# 5G Network Security Practices: An Overview and Survey ††thanks: Citation:
Bannat Wala. M Kiran. 5G Network Security Practices: An Overview and Survey.

Fatema Bannat Wala
ESnet
Lawrence Berkeley National Laboratory
Berkeley, CA
<EMAIL_ADDRESS>
Mariam Kiran
Quantum Communications and Networking
Oak Ridge National Laboratory
Oak Ridge, TN
<EMAIL_ADDRESS>

###### Abstract

This document provides an overview of 5G network security, describing various
components of the 5G core network architecture and what kind of security
services are offered by these 5G components. It also explores the potential
security risks and vulnerabilities presented by the security architecture in
5G and recommends some of the best practices for the 5G network admins to
consider while deploying a secure 5G network, based on the surveyed documents
from the European government’s efforts in commercializing the IoT devices and
securing supply chain over 5G networks.

_K_ eywords 5G security $\cdot$ best practices $\cdot$ 5G core architecture
$\cdot$ risks and vulnerabilities

##  1 5G Security Overview

The diagram in Figure 1 shows various network functions that play a part in
the overall 5G network security. The figure shows various elements that are
somehow involved in providing the security in a 5G network. Starting with
mutual authentication, it has been around in previous technologies before 5G
and is also available in 5G, which happens between the user equipment (UE) and
the Access and Mobility Management Function (AMF), but unified Data Management
(UDM) also plays crucial role in the overall authentication process, as it
hold the data associated with the UE identification. The detailed explanation
of how it works is provided in the next sub-sections. Next, various types of
signalling - RRC, NAS (non-access stratum) and User Plane traffic can be
protected by the encryption and integrity protection options available in 5G.
Encryption provides the confidentiality of the signalling messages transferred
between UE/gNB and AMF, and integrity is provided by verifying the sender’s
and receiver’s identity and making sure that the message is not forged in the
transit. Although in all the signalling use-cases, confidentiality protection
via encryption is an optional feature and not enabled by default. However,
integrity protection in RRC and NAS signalling is mandatory and can’t be
disabled. Apart from the interaction of UE and AMFs, there are other
components and network functions that come into picture when the UE tries to
keep connectivity between other 5G networks while roaming, hence the roaming
and inter-connections protection is provided by the security edge protection
proxy (SEPP) and user plane functions (UPFs). The details on how SEPP provides
protection for roaming connections is described in details in next sub-
sections. AS we move into the network, we have got IP connectivity between RAN
and the core, and sometimes that connectivity is not owned by the mobile
service provider but some third party service provider, and in that case
technologies like IPSEC can be used to protect that traffic.Lastly, the
service based interfaces can also be protected using OAuth 2.0 and the
necessary transport layer security, and can control which network functions
are able to access services from other network functions. Finally, for the
subscriber identification protection, 5G usually uses a temporary ID called 5G
Global Unique Temporary Identifier (GUTI), but if there is a need for the UE
to share it’s IMSI over the radio network, that can also be protected using
asymmetric encryption which is also called as SUCI (Subscriber Concealed ID).
Next sub-sections describe in detail these fundamental methods of protecting
5G environment.

Figure 1: 5G Security Overview

###  1.1 Network Functions Virtualization NFVs and Network-Slicing

The 3rd Generation Partnership Projects (3GPP) have defined 5G network to be a
Service-Based Architecture (SBA), consisting of the various network functions
(NFs) interconnected to deliver the control plane functionality and common
data repositories of a 5G network. Since 5G is considered to be a SBA, a lot
of it’s NFs will be realized to be running in a virtualised environment to
support the 5G core, the Radio Access Network (RAN) and Multi-access Edge
Computing (MEC). Concepts like Cloud RAN (CRAN), Virtualised RAN (VRAN) and
Open RAN (ORAN) to run baseband functions on commodity server hardware, are
based on the Network Function Virtualization (NFV) principles. This SBA model
further adopts principles like modularity, reusability and self- containment
of network functions to enable deployments to take advantage of the latest
virtualisation and software technologies. NFV Management and Orchestration
(MANO) covers the orchestration and lifecycle management of physical and/or
software resources that support the virtualisation of the infrastructure, and
the lifecycle management of VNFs.

Apart from NFVs being the base for the 5G service based architecture, network
slicing is also another key feature introduced by 3GPP in 5G architecture. A
network slice, within 5G scope, is considered as a set of 3GPP defined
features and functionalities that together form a complete Public Land Mobile
Network (PLMN) for providing services to UEs. Network slicing will allow
customised control plane and user plane NFs to cater to the needs of a subset
of users/UEs or related to the business customer needs. Because of which it is
possible to create multiple network slices, each one composed of a collection
of features, capabilities and services required to the needs of the slice. For
example, one network slice can include the NFs to support mobile broadband
services with full mobility support, and another one to support non-mobile,
latency-critical industry applications.

###  1.2 Mutual Authentication

Mutual Authentication is not a new technique for 5G, it was actually
introduced in 3G when it was recognized that the 2G algorithms had weaknesses
around authentication. In 5G, however, before device tries to answer any
security challenges coming from the network, it firstly verifies whether the
network it is trying to connect to is legitimate. Once the device verifies the
network, the device responds to the security challenge and allows the core
network to verify that the device is legitimate. Usually this initial
authentication verification happens between the UE and AMF, but the
Authentication Server Function (AUSF) plays an important part in the mutual
authentication and key agreement process, and it acts a liaison between the
AMF and UDM to get the security information from UDM. The UDM generates the 5G
authentication vector using the secret key of the device stored at the UDM,
the same secret key is stored at the UE, hence the keys are symmetrical. This
process of mutual authentication and the authentication vector creation are
the first steps before any other security procedures are conducted in 5G. This
mutual authentication process is shown in Figure 2.

Figure 2: Mutual Authentication

###  1.3 Encryption and Integrity

Encryption and Integrity checking has been around since 2G, and 5G also
provides the encryption and integrity checking for various types of signalling
messages namely RRC, NAS and user-plane. These signalling require separate
keys for the encryption and decryption processes and those keys are exchanged
between the UE and AMF during the 5G mutual authentication and key agreement
process which are distributed through the system where necessary. However not
all these message signalling have same expectations in regards of encryption
of messages. From the figure 3, it can be derived that for RCC and NAS
signalling messages, encryption is optional and only integrity checking is
mandatory and for the user-plane traffic both the encryption and integrity
checking are optional, assuming that it depends on what type of device is
connecting to the radio network and whether it would require encryption or
not. If correctly used, these options of encryption of signalling messages for
confidentiality protection between the user device and the network functions
will potentially protect against the man in the middle (MITM) and fake base
station attacks(Stingray/IMSI catcher).

Figure 3: Encryption and Integrity

###  1.4 Service based interface protection

To protect the service based interfaces 5G provides two different options to
consider. The first one is to protect the protocol stack for the service based
interface by using transport layer security protocol (TLS) between HTTP/2 and
TCP. Only TLS v 1.2 and above are allowed to be used, furthermore 3GPP also
specifies different cipher suits that are allowed to be used in the TLS
security. TLS provides integrity and encryption of all the service based
interface messages in transit. The other method available is to use OAuth 2.0,
which doesn’t necessarily provide protection of traffic in transit but
provides the protection of service producer against malicious service
consumers trying to inject malicious attempts to invoke a service at the
service producer. OAuth 2.0 is token based authentication method, where the
network repository function acts as OAth 2.0 token server and a network
function acting as an OAuth 2.0 client requests for the access token to be
granted to access a specific service at a network function acting as OAuth 2.0
server. The access tokens can be applicable to a specific type of network
function or can be much more granular than that, and can also be applied to a
very specific network function. The token can grant very granular permissions
to access to a single service or service operations or a group of services,
which can then be part of the service request sent by the client (part of the
HTTP/2 header) to the server. The server verifies the authenticity of the
token and checks if it is valid and then grants the access to the requested
service. These both methods of providing TLS security and token based
authentication are optional. Figure 4 shows the basic OAuth between the
service consumer and the service provider.

Figure 4: OAuth Token Based Authentication

###  1.5 Roaming and inter-connections security

In most cases we don’t have direct connectivity between the visited Public
land mobile network (PLMN) and home PLMN. Usually the network connection
transition between Visited PLMN and Home PLMN is handled by IPX (IP packet
exchange), and for the obvious reasons it is not owned by the mobile network
service provider, rather a third party connection provider. Hence, it’s
important to consider the security of the traffic as it traverses between
different network interfaces. In service based architecture that security is
provided by Security Edge Protection Proxy (SEPP) for the control plane
traffic. It has number of security features like topology hiding support,
where any sensitive addressing information is filtered out in any signalling
messages, message filtering supporting filtering of any specific messages and
traffic policing where traffic streams can be policed. As shown in the figure
5, these SEPPs are connected by N32 protocol, which is an enhancement over
older 4G S6a protocol, in a way that it provides the TLS protection under
HTTP/2, parts of which can also be protected using PRINS (Protocol for N32
Interconnect Security) which allows one to selectively protect parts of the
message, leaving other parts that are required for end to end routing in the
inter-connected network. For user plane function (UPF), Inter PLMN User Plane
Security (IPUPS) is used which provides N9 protocol connectivity between UPFs.
There will be no traffic passing across N9 unless it is associated with a pre-
configured GTP v-1u tunnel which prevents any unintentional/malicious traffic
from traversing one UPF to reach other UPF if a GTP tunnel hasn’t been setup
and is unavailable, hence that traffic is dropped. Both the UPFs (each end of
the connection) have to be aware of the traffic between each other.

Figure 5: Roaming and inter-connections security

###  1.6 Subscriber Identity protection

In 5G a mechanism called as Subscription Concealed ID (SUCI) is introduced,
which means that the device (UE) never has to send IMSI(in 5G terminology its
called subscription permanent ID) in clear text. The user element converts
IMSI to SUCI by encrypting the MSIN (Mobile Subscriber Identification Number)
using the public key derived from the 5G network itself, leaving MCC (Mobile
Country Code) and MNC (Mobile Network Code) in clear as it’s required for the
routing purposes. During the registration process the UE sends over the SUCI
to the UDM where it gets decrypted by the UDM using the corresponding private
key and the IMSI is derived. It is not mandatory however to use encryption of
MSIN in SUCI and the algorithm can be set to null for clear text SUCI. The
figure 6 shows how SUCI is transferred between UE and UDM.

Figure 6: Subscriber Identity Protection

##  2 5G security best practices and recommendations

Most of these recommendations are inspired by the European Union Agency for
Cybersecurity (ENISA). The ENISA is the Union’s agency dedicated to achieving
a high common level of cybersecurity across Europe.

The best practices can mainly be categorized into three important areas,
policy, technical and organizational. The details of these practices are
described in the Challenges and Best Practices document by ENISA. Below sub-
sections highlights only critical and most important ones. For the complete
list, please refer to the ENISA doc [1].

###  2.1 Policy based

Zero-Trust: 5G NFV deployments should build on mature cybersecurity standards
employed in enterprise and cloud environments. Examples include the NIST
Cybersecurity Framework and ISO Information Security Management System
series71, the ETSI GS NFV-SEC 00372 Security and Trust Guidance and the UK
NCSC Zero Trust Architecture Design Principles [2]. Zero Trust represents an
overarching access security model that deliberately avoids assuming implicit
trust between elements in a network. This is particularly important in 5G as
various external stakeholders may need to access infrastructure components or
services for management, maintenance, or monitoring purposes [3].

Vulnerability handling and patch management: NFV and MANO software components
will need to be monitored for vulnerabilities and patched as quickly as
possible to address evolving risks and ensure security and functionality.

Security assessment of new or changes to existing VNF service templates: New
or modified VNF service templates should be validated through proper risk
assessment by a security professional.

Security testing and assurance: Regular penetration and vulnerability testing
should be performed across the NFVi and MANO production environment to
identify any vulnerabilities (e.g. OSs, hypervisor, CIS, VMs or containers) or
compromise of the network zoning rules. 5G stakeholders should also leverage
internationally recognised product testing, assurance, and certification
regimes. Potential solutions to these requirements include, among others, the
Network Equipment Security Assurance Scheme (NESAS) [4], jointly defined by
3GPP and GSMA, and the ENISA EUCC[5], EUCS[6] schemes.

###  2.2 Organizational based

Secure physical environment and geographical location: Security is not just
about encrypting data streams but also about the physical deployment of
equipment. NFV devices and equipment location, type of equipment, and type of
services running on the equipment are all parts of the complete system and
must be protected in a secure physical environment.

Trust model: A trust model of relationships among various 5G stakeholders
should be built and be able to answer questions such as: ‘for what one does on
trust?’, ‘How much should one trust?’ and ‘How much anyone can trust?’.

SLAs establishment: SLAs established between 5G stakeholders should include
important security and compliance measures. In this respect, SLAs are like an
insurance policy for the 5G network services. SLAs set the expectations for
the performance of a service provider and for the security level. They also
establish penalties for missing targets. In that concept, NFV could not leave
unaffected the evolution of SLA models and their flexibility in adapting to
more demanding parameters. An SLA management framework should be defined in
order to fill the gap between 5G stakeholders.

###  2.3 Technical based

Security monitoring and filtering: This covers the use of virtual network
security appliances such as anti-virus, virtual firewalls or virtual IDS/IPS
to achieve a level of security comparable to traditional networks. Further,
machine learning (ML) assisted solutions can be used to detect attack traffic
(e.g. DoS attacks) and distinguish it from normal traffic, so that it can be
handled appropriately.

Data protection and privacy: To prevent disclosure of user data (e.g.
subscriber identifiers), 5G networks should use security mechanisms such as
the encryption cryptographic operation. It is important to pay attention to
the requirements of lawful interception60. Privacy and security, both of
individuals’ personal data and of critical infrastructure, are important
preconditions for GDPR. Encryption is a crucial tool to achieve these goals.
Any approach to weaken or grant backdoor access to encryption methods defeats
the entire purpose of encryption and undermines users’ trust, exposing 5G
systems to increased risks. At the same time, it remains vitally important
that companies and law enforcement authorities continue to work together,
ensuring that authorities have the best methods and access to electronic
evidence without weakening or putting strong encryption at risk.

Centralised log auditing: All the NFV, SDN and MANO elements should submit
information on security events to a centralised platform, which shall monitor
and analyse the logs in real time for possible attempts at intrusion.

##  3 Security risks and vulnerabilities

There are two major 5G deployment models so far, although there will be more
options added in future. The currently deployed model is called non-standalone
(NSA) mode, more precisely referred to as EN-DC. In this model, the 5G
stations are integrated with an existing 4G network working in tandem with LTE
base stations and hence connected to the LTE core, which relies on the
measures and security protections of 4G/LTE network. Again, this cause a major
security risk implication in 5G because of the reliance on the 4G core
network. This backward compatibility with current deployment models of 5G will
pose a security risk in terms of the security vulnerabilities inherited from
4G network. The next phase of 5G deployment will most likely be Stand Alone
(SA) mode, more precisely referred to as SA-NR. As the name suggest, in this
model, 5G new radio network (NR) will be connected directly to the 5G core,
which will allow the full security features of 5G specifications to be
realised. Although it is recognised that new paradigms (cloud native, service-
based architecture) will introduce new security challenges.

Apart from the enhanced security features supported by 5G, there will be new
attack vectors and threat models which will be realized as 5G’s adaptation
will expand to the AI devices (intelligent cars etc.), Industrial control
devices, and most importantly the IoT sector, which will include every corner
case of a low level device capable of connecting to the internet.
Internationally, 5G is already being deployed commercially to the supply-chain
industry. Because of this broad paradigm that 5G open up to the internet, and
given the fact that historically the telecommunication and mobile network
service provides paid less attention to the security while building custom
protocol and software for the mobile/wireless communication stack, it will be
daunting task for 5G security to keep up with the industry grade standard of
providing security and privacy services. Also, not to mention that wireless
being it’s own bubble, developing custom software stacks and protocol, the
attackers might not be very well versed to exploit it efficiently. Hence,
there is a learning curve there as well, which balances out the 5G security
risks, but it won’t be long until adversaries would be able to catch up with
the wireless technology and will start exploiting potential vulnerabilities.
Therefore, precautions should be taken from the start to build a better and
more secure 5G network and close as many gaps as possible.

##  4 Conclusion

5G is an opportunity for the mobile industry to enhance network and service
security. New authentication capabilities, enhanced subscriber identity
protection and additional security mechanisms will result in significant
security improvements over legacy generations.

Experience has shown that 2G/3G networks make use of insecure, un-managed
protocols and are subject to fraud and threats on a regular basis. Many of
these attacks have been mitigated with 4G and 5G. However, due to the backward
compatibility of 4G with 3G/2G they will not disappear until the legacy
technology or backward compatibility is ceased.

When defining 5G roll-outs operators will have to consider how these legacy
networks will impact them over time; considering how attacks could be
prevented if legacy generations are either isolated or removed from the
ecosystem.

With the advent of new transport protocol for fast and secure communication,
QUIC - Quick UDP Internet Connections, and 5G’s stepping into the network
protocol stack more, it would be worth while to investigate the performance
impact of implementing 5G over QUIC, which might be used as default if 5G will
run HTTP3 for service based interfaces. This area is currently being explored
and the performance and security enhancement will be tested for this
experiment.

Lastly, since the industry is moving towards Zero-Trust implementation, we are
also currently exploring and researching how 5G deployment models will fit in
the Zero Trust Architecture, as 5G once deployed will constitute a part of the
network where users will be connecting to network to access the internet and
and authentication and authorization will come into play together with other
major pillars of ZTA.

## Acknowledgments

This work is funded under DOE Advanced Scientific and Computing Research
Program contract number ERKJ437 for Self-Driving 5G for Science.

## References

  * [1] Evgenia Nikolouzou, Goran Milenkovic, Georgia Bafoutsou, and Slawomir Bryska.  Nfv security in 5g challenges and best practices.  https://www.enisa.europa.eu/publications/nfv-security-in-5g-challenges-and-best-practices, 2022.
  * [2] Introduction to zero-trust.  https://www.ncsc.gov.uk/collection/zero-trust-architecture/introduction-to-zero-trust, 2021.
  * [3] Jonathan Olsson, Andrey Shorov, Abdelrazek, and Jorden Whitefield.  Zero-trust and 5g.  https://www.ericsson.com/en/reports-and-papers/ericsson-technology-review/articles/zero-trust-and-5g, 2021.
  * [4] Network equipment security-assurance scheme.  https://www.gsma.com/security/network-equipment-security-assurance-scheme/, 2022.
  * [5] ENISA-EU Union Agency For Cybersecurity.  Cybersecurity certification eucc candidate scheme v1-1.1.  https://www.enisa.europa.eu/publications/cybersecurity-certification-eucc-candidate-scheme-v1-1.1, 2021.
  * [6] ENISA-EU Union Agency For Cybersecurity.  Eucs cloud-service scheme.  https://www.enisa.europa.eu/publications/eucs-cloud-service-scheme, 2020.


# Kuperberg dreams

Steven Hurder  Steven Hurder, Department of Mathematics, University of
Illinois at Chicago, 322 SEO (m/c 249), 851 S. Morgan Street, Chicago, IL
60607-7045<EMAIL_ADDRESS>and  Ana Rechtman  Ana Rechtman, Institut de
Recherche Mathématique Avancée, Université de Strasbourg, 7 rue René
Descartes, 67084 Strasbourg, France<EMAIL_ADDRESS>
###### Abstract.

The “Seifert Conjecture” stated, “Every non-singular vector field on the
3-sphere ${\mathbb{S}}^{3}$ has a periodic orbit”. In a celebrated work,
Krystyna Kuperberg gave a construction of a smooth aperiodic vector field on a
plug, which is then used to construct counter-examples to the Seifert
Conjecture for smooth flows on the 3-sphere, and on compact 3-manifolds in
general. The dynamics of the flows in these plugs have been extensively
studied, with more precise results known in special “generic” cases of the
construction. Moreover, the dynamical properties of smooth perturbations of
Kuperberg’s construction have been considered. In this work, we discuss the
genesis of Kuperberg’s construction as an evolution from the Schweitzer
construction of an aperiodic plug. We discuss some of the known results for
Kuperberg flows, and discuss some of the many interesting questions and
problems that remain open about their dynamical and ergodic properties.

###### Key words and phrases:

Kuperberg flows, aperiodic flows, dynamical plugs; Scottish Book Problem 110

Preprint date: December 8, 2021; to appear online at Celebration.org

2020 Mathematics Subject Classification. Primary 37C10, 37C70, 37B45

##  1\. Birth of a method

In August 1993, Krystyna Kuperberg wrote out a 3 page sketch, of a
construction of what she thought would be a $C^{3}$-counterexample to the
Seifert Conjecture. The idea for this construction had occurred to her earlier
in the month, while attending the _Georgia Topology Conference_ in Athens,
Georgia. She faxed these notes to her son Greg, with the admonition that he
wait for a more final draft. In the following days, she wrote out the precise
construction, which expanded to an 8 page note, and realized that it was a
$C^{\infty}$-counterexample. Greg then sent an email to several leading
topologists, announcing the exciting news, that his mother had found a smooth
counterexample to the Seifert Conjecture!

The announcement of Krystyna’s work caused a sensation. William Thurston
lectured on her construction at the MSRI, Berkeley in September, and observed
that the construction could be realized as a _real analytic_ aperiodic flow.
Krystyna mailed around 60 copies of the manuscript to mathematicians around
the world. Then in October, she gave a plenary lecture on her work at the
conference in honor of Morris Hirsch in Berkeley California. Elsewhere, a
seminar was held in Tokyo in November, as part of the _International
Symposium/Workshop: Geometric Study of Foliations_. The participants in this
informal seminar included, among many others, Shigenori Matsumoto, Étienne
Ghys, Paul Schweitzer, and the first author.

The manuscript by Kuperberg was published almost immediately in the Annals of
Mathematics [27]. Moreover, soon afterwards, Ghys presented Kuperberg’s work
in a Séminaire Bourbaki [10] in June 1994. Matsumoto wrote a report in
Japanese [35], also in 1994, which provided more results about the dynamical
properties of these “Kuperberg flows”, with detailed proofs of their
properties. The joint paper by Greg and Krystyna [28] contained many new ideas
and variations on the basic construction. The brief note [31] gives an
overview of Krystyna’s work on aperiodic flows. Finally, Krystyna Kuperberg
[30] gave a report to the _International Congress of Mathematicians_ in 1998.

Krystyna commented on her work: “once you see the construction, you see it”.
In addition, her original sketches of the construction, involving a thickened
annulus with “rabbit ears”, were replaced in the published version by the now
very familiar images drawn by her husband, Włodzimierz (Włodek), which
conveyed a strong intuitive feel for the construction. Krystyna Kuperberg was
a student of Karol Borsuk in Warsaw, and many of her works correspondingly use
a strong geometric approach to analyzing problems in dynamical systems, and
this is more than true for her construction of the Seifert counter-examples.

The goals of this essay are to discuss the antecedents to the Kuperberg
construction, and give some inkling of the ideas that led Krystyna to her
construction. In addition, we speculate on extensions of the construction and
its future, especially in the context of other aspects of the theory of smooth
dynamical systems. First, in Section 2 we recall the radical idea that made
the Kuperberg construction possible, as it was a break from the past
approaches to constructing counterexamples in a crucial manner. Section 3
gives a concise overview of known results about the Kuperberg flow. We then
discuss in Section 4 questions about alternate flow dynamics for non-generic
constructions of Kuperberg flows. Section 5 discusses speculations and open
problems for flows that preserve some additional geometric structure.

##  2\. Shape of Plugs

In his 1950 work [45], Seifert introduced an invariant for deformations of
non-singular flows with a closed orbit on a $3$-manifold, which he used to
show that every sufficiently small deformation of the Hopf flow on the
$3$-sphere ${\mathbb{S}}^{3}$ must have a closed orbit. He also remarked in
Section 5 of this work:

> _It is unknown if every continuous (non-singular) vector field on the three-
> dimensional sphere contains a closed integral curve._

This remark became the basis for what is known as the “Seifert Conjecture”:

> _Every non-singular vector field on the $3$-sphere ${\mathbb{S}}^{3}$ has a
> periodic orbit._

There must exist at least one minimal set for a flow on a compact manifold - a
closed invariant subset which is minimal for this property. A periodic orbit
is a minimal set, but if there are no periodic orbits for a flow, then a
minimal set for the flow will have a possibly far more complicated topology.
So one can ask if there are restrictions on the _shape_ of a minimal set for
an aperiodic flow? Karol Borsuk defined the shape of a topological space as an
inverse limit continuum defined by a sequence of ANR approximations of the
space [2, 3, 33, 34]. For an aperiodic flow on a 3-manifold, it seems that the
minimal set must have a complicated shape, though just how complicated is an
open question.

Kuperberg’s strategy for the construction of an aperiodic flow was not based
on the choice of an invariant minimal set with prescribed shape, but was
instead to create a dynamical framework which avoided the implications of the
Brouwer Fixed-Point Theorem. We explain this remark in the following
discussions.

The first general approach to the Seifert Conjecture was made in the 1966
paper by Wesley Wilson [49], where he showed that every closed $3$-manifold
$M$ has a flow with only finitely many closed orbits. Wilson introduced the
use of a special kind of “plug” to modify a given flow, and this idea
profoundly influenced the thinking about the constructions of flows with
special properties, and the Seifert conjecture in particular.

A _Wilson plug_ for a flow on the 3-manifold $M$ is a neighborhood $P\subset
M$ diffeomorphic to an annulus cross an interval,
$P\cong{\mathbb{A}}^{2}\times[-1,1]$, where orbits enter one end of the plug,
$\partial^{-}P={\mathbb{A}}^{2}\times\\{-1\\}$, and exit the other end,
$\partial^{+}P={\mathbb{A}}^{2}\times\\{+1\\}$. The key idea introduced by
Wilson was that the plug flow should be symmetric: an orbit which traverses
the plug should enter and exit at symmetric points on the faces. No such
restriction is placed on an orbit which enters the plug and never exits, of
course, and one assumes there are such orbits. Wilson introduced the technique
of using a plug reflected on itself to achieve this symmetry property. As a
result, the plugs he uses have two periodic orbits, as illustrated in Figure
1.

A plug can be inserted into any non-singular flow on a manifold, not just once
but any finitely number of times as needed, to modify a given non-singular
flow. Wilson showed that any flow can be modified using finitely many plugs to
obtain a flow with only isolated periodic orbits. It is an observation in [40]
that by appropriately arranging the insertion of Wilson plugs, one can obtain
a flow with exactly two periodic orbits.

Figure 1.  An example of a Wilson plug with two periodic orbits

Subsequently, in the early 1970’s, Paul Schweitzer had the inspired idea to
construct a modified Wilson-type plug, but instead of the top and bottom
boundaries of the plug being annuli, they were diffeomorphic to a 2-torus with
an open disk removed, as illustrated in Figure 2. The paradigm shift of a
Schweitzer plug is that the faces of the plug are immersed in the transverse
space, but still transverse to the flow. Using a plug with this shape, the
second key point was to double the construction using a mirror symmetry as
with the Wilson plug, so that the flow in the plug has two Denjoy-type minimal
sets, instead of two circles.

A Denjoy minimal set, as introduced by Denjoy in [8], has the shape of a wedge
of two circles. The Schweitzer plug has the celebrated “double doughnut” shape
as seen in Figure 2. This plug can then be inserted into flows on 3-manifolds
to create new flows which have all minimal sets of Denjoy type. In particular,
using this plug one can construct flows with no periodic orbits. Thus,
Schweitzer showed that every 3-manifold carries a $C^{1}$-flow with no
periodic orbits! This celebrated result appeared in the Annals of Mathematics
[43], and in his talk [44] to the _International Congress of Mathematicians_
in 1974.

Figure 2. The Schweitzer plug

Mike Handel showed in his 1980 paper in the Annals of Mathematics [12] a
result which severely limits the shape of a minimal set for a smooth aperiodic
flow – it cannot be embedded as the minimal set for a flow on a surface. Thus,
a flow with a “Denjoy type” minimal set carried by a surface cannot be
$C^{2}$. This was an attempt to prove that plugs were not the method to find
smooth aperiodic flows on ${\mathbb{S}}^{3}$.

Jenny Harrison showed in [13] that the Schweitzer plug could be modified to
obtain an aperiodic plug with a $C^{2+\delta}$-regular flow, by embedding the
Denjoy minimal sets in a “3-dimensional” fashion. However, this result
required delicate arguments, and obtaining any further improvement in the
differentiability of the flow using this method, seemed completely out of
reach. The Handel and Harrison works suggested that any counter-example to the
Seifert Conjecture was going to require a radically new approach.

Meanwhile, in 1979, the Kuperbergs were attending the _Scottish Book
Conference_ in Denton, Texas, organized by Dan Mauldin [36]. At this
conference, Stanislav Ulam told Krystyna and Wlodek about an unsolved problem,
posed by Ulam:

> [36, Problem 110] _Let $M$ be a manifold. Does there exist a numerical
> constant $K$ such that every continuous mapping $f\colon M\to M$ which
> satisfies the condition $|f^{n}(x)-x|<K$ for $n=1,2,\cdots$ must possess a
> fixed point: $f^{n}(x_{0})=x_{0}$? _

The mathematician/inventor Coke Reed was a colleague of Ulam at Los Alamos,
and later a colleague of Krystyna at Auburn University. Kuperberg and Reed
discussed this problem, and subsequently solved it. They gave counter-examples
to the assertion in Problem 110, as published in the 1981 work [24]. The
methods they developed are analogous to the methods introduced by Wilson in
[49]. Kuperberg and Reed improved their result in the 1989 paper [25] via an
application of the Schweitzer plug. An analysis of the solutions to the Ulam
Problem 110 were recalled and analyzed in the paper [26] by K. Kuperberg, W.
Kuperberg, P. Minc and C. Reed. There is a connection between the solutions of
the Ulam Problem and the construction of counter-examples to the Seifert
Conjecture, in that the plug technique can be applied to construct counter-
examples to both, as remarked in the joint work of Greg and Krystyna [28,
Theorem 8].

Following the successful solution to the Ulam Problem 110, Kuperberg began to
consider what was required to construct a smooth plug without periodic points
for the flow. As she remarked [32], the problem was always the _Brouwer Fixed-
Point Theorem_ : any map of a (transverse) disk to itself must have a fixed
point, and so a corresponding suspension flow will have a periodic orbit.
Thus, to build a flow in a plug without periodic orbits, the transverse
holonomy for the flow must be a “translation” on an infinite line or region.

The inspired geometric observation is that such a translation is already
available in the Wilson plug, as the flow on the Reeb cylinder, as pictured in
Figure 3 below.

Figure 3. Reeb flow on cylinder

The flow of a point below the orbit ${\mathcal{O}}_{1}$ will climb up to the
periodic orbit ${\mathcal{O}}_{1}$ and the return map of this flow is a
translation on a vertical interval in the cylinder transverse to the flow. A
similar dynamical behavior happens for points in the cylinder above the
periodic orbit ${\mathcal{O}}_{2}$, but in reverse time.

With this point of view, one can then ask: how to get the flow of the periodic
orbit to “break open” and “start climbing the holonomy staircase”? That is,
break open the periodic orbit using its own holonomy! The solution is the
second inspired idea behind Kuperberg’s construction, to insert the flow on
the Wilson cylinder into itself, so that the orbit ${\mathcal{O}}_{1}$ is
inserted into one of the flow lines climbing up to ${\mathcal{O}}_{1}$. This
is precisely what the insertion maps illustrated in Figure 5 below accomplish.
To make this work and keep the plug properties, one must first do a “double
twist” of the embedding of the Reeb cylinder, as illustrated in Figure 4
below. (See also the illustration on the left side of Figure 8.) According to
Krystyna, the idea for doing this construction occurred to her during a beer
party at the Georgia Topology Conference.

Figure 4. Embedding of Wilson Plug ${\mathbb{W}}$ as a folded figure-eight
Figure 5.  The Kuperberg Plug ${\mathbb{K}}_{\epsilon}$

The essence of the novel strategy behind the aperiodic property of the flow
$\Phi_{t}$ on a Kuperberg Plug is perhaps best described by a quote from the
paper by Matsumoto [35]:

> We therefore must demolish the two closed orbits in the Wilson Plug
> beforehand. But producing a new plug will take us back to the starting line.
> The idea of Kuperberg is to _let closed orbits demolish themselves_. We set
> up a trap within enemy lines and watch them settle their dispute while we
> take no active part.

The reader is extolled to read the proof that these flows are aperiodic in
[27], or in the reports by Ghys [10] or Matsumoto [35], as the elegance of
Kuperberg’s idea is revealed in the simplicity of this proof.

Note that there are many choices of the vector field that can be made for the
flow in a Wilson plug. What is essential for Kuperberg’s construction is that
there are the two closed orbits which attract/repel a set of orbits entering
or leaving a face of the plug.

A Kuperberg plug can also be constructed for which its flow ${\mathcal{K}}$ is
real analytic. An explicit construction of such a flow is given in Section 6
of the paper [28] by Greg and Krystyna Kuperberg. There is the added
difficulty that the insertion of the plug in an analytic manifold must also be
analytic, which requires some subtlety. Analytic aperiodic flows are discussed
in the second author’s Ph.D. Thesis [41].

The construction of smooth aperiodic flows on compact manifolds using the
Kuperberg method produces orbits that never close up, so they wander around a
compact region of a plug that has been inserted. There is much that can be
said about the dynamical properties of these flows, deduced from the way the
flows in the plugs are constructed. These are the _known knowns_ about
Kuperberg flows, as discussed in Section 3.

There are also many _known unknowns_ about Kuperberg flows, which generate
interesting open questions about the dynamics of the Kuperberg flows, as
discussed in Section 4. There are also the _unknown unknowns_ , which are
speculations of dynamical phenomena yet to be discovered for this class of
smooth flows.

##  3\. Aperiodic flow dynamics

Ghys observed in [10] that a Kuperberg flow has zero topological entropy. This
follows from the remark that for a smooth flow on a compact 3-manifold,
Katok’s results in [22] imply that if a smooth flow has positive entropy, then
it must have periodic orbits. Thus, aperiodic flows belong to a class of
smooth dynamical systems which might be considered “less than chaotic”, or
more precisely “at worst, slowly chaotic”.

The flow in the Wilson plug in Figure 1 has two periodic orbits, labeled
${\mathcal{O}}_{1}$ and ${\mathcal{O}}_{2}$. Both of these orbits are broken
open by the insertion surgery illustrated in Figure 5, to yield what are
called the _special orbits_ for the flow. Not as obvious without a closer
inspection, is that each of the special orbits is trapped inside of the
Kuperberg plug formed by the surgery, and they limit on each other. Thus,
their closures form a minimal set for the flow in the plug. Furthermore, every
orbit entering the plug either escapes from the plug on the opposite face, or
else it limits on the closure of a special orbit. Thus, there exists a unique
minimal set for the flow in the plug. If the aperiodic flow is constructed
using multiple Kuperberg plugs, then there may be multiple minimal sets, each
disjoint from the other. For simplicity of exposition, we assume there is only
one plug, and let $\Sigma$ denote the unique minimal set.

Another highly non-obvious result due to Matsumoto [35, Proposition 7.2], see
also the discussion in [10, Section 8], is that there is an open set of points
in the entrance of a Kuperberg plug whose forward orbits limit on the minimal
set $\Sigma$. As the minimal set $\Sigma$ is contained in the interior of the
plug, this open set consists of non-recurrent points for the flow. In
particular, this implies that there is no invariant probability measure in the
Lebesgue measure class, for the flow in the Kuperberg plug. It is remarked
after the proof of [28, Proposition 18] that the Matsumoto argument works for
$C^{1}$-flows, and some but not all piecewise-linear self-insertions [28,
Section 9].

Remarkably, Greg Kuperberg showed in Theorem 1 of [29] that on every closed
3-manifold there is a volume-preserving $C^{1}$-vector fielld with no closed
orbits, obtained by modifying the Schweitzer plug. He also proved in Theorem 2
of [29] that for every closed 3-manifold, there is a volume preserving
$C^{\infty}$-flow with a finite number of periodic orbits. Also in the volume
preserving setting, Viktor L. Ginzburg and Başak Gürel [11], constructed a
version of Schweitzer’s plug that allows to produce examples $C^{2}$-functions
on symplectic 4-manifolds, having a level set whose Hamiltonian vector field
has no periodic orbits. A Hamiltonian vector field always preserves volume and
in this case is $C^{1}$, hence it gives examples of aperiodic volume
preserving flows.

The final general remark about the geometry of the Kuperberg flows is more
complicated to explain, but has profound consequences for their dynamics. The
Reeb cylinder is an annular region bounded by the two periodic orbits for the
Wilson flow, as pictured on the right side of Figure 1. The process of doing
flow surgery as illustrated in Figure 5 cuts two “notches” out of the
cylinder. This is the region labeled by ${\mathcal{R}}^{\prime}$ on the left
side of Figure 6 below, called the “notched Reeb cylinder” as pictured on the
right side of Figure 1, before the cylinder is deformed to appear as in Figure
4, which is then twisted as in Figure 5.

Figure 6.  Notch and the flow of its boundary arc

The Kuperberg flow in the Wilson cylinder preserves the region
${\mathcal{R}}^{\prime}$, called the _notched Reeb cylinder_ , until it
reaches a notch, then it continues to flow but the orbits are not longer in
the cylinder, as in Figure 5. This produces a tongue-shaped region as
illustrated on the right side of Figure 6. This tongue region continues to
flow until it intersects the insertion regions created by the surgery, and the
pattern repeats ad infinitum, attaching tongues to the tongues.

The complete flow of the _notched Reeb cylinder_ is a surface with boundary
embedded in the Kuperberg plug. Figure 7 gives three illustrations of this
central object for Kuperberg flows, which Siebenmann called a “chou-fleur” in
his communication [47]. This was illustrated by Ghys as a “fractal-like
cluster” in [10], as pictured in the lower left side of Figure 7. In the
monograph [15], the authors called the surface as laid out on the plane, a
“propeller” , which is approximately illustrated by the thickened
2-dimensional, tree-like structure on the right side of Figure 7, but again
not to scale (all the branches have approximately the same width). We call
this infinite surface ${\mathfrak{M}}_{0}$.

Figure 7.  The surface ${\mathfrak{M}}_{0}$ \- as Serpent, Chou-fleur, or
Propeller

The basic observation is that the boundary of ${\mathfrak{M}}_{0}$ is the
union of the special orbits for the flow! Thus, to understand the dynamics of
the flow, one analyzes the structure of the propellers in the Kuperberg plug,
which are defined by a recursion process that is generated by the insertion
surgery. The recursion process which defines the branches of the propeller
depends in a sensitive manner on the precise surgery process used to construct
the plug. This makes the analysis very delicate, and more will be said about
that later.

The Kuperberg flow preserves the embedded infinite surface
${\mathfrak{M}}_{0}$, whose boundary contains the special orbits. The fact
that the flow is aperiodic corresponds to the fact that the propellers are
infinite.

The closure ${\mathfrak{M}}=\overline{{\mathfrak{M}}_{0}}$ is a type of
2-dimensional _lamination with boundary_ that contains the minimal set
$\Sigma$. Thus, the topological shape of the minimal set $\Sigma$ is dominated
by the shape of the lamination ${\mathfrak{M}}$, which can be estimated using
the recursion on the reinsertions of the notched Reeb cylinder for the
Kuperberg plug. This approach to the study of the shape of $\Sigma$ is taken
in [15]. Conversely, the inclusion $\Sigma\subset{\mathfrak{M}}$ suggests
possible relations between the shape of ${\mathfrak{M}}$ and the dynamical
properties of the flow, as discussed in Section 4.

We describe the technical setting for comparing $\Sigma$ and ${\mathfrak{M}}$.
The drawing on the left side of Figure 8 below gives an expanded view of the
insertions illustrated in Figure 5. A key point is that the image of a
boundary orbit, denoted by $\tau({\mathcal{O}}_{1})$ in the illustration, has
image under the insertion as a twisted curve that is tangent to the cylinder
at the boundary curve ${\mathcal{O}}_{1}$, but intersects transversally
$\tau({\mathcal{O}}_{1})$. This results in a picture as on the right side of
Figure 8, where the two thickened curves are the boundaries of the cylinder
and its insertion.

The graph on the right side of Figure 8 is called the _radius function_
(actually, what is illustrated is the inverse of this function) and the
“quadratic shape” of the graph is the basis for showing that the Kuperberg
flow is aperiodic. Understanding these remarks takes some effort, and is
explained in the papers [27, 10, 35, 15], but reveals the great beauty of the
Kuperberg construction.

The authors introduced the notion of a _generic_ Kuperberg flow in the work
[15], which formulates a collection of optimal conditions on the choices for
the construction of the flow. One of these conditions is that the radius
function for the insertion is actually “quadratic”, as pictured on the right
side of Figure 8. We show in [15] that for a generic Kuperberg flow, there is
equality $\Sigma={\mathfrak{M}}$, and thus one can study the shape of the
minimal set for the flow by studying the properties of the 2-dimensional
lamination with boundary ${\mathfrak{M}}$.

Figure 8. The image of $L_{1}\times[-2,2]$ under $\sigma_{1}$ and the radius
function

##  4\. Non-generic flows

Ghys wrote in his 1995 survey [10, page 302]:

> _Par ailleurs, on peut construire beaucoup de pièges de Kuperberg et il
> n’est pas clair qu’ils aient la même dynamique._

In this section, we discuss some ways in which the dynamical properties of an
aperiodic smooth Kuperberg plug vary with the choices made in its
construction. The properties considered include the _shape_ and the _Hausdorff
dimension_ of the minimal set $\Sigma$, and the _slow entropy_ of the flow.

The construction of a Kuperberg flow begins with the choice of a flow
${\mathcal{W}}$ on a cylinder, with an invariant cylinder bounded by the two
periodic orbits that is called the _Reeb cylinder_ , as illustrated on the
left side of Figure 1. This flow is then extended smoothly to a thickened
cylinder, or interval times an annulus, as on the right side of Figure 1.
There are further symmetry conditions imposed on the flow ${\mathcal{W}}$, as
given in [27, Section 3], or for example as specified by (P1) to (P4) in [15,
Section 2]. The symmetry conditions still allow for a wide variation of
choices for the flow ${\mathcal{W}}$.

The flow of the Kuperberg plug, that results from the insertion process,
consists of orbit segments of the gradient like flow ${\mathcal{W}}$ which are
“patched together”. This view of the orbits, as a union of finite flows, is
key to the analysis of the Kuperberg flow in the works of Ghys [10] and
Matsumoto [35]. For a different take on this, Section 5 of the paper [28]
describes the result of the self-insertion patching, as the analog of a
“stacking subroutine” for a computer program. Then the fact that no orbit is
periodic corresponds to showing that the routine does not terminate for the
given starting point. So in essence, the solution to the Seifert Conjecture is
realized by building a “dynamical computer program” that is non-terminating.

This patching of flow segments is occurring orbitwise, so if there is to be
any hope of organizing this process in a unified description for the entire
flow on the plug, one needs to impose assumptions on the flow ${\mathcal{W}}$
that its orbits are as uniform as possible. Such additional conditions on
${\mathcal{W}}$ are formulated in the works [27, 28], and even more
restrictive conditions are imposed in [15].

The second step in the construction of a Kuperberg flow is to make a choice
for each of the two insertion maps, as pictured in Figure 5 and Figure 8. In
this step, the embedding of the Reeb cylinder makes a transverse contact at
the periodic orbit, either the bottom or the top orbit, so as to break open
these closed orbits. The result is that the special orbits which result by
breaking open the closed orbits, they intersect the vertex of the parabola
pictured on the right side of Figure 8. They return infinitely often to a
neighborhood of the vertex of the parabola as a result of the translation
holonomy of the flow. Hence, the germinal shape at its vertex of the parabolic
map has a strong influence on the dynamics of the special orbit, and so on the
dynamics of the entire flow.

A basic question is, when does the minimal set satisfy
$\Sigma={\mathfrak{M}}$? Ghys gave a construction of a flow on a plug and
sketched the proof that $\Sigma={\mathfrak{M}}$ for this flow, at the end of
Section 7 in [10]. The Kuperbergs gave an explicit real analytic flow on a
plug for which $\Sigma={\mathfrak{M}}$ in their joint paper [28]. In both
cases, the idea of the proof is to show that the closure of a special orbit
contains the notched Reeb cylinder ${\mathcal{R}}^{\prime}$, hence contains
the flow of ${\mathfrak{M}}_{0}$ and thus equals its closure ${\mathfrak{M}}$.

The authors began our investigations of the shape of the minimal set in the
Kuperberg flows in 2010, and for several years after that, every time we met
Krystyna we had to admit we couldn’t answer one of the mysteries of the
Kuperberg construction:

######  PROBLEM 4.1.

Is the minimal set _always_ two dimensional for a smooth Kuperberg flow?

This is the conclusion for the examples by Ghys in [10] and in the joint paper
[28], but the situation in general was not known. In order to answer
Krystyna’s question, we introduced the notion of a “generic Kuperberg flow” in
[15], which assumes that the flow satisfies the conditions of Hypothesis 12.2
and Hypothesis 17.2 formulated in that work. Hypothesis 12.2 assumes in
particular that the vertical component of the vector field ${\mathcal{W}}$ on
the Reeb cylinder has quadratic germ near the two periodic orbits at which its
“vertical component” must vanish. This condition is used to show that the
closure of the special orbits contain ${\mathcal{R}}^{\prime}$. But it is
unknown, for example, that if the vector field vanishes to higher order at the
periodic orbits, can one still show that ${\mathcal{R}}^{\prime}$ is contained
in the closure of a special orbit?

The equality $\Sigma={\mathfrak{M}}$ can be considered as a type of _Denjoy
Theorem_ for 2-dimensional laminations. The usual Denjoy Theorem states that
the closure of an orbit for a $C^{2}$-flow without closed orbits on the
2-torus is all of ${\mathbb{T}}^{2}$. Then the equality
$\Sigma={\mathfrak{M}}$ can be considered as analogous, for it states that the
closure of an orbit is not a 1-dimensional submanifold of ${\mathfrak{M}}$,
but fills up its 2-dimensional leaves. The known proofs of this conclusion
cited above, they all make assumptions on both the flow ${\mathcal{W}}$ and on
the insertion map. Formulating general criteria which suffice to imply the
equality $\Sigma={\mathfrak{M}}$ has proven difficult.

If $\Sigma\subset{\mathfrak{M}}$ is a proper inclusion, then the topological
type of $\Sigma$ and ${\mathfrak{M}}$ will differ, for example they could have
distinct shapes. A $C^{1}$-flow on ${\mathbb{T}}^{2}$ without closed orbits
provides a good model for this difference. If the minimal set ${\mathfrak{D}}$
for the flow is not all of ${\mathbb{T}}^{2}$ then ${\mathfrak{D}}$ is a
1-dimensional continuum, as used in the construction of the Schweitzer plug.
If $p\in{\mathbb{T}}^{2}-{\mathfrak{D}}$ then there is a retract of the open
punctured torus ${\mathbb{T}}^{2}-\\{p\\}$ onto ${\mathfrak{D}}$, so that
${\mathfrak{D}}$ has the shape of a wedge of two circles. It seems a very
interesting problem to compare the shapes of the closed invariant sets for a
general aperiodic flow:

######  PROBLEM 4.2.

Suppose the Kuperberg flow in a plug has minimal set
$\Sigma\subset{\mathfrak{M}}$ which is a proper inclusion. What is the
relation between the shapes of $\Sigma$ and ${\mathfrak{M}}$?

Theorem 19 in [28] gives the construction of aperiodic PL plugs for which
$\Sigma$ is 1-dimensional. In addition, there is a discussion of the symbolic
dynamics for these special flows they construct. The consideration of the
topological type of $\Sigma$, in addition to the known unknowns about its
shape, one senses that its study leads into the realm of the unknown unknowns,
that new dynamical phenomena will be discovered.

We next consider another dynamical aspect of Kuperberg flows, their entropy
and the Hausdorff dimension of their minimal sets. While Ghys observed in [10]
that an aperiodic flow must have entropy equal to zero, using a well-known
deep result of Katok [22], none-the-less the dynamical behavior of the flow
appears to be chaotic upon closer inspection.

The geometry of the propeller as pictured in Figure 7, suggests that the flow
in the minimal set $\Sigma$ should exhibit exponential behavior, as it follows
the boundary of the propeller which appears to have exponentially growing
area. This intuition is flawed, though, as the flow cannot simply travel down
the core of the propeller, but must instead travel on the boundary to get to
the extremities of the tree. The time required to get to the extremes of the
tree is not linear, but grows approximately as the square of the distance to
be traveled. This observation is the basis for Theorem 21.10 in [15]:

######  THEOREM 4.3.

The “slow entropy” of a generic Kuperberg flow is positive for exponent
$\alpha=1/2$.

The notion of slow entropy was introduced by Katok and Thouvenot [23]. The
exponent $\alpha=1/2$ is essentially saying that the entropy would be positive
if the time variable is sped up by taking the square of time. We also note
that for this result, the generic hypotheses include an extra condition, the
assumption that the insertion maps for the construction of the plug have “slow
growth” themselves. This is a technical property, but its requirement
emphasizes that the proof of Theorem 4.3 is quite technically involved. In
particular, the proof uses an estimate on the growth rates of leaves in the
lamination ${\mathfrak{M}}$, that they grow at subexponential rate $1/2$ as
well. Define the growth rate of ${\mathfrak{M}}$ as the exponential growth
rate of the dense leaf ${\mathfrak{M}}_{0}$, so for a generic flow this rate
is $1/2$. Also, define the entropy dimension $HD({\mathfrak{M}})$ of
${\mathfrak{M}}$ as the least upper bound of the exponents $0\leq\alpha\leq 1$
such that the $\alpha$-slow entropy of the flow is positive [7]. Thus, for a
generic flow we have $HD({\mathfrak{M}})\geq 1/2$. There are many questions
one can ask about these invariants, for example:

######  PROBLEM 4.4.

What range of values for $HD({\mathfrak{M}})$ can be realized by aperiodic
flows on 3-manifolds? Is there an aperiodic flow with $HD({\mathfrak{M}})=0$?

For a Kuperberg flow, one can also ask about the Hausdorff dimension of the
lamination ${\mathfrak{M}}$. This problem seems almost intractable, but Daniel
Ingebretson developed an approach to calculating this dimension for a
particular class of flows in his thesis work [20, 21].

######  THEOREM 4.5.

[20] The Hausdorff dimension of the minimal set for generic Kuperberg flow
satisfies $2<d_{h}({\mathfrak{M}})<3$.

The dimension of ${\mathfrak{M}}$ must be at least $2$, as ${\mathfrak{M}}$ is
a union of $2$-dimensional “leaves” by the generic assumption. But the fact
that the dimension is greater than 2 is a measure of the transverse complexity
of ${\mathfrak{M}}$. The actual Hausdorff dimension appears to depend in a
sensitive manner on the choices made. The thesis work also gives a method to
numerically calculate this number. We can then formulate a question which
falls into the category of an unknown unknown about Kuperberg flows:

######  PROBLEM 4.6.

Is there a relation between the slow entropy of a Kuperberg flow, and the
Hausdorff dimension of its unique minimal set? Is there a relation if the flow
is assumed to be generic?

We should also mention a question that belongs to the class of “unknown
unknown” problems.

######  PROBLEM 4.7.

How do the dynamics of a real analytic Kuperberg flow in a plug differ from
the general smooth case? Are there further restrictions on the shape of the
minimal sets in the analytic case?

Another remarkable fact about the Kuperberg flows, is that they lie at the
“boundary of chaos” in the $C^{\infty}$-topology on flows. The idea behind
this remark is that when making the insertion as in Figure 8, one can stop the
insertion “too soon”. That is, if the two cylinders do not make contact, then
the insertion does not break open the periodic orbits for the Wilson flow
${\mathcal{W}}$. We show in [16] that the flow for such a truncated insertion
again has two periodic orbits. However, the lengths of these orbits tends to
infinity as the insertion limits to one which breaks open the orbits. Thus, we
obtain smooth families of variations of the Kuperberg plug with simple
dynamics, and exactly two periodic orbits, and the limit of the family is an
aperiodic flow.

In such a family, the period of the periodic orbits must blow up at the limit.
Palis and Shub in [38] asked whether this dynamical phenomenon can occur in
families of smooth flows on closed manifolds, and called a closed orbit whose
length “blows up” to infinity under deformation a “blue sky catastrophe”. The
first examples of a family of flows with this property was found by Medvedev
in [37]. The constructions in [16] show that deformations of Kuperberg flows
provide a new class of examples. The work of A. Shilnikov, L. Shilnikov and D.
Turaev in [46] further discusses “blue sky catastrophe” phenomenon.

On the other hand, when making the insertion as in Figure 8, one can take the
insertion “too far”. That is, the two cylinders intersect not along one arc,
but along two arcs. We show in [16] that the flow for these over-extended
insertions have countably many embedded horseshoes, so an abundance of
hyperbolic behavior. Moreover, as the insertion maps are deformed to one that
just makes contact at a single point, then these horseshoes increase in
number. The dynamics of the generic Kuperberg flow is thus the limit of the
dynamical chaos in a continuous family of horseshoes for neighboring smooth
flows.

The immediate conclusion is that the Kuperberg flows are not stable in the
$C^{\infty}$-topology on flows.

The recitation of remarkable properties of the class of flows created by
Kuperberg could continue, as they are zero entropy dynamical systems, which
are simply not boring [17]! Instead, we want to also discuss a variety of
other constructions and results following from Seifert’s Conjecture, and the
questions they engender.

##  5\. Decorated flows

As explained by Ghys [10], the construction of non-singular aperiodic flows is
particularly complicated in dimension 3. The Seifert conjecture, that is false
in full generality, can be decorated with invariant structures: we can ask, if
the flow preserves something, then must it have a periodic orbit? On one hand,
there are families of non-singular flows that always have periodic orbits, on
the other hand we have the examples discussed previously in this text. Here we
discuss this broader subject of flows on closed 3-manifolds, we review some
results on the existence of periodic orbits. We do not intend to make an
exhaustive list.

Let us start with the class of Reeb vector fields associated to a contact
form. It has been shown by Hofer in [14], that the flow of a Reeb vector field
on ${\mathbb{S}}^{3}$ must have a periodic orbit, and this was generalized to
every closed 3-manifold by Taubes in [48]. But we know more, in fact every
Reeb vector field on a closed 3-manifold has at least two periodic orbits as
proved by Cristofaro-Gardiner and Hutchings [6] and the examples with exactly
two periodic orbits are completely understood [5]. The second author, together
with Colin and Dehornoy, proved that if a Reeb field is non-degenerate then it
has either two or infinitely many periodic orbits [4].

A Reeb vector field always preserves a volume form. A big open question in the
subject, is whether a $C^{\infty}$ volume preserving flow on a 3-manifold has
a periodic orbit. It seems impossible to mimic Krystyna’s construction of a
plug in this setting, but we can do a Wilson plug for volume preserving flows.
The lack of examples of possible minimal sets, makes it also impossible to
mimic other plug constructions.

In the case of Reeb flows, Alves and Pirnapasov [1] proved recently the
existence of some knots such that if a Reeb flow has a periodic orbit
realizing the knot then it has positive entropy. In particular, by the already
mentioned theorem of Katok [22], such Reeb flows have lots of periodic orbits.
It is an observation that such a result cannot hold for categories of flows
for which we can construct plugs. Using a suitable Wilson plug one can prove
the following:

######  THEOREM 5.1.

Let $K$ be a given knot. Every 3-manifold that admits a zero entropy flow
without fixed points, admits a zero entropy flow without fixed points having
(at least) two periodic orbits whose knot type is $K$. Moreover, if the
original flow is generated by a Reeb vector field, then the new flow preserves
a plane field that fails to be a contact structure along the two periodic
orbits of knot type $K$.

The existence of periodic orbits was extended to geodesible volume preserving
flows (also known as Reeb vector fields of stable Hamiltonian structures) on
manifolds that are not torus bundles over the circle by Hutchings and Taubes
[18, 19], and Rechtman [42], and for real analytic geodesible flows by
Rechtman [42]. The existence of a periodic orbit is also established for real
analytic solutions of the Euler equation by Etnyre and Ghrist [9]. Geodesible
volume preserving flows are solutions to the Euler equation, but we do not
know if the last ones have periodic orbits. We do know that it is impossible
to construct plugs whose flow satisfies the Euler equation [39].

## References

  * [1] M. Alves and A. Pirnapasov,  Reeb orbits that force topological entropy, Ergodic Theory and Dynamical Systems, 2021.
  * [2] K. Borsuk,  Concerning homotopy properties of compacta,  Fund. Math., 62:223–254, 1968.
  * [3] K. Borsuk,  Theory of shape,  Monografie Mat., vol. 59, Polish Science Publ., Warszawa, 1975..
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# Squigglebot: a battery powered spherical rolling robot
as a model active matter system to measure its energetics

Soumen Das<EMAIL_ADDRESS>Anit Sane  Shankar Ghosh  Department of
Condensed Matter Physics and Materials Science,
Tata Institute of Fundamental Research, Mumbai 400005, India

###### Abstract

Active matter systems use their internal or ambient source of energy and
dissipate them at the scale of individual constituent particles to generate
motion. Direct measurement of the energy influx for individual particles has
not been achieved in the experiments. Here we present “Squigglebot” - a
battery powered spherical rolling robot based on open source hardware as an
artificial active matter system whose energy consumption as well as the energy
dissipation into different modes of motion both can be measured
experimentally. This can serve as a prototype system to study a number of
interesting problems in non-equilibrium statistical physics, where details of
the energetics are required.

††preprint: APS/123-QED

##  I Introduction

The study of active matter systems whose constituent particles consume and
dissipate energy in order to generate mechanical forces and move in a
collective manner has been an area of interest in the field of non-equilibrium
statistical physics [1, 2]. These are mostly observed in the biological
settings which include all living organisms; spatial organization of chromatin
in the nucleus [3], self organization of microtubules [4] and actin present in
the cyctoskeleton of living cells [5]; animal groups [6] (e.g. bird flocks,
fish schools, herd of sheep’s) etc. Artificial active matter include self
propelled particles [7], driven granular systems [8], swarming robots [9] etc.
Sustained consumption and dissipation of energy make these systems inherently
out of equilibrium. In most studies of active matter systems the energy input
at the level of individual particles are not monitored directly. Here we
introduce “Squigglebot” - a battery powered rotating spherical robot as a
model active matter particle whose energy intake as well as the energy
expenditure both can be measured experimentally.

This can be useful in situations where details of both input and output
energies are needed in non-equilibrium systems. For example one can study the
energy efficiency of an individual active particle, i.e., how it converts its
internal energy to achieve its desired motion. Similarly, how active matter
systems move away from equilibrium is an interesting question which has not
been studied beyond a few theoretical models [10]. These approaches use the
idea of entropy production based on the statistics of individual microscopic
trajectories which is hard to obtain in experiments. However one can construct
an equivalent measure of entropy production based on the distributions of the
input energy in the system.

##  II Design

###  II.1 Mechanical Construction

Figure 1: Schematic of the mechanical assembly with its different components
pointed out. X-Y-Z is the body frame and X’-Y’-Z’ is the lab frame. The motor
rotates about the body Y axis. Figure 2:  Schematics of the circuit diagram
used in the Squigglebot. JP indicate the pins for connecting jumper cables to
the pcb.

The experimental system, based on Squiggle-ball toy, consists of a plastic
spherical shell of diameter $80~{}\rm{mm}$ inside which a dc motor is mounted
along the axis. Shaft of the motor is connected to the shell. The body of the
motor is held in an acrylic enclosure. On one side of the motor, a solid block
of brass of dimension $10\times 20\times 40~{}\rm{mm}$ and mass $57~{}\rm{gm}$
is screwed to the acrylic. A $2\times\rm{AA}$ battery holder case and a
printed circuit board (pcb) is attached to the adjacent two sides of the
acrylic enclosure. Due to the asymmetric mass distribution of the brass load -
motor assembly, the center of mass of the ball changes as the motor rotates.
When placed on a solid surface with the motor turned on, the sphere rolls in a
chaotic manner [11]. When the sphere collides with a wall or any other rigid
object, it rebounds in a random direction. The schematic of the mechanical
assembly of the “Squigglebot” made using SolidWorks is shown in Fig. 1. The
moment of inertia of the assembly in the body frame can be calculated using
SolidWorks as

$J=\begin{pmatrix}730.58&12.02&-12.27\\\ 12.02&717.19&22.27\\\
-12.27&22.27&698.53\end{pmatrix}$ gm * $\rm{cm}^{2}$.

###  II.2 Electronics

The schematic diagram of the electronics circuit is shown in Fig. 2. The
micro-controller used for control, data collection and sensing is ESP-12F from
Ai-Thinker Technology [12]. It is an Wi-Fi enabled microchip based on ESP8266
SoC (System on Chip) module. ESP8266 integrates Tensilica’s L106 Diamond
series 32-bit RISC processor running at 80 MHz, 36 KB of RAM, 4 MB of external
SPI flash, 10 bit ADC (analog to digital converter), I/O and a PCB-on-board
antenna in the same chip. The module supports standard IEEE 802.11 b/g/n 2.4
GHz Wi-Fi and complete TCP/IP protocol stack. It also supports SDIO, SPI,
$\rm{I}^{2}\rm{C}$, $\rm{I}^{2}\rm{S}$ and UART protocols for serial
communication.

The recommended power supply voltage for its operation is 3 - 3.7 V. Two
decoupling capacitors of capacitance 0.1 $\mu\rm{F}$ (C5) and 10 $\mu\rm{F}$
(C6) are placed between the power supply (Vcc) pins and the ground close to
the chip to filter out the high and low-frequency noise respectively. A push
button switch is connected between the RST pin and ground for resetting the
chip manually. RST, EN, GPIO0 and GPIO2 pins are connected to power supply VS
via $10~{}\rm{k}\Omega$ pull-up resistors (R1, R2, R7, R6). GPIO15 is grounded
via a $10~{}\rm{k}\Omega$ pull-down resistor (R5). GPIO0 pin needs to be
shorted to ground temporarily to put it into the flash mode for uploading the
code via UART protocol using an external programmer chip.

To measure the instantaneous angular velocity as well as the linear
acceleration of the ball, MPU-6050 which is a MEMS based IMU (inertial
measurement unit), is used [13, 14]. MPU-6050 is an integrated 6-axis motion-
tracking device that combines a 3-axis gyroscope, 3-axis accelerometer, and a
Digital Motion Processor (DMP). A MEMS accelerometer consists of a proof mass
suspended on a spring. When the acceleration is applied on a particular axis,
it causes the proof mass to shift to one side. Due to this deflection the
capacitance between fixed plate and plate attached to the proof mass is
changed. This change in capacitance is proportional to the acceleration. The
sensor processes this change in capacitance and converts it into an analog
output voltage [15]. A MEMS gyroscope sensor is composed of a proof mass which
is kept in a continuously oscillating motion. When a rotation is applied, the
Coriolis force acting on the moving proof mass changes the direction of the
vibration. This causes a capacitance change proportional to the angular
velocity which is picked up by the sensing element and then converted to a
voltage signal [15].

MPU6050 can measure the angular velocities with four programmable full scale
ranges of $\pm 250^{\circ}\rm{/s}$, $\pm 500^{\circ}\rm{/s}$, $\pm
1000^{\circ}\rm{/s}$ and $\pm 2000^{\circ}\rm{/s}$. Similarly it can measure
the accelerations with full scale ranges of $\pm 2\rm{g}$, $\pm 4\rm{g}$, $\pm
8\rm{g}$ and $\pm 16\rm{g}$, where $\rm{g}$ is the acceleration due to
gravity. We use full scale ranges of $\pm 2000^{\circ}\rm{/s}$ and $\pm
2\rm{g}$ for gyroscope and accelerometer respectively. Following capacitors
are used - (i) 0.1 $\mu\rm{F}$ regulator filter capacitor (C2) between REGOUT
and ground, (ii) 0.01 $\mu\rm{F}$ bypass capacitor (C3) between VLOGIC and
ground, (iii) 2.2 $\rm{nF}$ charge pump capacitor (C10) between CPOUT and
ground and (iv) 0.1 $\mu\rm{F}$ bypass capacitor (C11) between power supply
VDD and ground pins. $\rm{I}^{2}\rm{C}$ [16] is used for communication between
MPU6050 and ESP-12F micro-controller. SCL is the $\rm{I}^{2}\rm{C}$ clock pin
used to carry the timing signal supplied by the bus master device (ESP-12F).
This pin is connected to the GPIO5 pin on ESP-12F. SDA is the
$\rm{I}^{2}\rm{C}$ data pin used for both transmitting and receiving data. It
is connected to the GPIO4 pin on ESP-12F. Both SCL and SDA pins are connected
to the power supply voltage VS via two $4.7~{}\rm{k}\Omega$ pull-up resistors.
The AD0 pin determines the $\rm{I}^{2}\rm{C}$ address of the module. This pin
is connected to ground using a $4.7~{}\rm{k}\Omega$ pull-down resistor, which
sets its $\rm{I}^{2}\rm{C}$ address as 0x68 in hexadecimal representation.
Data is collected from MPU6050 at 25 Hz which is then transmitted over Wi-Fi
by the micro-controller.

The maximum current that each GPIO pin of ESP8266 can supply is about 6 mA
which is not sufficient to drive the motor. Hence, DRV8833 dual H-bridge motor
driver is used to drive the motor [17]. Another reason for using the motor
driver is that without it, the noise from the motor can reset the control
circuitry or burn out internal components of the micro-controller. The
capacitors used are - (i) 10 $\mu\rm{F}$ ceramic bypass capacitor (C9) between
the device power supply (VM) and ground pins, (ii) 0.01 $\mu\rm{F}$ X7R
seramic capacitor (C4) between high-side gate driver VCP and VM pins and (iii)
2.2 $\mu\rm{F}$ bypass capacitor (C8) between VINT and ground pins. AIN1 and
AIN2 pins are connected to ground (Logic Low) and power supply VIN (Logic
High) respectively. GPIO2 pin of ESP-12F is connected to NSLEEP (enable) pin
of the motor driver. Therefore by setting the GPIO2 pin to logic High and Low,
the motor driver and in turn the motor can be turned on and off.

Figure 3:  Printed circuit board with the main components pointed out.

In order to measure the current drawn by the motor from the battery, a Hall
effect based current sensor WCS2702 is used [18]. A 0.01 $\mu\rm{F}$ capacitor
(C7) is connected between the output pin (VOUT) and the ground to reduce the
noise. The motor is connected between the IP- pin and AOUT1 pin of the motor
driver. IP+ pin is connected to the AOUT2 pin of the motor driver. Here the
current flows from IP+ to IP- pin.

The input power supply of the current sensor (VDD) is kept fixed at 3.3 V
using LT1963A-3.3 linear and low-dropout (LDO) voltage regulator [19]. A 10
$\mu\rm{F}$ bypass capacitor (C13) is connected between the input (IN) and
ground pins. Another 10 $\mu\rm{F}$ capacitor (C12) is connected between
output (OUT) and ground to prevent oscillations of the output voltage and make
it stable.

ADC pin of ESP-12F can handle a maximum voltage of 1 V. Since the output of
the current sensor (VOUT) is more than 1 V, a voltage divider (consisting of
resistors R3 and R4 - both $100~{}\rm{k}\Omega$) is used to decrease the
voltage of VOUT pin before feeding it to ADC pin. This sets the current
resolution of the present experimental setup at 2.3 mA. By reading the voltage
of the ADC pin, the current drawn by the motor can be estimated.

Any 3-6 V geared DC motor with a cross section of $10\times 12~{}\rm{mm}$ and
D-shaped gearbox output shaft of diameter $3~{}\rm{mm}$ can be used. For the
data presented in this paper, a 60 rpm motor is used with a output shaft of
length $9~{}\rm{mm}$. The no load and stall current are about $10~{}\rm{mA}$
and $1~{}\rm{A}$ respectively. The rated and stall torque are $200$ gm-cm and
$1600$ gm-cm respectively.

Two 3.7 V 800 mAh Li-ion batteries are used as power supplies. One (VIN) is
used to power the motor along with the motor driver and the other one (VS) is
used for powering the rest of the electronics including the micro-controller.
Ground pins of both batteries as well as all IC’s are connected together. The
designed printed circuit board is shown in Fig. 3.

Figure 4:  Calibration of accelerometers and gyroscopes in MPU6050s is done by
placing them on a horizontal plane.

##  III Measurements

We are interested in the energetics of the “Squigglebot”. Here input energy is
the electrical energy consumed by the motor from the battery. On the other
hand, output energy consists of the translational and rotational kinetic
energies of the ball as well the energy dissipated in the motor and that due
to the friction between the ball and the surface on which it is moving. During
a single run of the experiment ($\sim$ 2 h 15 min), voltage $V$ of the battery
powering the motor almost remains constant ($\sim$ 3.6-3.7 V). Hence
monitoring the current drawn by the motor, ${I(t)}$, is enough to estimate the
electrical energy which is ${E_{e}(t)}=\int V*I(t)~{}dt$.

Accelerometer measures the dynamic linear acceleration resulting from motion
of a body, as well as the static acceleration due to gravity acting on it.
Here the ball has a dynamic acceleration much smaller than the acceleration
due to gravity. Since the accelerometer attached to the ball itself is
rotating, it becomes difficult to filter out the gravity component and get
only the linear acceleration due to the motion of the ball. Hence to keep
things simple, we take images of the ball from the top of the arena where it
is moving and by detecting the position, we can calculate its linear velocity
$\vec{v}$ and the translational energy, $E_{T}=\frac{1}{2}m|\vec{v}|^{2}$,
where $m$ is the mass of the “Squigglebot” ($\sim 145~{}\rm{gm}$).

Gyroscope measures the angular velocity of the ball with respect to the body
frame. By fusing the acceleration and angular velocity from the IMU sensor,
one can estimate the rotation matrix $R$ describing the orientation of the
body frame with respect to the lab frame [20, 21]. It can be used to convert
physical quantities from body frame to lab frame. Since energy is invariant in
different frames, estimating the rotational kinetic energy in the body frame
is sufficient which can be calculated as
$E_{R}=\frac{1}{2}\bm{\vec{\omega}}^{T}\bm{\tilde{J}}\bm{\vec{\omega}}$, where
$\bm{\vec{\omega}}$ is the angular velocity column vector with the components
$\omega_{x},~{}\omega_{y},~{}\omega_{z}$.

Figure 5:  Calibration curve of the current sensor WCS2702. Figure 6:
Snapshot of the ball moving on a horizontal surface.

###  III.1 Gyroscope and Accelerometer Calibration

For calibrating the gyroscope and the accelerometer values, the pcbs are
sticked using double-sided tape to a flat acrylic plate placed on a horizontal
plane and their values are measured (Fig. 4). These values are then stored in
the code as offsets and during the experiment these offsets are subtracted
from the measured raw values before transmitting the data.

###  III.2 Current sensor Calibration

The current sensor generates a voltage proportional to the current flowing
through it. It is calibrated using Keithley 2410 sourcemeter as a constant
current source. The calibration curve is linear with
$V_{\rm{out}}=1.7047+0.8464*I$ as shown in Fig. 5. Here $V_{\rm{out}}$ is the
output voltage of the current sensor and $I$ is the current flowing through it
from IP+ to IP- pin (see Fig. 2 for the direction of current flowing). Thus it
has a sensitivity of 0.8464 V/A.

Figure 7:  Linear velocity of the ball during a typical experimental run.

###  III.3 Measurement of linear velocity

During a typical experiment, the “Squigglebot” is placed on a horizontal
surface (optical table) and its motor is turned on wirelessly. To detect its
position, a usb camera (Pixelink-PL-D734CU-T) [22] is fixed on top of the
arena and the images are taken at 25 fps. In the case of a single ball, by
tracking its position, one can compute its linear velocity. Whereas for
experiments with multiple balls, one can construct the trajectories of each
ball by using some kind of tracking algorithms [23] and then calculate their
linear velocities. Figure 6 shows an image of a single moving ball during an
experimental run. The ball is painted black for the ease of detection. The
linear velocity of a single ball as a function of time is shown in Fig. 7.

Figure 8:  (a) Time series of the current drawn by the motor in horizontal
orientation. The angle $\theta$ is close to $180^{\circ}$ as shown in the
inset. (b) Time series of $|\omega_{y}|$. $\omega_{x}$ and $\omega_{z}$ remain
close to zero, hence not shown. Figure 9:  (a) Time series of the current
drawn by the motor in vertical orientation. The angle $\theta$ is close to
$90^{\circ}$ as shown in the inset. (b) Time series of $|\omega_{y}|$.
$\omega_{x}$ and $\omega_{z}$ remain close to zero, hence not shown. Figure
10:  (a) Time series of the current drawn by the motor in tilted orientation.
The angle $\theta$ is shown in the inset which is close to $130^{\circ}$. (b)
Time series of $|\omega_{y}|$. $\omega_{x}$ and $\omega_{z}$ remain close to
zero, hence not shown. Figure 11:  (a) Time series of the current drawn by the
motor during a typical experimental run. The orientation of the plane of
rotation is shown in the inset. (b) Time series of the magnitude of the
different components of the angular velocity.

###  III.4 Measurement of the current drawn by the motor and its angular
velocity

As stated in II A, the brass load rotates along with the motor. If the plane
of rotation is not horizontal, the motor has to work against the gravity to
pull the load up and when the load comes down, motor is aided by the gravity.
Hence, the current drawn by the motor changes in time depending on the angular
position of the load in the plane of rotation as well as the orientation of
the plane of rotation itself. In the present setup, the motor rotates about
the body Y-axis which is perpendicular to the plane of rotation (Fig. 1).
Thus, the orientation is measured in terms of the angle between the gravity
vector and body Y-axis as
$\theta=\cos^{-1}\left(a_{y}/\sqrt{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}\right)$.
Before presenting the data during a typical experimental run where the ball
moves on a horizontal surface, we want to study the effects of the orientation
of the plane of rotation on the current drawn by the motor as well as its
angular velocities. This is achieved by holding the outer shell of the ball
fixed in different orientations, such that the body of the motor as well as
the load rotates along with it. The results are described in the next three
sections.

####  III.4.1 Horizontal rotation

The motor rotates in the horizontal plane. Hence, the gravity vector and the
axis of rotation is parallel to each other, i.e., $\theta\approx 180^{\circ}$
as can be seen from the inset of Fig. 8 (a). The corresponding time series of
the current drawn by the motor is shown in Fig. 8 (a). The magnitude of the
y-component of the angular velocity $|\omega_{y}|$ is shown in Fig. 8 (b).
$\omega_{x}$ and $\omega_{z}$ components are not shown as they are close to
zero.

####  III.4.2 Vertical rotation

Here the motor rotates in the vertical plane. Hence, the gravity vector and
the axis of rotation is perpendicular to each other, i.e., $\theta\approx
90^{\circ}$ (Fig. 9 (a) inset). The corresponding time series of the current
drawn by the motor is shown in Fig. 9 (a). Clearly the average current as well
as the variation in current is higher as compared to the case of horizontal
rotation due to the effect of gravity. The gravitational force $mg$ produces a
torque which opposes and aids the motor torque during upward and downward
motion respectively. For the same reason, $|\omega_{y}|$ shows a large
variation in time which can be seen in Fig. 9 (b). As before, $\omega_{x}$ and
$\omega_{z}$ components are not shown as they are close to zero.

####  III.4.3 Tilted rotation

Here the motor rotates in a plane that is tilted at an angle with respect to
the horizontal ($\theta\approx 130^{\circ}$). This is shown in Fig. 10 (a)
inset. The corresponding time series of the current drawn by the motor is
shown in Fig. 10 (a). Here, the average current as well as the variation in
current is higher than that of the horizontal rotation but less than that of
the vertical rotation. This is due to the fact that in the plane of rotation
the gravitational force has a component $mg\cos\theta$ acting on the load
which is less than $mg$. Figure 10 (b) shows the time series of
$|\omega_{y}|$. As before, $\omega_{x}$ and $\omega_{z}$ components are not
shown as they are close to zero.

####  III.4.4 Experimental run

During the experiment, the “Squigglebot” moves on a horizontal surface. Since
there are no external torque acting on the ball, the outer shell and the brass
load - motor body assembly both rotate in opposite direction to each other.
Since the coupling between the outer shell of the ball and the motor shaft is
not perfectly rigid, the ball has wobbling motion. The load can also fall in a
direction perpendicular to the plane of rotation causing the ball to topple.
Thus the plane of rotation has access to different angles between
$0^{\circ}~{}\rm{to}~{}180^{\circ}$ as can be seen from Fig. 11 (a) inset. The
corresponding time series of the current drawn by the motor is shown in Fig.
11 (a). Clearly all the three components of angular velocity are non-zero here
which can be seen from Fig. 11 (b).

###  III.5 Software

The source code for controlling the micro-controller, collecting data from the
IMU and current sensors as well as transmitting data over Wi-Fi is written in
the Arduino IDE. After booting up, ESP-12F connects to a local Wi-Fi network
whose SSID and password is put in the code. It then starts a HTTP web server
on port 80 and a Telnet server on port 23. The web server is used for the
following jobs - (i) it is used to send the instruction for starting the
measurement; (ii) it is used to toggle the state of the GPIO2 pin thereby
turning the motor on or off; (iii) it is used to upload the source code to the
micro-controller over Wi-Fi (OTA or over the air update) and (iv) it is used
to reboot the micro-controller. When ESP-12F receives the instruction to start
the measurement, it puts the GPIO2 pin to logic High and turns the motor on.
It then starts collecting data from the IMU sensor over $\rm{I}^{2}\rm{C}$ and
also from the current sensor. The data is transferred to a computer on the
same local Wi-Fi network using Telnet protocol which runs over the
Transmission Control Protocol (TCP) at $25~{}\rm{Hz}$.

The source codes for collecting the sensor data over Wi-Fi from the micro-
controller as well as the images from camera over USB are both written in
Python. In order to synchronize the sensor and image data, multirun plugin of
the Pycharm IDE is used to start both the codes simultaneously [24]. It has
been observed that during a typical experimental run, both the sensor and
image data stay almost synchronized to each other - only about 20 secs off
after 2 h 15 min. If one wants to get the simultaneous data from multiple
“Squigglebots”, then they can be executed parallelly with threads using
ThreadPoolExecutor subclass of the concurrent.futures module [25]. All the
source codes are provided in Github [26].

##  IV Conclusion

In this paper, we have presented the design of an artificial active matter
particle based on a battery operated rolling sphere with asymmetric mass
distribution. Along with the output energy, the input energy can also be
measured directly which is a non-trivial feature of this system since direct
measurement of input energies at the level of individual active matter
particles has not been done in the experiments previously. The mechanical
construction as well as the details of electronic instrumentation based on
open source hardware are provided, so are the source codes required to collect
the data. We believe this can serve as a prototype system to study a variety
of problems in non-equilibrium statistical physics, specially where details of
the energetics are required experimentally.

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