive prediction method is needed to estimate the
influence of political and social measures on all stakeholders. Public
participation could agree on the desired output of the measures. According to
this aim, a nonlinear dynamical network is needed to predict the best measures
to be taken to achieve this aim.

The Impulse Pattern Formulation (IPF) model proposed in the present paper to
arrive at this aim has first been developed for musical instrument
applicationsBader2013 ; Linke2021b ; Linke2019b . It has also been proven to
predict human-human interaction in rhythm perception and productionLinke2021a
. The IPF has also proven to work in brain dynamicsBader2022 , explaining
fundamental aspects like the excitatory/inhibitory neuron concentration or
typical brain time scales. The brain IPF also corresponds to EEG measurements
of brain synchronization as the physiological cause of musical large-scale
tension dynamicsBader2024 .

The IPF takes a viewpoint neuron of, e.g., neurons or musical instrument parts
from which an impulse is sent out to several other neurons, musical instrument
parts, or any object. This impulse is processed, damped, and returns to the
viewpoint objectBader2013 ; Linke2019b ; Bader2021 ; Linke2021b  leading to
convergent, bifurcating, chaotic, or diverging behavior. Although often
modeling a system with only very few nodal points, the IPF has already been
shown to be highly precise.

As the IPF already assumes a set of standpoints, which in the case of city
planning are stakeholders, and can simulate very different types of temporal
stakeholder developments as well as sudden phase changes, converging,
bifurcating, and chaotic behavior, it is used as a basis for a political and
social prediction process.

In the frappant case discussed above, six such stakeholders were identified.
These are the artists, local stores, local residents, residents of Altona, the
investor IKEA, and the City of Hamburg. Although the present paper outlines
the basic framework of the model discussing real-world cases in detail is
beyond its scope. Still, the number of stakeholders used in the model is fixed
to six throughout.

Such an iterative, nonlinear-dynamical process is scale-free, capable of
modeling sudden phase changes, and includes convergent, bifurcating, complex,
and chaotic states. Scaling is a problem in social sciences in general,
leading to so-called voltage drops when evidence found in one social group is
scaled up to a broader levelList2024 .

Like with the brain IPF, the social and political IPF suggested here needs
adaptation of stakeholders. Adaptation is a notion with many aspects; for an
overview of contemporary ideas, seeSawicki2023 . In the present case,
adaptation is understood as a change of the internal state of a network
participant or stakeholder according to its own state, and therefore, it is
called self-adaptation.

The interaction between stakeholders in the model is formulated in two
different versions: a direct and a mediated interaction. Therefore, the
structural balance theory discussed above is not used as a model input but
might emerge from it.

The IPF is an algorithm emerging from a Physical Culture TheoryBader2021 ,
assuming no difference between humans, nature, culture, or technology by
finding all to be electromagnetic fields when broken down to a physical basis.
This includes consciousness, which is found to be a tempo-spatial
electromagnetic field complex enough to arrive at conscious content of all
kinds. Returning to this introductions beginning, where the self-organization
nature of life in general was discussed, in this view, cultural artifacts like
musical instruments, which are self-organizing structures by their very nature
but made by humans, are viewed as enlarging the human body and mind with
interacting artifacts as a way to maintain life by reducing a system entropy.

Therefore, as the IPF has already proven to work with musical instruments and
the human brain, adding politics, social, and city planning issues to the
model might help arrive at a model that bases culture and nature on a common
ground.

##  II Method

The social IPF assumes a set of interacting stakeholders. Each stakeholder s
is represented by a system variable $g^{s}_{t}$ at discrete, adjacent time
steps t. This system variable represents the relevant parameters of the
stakeholder, which can be artistic freedom, health condition, financial
situation, or the like. The IPF is always formulated from a standpoint where
each standpoint has its own IPF. Here, each of $N$ stakeholders is a
standpoint.

The IPF furthermore assumes a self-reference of each standpoint upon itself
using a self-adaptation variable $\alpha^{s}_{t}$, adapting according to the
standpoint itself. The damping variable $\alpha^{s}$ determines the system is
stable, bifurcating, chaotic, and unconverting.

Such a system has extensively been studied with various forms of musical
instrument acoustics, human-machine interaction, or brain dynamics. It is
suitable for reconstructing the fundamental behavior of systems with only a
few parameters to very high precision.

Furthermore, the interaction of the different stakeholders is modeled using an
interaction variable $\beta^{s,i}_{t}$, of stakeholder i acting on the
viewpoint stakeholder s at time point t. Again, $\beta^{s,i}_{t}$ is adapting
at each time point.

In previous IPF formulation, e.g., with the classical guitar, each standpoint
IPF could be calculated separately, as the values of $\alpha$ and $\beta$ were
constant. In the case of the social IPF, where these constants are adaptive
and, therefore, variables, the IPF can only be calculated simultaneously for
all stakeholders or standpoints.

Social or political issues are often very complex and individual. To account
for this, the proposed social IPF consists of three equations with three
varying terms and a set of parameters.

###  II.1 Social IPF system state equations

The general form of the social IPF is

$g^{s}_{t+1}=g^{s}_{t}-\log_{e}\left(\frac{1}{\alpha^{s}}\
g^{s}_{t}-\sum_{i=1}^{N,i\neq
s}\beta^{s,i}e^{g^{s}_{t}-e^{s}_{t-d^{s,i}}}\right)\ +I^{s}_{t}.$ (1)

The time delay parameter $d^{s,i}$ it takes until the state $g^{i}$ is acting
on $\beta^{s,i}$ might vary according to the relation between stakeholders and
therefore need to be a variable.

To account for unexpected external input to the system, like unpredicted
ecological, political, or financial boundary condition changes, an external
function $I^{s}_{t}$ is introduced, directly acting on the stakeholders. This
implies adding or subtracting energy from the system, which is reasonable as
no system exists in isolation.

In many cases, no such impact exists, and for one or all stakeholders s

$I^{s}_{t}=0\ .$ (2)

Therefore, $I^{s}_{t}$ is the first varying term that might or might not be
included in a scenario.

###  II.2 Stakeholder self-adaptation equation

Adaptivity is needed due to the adaptive nature of living systems and has
already been successfully implemented in the IPF model family in the brain IPF
model. Adaptivity of a self-referencing system can only be implemented using
the system variable $g^{s}$ of the stakeholder itself and is suggested as a
self-adaptation variable

$\alpha^{s}_{t+1}=\left(1/\alpha^{s}_{t}-C^{s}(g^{s}_{t}-g^{s}_{t-1})\right)^{-1}\
.$ (3)

Here, $\alpha^{s}_{t}$ is time-dependent and altered according to the system’s
gradient $g^{s}$ with a constant $C^{s}$.

In some cases, a stakeholder might not adapt its self-reference, as in the
case of laws or regulations acting in the system, which are not subject to any
changes. In this case, for some stakeholder $s$, it is assumed that

$\alpha^{s}=const\ .$ (4)

Therefore, $\alpha^{s}_{t}$ is the second varying term that needs to be
decided when modeling a real social scenario.

###  II.3 Stakeholder interaction equation

The impact of stakeholder $i$ on stakeholder $s$ is according to the
interaction variable $\beta^{s,i}$. Changing $\beta^{s,i}$ during modeling is
well known from the literature (e.g., Linke2019a ; Linke2021a ). As shown in
Fig. 1, a sudden change of $\beta^{s,i}$ directly affects the system state
$g$. The system may (after a short transient) converge to a new stable limit.
However, it is also possible that the state will change to bifurcations,
chaos, and sometimes even diverging states.

Figure 1: different time series of $g$ when suddenly changing beta for an IPF
with $N=2$ stakeholders / refelction points: $\alpha=0.369$ is constant.
$\beta$ always starts at $0.2$ and is suddenly changed to the different
displayed values after 50 iteration steps

Nevertheless, as shown in Fig. 2, continuously changing $\beta^{s,i}$ keeps
the system close to its fixed point. Thus, the system appears to be stable
even in regions where the fixpoint is not. However, as soon as $\beta^{s,i}$
does not change anymore, the system gradually shows the expected behavior.

Figure 2: different time series of $g$ when continously changing beta for an
IPF with $N=2$ stakeholders / refelction points: $\alpha=0.369$ is constant.
$\beta$ always starts at $0.2$ and is linearly changed to the different
displayed values between the 40th and the 60th iteration step

The mutually changing of $\beta^{s,i}$ of different stakeholders can be
subject to several interaction processes. In direct interaction, the
interaction variable is proportional to the system variable of the impacting
stakeholder $i$ multiplied by a constant $W^{x,i}$ like

$\beta^{s,i}_{t+1}=W^{s,i}g^{i}_{t-d^{s,i}}\ .$ (5)

We refer to this version as direct impact in the following.

Again, the time delay parameter $d^{s,i}$ of Eq. 1 is introduced to account
for the delay one stakeholder acts on another.

Yet, another option is to assume that a stakeholder is actively altering the
impact of another stakeholder upon him with respect to the difference between
both system states, like

$\beta^{s,i}_{t+1}=\beta^{s,i}_{t}-V^{s,i}\left(g^{s}_{t}-g^{i}_{t-d^{s,i}}\right)\
.$ (6)

This accounts for the fact that stakeholders are in constant interaction and
try to balance their impact constantly in time from one to another. We refer
to this version as mediating impact in the following.

So $\beta^{s,i}_{t}$ is the third varying term to be decided for in a social
scenario.

The time delay can be formulated more generally like

$\beta^{s,i}_{t+1}=\beta^{s,i}_{t}-V^{s,i}\left(g^{s}_{t-d^{s,i}}-g^{i}_{t-d^{s,i}}\right)\
.$ (7)

Here, the system state $g^{s}$ is not compared with its present state to a
previous state of $g^{i}$ at $d^{s,i}$ but also in its previous state
$d^{s,i}$. This would imply that one stakeholder remembers a previous state or
that the impacting stakeholder i refers to a previous stake of s. This is a
more general version to be included, although it is not expected to be the
general case. Still, we will discuss the impact of such a case below in some
detail, too.

###  II.4 Workflow of the social, political, and city planning IPF

Therefore, the social IPF is a toolbox consisting of three equations, three
varying terms, and three parameters. The three equations have been outlined
above.

Tab. 1 gives an overview of the varying terms, again as discussed above. There
might or might not be an external impact (Varying Term I) for each stakeholder
$s$ separately. There might or might not be self-adaptation (Varying Term II)
for each stakeholder separately. And interaction might be direct or mediated
(Varying Term III). This last condition might not be exclusive, and both
interactions might take place at the same time. In this paper, we first
consider both terms alone to discuss their properties and leave a combined
version for future studies.

The fixed parameters of the model are summarized in Tabl. 2. The self-
adaptation weights determine the self-adaptation strength of the variable
$\alpha^{s}_{t}$, as are the direct and mediating interaction weight for the
interaction variables $\beta^{s,i}_{t}$. The delays of stakeholders
interacting with each other are fixed parameters.

| Varying term I | Varying terms II | Varying term III
---|---|---|---
| External impact | Self-adaptation | Interaction
| $I^{s}_{t}$ | $\alpha^{s}_{t+1}$ | $\beta^{s,i}_{t+1}$
Variation I | External impact | Self-adaptation | Direct interaction
| var. | $\alpha^{s}_{t+1}=\left(1/\alpha^{s}_{t}-C^{s}(g^{s}_{t}-g^{s}_{t-1})\right)^{-1}$ | $\beta^{s,i}_{t+1}=W^{s,i}g^{i}_{t-d^{s,i}}$
Variation II | No external impact | No self-adaptation | Mediating interaction
| 0 | $\alpha^{s}_{t+1}=const$ | $\beta^{s,i}_{t+1}=\beta^{s,i}_{t}-V^{s,i}\left(g^{s}_{t}-g^{i}_{t-d^{s,i}}\right)$
Table 1: Varying terms of social IPF Parameter | Symbol
---|---
Self-adaption weight | $C^{s}$
Direct interaction weight | $W^{s,i}$
Mediating interaction weight | $V^{s,i}$
Interaction delay | $d^{i,j}$
Table 2: Parameters of social IPF

In real-world situations, the social IPF is expected to predict a best
practice for political action, giving an initial state of all variables and
parameters and a desired time development of the system parameters for all
stakeholders. To achieve this, it seems inappropriate to evaluate all possible
combinations of varying terms, parameters, and initial variable states but
pre-select varying terms and initial variables first to determine the best
parameters to achieve the goal. In other words, political and social
constraints determine the varying terms and parameters, while political and
social actions are expressed in the parameters.

Modeling a social IPF takes the following steps:

####  II.4.1  Decision on stakeholders

First, the stakeholders need to be determined. In the Frappant case, there
would be six relevant parties. This also implies determining the system
variable $g^{s}$ for each stakeholder. In the Frappant case, this might be
artistic freedom and possibilities for the artists, well-being for the
surrounding public, financial interest of the investor, or the like.

####  II.4.2  Decision on varying terms

For each stakeholder, a choice of the varying terms needs to be decided. In
the case of Frappant, all stakeholders would be subject to self-adaptation;
therefore, varying terms II would be a self-adaptation case.

####  II.4.3  Decision on initial values

Each stakeholder’s system variable $g^{s}$ has an initial value at the
beginning of the process. This value needs to be decided in terms of the
interpretation of the system variable $g^{s}$, might it be health state,
artistic freedom, or social equality.

Sometimes, the initial values’ decision cannot be clearly determined
perfectly. This is accounted for in step 5).

####  II.4.4  Decision on delay parameter

Next, the delay parameter matrix $d^{s,i}$ of impacting all stakeholders onto
all others must be set. This also implies a choice of the time step of the
system for each iterative step. This time step will depend on the data
available, like traffic measured in intervals of months, house rent costs
known in steps of years, or traffic noise where data is available several
times a day or even hours.

The time step needs to be set to the smallest time interval necessary, which
will determine the interaction’s delays. Such interactions can be clearly set,
like monthly payments, regular concert activity, fiscal seasons, or regular
business meetings. Still, they can also be on a weekly or daily basis with
buying decisions or depending on seasonal changes over the year.

Again, some delays might not be clearly determined, which is addressed in the
next step.

####  II.4.5  Machine learning of possible scenarios

After making the above decisions, the task is to find the parameters best
suited to a desirable social or political output for all stakeholders. As the
parameter space of the three parameters left to determine can become large,
and as some of the above parameters might not be determined satisfactorily, a
machine learning approach is taken.

For each parameter, and for all initial values of step 3) and delay parameters
in step 4), reasonable upper and lower bounds are chosen. Also, for all
varying parameters, a simulation step size that is small enough to reasonably
discretize the parameter range from lower to upper bounds of each parameter is
chosen. Such a parameter space consists of N simulations and can contain
several thousand cases.

Each case is simulated, resulting in a time series over the desired total
simulation time with a respective amount of time steps T for all stakeholders.
Each stakeholder set of simulated system parameter $g^{s}_{t}$ values is taken
as a feature vector of size $T\times s$.

A machine learning approach is used. In the present paper, we use a Kohonen
self-organizing mapKohonen1995 Blass2019  for several reasons. First, after
training and best-fit matching of the feature vectors to the map, the reason
for such a fit can be found in the trained maps’ feature vectors. Many machine
learning algorithms, like deep-learning methods, do not easily allow to
determine the reason for successful training. So, Kohonen maps are used here
to understand the reason for possible cluster formation. Also, Kohonen maps do
work with small data sets. Although these are not too often expected in the
case of a social IPF, in some cases, this feature might be of benefit. Yet, a
third reason is the visualization in a 2D map, which is beneficial again for
estimating the amount and sizes of clustering.

####  II.4.6  Determining social and political strategies from social IPF
prediction

When a desired development of all stakeholders states $g^{s}$ is determined by
political means, the trained machine learning algorithm is fed with the
desired development. It gives best-fit parameters to fulfill the need. The
social IPF, in this case, then predicts a particular social or political
development and outcome for all stakeholders when the respective parameters
are set.

Such a parameter setup also needs to be determined according to uncertain
input spaces in initial values (step 3) or delay parameters (step 4).
Therefore, stable clusters are required where, at best, all simulated cases
within the uncertain parameter range meet the desired output.

####  II.4.7  Interpreting social or political actions from predicted
parameters

The parameters of self-adaptation weight $C^{s}$ and direct or mediated
interaction weight $W^{s,i}$ or $V^{s,i}$ respectively, need to be translated
into action or decisions.

With best-fit results in the case of mediated interactions between several
stakeholders demanding a high weight, stakeholders need to make sure both
sides interact closely to achieve the appropriate goal. This might also
determine a best-fit interaction delay, which could imply regular meetings,
financial benefits, a suggested time interval for artistic or cultural
activity, or the like.

##  III Results

###  III.1 Single stakeholder without self-adaptation with external
disturbance

To get familiar with the model, at first, the behavior of a single
stakeholder, s = 1, with and without self-adaptation, with varying term II in
version I and version II, is discussed, also including the disturbance of an
external impact, so with a varying term I in version I. As an example, 50
iteration steps were performed with $\alpha=0.56$, ensuring a converging
system variable $g^{1}$ (first stakeholder). An initial $g_{1_{0}}=1$ is set
as boundary conditions.

Five cases are shown in Fig. 3. The top blue curve shows $g^{1}$ over 50
iterations with $I_{t}=0$ without impact. After an initial transient, $g^{1}$
becomes stable and converges.

Figure 3: System variable $g^{1}$ of stakeholder one for $\alpha=0.56$ and
different impact $I_{20}=\\{0,1,5,-0.5,-1\\}$ at iteration point 20. Blue: no
impact (varying term I version II), yellow positive impact $I_{20}=1$ (varying
term I version I) leading to a disturbance with returning convergence. Green:
even stronger impact of $I_{20}=5$ where convergence again appears only after
a larger time interval. Red: negative impact $I_{20}=-0.5$ showing a strong
positive reaction of the system followed by convergence. Purple: negative
impact $I_{20}=-1$ leading to a negative $g^{1}$ and therefore a break down of
the system.

The yellow curve shows the reaction to a positive $I_{20}=1$, so an impact of
one at a single iteration time 20. The system is disturbed and returns to its
converging state. This is expected as such a distortion corresponds to a new
initial $g_{0}$ after which convergence is ensured due to the choice of
$\alpha=0.56$ for which such a system is stable.

The green curve shows a stronger impact of $I_{20}=5$. Although the system
becomes stable again, it takes more time to recover.

The red curve shows an impact of $I_{20}=-0.5$, so a negative impact. Contrary
to the positive impact, the system reacts with a strong positive $g^{1}$ at
the next time step, from which it again converges.

Yet the purple curve shows an even larger negative impact of $I_{20}=-1$. Then
$g^{1}$ becomes negative at the next time step, leading to a breakdown of the
system.

All these behaviors represent reasonable reactions of a social system to an
impact; it might converge to a stable value, even with a negative impact, or
it might break down and leave the stage.

Decreasing $\alpha$ does not change the system’s reaction to an external
impact considerably. Still, with low $\alpha$ around 0.37 for which chaotic
behavior is reached, only a very small negative impact $I_{t}$ is needed to
break the system down. A chaotic time series easily reaches low values of g
where only a small negative impact makes g negative, leading to the breakdown.

In the Frappant case, the impact of the stakeholder City of Hamburg decided to
tear down the building, representing a heavy impact on the artist’s
stakeholders. The City of Hamburg offered the artists a new building, the
Victoria Kaserne, which allowed the artists to self-organize the building and
its content. About 130 artists then founded a Fux e.G. and organized the
artistic work with the low rental cost of the new building. The offering of
the new building by the City of Hamburg only came into place due to strong
political pressure implemented by the artists on the City of Hamburg to
maintain artistic content and social life.

Therefore, the strong impact of one City of Hamburg stakeholder led to a
strong movement of the artist’s stakeholders in the City of Hamburg, which
reacted to the artists again, and the whole system reached a new stability.

###  III.2 Single stakeholder with self-adaptation with external impact

To estimate the effect of self-adaptation, the model again with s=1 and a
strong impact at iteration time 20 of $I_{20}=10$ is used, but now including
self-adaptation, so using varying term II in version I. Fig. 4 compares
$g^{s}_{t}$ for both cases, starting at $\alpha=0.56$. In the above case,
$\alpha$ = const, while now $\alpha$ is adaptive, therefore time-varying. Also
now, $C^{s}=-1$ is used.

The green curve in Fig. 4 is $g^{s}_{t}$ of varying term I version II already
shown in Fig. 3. The blue curve is the corresponding curve of varying term II
version I with adaptive $\alpha$, shown in yellow. Both, $g^{s}_{t}$ and
$\alpha_{t}$ are bifurcating but are stable in this respect. The impact at
iteration time point 20 disturbs both curves considerably. Still, $g^{s}_{t}$
of version I returns to its bifurcating state much faster than $g^{s}_{t}$ of
version II.

So, as the adaptive version I of varying term II is much more active all
through time, it is more feasible to return to a stable oscillation after a
strong impact compared to the non-adaptive case. When reducing $C^{s}$, the
bifurcating $g^{s}_{t}$ and $\alpha_{t}$ weaken their bifurcating amplitudes,
and adaptation to a previous state is slower; the basic behavior remains the
same.

Figure 4: Adaptive version I of varying term II compared to non-adaptive
version II concerning reaction of external input at iteration time point 20.
Green: Version II time series from Fig. 3 with constant $\alpha=0.56$. Blue:
Version I time series with adaptive $\alpha$. Compared to Version II, the
Version I time series is bifurcating due to adaptivity; still, it is returning
to this state faster compared to Version II with static $\alpha$. Yellow:
Version I adaptive $\alpha$ corresponding to Version I system parameter
$g^{s}_{t}$. After the impact on time point 20 $\alpha$ is way above the
diverging limit of $\alpha=0.37$, the system remains stable due to adaptation.

To investigate this systematically, a parameter space is calculated for
varying $C^{s}$ and $\alpha_{0}$ for three external impacts at iteration point
20 of $I^{s}_{2}0=\\{0,1,10\\}$. The system runs over 100 iterations, and
convergence is calculated, taking the mean of iterations 90 - 100 and only
accepting real means < 100 and $\Im{g_{t}}=0$ for all t.

All calculations were performed using an initial system value $g^{s}_{0}=0.1$.
This ensures $\Im{g_{1}}=0$ for $\alpha_{0}>0.09048$ within the first
iterative step, a value of $\alpha$ much lower than expected in the system at
all. Using $g_{0}=1$ would lead to $\Im{g_{1}}\neq 0$ for $alpha_{0}\geq e$.
Therefore, any adaptation algorithm would no longer be able to converge
$\alpha_{0}<0.3679$, and extended stability by including any adaptation
algorithm could not be tested. Using any $g_{0}$ is artificial in any case, as
the initial time point of a social or political situation is arbitrarily set
by an investigator while the stakeholders taking part in the social action
most often existed before the chosen model starting time point.

Figure 5: Convergence of adaptive Version I of varying term II for varying
$C^{s}$ like $-3\leq C^{s}\leq 7$ and initial $\alpha_{0}$ like 0.1961
$\leq\alpha_{0}\leq$ 10 for three impact strength I. Top: $I_{20}$ = 0, no
impact, $\alpha_{0}$ can be decreased way above the lowest value of non-
adapting Version II with suitable $C^{s}$. Middle: reduced variability of
$\alpha_{0}$; still, again, lower values are possible. Bottom: only negative
$C^{s}$ lead to a stable solution and $\alpha_{0}$ only leads to conversion
for values up to $\alpha_{0}\sim 0.370$ as in the non-adaptive case.

Fig shows the three cases for I = 0, so without adaptation, I = 1, and I = 10.
The adaptation constant was varied from -3 $\leq C^{s}\leq 7$ and the initial
$\alpha_{0}$ like 0.1961 $\leq\alpha_{0}\leq$ 10.

In the top plot of Fig. 5, colors denote converging cases of $C^{s}$ vs.
$\alpha_{0}$ combinations where the color corresponds to the mean converged
value. A tail appears starting from $\alpha_{0}$ = 10 and 0 $\leq C^{s}\sim$
-2 which is getting narrower towards $\alpha_{0}\sim$ and $C^{s}\sim$ 0.2.
Therefore, choosing a suitable $C^{s}$ allows for much higher $\alpha_{0}$ to
converge compared to the non-adaptive Version II with a minimum of $\alpha\sim
0.37$. Indeed, with decreasing $\alpha_{0}$, the range of $C^{s}$ for
converges is getting smaller, as expected. Still, it is interesting that the
convergence value is smaller towards higher $\alpha$ displayed by the lighter
color.

The convergence area converges for all parameter values within the area and is
not fragmented in the sense of non-converging islands. So, convergence is
ensured within the displayed regions.

Convergence is also present with negative $C^{s}$, as already discussed above.
Still, negative $C^{s}$ allows only reduced $\alpha_{0}$ for convergence.

An unsystematic convergence area also exists for $C^{s}<-2$, which is quite
unstable. A stakeholder in this region might consider it quite random to
maintain himself stable.

The middle plot of Fig. 5 shows convergence for a slight impact $I_{20}=1$.
Again, lower $\alpha_{0}$ can be used to arrive at convergence compared to
non-adaptive Version II., still slightly reduced compared to the case of
$I_{20}=0$.

Yet in the bottom plot with the case of $I_{20}=10$, so strong impact,
$\alpha_{0}$ can no longer be extended over the limit of Version II. This case
also no longer allows for positive $C^{s}$.

Therefore, we can conclude that the adaptive Version I of varying term II is
more stable over a wider range of $\alpha_{0}$ compared to Version II and,
therefore, more flexible, as an expected consequence when including adaptation
to the model.

Furthermore, with high external impact, the adaptation strength needs to be
negative to have a robust, wide-range convergence.

Therefore, the adaptive varying term II version I is superior to the non-
adaptive version II as it covers all convergence of the non-adaptive with
additional convergence capabilities with a suitable choice of parameters.

###  III.3 Multiple stakeholders with direct impact and no external
disturbance

Adding the interaction between multiple stakeholders leads to a temporal
description of the system parameter $g^{s}$ for all stakeholders in the
process. This is implemented by the interaction parameter $\beta^{s,i}_{t}$ of
stakeholders i impacting on stakeholder s. First, direct impact is discussed,
which means including the Variation I of Varying term III as shown in Tab. 1.

When modeling city planning, a starting time point needs to be defined, and
the initial values for all stakeholders $g^{s}$ and the parameters need to be
decided. Different scenarios can happen. There might be a running system, like
with the Frappant case, where many stakeholders have already reached
convergence, and only one or two new stakeholders come in, or a new impact
disturbs the system. In present-day city planning, this might be the more
likely case, with upcycling or densifying of cities being the main focus.
However, a system might also be set up from scratch, like when planning a new
area of construction.

In this simulation, we assume all stakeholders are already present in the
system before the simulation starts at $t=0$. Still, couplings $\beta^{s,i}$
are often not known or hard to determine, which leads to the simplification of
uncoupled stakeholders before $t=0$. Thus Eq. 1 and 5 are valid $\forall\
t>=0$. $\forall\ t<0$ the system can be described by the IPF with
$\beta^{s,i}=0\forall\ s,i$. As we assume the system to be running for a
sufficiently long time, all stakeholders have already converged to their fixed
point. When assuming that no $\beta$ are involved before $t=0$ this fix point
is $g^{s}_{f}=\alpha$ according to Eq. 1 and thus $g^{s}_{t}=\alpha^{s}\
\forall t<0$. Then, all necessary initial values of the simulation are set.
Further, the time delay $d^{s,i}=i$ directly depends on the couped
stakeholder.

In the following, IPFs with $N=6$ stakeholders are used throughout, as taken
from the Frappant example. Each IPF is iterated for 500 steps using Equations
1 and 5. $d^{s,i}=1$ is used throughout. By randomly varying $\alpha^{s}$,
IPFs were calculated until 1000 non-diverging time series were obtained. This
amount was reached after 3523 attempts. Thereby, $\alpha^{s}$ was chosen
within a range of $\alpha^{s}\in[0.2,1]$. Also, the coupling constant
$W^{s}_{i}=W$ was fixed for all stakeholders but varied for each IPF again
randomly in the range of $W\in[0.01,1/6]$.

According to the process of predicting a best practice of political action
according to stakeholders’ needs in a public participation process, possible
scenarios need to be identified, as discussed above. Therefore, a clustering
algorithm is needed to determine typical temporal developments of stakeholders
$g^{s}$. If this is successfully implemented, the inverse problem is solved,
starting from a desired output and deriving parameters of the system leading
to this output. From these parameters, political actions are derived as part
of the workflow discussed above in Section II.4 of city planning or political
action.

The clustering algorithm used is a self-organizing Kohonen map (SOM) for the
reasons discussed above. The SOM is trained with the 1000 converging IPFs.
Thereby, the time series of each stakeholder $s$ is normalized to mean=0 and
std=1. Thus, the SOM organizes the data according to the overall shape of
$g_{t}$ but not on the absolute values, as they are highly dependent on
$\alpha$. We, therefore, arrive at types of $g^{s}$ developments.

Figure 6: Trained SOM of 1000 IPF cases of six stakeholders using direct
interaction with Variation I of Varying terms II. The background color is the
u-matrix displaying similarity (dark colors) of differences (light colors) of
neighbouring neurons. Therefore, the yellow ridges are boundaries between
clusters present in darker regions. Each dot is an IPF $g^{s}$ iteration’,
which is placed according to its best fit to the feature vectors of the
trained SOM. The colors of the dots refer to the coupling strength $W$; darker
colors are stronger, and lighter colors are weaker coupling. For each of the
clusters or regions I to VIII, an assigned time series $g^{s}_{t}$ is picked
and plotted aside the SOM as a typical example of stakeholders $g^{s}$
development of the last 20 values of the 500 iteration steps for the
respective regions I to VIII.

Fig. 6 shows the trained SOM for Variation I of Varying term III. The
background color is the so-called u-matrix, displaying similarities and
differences of neighboring neurons. Lighter yellow colors have strong
differences, and darker blue colors have small differences. Therefore, the
yellow ridges are bounders between blue regions of very similar neurons. The
dots represent the 1000 IPF cases the map was trained with, and they are
positioned after training at the best-fit neurons for each IPF. The colors of
the dots show varying coupling strengths $W$, where lighter colors are weaker
and darker colors have stronger coupling.

In this SOM, clearly, eight distinct regions of the u-matrix are visible and
named I to VIII, with I being the largest region. In region I, all
stakeholders converge to their fixed point. A typical $g^{s}$ time series of
all six stakeholders is shown in Figure 7. After an initial short transient
phase, all $g^{s}$ converge. Region I is the only region where such
convergence takes place.

Such a situation could be after construction of a new housing area. Although
at the planning stage negotiations will happen, leading to the complex initial
state in the simulation, it is expected that for some time after construction
ends, the inhabitants and all other stakeholders are satisfied.

Figure 7: Example IPF time series of region I of Fig. 6 of $g^{s}$ converging
for all s=6 stakeholders, calculated from Eq. 5. This example was initial
$\alpha^{s}\approx\\{0.98,0.71,0.79,0.75,0.55,0.77\\}$ and $W\approx 0.011$.
Therefore, the largest region I of all VIII regions is that of convergence of
all stakeholders.

The region VIII in Fig. 6 is also unique. Here, the stakeholders $g_{s}$ show
chaotic time series and complex patterns. An example IPF $g^{s}$ time series
is shown in Fig. 8.

Typical recent examples are big projects within cities where construction
already began and due to financial or political struggle progress only very
slowly or not at all. This leads to uncertainties for investors,
entrepreneurs, local residents, politics, city planners, etc. The IPF could
help here to determine in such cases possible solutions.

Figure 8: Example IPF time series of region VIII of Fig. 6 of a chaotic time
series of stakeholders $g^{s}$, as calculated from Eq. 5 with initial
stakeholder $\alpha^{s}\approx\\{0.61,0.73,0.36,0.88,0.93,0.63\\}$ and
$W\approx 0.016$.

The remaining regions II to VII are rather similar. All lead to bifurcations.
However, they differ in the order of bifurcation as well as in the magnitude
of fluctuation of the different stakeholders. Another difference between those
regions is that different stakeholders dominate in terms of their stability or
their bifurcations. An example of such a time series is shown in Fig. 9.
There, five stakeholders converge while one stakeholder is bifurcating.

Typical such cases are ecology left in a very uncertain or devastated state,
although the other stakeholders are satisfied. As discussed above, this is one
of the main challenges of our times.

Figure 9: Example IPF time series of region VI as a bifurcating time series of
$g^{s}$ as calculated by Eq. 5 with initial stakeholders
$\alpha^{s}\approx\\{0.62,0.44,0.93,0.65,0.79,0.98\\}$ and $W\approx 0.014$.

A more systematic view of regions II \- VII obtained when looking at Fig. 10.
Here, for each node of the SOM, the last ten values of the time series
$g^{s}_{t}$ are inspected in terms of the number of different values. Values
are assumed to be different when they differ more than 5 % from each other.
One can clearly distinguish the stable region I and the chaotic region VIII.
Furthermore, II \- VII, while looking rather similar in Figure 6, are found to
differ in terms of the order of bifurcation.

Figure 10: Amount of different values of neurons of trained SOM of Fig. 6,
when inspecting the last ten values of each stakeholder time series $g^{s}$.
Region I clearly has the lowest values as it is converging throughout. Also,
the chaotic region VIII has the highest values. Still, region II - VII can now
be differentiated with respect to the number of different values and,
therefore, the order of bifurcation.

Using only version I of varying terms III of direct interaction between
stakeholders shows all kinds of possible scenarios for city planning. These
include

  * •

Convergence: All stakeholders arrive at their desired value.

  * •

Bifurcation: Complex output of very different kinds appear, like convergence
of many stakeholders, while one or two stakeholders show complex development.
All stakeholders bifurcate to a certain extent with all kinds of bifurcating
orders.

  * •

Chaotic: Stakeholders are highly chaotic, and system output is unpredictable.

  * •

Non-converging: City planning has failed as the system breaks down and
stakeholders disappear.

So, the inverse problem of finding parameters and therefore, political actions
to arrive at a desired output for all stakeholders is solved using the
proposed method. Still, a second interaction process will appear between
stakeholders, that of mediated interactions discussed below.

###  III.4 Multiple stakeholders with mediating impact and no external
disturbance

Along the method of direct interaction discussed above, for mediated
interaction again IPFs for $N=6$ stakeholders with 500 iteration steps each
were calculated, using Equations 1 and now Version II of Varying terms III of
Eq. 6.

This time, we also take an even closer look at the initial conditions. Also,
the delay term $d^{s,i}$ of stakeholder i acting on stakeholder s as amount of
iterations is inspected more closely.

In general, four different cases are possible:

  * •

I1: Same as the previous modeling example of direct impact, it is assumed that
no coupling is present $\forall t<0$ and that the system has already settled
itself, and thus $g^{s}_{t}=\alpha^{s}\ \forall t<0$. At $t=0$, suddenly, all
stakeholders are coupled to each other.

  * •

I2: Similar to I1, but as the system is supposed to be uncoupled $\forall
t<0$, it is not realistic that all stakeholders are suddenly coupled to each
other. The IPF models the influence of coupled stakeholders by the damped
influence of a preceding system states $g^{s}_{t-d^{s,i}}$. As coupling is
only present after $t=0$, previous system states can not act on future ones.
Thus, the different $\beta^{s,i}_{t}$ are only taken into account if $t\geq
d^{s,i}$.

  * •

I3: It is assumed that the system is not settled at the start. Thus
$g^{s}_{t}$ my have an arbitrary value $\forall t<0$. Here, for simplicity, we
chose $g^{s}=1$. Like in I1 it is assumed that all stakeholders are suddenly
coupled to each other at $t=0$.

  * •

I4: Again, like in I3 $g^{s}_{t}=1\ \forall t<0$. However, similar to I2 the
different $\beta^{s,i}_{t}$ are only taken into account if $t\geq d^{s,i}$.

For each of the above initial conditions, 125 different, non-diverging time
series are calculated, where again $\alpha^{s}\in[0.5,1]$ are chosen randomly.
The coupling $V^{s}_{i}=V$ is supposed to be the same for all stakeholders.
Values are also chosen randomly in the range of $V\in]0,\ 0.01]$. The time
delays are different for each stakeholder

$d^{s,i}=C_{dt}^{s}\times i\,$ (8)

where the scaling factor is the same for all stakeholders and chosen randomly
$\\{C_{dt}^{s}=C_{dt}\in\mathbb{N}:0<d\leq 16\\}$.

The resulting time series are shown in Fig. 13 to 16 as examples for each of
the four initial conditions. They are very much different from those of the
direct interactions discussed above. Bifurcations and chaos can rarely
present, and, due to the mutual adaption of $\beta^{s,i}_{t}$, all coupled
IPFs converge to the same fixed point. This fixed point might be reached in an
exponential development, like shown in Fig.15, it might show vibrations before
convergence, like in Fig. 13 \- 14, or it might show very slow convergence, so
basically each stakeholder keeps its own state.

An example is a legally binding land-use plan process (B-Plan Verfahren) in
which all stakeholders negotiate over time on a planning of a area. The result
is planning certainty. Due to negotiations, a mediating and self-reference
process is leading to perfect convergence of all stakeholders needs. Of
course, this process is not the realization, which again can lead to complex
situations like financial shortages or the like.

To analyze the models’ behavior systematically and for an inverse problem
solution, again, a self-organizing Kohonen map (SOM) is trained. This time, no
distinguished regions or clusters can be detected, as was the case with direct
interaction when training the map with the simulated time series of $g^{s}$.
This is expected when examining Fig. 13 \- 16. The differences in the time
series are gradual. Still, different phenotypes appear, like exponential
convergence, vibrations, or slow convergence, with practically different
values for each stakeholder.

Therefore, to examine these types, a few meaningful parameters must be derived
from the modeled time series $g^{s}$. At first, each series is divided into
three windows of equal length to distinguish between its start, its end, and
its middle part. For each time window, three values are calculated:

  * •

std(G)

Convergence detection: The overall standard derivation of all $g^{s}$ is
calculated to detect systems convergence and differences between stakeholders
$g^{s}$ still present.

  * •

std(G-gs)

Bifurcation detection: The ratio between the actual $g$ and the theoretical
fixed point Linke2019a  $g_{f}=\alpha+\sum\beta_{k}$ is calculated for each
stakeholder individually. When the standard deviation of this ratio is
calculated, a first impression of whether bifurcations occur can be derived.
The SOM is then trained by the mean of this ratio over all stakeholders.

  * •

Pears2Lin

Convergence time detection: The absolute value of Pearson Correlation $r$, as
calculated from $g^{s}_{t}$ and a straight line ($f(t)=t$), describes whether
the $g^{s}_{t}$ is monotonous falling or growing, or if wave-like fluctuations
occur, as shows up in Fig. 13.

For each of the 4 $\times$ 125 time series, these three values were calculated
and fed as feature vectors into a SOM.

Fig 11 shows this trained SOM using the values as input feature vectors to the
neural net. Again, the background color is the u-matrix, which now does not
show clear regions or clusters. This points to the continuous nature of the
$g^{s}$ in all conditions.

Figure 11: Trained SOM for mediated interaction between stakeholders using
Version II of Varying term III of Eq. 6: a) and b) show the same u-matrix, but
the dots, representing the sorted training data are colored following
different criteria: In a), the colors are chosen due to the initial
conditions. While green dots refer to I1, red dots refer to I2, blue dots
refer to I3, and orange dots refer to I4. In b), the colors represent the
coupling strength $V$, while darker values refer to weaker coupling.

Fig. 11 has the same u-matrix in plots a) and b). Also, the $g^{s}$ of all
cases are best-fit to respective neurons in the same way. The difference
between both plots is in the coloring of the dots. While a) colors are
according to initial conditions I1 - I4, in b) the colors are chosen according
to the coupling strength $V$ of the mediating terms of $\beta^{s,i}$ in Eq. 6.

Differentiating with respect to initial conditions I1 \- I4 in Fig. 11 a)
shows systematic differences between initial conditions. Only arbitrary
$g_{0}$ of I3 and I4 show chaotic transients, with an examples shown in Fig.
15 and Fig. 16). As soon as the $g$ is close to $g_{f}$, the system will
behave initially stable. As Eq. 6 changes $\beta$ iteratively, fluctuations
are prevented, and the system always shows smooth behavior as already
introduced in Fig. 2.

This phenomenon is also strongly influenced by the coupling strength $V$. Fig.
11 b) shows that the SOM also sorts training data according to $V$. The reason
for this becomes clear when examining the component planes of the trained SOM,
shown in Fig. 12. Columns show the distribution of one feature vector entry
and one component over the trained SOM of the three analysis parameters:
convergence, bifurcation, and convergence time detectors. By correlating the
component planes to the dot distribution, the reason for sorting the $g^{s}$
on the SOM can be found.

In the SOM in Fig. 11 b), the dots in the upper part of the plot are darker,
pointing to a weak coupling. The Pears2Lin component plane on the right of
Fig. Fig. 12 shows bright values in its upper parts, corresponding to large
values of the convergence time detector. Fig. 16 shows that, due to weak
coupling, changes of $\beta$ are small, where then only minor changes in $g$
appear. The system is very stable but takes a long time to converge. When the
coupling is strong, like in Fig. 13 or 14, the $g^{s}$ approach each other
very fast, but might overshoot, leading to vibration behavior. This is
detected by the convergence time analysis method Pears2Lin.

The length of the vibration also depends on the use of delay Fig. 14: When the
delay $d^{s,i}$ is large, it takes several iteration steps until the vibration
starts, and usually even more for it to end. Further, a large $d^{s,i}$ allows
a more complex and chaotic time series.

Figure 12: Component planes of trained SOM of Fig. 11 of the mediating
interaction case, Version II of Varying term III when using Eq. 6. The rows
refer to the different time windows. The columns denote the three detection
methods of convergence, bifurcation, and convergence time as input feature
vectors to the SOM and are displayed as magnitudes of these different
features. Figure 13: Example time Series of $g^{s}$, for each of the s=6
stakeholders, resulting from an IPF model with initial condition I1 and
$\alpha^{s}\approx\\{0.89,0.87,0.91,0.81,0.92,0.56\\}$, $V\approx 0.01$ and
$C_{dt}=12$. On the SOM shown in Figure 11, this time series is sorted into
the lower left corner. Figure 14: Example time Series of $g^{s}$, for each of
the s=6 stakeholders, resulting from an IPF model with initial condition I2
and $\alpha^{s}\approx\\{0.76,0.73,0.80,0.79,0.68,0.84\\}$, $V\approx 0.008$
and $C_{dt}=15$. On the SOM shown in Figure 11, this time series is sorted
into the lower right corner. Figure 15: Example time Series of $g^{s}$, for
each of the s=6 stakeholders, resulting from an IPF model with initial
condition I3 and $\alpha^{s}\approx\\{0.89,0.87,0.90,0.81,0.92,0.56\\}$,
$V\approx 0.003$ and $C_{dt}=2$. On the SOM shown in Figure 11, this time
series is sorted into the small island of orange and blue dots, surrounded by
red and green dots, on the upper right of the map. Figure 16: Example time
Series of $g^{s}$, for each of the s=6 stakeholders, resulting from an IPF
model with initial condition I4 and
$\alpha^{s}\approx\\{0.86,0.62,0.98,0.91,0.61,0.73\\}$, $V\approx 4.7\times
10^{-}5$ and $C_{dt}=2$. On the SOM shown in Figure 11, this time series is
sorted into the upper left corner.

####  III.4.1 Complex delay interactions

In the methods, we presented a more complex version of the mediating
interaction. Eq. 7 describes the change of $\beta^{s,i}$ as caused by a
previous state of stakeholder $i$ on a previous state of stakeholder $s$, not
on the present state of stakeholder $s$ like in Eq. 6. This adds a memory to
the stakeholders as they remember their previous states and take this into
account while judging the impact of a coupled stakeholder.

Calculating again 125 time series for each of the four initial conditions I1
\- I4 and using them to train a SOM leads to results shown in Fig. 17.

Figure 17: Trained SOM of mediated interaction but with more complex delay
term using Eq. 7: a) and b) show the same u-matrix, dots represent the best-
fit IPF cases onto the map. a) Colors are chosen according to initial
conditions. Red dots: I1 and I2, white dots: I3 and I4. b) Colors represent
coupling strength $V$, with darker colors representing weaker, and lighter
colors representing stronger coupling.

Again, the different models are ordered according to the coupling strength
$V$, as visible in Fig. 17 b). Furthermore, Fig. 17 a) shows that a general
difference between initial conditions I1 & I2 and I3 & I4 remains. Therefore,
the complex delay does not change the system’s behavior fundamentally.

Still, the system is more unstable. Fig. 18 shows that using Eq. 7, the
chaotic transients can be significantly extended, and the IPF must not
converge anymore.

Figure 18: Time series of $g^{s}$, for each of the s=6 stakeholders of the
mediated interaction Version II of Varying terms III but here with complex
delay of Eq. 7 with initial condition I3,
$\alpha^{s}\approx\\{0.57,0.94,0.50,0.71,0.94,0.97\\}$, $V\approx 0.008$, and
$C_{dt}=8$. The case is mapped to the upper left corner of the SOM shown in
Figure 17.

##  IV Conclusions

The proposed algorithm is able to model all typical kinds of city planning,
social, or political time developments of interacting and self-adapting
stakeholders. These include conversion to a common state, conversion to
different states, bifurcations of many kinds, as well as chaotic behaviour.
The reaction to an external impact is also realistic with adaptation,
convergence, or breakdown.

The workflow of city planning includes the solution of the inverse problem,
which is finding a set of parameters needed to arrive at a desired output. The
model suggests different such scenarios, estimates their stability in terms of
changing external conditions, and points to problems for certain stakeholders
within the project.

Using a self-organizing Kohonen map to arrive at such suggestions enables
modelers to find the reasons for successful parameter estimations. Also, the
whole field of possible parameters and interactions is described in the two-
dimensional map representation, giving an overview of potential alternatives
to a proposed output or parameter space.

The model is flexible in terms of using different varying Terms in different
variations. This allows flexible modification of the model according to real-
world situations. Also, by adding or omitting terms, certain kinds of
interactions or self-adaptations might be suggested by the system, therefore
making aware of critical stakeholder interaction processes needing more care
during a planning process.

The proposed city planning model is only exemplified in this paper using
typical city planning situations. As it has now proven to be powerful, robust,
and flexible for planning processes, the next step is to validate the model
with respect to a complex real-world situation. Such a case could be a
finished project where the stakeholder developments are known. It might also
be a running project where the IPF is performed in parallel. This is beyond
the scope of the present paper, which first needs to outline the basic
behavior of such a system. Still, this IPF has been developed in cooperation
with city planners and is, therefore, close to meeting their constraints and
demands. Future work will be conducted, giving examples of real-world models.

Then, a vision for the future would be to implement an IPF in all big planning
processes, leading to much more efficient and robust plannings that take care
of all stakeholder’s demands and needs.

## References

  * (1) Abas, A., Arifin, K., Azhar, M., Ali, M., & Khairil, M.: A systematic literature review on public participation in decision-making for local authority planning: A decade of progress and challenges, Environmental Development, 46, 2023, doi.org/10.1016/j.envdev.2023.100853.
  * (2) Bader, R.: Modeling Temporal Lobe Epilepsy during Music Large-Scale Form Perception Using the Impulse Pattern Formulation (IPF) Brain Model. Electronics 2024, 13(2), 362; https://doi.org/10.3390/electronics13020362
  * (3) Bader, R.: Impulse Pattern Formulation (IPF) Brain Model. arXiv:submit/4660893 [q-bio.NC] 21 Dec 2022
  * (4) Bader, R.: How music works. A Physical Culture Theory. Springer Nature 2021.
  * (5) Baumann, F., Lorenz-Spreen, Ph., Sokolov, I. M., & Starnini, M.: Modeling echo chambers and polarization dynamics in social networks, 2020, opinarXiv:1906.12325v2 [physics.soc-ph].
  * (6) R. Bader, _Nonlinearities and Synchronization in Musical Acoustics and Music Psychology,_ Springer-Verlag, Berlin, Heidelberg, Current Research in Systematic Musicology, vol. 2, (2013).
  * (7) Blaß, M. & Bader, R.: Content Based Music Retrieval and Visualization System for Ethnomusicological Music Archives. In: R. Bader (ed.): _Computational Phonogram Archiving_ , Springer Series: Current Research in Systematic Musicology, Vol. 5, 145-174, Springer, Heidelberg, 2019.
  * (8) Chaturvedi, V., de Vries, W.T. Machine Learning Algorithms for Urban Land Use Planning: A Review. Urban Sci. 2021, 5, 68. https://doi.org/10.3390/ urbansci5030068
  * (9) Coskun, H.: Eco-City Planning; concepts and contemporary models. Glob Journal of Ecology 8(2): 110-119. DOI: https://dx.doi.org/10.17352/gje.000090
  * (10) Flack, J. C. & Krakauer, D. C.: Challenges for complexity measures: A perspective from social dynamics and collective social computation. Chaos 21, 037108, 2011.
  * (11) Heider F.: Attitudes and cognitive organization. J. Psychol. 21, 107–112, 1946, doi:10.1080/00223980.1946.9917275
  * (12) Jedrzejewski, A., Nowak,B., Abramiuk, A., & Sznajd-Wero, K.: Competing local and global interactions in social dynamics: How important is the friendship network? Chaos 30, 073105, 2020, doi: 10.1063/5.0004797
  * (13) Kohonen, T.: _Self-organizing maps._ Springer, Berlin 1995.
  * (14) Latour, B.: Politics of Nature: How to Bring the Sciences into Democracy, Cambridge MA, Harvard university Press, 2022.
  * (15) Latour, B.: Facing Gaia: eight lecutres on the new climate regime, Cambridge, UK Medford, MA, Polity 2017.
  * (16) Linke, S., Bader, R., & Mores, R.: Modeling synchronization in human musical rhythms using Impulse Pattern Formulation (IPF). arXiv:submit/4062462 [q-bio.NC] 6 Dec 2021
  * (17) Linke, S., Bader, R., & Mores, R.: Influence of the supporting table on initial transients of the fretted zither: An impulse pattern formulation model. Proc. Mtgs. Acoust. 43, 035003 (2021); doi: 10.1121/2.0001494
  * (18) Linke, S., Bader, R. & Mores, R.: The impulse pattern formulation (IPF) as a model of musical instruments—Investigation of stability and limits, Chaos 29, 103109-1, 2019.
  * (19) Linke, S., Bader, R. & Mores, R.: Measurements and impulse pattern formulation (IPF) model of phase transitions in free-reed wind instruments. JASA 146, 2779, 2019, doi.org/10.1121/1.5136626
  * (20) List, J. A.: Optimally generate policy-based evidence before scaling. Nature 626, 491-497, 2024.
  * (21) Lovelock, J. (1988). The Age of Gaia. A Biography of Our Living Earch. W.W. Norton & Co. NY, London.
  * (22) Margulis, L. (1998). Symbiotic Planet : A New Look at Evolution. Basic Books.
  * (23) Marvel, S. A., Kleinberg, J., Kleinberg, R. D., Strogatz, S. H.: Continuous-time model of structural balance. Proc. Natl Acad. Sci. USA 108, 1771–1776, 2011. (doi:10.1073/pnas.1013213108
  * (24) Necker, S.: Das "frappant", moderneREGIONAL, 14/2, 2014, https://www.moderne-regional.de/fachbeitrag-frappant-hamburg/
  * (25) Minh Pham, T, Kondor, I, Hanel, R, Thurner, S.: The effect of social balance on social fragmentation. J. R. Soc.Interface 17: 20200752, 2020. http://dx.doi.org/10.1098/rsif.2020.0752
  * (26) Rajaan, R., Baishya, B. K., Rao, T. V., Pttanaik, B., Tripathi, M. A., & R, A.: Efficient Usage of Energy Infrastructure in Smart City Using Machine Learning, EAI Endorsed Transactions on Internet of Things, 10, 2024, doi: 10.4108/eetiot.5363
  * (27) Sawicki, J., Berner, R., Loos, S.A.M., Anvari, M., Bader, R., Barfuss, W., Botta, B., Brede, N., Franović, I., Gauthier, D.J., Goldt, S., Hajizadeh, A., Hövel, Ph., Karin, O., Lorenz-Spreen, Ph.,Miehl, Ch., Mölter, J., Olmi, S., Schöll, E., Seif, A., Tass, E.A., Volpe, G., Yanchuk, S. & Kurths, J.: Perspectives on adaptive dynamical systems. Chaos 33, 071501 (2023) https://doi.org/10.1063/5.0147231
  * (28) Xinghui Lei, J. Z.: A System Dynamics Model for Urban Low-Carbon Transport and Simulation in the City of Shanghai, China. International Journal on Advances in Information Sciences and Service Sciences, 239-246, 2012.
  * (29) Smith, E. & Morowith, H. J.: The origin and nature of life on earth. The emergence of the fourth geosphere, Cambridge University Press 2016.
  * (30) Taylor, H. F.: Balance in small groups. Sociological Concepts, Methods, and Data Series. New York, USA: Van Nostrand Reinhold Company, 1970.
  * (31) Titz, M., Kaiser, F., Kruse, J., & Witthaut, W.; Predicting dynamic stability from static features in power grid models using machine learning. Chaos 1 January 2024; 34 (1): 013139. https://doi.org/10.1063/5.0175372
  * (32) Tucci,G. & Ratti, C.: Technology as an enabler of a new ecosystem responsive urbanism. AGATHÓN - International Journal of Architecture, Art and Design 12, 190-201, 2022.
  * (33) Watson, A. J. & Lovelock, J. E. (1983). Biological homeostasis of the global environment. The parable of Daisyworld. Tellus 35/B, 286–289.
  * (34) Wilkinson, R. G. & Pickett, K. E.: Why the world cannot afford the rich. nature 627, 2698-270, 2024.

## Data Availability Statement

Data available on request from the authors.

###### Acknowledgements.

This research was funded under the Program "Innovative Hochschule" (innovative
university) by the Federal Ministry of Education and Research (BMBF) of
Germany (Grant No.13IHS232C) and the City of Hamburg.


# Broken Symmetries in Microfluidic Pillar Arrays are Reflected in a Flowing
DNA Solution across Multiple Length Scales

Jason P. Beech  Oskar E. Ström  Jonas O. Tegenfeldt  Division of Solid State
Physics, Department of Physics and NanoLund, Lund University, P.O. Box 118,
22100 Lund, Sweden

###### Abstract

Unlike Newtonian fluids, viscoelastic fluids may break time-reversal symmetry
at low Reynolds numbers resulting in elastic turbulence. Furthermore, under
some conditions, instead of the chaotic turbulence, large-scale regular waves
form, as has been shown for DNA flowing in microfluidic pillar arrays. We here
demonstrate how the symmetry of the individual pillars influences the symmetry
of these waves, thereby contributing to the understanding of the origin of the
waves and opening up for better control of the waves with relevance to
applications such as microfluidic sorting and mixing. The onset of waves
occurs at different Deborah numbers for flow in different directions through
the same array. Because the onset of waves leads to an increase in flow rate
for a given driving pressure, we observe an increase in diodicity within this
range.
* Authors contributed equally.
$\dagger$ Corresponding author<EMAIL_ADDRESS>

††preprint: APS/123-QED

Viscoelastic effects in the flows of polymer solutions through and around
structures at the microscale are believed to have practical importance in
diverse areas ranging from the extraction of oil from mineral matrices [1] to
the understanding of the transport of biological fluids, both in their native
environments [2] and through novel types of microfluidics devices [3]. An
example of the latter are microfluidics devices for the separation of DNA
molecules by length that use deterministic lateral displacement (DLD) [4, 5,
6, 7, 8, 9] as well those relying on pulsed electrical fields combined with a
pillar array arranged in an hexagonal pattern [10, 11]. While the underlying
mechanisms are entirely different, both are based on arrays of pillars through
which the molecules follow size dependent trajectories. Since solutions of DNA
are viscoelastic fluids, an understanding of their behavior in pillar arrays
is essential for the development and improvement in functionality of these
devices. Indeed, during recent efforts to improve the separation of long ($>$
20 kbp) DNA fragments in DLD devices [9] we discovered that DNA can assemble
into ordered waves of high concentration and molecular stretching and what is
more that the local flow direction inside these waves varies from the bulk
flow direction, an effect which both increased and decreased the quality of
separation dependent on the specific conditions. We have subsequently shown
that the formation of these waves depends to large degree on the viscoelastic
properties of the fluid and the order of the pillar array, with large scale
flow patterns or waves entirely absent in arrays where the position of each
pillar has been randomly perturbed [12].

Previous research points to the onset of instabilities in the flow of
viscoelastic fluids through porous media (such as arrays of pillars) as flow
velocities increase with pillars symmetric along the flow direction [13] as
well as with triangular pillars that break the symmetry along the flow
direction [14]. Triangular pillars [15, 16] and pillars of other shapes are
also of interest for DLD where they have been demonstrated to improve the
performance of the separation with respect to decrease of the critical size as
well as for minimizing propensity for clogging. Here we show that by breaking
the symmetry of the pillars, a microscopic property of the array, and doing so
both parallel with and perpendicular to the flow direction, we observe
symmetry breaking in the flow behavior of the DNA solution at macroscopic
scales. Large differences in both micro and macroscale flow behavior are made
clearly visible using fluorescence microscopy and can be observed across a
range of spatial and temporal scales.

Our device consists of a straight channel, 0.8 mm wide and 8 mm long
containing an array (22 $\times$ 222) of right angled isosceles triangular
pillars with a pitch of 36 µm. The legs of the triangles are 28 µm, the gaps
between are 8 µm and the depth of the channel 11.5 µm. The pillars are
oriented such that one leg is parallel with the side walls of the channel.
This means that flow in one direction along the channel impinges on a leg of
the triangular pillars (which we will call flatwise flow), and that flow in
the other direction along the channel impinges on one of the acute vertices
(which we will call pointwise flow). We refer to lines of pillars along the
direction of the flow as “columns” and lines of pillars perpendicular to the
flow direction as “rows”. More details about the device can be found in
Supplemental Fig. S1 [17].

Figure 1: Low-magnification snapshots of fluorescent DNA moving through an
array of triangular pillars in the pointwise (a) and the flatwise (b)
directions. $u\approx$ $8.7\text{\,}\mathrm{mm}\text{/}\mathrm{s}$, $De\approx
350$, and $u\approx$ $10\text{\,}\mathrm{mm}\text{/}\mathrm{s}$, $De\approx
400$ for the pointwise and flatwise flow, respectively. In both flow
directions periodic waves with well-defined orientations, highlighted with
white dashed lines, are observed but these differ, both in the orientation of
the wavefronts and in the behavior of the flow pattern inside the waves.
Dynamic phenomena such as these are clearly seen in Supplemental Movie S1
[17]. The intensity and brightness have been set to enhance the visibility of
the waves. Scale bars are 500 $\mu$m for the low-magnification snapshots and
20 $\mu$m in the inset.

The devices were fabricated using standard replica molding techniques.
$\lambda$ phage DNA (48.5 kbp, New England Bio-labs, Ipswich, MA, USA) was
stained with the bisintercalating dye YOYO-1 at a DNA basepair to dye ratio of
200:1 in a 5$\times$ Tris EDTA (TE)-buffer. The flow was generated by applying
nitrogen gas overpressure and the flow rate measured using a flow sensor. The
flowing DNA was imaged using standard epifluorescence microscopy. Image
artifacts due to the pillar arrays could be interpreted as waves. However,
they do not not move and they appear only for certain illumination angles.
They were characterized using devices without any DNA solutions (see Section 5
in the Supplemental Material [17]). Full details of the materials and methods
are described in the Supplemental Material [17].

As a solution of $\lambda$ phage DNA at
$400\text{\,}\mathrm{\SIUnitSymbolMicro g}\text{/}\mathrm{ml}$ is forced to
flow through the pillar array at various applied pressures and in the two
different directions, several phenomena are observed with large qualitative
differences immediately apparent. Fig. 1 shows the two flow directions
generated under a mean flow velocity
$10\text{\,}\mathrm{mm}\text{/}\mathrm{s}$ at the device scale and Figs. 3 and
4 at the scale of the individual pillars under a range of flow velocities. See
Supplemental Figs. S2 and S3 [17] for low-resolution micrographs over a larger
range of flow velocities.

To be able to relate our observations to the prevailing flow conditions we
derive the flow velocities from the measured volumetric flow rates, $Q$,
0.01–$8\text{\,}\mathrm{nl}\text{/}\mathrm{s}$, dividing by the total cross-
sectional area of the smallest gaps between the pillars, $A$. Using the fact
that the maximum fluid velocity is twice the mean for a parabolic flow profile
(a reasonable approximation), we calculated the range of maximum flow
velocities to be $u_{\rm max}~{}=~{}1.0~{}\times~{}10^{-5}$ –
$7.2\text{\times}{10}^{-3}\text{\,}\mathrm{m}\text{/}\mathrm{s}$. With $\rho$
= $997\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{3}$, $\mu$ =
$8.90\text{\times}{10}^{-4}\text{\,}\mathrm{Pa}\text{\,}\mathrm{s}$, and the
gap between pillars $G$ = $8\text{\times}{10}^{-6}\text{\,}\mathrm{m}$ and
using Eq. (1):

$\displaystyle Re=\frac{\rho u_{\rm max}G}{\mu},$ (1)

we estimate Reynolds numbers for all experiments to be in the range
$9.3~{}\times~{}10^{-5}$ to $6~{}\times~{}10^{-2}$, such that we can neglect
any inertial contributions. The Deborah number can be estimated with equation
2:

$\displaystyle De=\frac{\tau u}{L}.$ (2)

Here $\tau$ = 1.43 s is the relaxation time for $\lambda$ DNA (measured with a
stress-controlled rheometer with a 25 mm cone plate geometry at 25∘C, see
details of the measurement in [12]),
$L=~{}$36\text{\times}{10}^{-6}\text{\,}\mathrm{m}$$ is the array pitch. $u$
is the mean flow velocity between pillars (half of $u_{\rm max}$ as described
above, see detailed description of $u$ in section 2 in the Supplemental
Material [17]) resulting in a $De$ in the range 1.4 – 400. Elastic effects are
thus important.

These dimensionless numbers are calculated using estimates of the maximum flow
rate between pillars and using the bulk properties of the DNA solution and
should be considered as nominal values and guides for comparisons only since
the presence of vortices and the variation in the local concentration of DNA
(as easily seen in Fig. 3) leads to large variations in these numbers in time
and space. What is more, the measured flow rates for given applied driving
pressures differ between the flatwise and pointwise directions, explaining the
variation between compared $De$ in the figures (see discussion on diodicity
below).

While an array of circular pillars generates two types of waves with
orientations mirrored around the axis along the device channel with equal
probability, as previously reported [12], only one type of wave dominates for
each flow direction when we break the lateral symmetry using the triangular
pillars. Not only do the waves differ in their angle, see Fig. 1 but there are
other striking differences. In both directions, waves are preceded by an
increase in fluctuations, see Fig. 3. In the flatwise direction vortex pairs
form that appear to stabilize the flow between waves. The stabilizing effect
of the vortex pairs may explain why waves appear at lower flow rates for the
pointwise flow than for the flatwise flow. Other differences in the
distributions of DNA within waves can be seen in Fig. 4 as we discuss further
below.

Figure 2:  Low frequency region of Fourier amplitude spectra difference for
pointwise minus flatwise flows. Before subtraction, the mean spectra for all
the frames of each videograph were computed, see Supplemental Material,
section 4 [17] for details of the analysis. The data is based on fluorescence
videos captured at a low magnification (2$\times$ objective) of various
lengths in the range of 100 to 300 s as specified in the Supplemental Material
[17]. The panels show only the positive wavelengths in the y-direction
(perpendicular to the flow direction) and not the negative ones. The image
brightness and contrast has been set so that the absolute amplitude limits are
the same for all pressure values. Guides for the eye are added as follows. The
thin red dashed line to the left at the lowest pressure correspond to an
artefact in the device. This line is repeated for all panels. The thick dashed
green line to the right corresponds to the waves for pointwise flow. Note that
it starts for $De=100$ or the second panel from the left. The thick red dashed
line corresponds to waves for flatwise flow. Note that it starts for $De=300$
or the fourth panel from the left. This line is mirrored around a vertical
center line in each panel except the first to enable a comparison with the
green line.

Analyzing the spatial, two-dimensional (2D) frequency components of the flow
patterns seen in low magnification images such as those shown in Fig. 1
reveals a distinct difference in the Fourier transform amplitude spectra of
the two flow directions above approximately
$3\text{\times}{10}^{-3}\text{\,}\mathrm{m}\text{/}\mathrm{s}$, see Fig. 2 for
the low frequency components. The angles corresponding to a high amplitude for
each flow direction in the high velocity spectra
($3\text{\times}{10}^{-3}\text{\,}\mathrm{m}\text{/}\mathrm{s}$,
$6\text{\times}{10}^{-3}\text{\,}\mathrm{m}\text{/}\mathrm{s}$, and
$1\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{/}\mathrm{s}$) correspond to
the orientations of the waves of the two directions visible in Fig. 1. It is
very clear that the waves corresponding to the pointwise flow (green) appear
for lower flow velocities than the waves corresponding to the flatwise flow
(red). Although the data is not sufficient for any quantitative
characterization of the change of the angle with flow rate, there is a clear
tendency of the pointwise waves to change angle with flow velocity. See
Supplemental Figs. S7 and S8 [17] for amplitude spectra of the full frequency
range as well as a large number of flow velocities.

Figure 3: Observed at the microscale, the flow behavior differs significantly
between the two flow directions. The pairs of images in each panel consist of
micrographs ($x$,$y$) and kymographs ($y$,$t$) taken along the red line shown
in the first panel. Panels (a-f) show flatwise flow and (g-l) pointwise flow.
Cyan bars found at the bottom of each kymograph indicate the relative time
points of the micrograph in each pair. The red arrows highlight waves. The
brightness and contrast settings have been adjusted so that the settings are
the same for (a-l). Each kymograph spans a total of 7 s. See Supplemental
Movie S2 [17] for videos corresponding to the data of all the panels. Scale
bars are 20 $\mu$m.

For flow in both directions, as the flow velocity is increased from zero, the
first observation we make is that areas depleted of DNA form and that their
distributions differ between the two directions, Fig. 3(a and g). These
deadzones correspond to what has been reported in viscoelastic flows in pillar
arrays of different shapes [13, 14], and is similar to what we see for
circular pillars [12]. As the flow velocity is increased the asymmetry becomes
more pronounced. In both directions vortices also form but their positions,
fluctuations, and interactions with each other are very different, see Fig.
3(b-d and h). Finally, as can be seen in Fig. 1 and Fig. 3(e, f and i-l) waves
form that are very different on both the microscopic and macroscopic scale. In
the following, we consider the emerging patterns in the flow in each direction
in more detail.

Flow in the flatwise direction is dominated by the formation of non-symmetric
vortex pairs between pillars along the columns, with a smaller clock-wise
vortex near the vertex of the pillar and a larger counter clock-wise brighter
vortex below. The formation of these vortex pairs can be clearly seen in Fig.
3(d) at $u$ $4.7\text{\,}\mathrm{mm}\text{/}\mathrm{s}$ and they persist
between waves at higher flow rates, Fig. 3(e and f). Prior to the passage of a
wave the vortices mix, see fig. 4(c and d) and are then flushed away within
the wave.

Figure 4:  Microscopic flow behavior (a-h) and fluidic characteristics (i and
j) for flatwise and pointwise directions. (a) and (e) show the mean
fluorescence intensity over time in a unit cell, indicated by the square in
(h), for flatwise and pointwise flow, respectively. The corresponding flow
velocity is noted above each graph. $u_{1}$ =
$2.8\text{\,}\mathrm{mm}\text{/}\mathrm{s}$, $u_{2}$ =
$6.3\text{\,}\mathrm{mm}\text{/}\mathrm{s}$, $u_{3}$ =
$4.8\text{\,}\mathrm{mm}\text{/}\mathrm{s}$ and $u_{4}$ =
$7.2\text{\,}\mathrm{mm}\text{/}\mathrm{s}$, respectively. (b-d), (f-h) show
micrograph snapshots from the corresponding plots to the left-hand side. (c,
d, f and g) are marked in insets with a red frame. The red arrows indicate the
local flow direction in the pillar gap which is more easily seen in
Supplemental Movie S3 [17]. The intensity and brightness have been set to be
the same for all the micrographs. Scale bar is 20 $\mu$m. (i) Mean flow
velocity dependence of applied pressure and (j) Diodicity, $Di=Q_{\rm
pointwise}/Q_{\rm flatwise}$. The vertical arrows in (i) mark the earliest
wave observations in pointwise (150 mbar, left arrow) and flatwise (330 mbar,
right arrow). The error bars in (i) represent one standard deviation and those
in (j) represent the fractional standard deviations added in quadrature.

In the pointwise direction, stable vortices are not observed. Instead, the
flow of DNA exhibits two distinct phenomena. The unstable, turbulent-like flow
between the pillars, has a tendency to move, on average, to the right of the
downstream pillar, as seen from the perspective of the flow. The result is a
flow with a significant component perpendicular to the mean flow direction,
which is along the device. This perpendicular component of the flow leads to a
perpendicular pressure gradient and accumulation of elastic stress that is
released by a significant switching of the flow direction within a wave.
Within a wave the flow becomes bifurcated around the pillars and downstream
wakes containing a high concentration of DNA form as described in [18] and can
be seen in Fig. 4(f-h). The wakes, which have been shown to be stable and non-
symmetric along the flow direction for individual, circular pillars [18],
stretch across several pillars in our array. The non-symmetric wake that forms
around circular pillars is bi-stable and falls into one of the two stable
positions. We propose that in our array the non-symmetric pillars form a
stable wake in one direction only and that due to their proximity, they
interact. Simultaneously, vortices form in the columns that also destabilize
the wakes. There is an accumulation of DNA in these vortices that is released
at the onset of a wave. These mechanisms together explain both the orientation
of the DNA strands within the pointwise wave and the increased concentration
of DNA within the wave as seen in Fig. 3(i-l).

An analysis of the intensity in a unit cell of the array further highlights
the differences in flow behaviour in the two flow directions. As can be seen
in Fig. 4(a and e), waves appear at lower flow velocities ($De$) for pointwise
flow than for flatwise flow. Fig. 4(e and i) shows the clear difference of the
time evolution of the intensities within wavefronts for the two directions as
can be seen in the insets (red boxes). A Fourier analysis of the time signals
(Fig. S5) from a unit cell (as defined in Fig. 4 (h)) identifies a clear
frequency peak for flatwise flow in strong contrast to what is seen for
pointwise flow.

We observe a rapid increase in the measured flow rate at the onset of waves.
While the measured flow rate between the two flow directions for the same
applied pressure [see e.g., Fig. 4(i)] is larger for the pointwise flow than
for the flatwise flow across the range of driving pressures observed, the
difference is at a maximum in the range where waves occur for the pointwise
flow, but not for the flatwise flow. We ascribe this measured difference to
the difference in the onset of waves in the two directions. Movie S4 in the
Supplemental Material [17] shows this difference nicely during the application
of an oscillating driving pressure. We can define a diodicity, $Di$, as a
ratio between the two measured flow rates at the same driving pressure, see
Fig. 4(j). A potential application of the observed flow phenomena could then
be a fluidic rectifier [19], and will be studied in the future.

We have shown that the patterns that emerge in a flow of DNA through a pillar
array depend on the symmetry of the pillars. When the symmetry of the pillars
is broken both in the flow direction and perpendicular to the flow direction
the emerging flow patterns are not only non-symmetric with regard to flow
direction at the scale of the pillars, but the traveling waves that form and
propagate through these microscopic states above a threshold in $De$ are also
highly non-symmetric, travel with different angles to the array and are
qualitatively different in their underlying mechanisms.

We believe that knowledge of these phenomena will aid the design of effective
low $Re$ mixers and in improvement of devices that use pillar arrays for the
separation of DNA. For example, in separation systems with symmetric pillars
that have been used for the separation of DNA, waves are observed to be
equally prevalent in two symmetric orientations leading to considerable
dispersion of molecular fractions [9]. We propose that the ability to control
the nature and directions of waves will allow improved DNA separations at high
concentrations and throughputs. Lastly, the rectifying abilities of these
arrays holds potential for application within microfluidics and will be
further studied.

This research was funded by the European Union, grant number 634890 (project
BeyondSeq /Horizon2020), EuroNanoMed (NanoDiaBac), by the Swedish Research
council, grant number 2016-05739 and NanoLund. All device processing was
conducted within Lund Nano Lab.

The author contribution statements have been written according to CRediT
(Contributor Roles Taxonomy, see credit.niso.org for role descriptions):
Conceptualization, O.E.S., J.P.B., and J.O.T.; Data Curation, O.E.S. and
J.P.B.; Formal Analysis, O.E.S. and J.P.B.; Funding Acquisition, J.O.T.;
Investigation, O.E.S., and J.P.B.; Methodology, O.E.S., J.P.B., and J.O.T.;
Project Administration, J.O.T.; Resources, J.O.T.; Software, O.E.S. and
J.P.B.; Supervision, J.P.B., and J.O.T.; Validation, O.E.S., J.P.B., and
J.O.T.; Visualization, O.E.S., J.P.B., and J.O.T.; Writing - Original Draft,
O.E.S., J.P.B., and J.O.T.; Writing - Reviewing & Editing, O.E.S., J.P.B., and
J.O.T.

## References

  * Sorbie [1991] K. S. Sorbie,  _Polymer-improved oil recovery_ (Blackie ; CRC Press, Glasgow Boca Raton, Fla., 1991) pp. xii, 359 p.
  * Carrel  _et al._ [2018] M. Carrel, V. L. Morales, M. A. Beltran, N. Derlon, R. Kaufmann, E. Morgenroth, and M. Holzner, Biofilms in 3d porous media: Delineating the influence of the pore network geometry, flow and mass transfer on biofilm development, Water Research 134, 280 (2018).
  * Groisman and Steinberg [2001] A. Groisman and V. Steinberg, Efficient mixing at low reynolds numbers using polymer additives, Nature 410, 905 (2001).
  * Hochstetter  _et al._ [2020] A. Hochstetter, R. Vernekar, R. H. Austin, H. Becker, J. P. Beech, D. A. Fedosov, G. Gompper, S. C. Kim, J. T. Smith, G. Stolovitzky, J. O. Tegenfeldt, B. H. Wunsch, K. K. Zeming, T. Kruger, and D. W. Inglis, Deterministic lateral displacement: Challenges and perspectives, Acs Nano 14, 10784 (2020).
  * Huang  _et al._ [2004] L. R. Huang, E. C. Cox, R. H. Austin, and J. C. Sturm, Continuous particle separation through deterministic lateral displacement, Science 304, 987 (2004).
  * Chen  _et al._ [2015] Y. Chen, E. S. Abrams, T. C. Boles, J. N. Pedersen, H. Flyvbjerg, R. H. Austin, and J. C. Sturm, Concentrating genomic length dna in a microfabricated array, Physical Review Letters 114, ARTN 198303 10.1103/PhysRevLett.114.198303 (2015).
  * Wunsch  _et al._ [2019] B. H. Wunsch, S. C. Kim, S. M. Gifford, Y. Astier, C. Wang, R. L. Bruce, J. V. Patel, E. A. Duch, S. Dawes, G. Stolovitzky, and J. T. Smith, Gel-on-a-chip: continuous, velocity-dependent dna separation using nanoscale lateral displacement, Lab on a Chip 19, 1567 (2019).
  * Wunsch  _et al._ [2021] B. H. Wunsch, K. Y. Hsieh, S. C. Kim, M. Pereira, S. Lukashov, C. Scerbo, J. M. Papalia, E. A. Duch, G. Stolovitzky, S. M. Gifford, and J. T. Smith, Advancements in throughput, lifetime, purification, and workflow for integrated nanoscale deterministic lateral displacement, Advanced Materials Technologies 6, ARTN 2001083 10.1002/admt.202001083 (2021).
  * Ström  _et al._ [2022] O. E. Ström, J. P. Beech, and J. O. Tegenfeldt, High-throughput separation of long dna in deterministic lateral displacement arrays, Micromachines 13, 1754 (2022).
  * Bakajin  _et al._ [2001] O. Bakajin, T. A. J. Duke, J. Tegenfeldt, C. F. Chou, S. S. Chan, R. H. Austin, and E. C. Cox, Separation of 100-kilobase dna molecules in 10 seconds, Analytical Chemistry 73, 6053 (2001).
  * Huang  _et al._ [2002] L. R. Huang, J. O. Tegenfeldt, J. J. Kraeft, J. C. Sturm, R. H. Austin, and E. C. Cox, A dna prism for high-speed continuous fractionation of large dna molecules, Nature Biotechnology 20, 1048 (2002).
  * Ström  _et al._ [2022] O. E. Ström, J. O. Tegenfeldt, and J. P. Beech, Short and long-range cyclic patterns in flows of dna solutions in microfluidic obstacle arrays, ArXiv e-prints arXiv:2211.16294 [physics.flu-dyn], https://doi.org/10.48550/arXiv.2211.16294 (2022).
  * Kawale  _et al._ [2017] D. Kawale, E. Marques, P. L. Zitha, M. T. Kreutzer, W. R. Rossen, and P. E. Boukany, Elastic instabilities during the flow of hydrolyzed polyacrylamide solution in porous media: effect of pore-shape and salt, Soft matter 13, 765 (2017).
  * Kawale  _et al._ [2019] D. Kawale, J. Jayaraman, and P. E. Boukany, Microfluidic rectifier for polymer solutions flowing through porous media, Biomicrofluidics 13, Artn 014111 10.1063/1.5050201 (2019).
  * Loutherback  _et al._ [2009] K. Loutherback, J. Puchalla, R. H. Austin, and J. C. Sturm, Deterministic microfluidic ratchet, Physical Review Letters 102, 045301 (2009).
  * Loutherback  _et al._ [2010] K. Loutherback, K. S. Chou, J. Newman, J. Puchalla, R. H. Austin, and J. C. Sturm, Improved performance of deterministic lateral displacement arrays with triangular posts, Microfluidics and Nanofluidics 9, 1143 (2010).
  * [17] See Supplemental Material at [URL will be inserted by publisher] for details about experimental methods, data analysis, supplemental figures and movies.
  * Haward  _et al._ [2021] S. J. Haward, C. C. Hopkins, S. Varchanis, and A. Q. Shen, Bifurcations in flows of complex fluids around microfluidic cylinders, Lab on a Chip 21, 4041 (2021).
  * Groisman and Quake [2004] A. Groisman and S. R. Quake, A microfluidic rectifier: Anisotropic flow resistance at low reynolds numbers, Physical Review Letters 92, ARTN 094501 10.1103/PhysRevLett.92.094501 (2004).