# Data-Driven Operator Theoretic Methods for Phase Space Learning and Analysis

Sai Pushpak Nandanoori<EMAIL_ADDRESS>Subhrajit Sinha
<EMAIL_ADDRESS>Enoch Yeung<EMAIL_ADDRESS>
###### Abstract

This paper uses data-driven operator theoretic approaches to explore the
global phase space of a dynamical system. We defined conditions for
discovering new invariant subspaces in the state space of a dynamical system
starting from an invariant subspace based on the spectral properties of the
Koopman operator. When the system evolution is known locally in several
invariant subspaces in the state space of a dynamical system, a phase space
stitching result is derived that yields the global Koopman operator.
Additionally, in the case of equivariant systems, a phase space stitching
result is developed to identify the global Koopman operator using the symmetry
properties between the invariant subspaces of the dynamical system and time-
series data from any one of the invariant subspaces. Finally, these results
are extended to topologically conjugate dynamical systems; in particular, the
relation between the Koopman tuple of topologically conjugate systems is
established. The proposed results are demonstrated on several second-order
nonlinear dynamical systems including a bistable toggle switch. Our method
elucidates a strategy for designing discovery experiments: experiment
execution can be done in many steps, and models from different invariant
subspaces can be combined to approximate the global Koopman operator.

###### keywords:

Dynamical systems, Koopman operator, data-driven, invariant subspaces, phase
space stitching.

fn1fn1footnotetext: Sai Pushpak, Subhrajit Sinha are with the Pacific
Northwest National Laboratory, Richland, WA 99354fn2fn2footnotetext: Enoch
Yeung is with the Department of Mechanical Engineering, University of
California Santa Barbara, CA 93106

##  1 Introduction

Multiple equilibria, periodic orbits, limit cycles, chaotic attractors, and
other complicated behaviors are common in nonlinear dynamical systems, and
identifying such systems from data can be difficult at times. Furthermore,
most known approaches for studying or analyzing nonlinear systems theory
necessitate knowledge of mathematical models governing the system of interest,
and most methods necessitate the computing of a unique energy function for the
underlying system. It is difficult to compute such an energy function,
especially for a higher-dimensional system. Recent research suggests that
nonlinear systems can be examined using data alone, without the use of any
mathematical models.

The data from a nonlinear system is mapped to a higher-dimensional observable
space, where it may be examined successfully utilizing transfer operators such
as the Perron-Frobenius and Koopman operators [1, 2, 3, 4, 5, 6]. Because both
operators are adjoint, a dynamical system can theoretically be analyzed using
any of them. However, in terms of application to real-world circumstances,
each of the operators has its own set of advantages and disadvantages. In
comparison to the Perron-Frobenius operator, Koopman operators are more
commonly utilized for data-driven analysis.

Bernard Koopman introduced the Koopman operator in [7] and this seminal work
became a popular tool with [1] and several other works followed where Koopman
operator theory is used in the system identification [3, 8], for control
design [9, 10, 11], sensor fusion [12] and analysis of spectrally conjugate
systems [13], finding observability gramians or observers [14, 15, 16], study
of chaotic systems [17, 18], data-driven information transfer [19, 20, 21],
data-driven based causal inference in dynamical systems [22], in data-driven
classification of power system stability [23], and in power system coherency
identification, power system stability monitoring [24, 25, 26, 25, 27],
dynamic state estimation in power networks [28], fault, cyber-attack
localization in cyber-physical systems [29, 30], computational neuroscience
applications [31] and data assimilation for climate forecasting [32].

The Koopman operator definition is an infinite-dimensional linear operator
that yields the evolution of a dynamical system on an infinite-dimensional
space of functions. Finding an infinite dimensional (linear) operator is
computationally infeasible, hence many approaches are devised to best
approximate the Koopman operator in finite dimensional space [3, 33, 34, 12,
35, 36, 37, 38, 39, 40, 41]. Most popular methods include dynamic mode
decomposition (DMD) [3, 33]; extended dynamic mode decomposition (E-DMD) [12,
42]; kernel dynamic mode decomposition (K-DMD) [34], naturally structured
dynamic mode decomposition (NS-DMD) [43], Hankel-DMD [17], deep dynamic mode
decomposition (deep-DMD)[6, 42, 44, 45, 46]. All these methods are data-driven
and accuracy of some such approximations are discussed in [47]. The authors of
[48] look at the space of Koopman observables, which includes the underlying
system’s states. The authors of [48] also demonstrate the fidelity of this
class of observable functions and discuss performance tuning options.

###  1.1 Problem Statement

The major goal of this research is to use data to examine the global phase
space of nonlinear dynamical systems. We identified a few natural questions
that occur during experimental phase space analysis when there is less
knowledge about the underlying system and the experiments are costly. Given
the time-series data corresponding to an invariant subspace of a nonlinear
system, it is of interest to know as additional time-series data emerges, can
the other invariant subspaces be discovered? Is it possible to uncover the
global phase space? On the contrary, suppose if the evolution of a system in
different subspaces is known, can they be combined to analyze the global phase
space? Is it possible to examine the global phase space with only one
invariant subspace and knowledge of the nature of underlying symmetry?
Finally, can the phase space properties of a topologically conjugate system be
determined if the phase space properties of a system are known?

This work attempts to address these questions by leveraging the tools from
Koopman operator theory. The experimental phase space discovery procedure of a
classic biological nonlinear network, the bistable toggle switch, inspired
this work.

###  1.2 Summary of Contributions

The Koopman operator and its spectral properties are utilized to investigate
the global phase space of nonlinear dynamical systems in this work. Our
contributions are as follows:

  * 1.

We provided conditions for updating the local Koopman operator when new
spatial time-series data emerges.

  * 2.

We developed conditions for discovering invariant subspaces from a global
phase space, and vice versa.

  * 3.

We presented a phase space stitching result in which many local Koopman
operators were fused to produce a (single) global Koopman operator.

  * 4.

We extended our phase space stitching result to obtain the global Koopman
operator for equivariant dynamical systems when data is available only in any
one of the invariant subspaces but the knowledge about the underlying symmetry
between the invariant subspaces is known.

  * 5.

Results were developed in order to comprehend the global phase space of
topologically conjugate systems.

We hope the phase space learning and analysis from this work serves as a basis
for designing and planning future experiments. This work is an extension to
our previous works [40, 49]. In particular, the partitioning of the state
space into multiple invariant sets, and the phase space stitching result are
introduced in [49]. The Koopman operator theory for equivariant systems is
developed in [40]. The rest of the contributions are exclusive to this work.

The rest of the paper is organized as follows. Section 2 describes the
preliminaries for operator theoretic methods and consists of a brief overview
on the computation of finite-dimensional approximation of Koopman operators.
Some thoughts on the relation between the state space and the observable space
are discussed in Section 3. Discovery of new invariant subspaces from the
spectral properties of the Koopman operator and the phase space stitching
results along with a use case for equivariant dynamical systems is presented
in Section 4. Using Koopman operators, global phase space learning and
analysis for topologically conjugate dynamical systems is presented in Section
5. The proposed Koopman operator theoretic methods are illustrated on second-
order dynamical systems in Section 6 and the paper concludes with final
remarks in Section 7.

##  2 Mathematical Preliminaries and Dynamic Mode Decomposition (DMD) Variants

We recall the necessary concepts that forms a base for the results in this
work from [1, 2, 5, 12, 50]. Consider the following discrete-time nonlinear
system

$\displaystyle x_{t+1}=T(x_{t})$ (1)

where $x\in{\cal M}\subseteq\mathbb{R}^{n}$ and $T:{\cal M}\to{\cal M}$. The
phase space ${\cal M}$ is assumed to be a compact manifold with Borel $\sigma$
algebra $\mathcal{B({\cal M})}$, $\mu$ denotes the measure [1, 2] and the
nonlinear map $T$ is assumed to be $\mathcal{C}^{1}$ continuous.

Let $\mathbb{N}$ denote the set of natural numbers. Recall the definition of
invariant set from the classical dynamical systems theory as follows.

######  Definition 1 (Invariant Subspace)

Let $M\subseteq{\cal M}$ be a subspace of the dynamical system, Eq. (1). Then,
the subspace $M$ is said to be (positively) invariant if for every $x_{0}\in
M$, $T^{n}(x_{0})\in M$ for all $n\in\mathbb{N}$.

Define an observable function on the state space to be a scalar valued
function ${\psi}:\mathcal{M}\rightarrow\mathbb{C}$, where $\mathbb{C}$ denotes
the set of complex numbers. The scalar valued function ${\psi}$ belongs to the
space of functions ${\cal G}$ acting on elements of $\cal M$. Usually for any
dynamical system, the output function describes the evolution of the states
and this output function can be considered as an observable function.
Moreover, any differentiable function of the states can be chosen as an
observable function. Normally, square integrable functions defined on ${\cal
M}$ forms a good choice of observable functions. Hence, the space of
functions, ${\cal G}=L_{2}({\cal M},{\cal B},\mu)$ is infinite dimensional.
The Koopman operator can now be defined as follows.

######  Definition 2 (Koopman operator)

For a dynamical system $x\mapsto T(x)$ and for ${\psi}\in\mathcal{G}$, the
Koopman operator $({\mathbb{U}}:{\cal G}\to{\cal G})$ associated with the
dynamical system is defined as

$\left[\mathbb{U}{\psi}\right](x)=\psi(T(x)).$

The Koopman operator $\mathbb{U}$ is defined on the space of functions and is
a linear operator. The function space ${\cal G}$ is invariant under the action
of the Koopman operator. Essentially, instead of studying the time evolution
of nonlinear system (1) on the state space ${\cal M}$, we study the evolution
of observables in an infinite dimensional space where the evolution is linear.
Apart from being linear, the Koopman operator is also a positive operator: for
any $g\geq 0$, $[\mathbb{U}g](x)\geq 0$.

The infinite dimensional Koopman operator admits discrete and continuous
spectrum. In the scope of this work, we are interested in the discrete
spectrum of the Koopman operator. The following section describes the popular
methods to identify a finite dimensional approximate of the Koopman operator
from the time-series data.

###  2.1 DMD Variants

This subsection discusses the computation of the approximate Koopman operator
from the time-series data using prominent methods such as dynamic mode
decomposition (DMD), extended dynamic mode decomposition (EDMD) and deep
dynamic mode decomposition (deepDMD).

Schmid and others introduced DMD in [8] to approximate the Koopman operator
that describes the coherent features of fluid flow and extract dynamic
information from flow fields through the spectrum of the approximate Koopman
operator. The authors in [12] generalized the idea of DMD to introduce EDMD.
In EDMD, an extended dictionary of functions is used to approximate the action
of the infinite dimensional Koopman operator.

Consider the time-series data $X=[x_{0},\;x_{1},\cdots,x_{k}]$ from an
experiment or simulation of a dynamical system and stack them as shown below.

$\displaystyle X_{pr}=[x_{0},x_{1},\dots,x_{k-1}],\qquad
X_{fw}=[x_{1},x_{2},\dots,x_{k}],$

where for every $i$, $x_{i}\in{\cal M}$. Assuming knowledge of the space
${\cal G}=L_{2}({\cal M},{\cal B},\mu)$ we can define the set of dictionary
functions ${\cal D}=\\{\psi_{1},\dots,\psi_{N}\\}$ where $\psi_{i}\in
L_{2}({\cal M},{\cal B},\mu)$ and $\psi_{i}:{\cal M}\to\mathbb{C}$ are scalar
valued functions. Each observable function in the dictionary ${\cal D}$ are
assumed to be bounded. Let the span of these $N$ dictionary functions in
${\cal D}$ be denoted by ${\cal G}_{\cal D}$ such that ${\cal G}_{\cal
D}\subset{\cal G}$.

Since we are considering either time-series data from an experiment or
simulation in the scope of this work, it is natural to describe their
evolution as a discrete-time system, and so we describe the Koopman operator
theory in discrete-time setting. All of the results presented in this paper,
however, are valid in the continuous-time setting as well.

Define a vector valued observable function ${\Psi}:{\cal M}\to\mathbb{C}^{N}$
such that,

$\displaystyle{\Psi}(x):=\begin{bmatrix}\psi_{1}(x)&\psi_{2}(x)&\cdots&\psi_{N}(x)\end{bmatrix}.$

Then by definition for a function $\hat{\psi}_{i}\in{\cal G}_{\cal D}$,
$\hat{\psi}_{i}$ can be expressed as an inner product of a coefficient vector
${a}_{i}$ with the dictionary functions as shown below.

$\displaystyle\hat{\psi}_{i}(x)=\sum_{\ell=1}^{N}a_{i\ell}\;\psi_{\ell}={\Psi}(x){a}_{i}$

for every $x\in X$. Similarly for
$\hat{{\Psi}}(x)=\begin{bmatrix}\hat{\psi}_{1}(x)&\hat{\psi}_{2}(x)&\cdots&\hat{\psi}_{N}(x)\end{bmatrix}$,
we obtain,

$\displaystyle\hat{{\Psi}}(x)=\Psi(x)A$

where
$A=\begin{bmatrix}a_{1}&a_{2}&\dots&a_{N}\end{bmatrix}\in\mathbb{R}^{N\times
N}$.

For every snapshot pair, $x_{pr},x_{fw}$ such that $x_{fw}=T(x_{pr})$, we
obtain,

$\displaystyle\hat{\Psi}(x_{fw})$
$\displaystyle=\left[\mathbb{U}\hat{\Psi}\right](x_{pr})$
$\displaystyle=\Psi(x_{pr})\hat{\mathbb{U}}A+r_{X}(x)$

where $\hat{\mathbb{U}}$ is a projection of $\mathbb{U}$ on the snapshot
space. The residual, $r_{X}(x)\in\mathbb{C}^{\mathbb{N}}$ is uniformly bounded
on the snapshot space $X$ and
$r_{X}(\cdot)\in\mathcal{G}\setminus\mathcal{G}_{\cal D}$. Therefore the
Koopman learning problem now results in minimizing the residual, $r_{X}$ which
leads to the following optimization problem.

$\displaystyle\min_{A,\hat{\mathbb{U}}}\;\;\parallel\left(\Psi(x_{fw})-\Psi(x_{pr})\right)\hat{\mathbb{U}}A\parallel$
(2)

The above optimization problem in (2) is equivalent to

$\min_{\hat{\mathbb{U}}}\;\;\parallel\Psi(x_{fw})-\Psi(x_{pr})\hat{\mathbb{U}}\parallel.$

Minimizing the residual over the given snapshot space, $X_{pr}$ and $X_{fw}$
leads to the following minimization problem.

$\displaystyle\min_{\tilde{\mathbb{U}}}\;\;\left|\left|\left(\begin{bmatrix}{\Psi}(x_{1})\\\
\vdots\\\ {\Psi}(x_{k})\end{bmatrix}-\begin{bmatrix}{\Psi}(x_{0})\\\ \vdots\\\
{\Psi}(x_{k-1})\end{bmatrix}\tilde{{\mathbb{U}}}\right)\right|\right|$

where

${\tilde{\mathbb{U}}}=\begin{bmatrix}{\cal K}_{1}&0&\ldots&0\\\ 0&{\cal
K}_{2}&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&{\cal
K}_{k}\end{bmatrix}\in\mathbb{C}^{kN\times kN}$

Note that we could thus parameterize $\tilde{\mathbb{U}}$ in terms of each
snapshot, thus mapping each snapshot to the diagonal entries ${\cal K}_{j}$
for $j=1,...,k.$ In practice, with standard DMD algorithms, it is common to
assume that the action of $\tilde{\mathbb{U}}$ is homogeneous across snapshot
space and that all ${\cal K}_{j}={\cal K}\in\mathbb{C}^{N\times N}.$ The
precise implications of this assumption are not immediately clear, though it
is a widely employed practice within the DMD community. This results in the
following classical and frequently formulated EDMD problem:

$\min_{\cal K}\parallel{Y_{fw}}-{Y_{pr}}{\cal K}\parallel_{2}^{2},$ (3)

where

$\displaystyle{Y_{pr}}=\begin{bmatrix}{\Psi}(x_{0})\\\ {\Psi}(x_{1})\\\
\vdots\\\
{\Psi}(x_{k-1})\end{bmatrix},\qquad{Y_{fw}}=\begin{bmatrix}{\Psi}(x_{1})\\\
{\Psi}(x_{2})\\\ \vdots\\\ {\Psi}(x_{k})\end{bmatrix}$

and ${\cal K}$ is the finite dimensional approximation of the Koopman operator
$\mathbb{U}$. For convenience, we minimize the upper bound of the induced
matrix 2-norm, which is the Frobenius norm

$\min_{\cal K}\parallel{Y_{pr}}{\cal K}-{Y_{fw}}\parallel_{F}^{2},$ (4)

with ${\cal K}\in\mathbb{C}^{N\times N}$. The solution to the optimization
problem (4) can be obtained explicitly and is given by

$\displaystyle{\cal K}={Y_{pr}}^{\dagger}{Y_{fw}}$

where ${Y_{pr}}^{\dagger}$ is the Moore-Penrose inverse of $Y_{pr}$. Note that
DMD is a special case of the EDMD algorithm with ${\Psi}(x)=x$. Hereafter, the
infinite dimensional Koopman is denoted by $\mathbb{U}$ and the finite
dimensional approximation of the Koopman operator is denoted by ${\cal K}$.

EDMD is formulated under the assumption that the observable functions for the
underlying system are available. As a result, using EDMD to compute the
approximation Koopman operator simply comes down to solving a least squares
problem (similar to DMD however with a richer observable space that could
better capture the properties of the underlying nonlinear system such as an
attractor set). It’s natural to wonder if considering the state inclusive
observable space, that is, integrating the observable functions defined in
EDMD with observables (measured functions of states) considered in DMD, has
any benefit. The authors in [48] show the state inclusive observables defined
on certain classes of dictionaries yield approximately Koopman invariant
subspaces and the corresponding Koopman can simultaneously learn the
underlying system dynamics and its corresponding flow.

Identifying the observable functions for a large scale system (perhaps with
thousands or millions of states) is difficult. Furthermore, computing the
Moore-Penrose inverse corresponding to this large-scale system for data
acquired under a variety of initial conditions adds a new level to the
challenge. DeepDMD approaches, which were recently developed in [6, 42, 44],
provide an alternative in this regard, as they make no assumptions on the
choice of observable functions. Furthermore, because deepDMD solves for
minimizing Eq. (4) as an optimization problem with decision variables as
observable functions and the approximate Koopman operator, it does not require
an explicit computation of Moore-Penrose inverse. For further implementation
details and detailed discussion on deepDMD, we refer the readers to the works
[6, 42, 44].

Corresponding to the time-series data $X_{pr},X_{fw}$, if the dictionary
functions are chosen in such a way that the optimal cost in Eq. (4) is in the
acceptable range of accuracy, then the spectral properties of the finite
dimensional Koopman operator and in particular, the dominant eigenvalues and
their respective eigenfunctions unfold the nonlinear system properties.

##  3 Discussion on the Observable Functions

Consider the dynamical system given in (1). In general, it can have multiple
attractors for example, in the case of a bistable toggle switch which is a
second-order system (as shown in Eq. (10) in Section 6). Assume that the
coordinate space defined by these state inclusive dictionary functions is
topologically conjugate to a linear system when the dictionary functions are
evaluated alongside the states. As a result, the first two elements of the
Koopman equation correspond to the original dynamical system in the case of
the bistable toggle switch. While the first two elements produce a multistable
phase portrait, the appearance is linear when raised into higher dimensional
space.

In the following we briefly recall the relation between the observable
functions and the Koopman eigenfunctions and introduce the Koopman modes.

###  3.1 Relation Between the Observables and the Koopman Eigenfunctions

Let $\lambda_{j}$ and $\phi_{j}$ denote the eigenvalues and eigenfunctions
corresponding to the point spectrum of the infinite dimensional Koopman
operator such that

$\mathbb{U}\phi_{j}=\lambda_{j}\phi_{j},\quad j=1,2,\dots$

Let ${f}$ be a vector-valued observable such that

${f}=\begin{bmatrix}f_{1}\\\ f_{2}\\\ \vdots\\\ f_{m}\end{bmatrix}$

where each $f_{i}\in{\cal G}$. Suppose if each $f_{i}$ lies in the span of
eigenfunctions of the Koopman operator, then ${f}$ can be expressed as a
linear combination of the eigenfunctions as follows.

$\displaystyle{f}(x)=\sum_{i=1}^{\infty}\phi_{i}(x){\vartheta}_{i}=\sum_{i=1}^{\infty}\lambda_{i}^{j}\phi_{i}(x_{0}){\vartheta}_{i}.$
(5)

The vector coefficients, ${\vartheta}_{i}$ in Eq. (5) are the Koopman modes
and this expansion is usually referred to as the Koopman mode decomposition
(KMD). The magnitude and frequency of each Koopman mode is given by the
magnitude and phase of its corresponding eigenvalue. The spatio-temporal
aspects in the dynamics are encoded in the eigenvalues (temporal signatures)
and the term $\phi_{i}(x_{0})\vartheta_{i}$ capture the spatial signatures.
Notice that the Koopman modes are indeed a function of the initial condition.
There are several methods to compute these Koopman modes and interested
readers can refer to [24, 50, 51, 52, 18, 53]. The Koopman eigenvalues,
Koopman eigenfunctions and the Koopman modes are usually referred to as the
Koopman tuple.

Recall that the learnt linear dynamics applying Koopman operator theory
evolves on the space of observables. The next section describes the relation
between the space of observables and the state space.

###  3.2 Relation between the Observable Space and the State Space

For the nonlinear dynamical system, Eq. (1), let $\mathbb{U}$ be the
corresponding Koopman operator such that

$\mathbb{U}\Psi(x_{t})={\Psi}(x_{t+1}).$

If $\phi(x)$ is an eigenfunction of $\mathbb{U}$ with eigenvalue $\lambda$,
then

$\mathbb{U}\phi(x_{t})=\lambda\phi(x_{t})=\phi(x_{t+1}).$

Among the eigenfunctions of the Koopman operator $\mathbb{U}$, the ones with
unit eigenvalue are of special interest as they correspond to the attractor
sets (like equilibrium points, limit cycles) of the underlying system [50]. In
particular, the number of unit eigenvalues of the Koopman operator gives the
number of attractor sets for the dynamical system. Hence, to find all the
attractor sets of an invariant subspace ${\cal M}$, it suffices to solve

$\phi(x)=\mathbb{U}\phi(x)$

Moreover, note that for an equilibrium point $x\in\mathcal{M}$,
$\phi(x)={\mathbb{U}^{n}}\phi(x)$ for all $n\in\mathbb{N}$, while for any
point on a limit cycle there exists some $n_{0}\in\mathbb{N}$ such that

$\phi(x_{*})={\mathbb{U}^{kn_{0}}}\phi(x_{*}),\quad\forall\quad
k\in\mathbb{N},$

such that corresponding to the eigenvalue $\lambda=1$, the eigenfunction,
$\phi$ satisfies

$\phi(T^{n_{0}}(x_{*}))=\mathbb{U}^{n_{0}}\phi(x_{*})=\lambda^{n_{0}}\phi(x_{*})$

Next, we briefly discuss the relation between the evolution of the original
nonlinear system on the state space and the observable space. Consider the
discrete-time nonlinear system as shown in Eq. (1). Suppose we have
$x_{0}^{1},..,x_{0}^{k}$ initial conditions of the dynamical system, then for
every initial condition, we have

$\displaystyle x_{n}^{i}=T^{n}(x_{0}^{i})\qquad\mbox{for}\;\;i=1,...,k.$

The corresponding Koopman operator, trained on these trajectories will give
the result

$\Psi(x_{n})=\mathbb{U}\Psi(x_{n-1})=[\mathbb{U}\circ\mathbb{U}]\Psi(x_{n-2})=\cdots=\underbrace{[\mathbb{U}\circ\cdots\circ\mathbb{U}]}_{n\;\mbox{
times}}\Psi(x_{0})=\mathbb{U}^{n}\Psi(x_{0})$

Then the flow map is approximated by the Koopman projection on the state-
observable as

$T^{n}(x_{0}^{i})=P_{x}\Psi_{n}(x_{0}^{i}),$

where $P_{x}$ is the projection operator which maps from the observable space
on to the state space.

In the next section, we derive conditions to search for invariant subspaces
starting locally from an invariant subspace of a dynamical system and develop
the phase space stitching result to identify the global Koopman operator.

##  4 Global Phase Space Exploration Using Approximate Koopman Operator

This section discusses the phase space analysis of a nonlinear system using
the approximate Koopman operator obtained using time-series data from either a
simulation or an experiment.

###  4.1 Phase Space Discovery

We begin the analysis by considering the time-series data in an invariant
subspace and computing the corresponding local Koopman operator. We define
conditions under which the Koopman operator must be updated as new spatial
data is added. The Koopman operator for the full state space is eventually
computed under the assumption that the spatial data is available from all the
invariant subspaces in the system. Otherwise, constructing the global Koopman
operator or identifying the global phase space is impossible without some
understanding about the underlying system (refer to the discussion in
Subsection 4.3 where the symmetry information is assumed to be known).

###### Assumption 3

Let $M_{p}\subset{\cal M}$ be an invariant subset of the phase space of the
dynamical system (1). We suppose that each invariant subset $M_{p}$ has
compact closure.

Then for the time-series data in $M_{p}$, the Koopman operator is given by
${\cal K}_{p}$. Let,

$\displaystyle{\cal E}_{p}({\Psi}_{p},n):=\max_{x_{0}\in
M_{p}}||{\Psi}_{p}(T^{n}(x_{0}))-\mathcal{K}_{p}^{n}{\Psi}_{p}(x_{0})||$

where $n$ denote the number of time-steps for which the initial condition is
evolved. The notation, ${\cal K}_{p}^{n}$ indicates that Koopman operator
${\cal K}_{p}$ is raised to the power $n$. The observable functions denoted by
$\Psi_{p}$ with an additional subscript $p$ to imply that these observables
are defined on the subspace, $M_{p}$ of the state space. Since $M_{p}$ has a
compact closure, ${\cal E}_{p}({\Psi}_{p},n)$ exists and is finite for all
$n\in\mathbb{N}$. Hence, there exists an $\varepsilon_{p}({\Psi}_{p})\geq 0$
such that

$\displaystyle\lim_{n\to\infty}{\cal
E}_{p}({\Psi}_{p},n)\leq\varepsilon_{p}({\Psi}_{p})$ (6)

The notation, ${\cal E}_{p}$ denote that the initial condition to compute this
error is chosen from the subspace $M_{p}$. In essence, for each subspace
$M_{p}$ there is a finite upper-bound for the steady-state prediction error
${\cal E}_{p}({\Psi}_{p},n).$ The magnitude of $\varepsilon_{p}({\Psi}_{p})$
indicates the fitness of the Koopman operator for the underlying time-series
data generated from a particular subspace $M_{p}$.

###### Assumption 4

The invariant subspace, $M_{p}$ is a Koopman-invariant subspace, that is, the
subspace formed by the span of observables is invariant under the action of
the Koopman operator, ${\cal K}_{p}$.

It follows from Assumption 4, $\varepsilon_{p}(\Psi_{p})=0$. Assume the time-
series data from the invariant subspace $M_{p}$ is given. Then, using one of
the DMD variants explained in Section 2 the associated local approximate
Koopman is computed from the time-series data. When new spatial time-series
data emerges, let’s assume it belongs to the subspace $M_{q}$. Now, it is
natural to wonder how $M_{q}$ is related with $M_{p}$. Furthermore, under what
conditions is a new approximate Koopman computed corresponding to the new
spatial time-series data? The following result formally specifies when the
Koopman operator must be updated.

###### Lemma 5

Let the dynamical system (1) have $v$ invariant subspaces,
$\\{M_{i}\\}_{i=1}^{v}$ such that $M_{i}\cap M_{j}=\Phi$ for $i\neq j$, where
$\Phi$ is the null set. Suppose $M_{p}\subset{\cal M}$ be a Koopman-invariant
subspace with the corresponding Koopman operator ${\cal K}_{p}$ trained on
$\Psi_{p}$ such that span of $\Psi$ is invariant under ${\cal K}_{p}$. Then,
given a set $M_{q}\subset M$ we have

${\cal E}_{q}(\Psi_{p},n)\geq{\cal E}_{p}(\Psi_{p},n).$

Proof. Depending on how the subspace $M_{q}$ is related to $M_{p}$, we have
the following two cases.

Case 1: $M_{q}\subset M_{p}$. In this case, ${\cal E}_{q}(\Psi_{p},n)={\cal
E}_{p}(\Psi_{p},n)$ since $M_{q}\subset M_{p}$ and span of $\Psi$ is invariant
under ${\cal K}_{p}$.

Case 2: $M_{q}\not\subset M_{p}$. In this case, either

  1. 1.

By Assumption 4, ${\cal E}_{q}(\Psi_{p},n)\leq{\cal E}_{p}(\Psi_{p},n)\equiv
0$ would imply that ${\cal E}_{q}(\Psi_{p},n)\equiv 0$ and in this case, the
dictionary functions $\Psi_{p}$ is valid for the subspace $M_{q}$ (or)

  2. 2.

${\cal E}_{q}(\Psi_{p},n)>{\cal E}_{p}(\Psi_{p},n)$ and in this case, a new
dictionary needs to be defined to compute ${\cal K}_{q}$ for subspace $M_{q}$.

###### Remark 6

If additional temporal time-series data is obtained in contrary to the spatial
time-series data as discussed in Lemma 4.1, then recursive least squares is
used to update the Koopman operator as shown in [39].

Hitherto we have seen how to update the Koopman operator when new spatial or
temporal time-series data emerges. In the following, we present conditions
under which a new invariant subspace is discovered using the spectral
properties of the updated approximate Koopman operator and the local
approximate Koopman operator corresponding to $M_{p}$.

###### Proposition 7

Let the subspace ${M}_{p}\subset{\cal M}$ be the smallest invariant subspace
of ${\cal M}$ and the corresponding Koopman operator is given by ${\cal
K}_{p}$. Choose ${M}_{q}$ such that ${M}_{p}\subset{M}_{q}\subseteq{\cal M}$
and denote the Koopman operator corresponding to ${\cal M}_{q}$ by ${\cal
K}_{q}$. Then the discovery of new invariant subspaces in the phase space of
the original dynamical system is such that the difference $||{\cal
K}_{q}||_{\rho}-||{\cal K}_{p}||_{\rho}$ increases monotonically where
$\parallel\cdot\parallel_{\rho}$ denote the geometric multiplicity of the
unitary eigenvalue, $\lambda=1$, that is, number of distinct eigenvectors
associated with the stable eigenvalue, $\lambda=1$.

Proof. The proof follows by noting that for a subspace ${\cal
M}_{q}\supset{\cal M}_{p}$, the inequality $\parallel{\cal
K}_{q}\parallel_{\rho}\geq\parallel{\cal K}_{p}\parallel_{\rho}$ is always
satisfied.

By continually applying the result, Proposition 7, to data across the state
space, the Koopman operator corresponding to the complete state space may be
computed. When the invariant subspaces are unknown but the data is available,
this discovery is especially valuable. In the following section, we will start
with the global phase space and show how the spectral properties of the
Koopman operator can be utilized to partition the global phase space into
invariant subspaces.

###  4.2 Invariant Phase Space Decomposition Using Global Koopman Operator

In this section, we will explore how to use the global Koopman operator to
investigate a dynamical system _locally_. For the self-containment of this
work, we recollect certain relevant concepts from [50, 54]. The findings below
are presented in terms of a finite dimensional approximate Koopman operator,
although they are universal and can be applied to infinite dimensional Koopman
operators. Interested readers may refer [50, 54] and the references contained
therein for more information.

Suppose $T:{\cal M}\to{\cal M}$ is an ergodic transformation, that is, for any
measurable set ${\cal S}\subset{\cal M}$ invariant under $T$, that is,
$T^{-1}({\cal S})={\cal S}$, then we either have $\mu({\cal S})=0$ or
$\mu({\cal S})=1$, and all the eigenvalues of ${\cal K}$ are simple (that is,
the algebraic multiplicity, $m_{a}$ of eigenvalues is 1) [50, 54]. However, if
$T$ is not ergodic, then the state space can be partitioned into subsets
${\cal S}_{i}$ (minimal invariant subspaces) such that the restriction
$T|_{{\cal S}_{i}}:{\cal S}_{i}\to{\cal S}_{i}$ is ergodic. A partition of the
state space into ergodic sets is called an ergodic partition or stationary
partition [50].

Furthermore, all ergodic partitions are disjoint and they support mutually
singular functions from ${\cal G}$ [50]. Therefore, the number of linearly
independent eigenfunctions of ${\cal K}$ corresponding to an eigenvalue
$\lambda$ is bounded above by the number of ergodic sets in the state space
[50]. Furthermore, for non-ergodic systems with attractor sets (equilibrium
points and limit cycles), the number of unit eigenvalues of $\cal K$ is equal
to the number of attractor sets of the underlying system [50]. With this we
have the following.

###### Lemma 8

Let $\lambda$ be an eigenvalue of the Koopman operator ${\cal K}$. Suppose the
algebraic multiplicity ($m_{a}$) of the eigenvalue $\lambda$, is equal to the
geometric multiplicity ($m_{g}$), $v$, then the corresponding $v$
eigenfunctions are linearly independent. Moreover the $v$ linearly independent
functions map to at most $v$ invariant subspaces in the state space.

Proof. Corresponding to the Koopman eigenvalue, $\lambda$, if $m_{a}=m_{g}=v$,
then there exists $v$ linearly independent eigenvectors111Without loss of
generality, the terms eigenfunctions and eigenvectors are used interchangeably
in this manuscript unless otherwise mentioned. (follows from standard results
on matrices [55]). The rest of the proof follows from [50].

Figure 1: Summary of the invariant phase space decomposition using the global
Koopman operator. The nonlinear system is denoted by $T$ and the spectral
properties of its corresponding Koopman operator are studied. The eigenvalues
of the Koopman operator are denoted by $\lambda$ where the unit eigenvalue
appears with $m_{a}=m_{g}=v$ and $\phi_{1},\phi_{2},\dots,\phi_{v}$ denote the
respective eigenfunctions.

The results discussed in this subsection to identify the invariant subspaces
are summarized in Fig. 1. The next section describes how the evolution of
system on the complete phase space (i.e., a global Koopman operator) can be
identified if the system evolution on invariant subspaces is known (i.e.,
local Koopman operators are known).

###  4.3 Coupling Distinct Phase Representations Assuming Approximate Koopman
Operators

Consider the dynamical system (1) and let $\\{M_{i}\\}_{i=1}^{v}$ be the
invariant sets of the dynamical system. For any $M_{p}$, let

${\Psi}_{p}(x)=\left[\psi_{1}^{p}(x),\cdots,\psi_{n_{p}}^{p}(x)\right]$

be the set of dictionary functions used for computation of the local Koopman
operator ${\cal K}_{p}$ where $\psi_{j}^{p}:{\cal M}\to\mathbb{C}$ is a
scalar-valued function for every $j\in\\{1,2,\dots,n_{p}\\}$. Then these local
Koopman operators can be combined to form a single Koopman operator which we
refer to as the stitched Koopman operator and it is given by

$\displaystyle{\cal K}_{\cal S}=\mbox{diag}({\cal K}_{1},{\cal
K}_{2},\cdots,{\cal K}_{v})$ (7)

with

$\displaystyle{\Psi}(x)=\begin{bmatrix}\chi_{1}(x){\Psi}_{1}(x)\\\
\chi_{2}(x){\Psi}_{2}(x)\\\ \vdots\\\ \chi_{v}(x){\Psi}_{v}(x)\end{bmatrix}$
(8)

where $\chi_{i}(x)$ is the characteristic function corresponding to the
(invariant) subspace $M_{i}$ and it is defined as follows

$\displaystyle\chi_{i}(x)=\begin{cases}1,&\mbox{if}\;\ x\in M_{i}\\\
0,&\mbox{otherwise}.\end{cases}$

for $i=\\{1,\cdots,v\\}$. The order of these local Koopman operators while
forming the stitched Koopman operator doesn’t matter as long as the local
Koopman operators are stacked in accordance with their corresponding
observable functions. Note that the intersection of any two invariant sets is
a set of measure zero and such sets have not been considered in the above
result. The stitched Koopman operator is a block diagonal matrix with ${\cal
K}_{\cal S}\in\mathbb{R}^{L\times L}$ where $L=\sum_{i=1}^{v}n_{i}$. The idea
behind the phase space stitching algorithm is depicted in Fig. 2.

Figure 2: Summary of phase space stitching algorithm using Koopman operators.
For every invariant subspace $M_{i}$, $i\in\\{1,2,\dots,v\\}$, the
corresponding obervable functions and the (local) Koopman operator are denoted
by $\Psi_{i},{\cal K}_{i}$. The local Koopman operators from each invariant
subspace are combined to form the stitched Koopman, ${\cal K}_{\cal S}$ with
the corresponding observables $\Psi$ using Algorithm 1.

One of the key differences between identifying the global Koopman operator as
described in Section 4.1 and 4.3 is summarized in the remark given below.

###### Remark 9

In Section 4.1, the global Koopman operator is identified by starting locally
in an invariant subspace and with new time series data, Lemma 4.1 dictates
when a new Koopman operator needs to be identified and Proposition 7 shows
when a larger invariant space is discovered. This process is repeated until
the time-series data from the entire space is seen and the corresponding
global Koopman operator has been determined. This is however different from
the (global) stitched Koopman operator (Section 4.3) which assumes knowledge
of a local Koopman operator that defines an Koopman invariant subspace for
each invariant subspace of the phase space. These results can be readily
applied for time-series data generated by multiple experiments as shown in
second-order system examples in Section 6.

So far, it has been assumed that every invariant subspace’s time-series data
is known. This assumption is relaxed in the next subsection thanks to the
knowledge of underlying dynamical system’s symmetry properties. In particular,
we explore the global phase space analysis of a class of dynamical systems
with symmetry known as equivariant systems.

###  4.4 Global Koopman Operator for Equivariant Dynamical Systems:

In this work, we show how to leverage the phase space stitching result to
compute the global Koopman for equivariant systems. We begin with the
definition of equivariant dynamical systems. algorithm for any system and in
particular the equivariant dynamical system.

######  Definition 10 (Equivariant Dynamical System)

Consider the dynamical system (1) and let $G$ be a group acting on ${\cal M}$.
Then the system (1) is called $G$-equivariant if $T(g\cdot x)=g\cdot
T(x)\;\text{for}\;g\in G.$

Koopman operator theory for equivariant systems is introduced in [40] and we
only recall a key result that is relevant to the global Koopman identification
for equivariant systems. In an equivariant system, if the finite dimensional
approximation of the Koopman operator is known in one invariant subspace, the
Koopman operator for other invariant subspaces can be identified provided the
symmetry between the invariant subspaces is known. The ensuing result
establishes this.

###### Theorem 11

Let ${\cal M}_{p}$ be an invariant set of the $G$-equivariant system (1) and
let ${\cal K}_{p}\in\mathbb{R}^{n_{p}\times n_{p}}$ be the local Koopman
operator on ${\cal M}_{p}$ with dictionary function ${\Psi}_{p}(x)$,
$x\in{\cal M}_{p}$. Suppose for $g\in G$, $g\cdot{\cal M}_{p}\subset{\cal
M}_{q}$ and ${\cal K}_{q}$ be the local Koopman operator on ${\cal M}_{q}$
with dictionary functions ${\Psi}_{q}={\Psi}_{p}$. Then

${\cal K}_{p}=\gamma{\cal K}_{q}\gamma^{-1},$

where $g\mapsto\gamma\in\Gamma$ and $\Gamma$ is the $n_{p}$ dimensional matrix
representation of $G$ in $\mathbb{R}^{n_{p}}$.

Proof. We refer the readers to [40] for the proof.

Under the assumption that the symmetry of the system is known, the local
Koopman operator in every invariant subspace of the system can be identified
starting with the knowledge of time-series data in only one invariant
subspace. This idea of computing the global Koopman for equivariant systems is
portrayed in Fig. 3. In the following, we present the phase space stitching
result as an algorithm for any system and in particular the equivariant
dynamical system.

Figure 3: Phase space stitching for equivariant systems. The symmetry
information between the invariant subspaces $M_{i}$, $i\in\\{1,2,\dots,v\\}$
is known and the local Koopman operator is known only in $M_{1}$. Theorem 11
is applied to identify the local Koopman operator in every invariant subspace
and they are now combined to form the stitched Koopman, ${\cal K}_{\cal S}$
with the corresponding observables $\Psi$ using Algorithm 1. Algorithm 1 Phase
space stitching to compute global Koopman operator:

  1. 1.

Input: For every invariant subspace ${\cal M}_{p}$, where
$i\in\\{1,2,\ldots,v\\}$, the observables $\Psi_{p}(x)$ and their
corresponding approximate Koopman operator ${\cal K}_{p}$

     * (a)

For an equivariant system: Only one invariant subspace (${\cal M}_{p}$), its
corresponding observable functions ($\Psi_{p}(x)$), the approximate local
Koopman operator (${\cal K}_{p}$) and the symmetry between the given invariant
subspace with the other invariant subspaces.

  2. 2.

Define the observables for the entire space (obtained from the union of all
invariant subspaces) and form the stitched Koopman operator

$\displaystyle{\Psi}(x)=\begin{bmatrix}\chi_{1}(x){\Psi}_{1}(x)\\\
\chi_{2}(x){\Psi}_{2}(x)\\\ \vdots\\\
\chi_{v}(x){\Psi}_{v}(x)\end{bmatrix},\qquad{\cal K}_{\cal
S}=\mbox{diag}({\cal K}_{1},{\cal K}_{2},\cdots,{\cal K}_{v})$

     * (a)

For an equivariant system, the local Koopman operators corresponding to each
invariant subspace are computed applying Theorem 11.

  3. 3.

Output: The stitched Koopman now gives the global approximate Koopman
operator.

In this section, we explored the phase space of a dynamical system using the
spectral properties of the approximate Koopman operator to discover invariant
subspaces, partition the global phase space into multiple invariant subspaces,
and fuse local Koopman operators to construct the global Koopman. However, in
the following section, we explore the phase space of one of two topologically
conjugate dynamical systems given knowledge of the phase space of the other.

##  5 Topologically Conjugate Dynamical Systems

If the global phase space has a topological conjugacy, then the global Koopman
for the topological conjugate system may be determined, as well as the
system’s characteristics such as the number of attractors. We begin with the
definition of topologically conjugate dynamical systems.

######  Definition 12 (Topological Conjugacy)

Let $T_{1}:{\cal M}\to{\cal M}$ and $T_{2}:{\cal N}\to{\cal N}$ describe two
nonlinear dynamical systems. Then $T_{1}$ and $T_{2}$ are topologically
conjugate if and only if there exists a homeomorphism $h:{\cal N}\to{\cal M}$
such that $T_{1}=h\circ T_{2}\circ h^{-1}$. Then the homeomorphism $h$ is
called a topological conjugacy between $T_{1}$ and $T_{2}$.

The following result establishes the relation between the observables of
topologically conjugate systems.

###### Theorem 13

Let $T_{1}:{\cal M}\to{\cal M}$ and $T_{2}:{\cal N}\to{\cal N}$ be two
topologically conjugate dynamical systems with the homeomorphism $h:{\cal
N}\to{\cal M}$. Let $\Psi$ be the basis function on $\cal M$, such that
$\mathbb{U}_{\Psi}$ is the representation of the corresponding Koopman
operator for $(T_{1},{\cal M})$ with respect to $\Psi$. Suppose if the
observables on $(T_{2},{\cal N})$ are $\Theta={\Psi}\circ h$, then the
representation of the Koopman operator for $T_{2}$ with respect to $\Theta$
(denoted by $\mathbb{U}_{\Theta}$) satisfies
$\mathbb{U}_{\Psi}=\mathbb{U}_{\Theta}$.

Proof. Given a $x\in{\cal M}$ and $y\in{\cal N}$, then they are related by the
homeomorphism as $x=h(y)$. Suppose $\mathbb{U}_{\Psi}$ be the Koopman operator
trained on the observables $\Psi$ defined on the state space of $T_{1}$. Then,
we have

$\displaystyle[\mathbb{U}_{\Psi}\Psi](x)$
$\displaystyle=\Psi(T_{1}(x))=\Psi(T_{1}(h(y)))=\Psi(h(T_{2}(y)))=[\mathbb{U}_{\Theta}(\Psi\circ
h)](y)=[\mathbb{U}_{\Theta}\Theta](y)$

where $\mathbb{U}_{\Theta}$ is the Koopman operator corresponding to the
system $T_{2}$ defined on the space of observables given by $\Theta=\Psi\circ
h$. Since we have $\Psi(x)=\Psi(h(y))=\Theta(y)$, we obtain
$\mathbb{U}_{\Psi}=\mathbb{U}_{\Theta}$ and hence the proof.

Figure 4: Overview of the study of topologically conjugate systems using
Koopman operator theory. The bottom sketch shows the evolution of nonlinear
system on the state space and how homeomorphic function connects the systems
$T_{1}$ and $T_{2}$. The top sketch illustrates the evolution of the system on
the space of observables given by the Koopman operator and shows the relation
between the spaces ${\cal F}_{\cal M}$ and ${\cal F}_{\cal N}$. The space
${\cal F}_{\cal M}$ and ${\cal F}_{\cal N}$ denote the space of observable
functions defined on ${\cal M}$ and ${\cal N}$ respectively.

The above theorem shows how the Koopman operators corresponding to the
topologically conjugate systems are related in the special case where the
observables are defined using the homeomorphism as shown in Theorem 13.
However, in general a natural question that arises here is to understand the
relation between the Koopman eigenvalues and Koopman eigenfunctions for the
topologically conjugate systems. The following result formalizes this
relation.

###### Proposition 14

Suppose $T_{1}:{\cal M}\to{\cal M}$ and $T_{2}:{\cal N}\to{\cal N}$ describe
two nonlinear dynamical systems that are topologically conjugate with the
homeomorphism $h:{\cal N}\to{\cal M}$. The corresponding Koopman operators on
$T_{1}$ and $T_{2}$ are given by $\mathbb{U}_{1}$ and $\mathbb{U}_{2}$. Then
the eigenvalues and eigenfunctions of $\mathbb{U}_{1}$ and $\mathbb{U}_{2}$
are in one-to-one correlation with one another via the homeomorphism $h$.

Proof. Let $\Lambda_{k}$ and $F_{k}$ respectively denote the set of
eigenvalues and set of eigenfunctions respectively for $\mathbb{U}_{k}$ where
$k=\\{1,2\\}$. Suppose $x\in{\cal M}$, $y\in{\cal N}$, and they are related by
the homeomorphism by $x=h(y)$.

Consider an eigenvalue $\lambda\in\Lambda_{1}$, and the corresponding
eigenfunction $\phi\in F_{1}$. Then,

$\displaystyle\lambda(\phi\circ
h)(y)=\lambda\phi(x)=[\mathbb{U}_{1}\phi](x)=\phi(T_{1}(x))=\phi(T_{1}(h(y)))=\phi(h(T_{2}(y)))=[\mathbb{U}_{2}(\phi\circ
h)](y)$

It now follows that $\lambda\in\Lambda_{2}$ and $\phi\circ h\in F_{2}$.
Similarly for $\bar{\lambda}\in\Lambda_{2}$ and $\bar{\phi}\in F_{2}$, it can
be shown that, $\bar{\lambda}\in\Lambda_{1}$ and $h^{-1}\circ\bar{\phi}\in
F_{1}$ and hence it can be seen that the eigenvalues and eigenfunctions for
topologically conjugate systems are in one-to-one correlation.

Note that the above result under the implicit assumption of eigenvalues being
same for the Koopman operators corresponding to topologically conjugate
systems is presented in [50]. More generally, in Proposition 14, we establish
the relation between the eigenvalues and the eigenfunctions of the
topologically conjugate systems. The ensuing result shows that the Koopman
modes for both the topologically conjugate systems remain same.

###### Proposition 15

Suppose $T_{1}:{\cal M}\to{\cal M}$ and $T_{2}:{\cal N}\to{\cal N}$ describe
two nonlinear dynamical systems that are topologically conjugate with the
homeomorphism $h:{\cal N}\to{\cal M}$. Then the Koopman modes for both the
topological conjugate systems are same.

Proof. Let $x_{0}\in{\cal M}$, $y_{0}\in{\cal N}$ and they are related to each
other through the homemorphic function, such that, $x_{0}=h(y_{0})$. Suppose
${\cal G}$ be the space of scalar-valued observable functions defined on the
state space ${\cal M}$. Let ${f}=\left[f_{1}\;f_{2}\;\ldots\;f_{m}\right]$ be
a vector-valued observable function where each $f_{i}\in{\cal G}$ for
$i\in\\{1,2,\dots,m\\}$, then ${f}$ can be expressed in terms of the Koopman
tuple as follows.

$\displaystyle{f}(x_{t})=$
$\displaystyle\sum_{j=1}^{\infty}\lambda_{j}^{t}\phi_{j}(x_{0}){\vartheta}_{j}\quad\mbox{for}\;x_{t}\in{\cal
M}$ (9)

The KMD shown in Eq. (9) can be rewritten as

$\displaystyle{f}(h(y_{t}))=$
$\displaystyle\sum_{j=1}^{\infty}\lambda_{j}^{t}\phi_{j}(h(y_{0})){\vartheta}_{j}$
$\displaystyle({f}\circ h)(y_{t})=$
$\displaystyle\sum_{j=1}^{\infty}\lambda_{j}^{t}(\phi_{j}\circ
h)(y_{0}){\vartheta}_{j}$

From Theorem 13 and Proposition 14, we know that the observables for the
topologically conjugate system are given by ${f}\circ h$ and the eigenfunction
for the topologically conjugate system is given by $\phi\circ h$. Therefore,
the Koopman modes, ${\vartheta}_{j}$ remain same for both the topologically
conjugate systems.

Note the Koopman modes in Eq. (9) is a function of the choice of the vector-
valued observables, $f$. We observe that if time-series data from one system
and its topological conjugacy are known, the global phase space of the
topologically conjugate system can be investigated, that is, the number of
invariant subspaces, attractor sets, spatiotemporal modes, and so on. It’s
important to note that the symmetry properties of an equivariant system are
shared by the system’s invariant subspaces, while topological conjugacy is
shared between separate systems. A special case arises in an equivariant
system if each of the invariant subspaces can be related by a homeomorphic
function with other invariant subspaces. We may thus claim that the invariant
subspaces are topologically conjugate in this scenario.

##  6 Simulation Study

We consider several second-order nonlinear dynamical systems in this section
and use the results from Section 4 and Section 5 to demonstrate the invariant
phase space identification, phase space stitching, global Koopman operator
computation with and without symmetric properties and phase space
identification of topologically conjugate dynamical systems. The phase space
study of the bistable toggle switch comes first.

###  6.1 Bistable Toggle Switch

The genetic toggle switch is a seminal memory device developed by Collins and
Gardner [56] to simulate binary logic and memory inside of living cells. The
design and analysis of a toggle switch model has been the subject of many
studies [57, 58, 59]. In our case, we use the toggle switch as simple example
of a two state nonlinear system with multiple equilibria and a non-trivial
invariant subspace partition. In it’s most basic form, the toggle switch
consists of two biological states, usually proteins, that repress or attenuate
each other to apply mutual negative feedback. The toggle switch’s dynamics can
be described by the following governing equations [56]:

$\displaystyle\dot{x}_{1}=$
$\displaystyle\frac{\alpha_{1}}{1+x_{2}^{\beta}}-\kappa_{1}x_{1}$ (10)
$\displaystyle\dot{x}_{2}=$
$\displaystyle\frac{\alpha_{2}}{1+x_{1}^{\gamma}}-\kappa_{2}x_{2}$

where the states $x_{1},x_{2}\in\mathbb{R}$ indicate the concentration of the
repressors $1$ and $2$. The effective rate of synthesis for repressors 1 and 2
are denoted by $\alpha_{1}$ and $\alpha_{2}$. The self decay rates of the
concentration of repressors 1 and 2 are given by $\kappa_{1}>0$ and
$\kappa_{2}>0$. The cooperativity of repression of promoter 1 and 2 are
respectively $\gamma$ and $\beta$. The bistable toggle switch is
mathematically modelled to investigate a bistable gene-regulatory network
[56]. There are two stable equilibrium points in this system (indicating
bistability). When the parameters like $\alpha_{1},\alpha_{2}$ for gene
concentrations, $x_{1},x_{2}$, are different, a monostable equilibrium is
observed [60].

Data Generation: We consider $81$ initial conditions in the global state space
and for each initial condition, system is evolved for $21$ time points. The
complete dataset is considered and the centers ($c_{i}$) of each cluster are
determined using k-means clustering. The Gaussial radial basis functions are
then defined using the centers, $c_{i}$. As indicated in Eq. (4), the related
Koopman operator is computed using the EDMD algorithm by minimizing the
residue function. The size of the dictionary is chosen to be $30$ and each
dictionary function is of the form, $\psi_{i}(x)=\exp(-{\parallel
x-c_{i}\parallel^{2}/\sigma^{2}})$ for $i\in\\{1,2,\dots,30\\}$ where $\sigma$
is chosen to be $0.4$. Figure 5 shows the phase portrait of this system with
two stable equilibrium points (i.e., two stable attractors). The equilibrium
points for the bistable toggle switch system are given by $(2,0.16)$ and
$(0.16,2)$.

Figure 5: Phase portrait of the bistable toggle switch. This system has two
stable attractors for device parameters: $\beta=3.55$, $\gamma=3.53$,
$\alpha_{1}=\alpha_{2}=1$ and $\kappa_{1}=\kappa_{2}=0.5$. Circles indicate
the initial conditions, green arrow indicate the direction of the vector field
at the initial condition and the red dots denote the equilibrium points.

Phase Space Exploration (Global to Local): The eigenvalues of the global
Koopman operator ${\cal K}$ are shown in Fig. 6 and there are two (dominant)
eigenvalues at $\lambda=1$. The eigenvectors corresponding to these
eigenvalues are linearly independent and hence, it shows that there are two
invariant subspaces on the state space (follows from Lemma 8). Moreover, these
eigenvectors associated with the unitary eigenvalues when evaluated on the
state space captures the attractors whose centers are the equilibrium points
as seen in Fig. 7.

Figure 6: Eigenvalues of the Koopman operator ${\cal K}$ corresponding to the
bistable toggle switch system given in Eq. (10).

Figure 7: (a) and (b) Eigenvectors of the Koopman operator corresponding to
the eigenvalue $\lambda=1$ on the state space. The region around the
equilibrium point can be seen inside the (blue) colored ellipses.

Phase Space Exploration (Local to Global): Assuming knowledge of the phase
space on the local invariant subspaces, local Koopman operators are computed
and the stitched Koopman operator corresponding to the global phase space is
computed using the phase space stitching result (see Algorithm 1 and Section
4.3). Recall, this stitched Koopman is equivalent to the global Koopman
operator but evolves on a different observable space. For computing each of
the local Koopman operators, namely, ${\cal K}_{left}$ and ${\cal K}_{right}$,
$30$ Gaussian radial basis functions are used. Clearly there is an eigenvalue
at $\lambda=1$ for ${\cal K}_{left}$ as well as ${\cal K}_{right}$ and their
corresponding eigenvectors on the state space reveals the attractor sets
around the equilibrium points. The stitched Koopman computed using Algorithm 1
is given by ${\cal K}_{\cal S}:=$diag$({\cal K}_{left},{\cal K}_{right})$. The
eigenvalues of the stitched Koopman operator are shown in Fig. 8. It is
important to note that the size of ${\cal K}_{\cal S}$ is $60\times 60$
whereas the size of the global Koopman (computed above), ${\cal K}$ is
$30\times 30$. Moreover the sparse structure of both of these Koopman
operators can be seen in Fig. 9.

Figure 8: Eigenvalues of the stitched Koopman operator ${\cal K}_{\cal S}$
corresponding to the bistable toggle switch system given in Eq. (10).

Figure 9: (a) Sparse structure of the Koopman operator ${\cal K}$ (b) Sparse
structure of the stitched Koopman operator ${\cal K}_{\cal S}$.

Stitched Koopman Operator Validation: It is crucial to validate the stitched
Koopman operator, ${\cal K}_{\cal S}$ if it captures the attractor sets on the
state space of the dynamical system. To validate this, the eigenvalues are
computed and their eigenfunctions corresponding to the dominant eigenvalues
are shown in Fig. 10. Clearly Fig. 10 shows that each of the attractor sets on
the state space are captured by the eigenfunctions associated with the first
two leading eigenvalues of ${\cal K}_{\cal S}$.

Figure 10: (a) and (b) Eigenvectors of ${\cal K}_{\cal S}$ associated with
dominant eigenvalues $\lambda=1$ on the state space. Eigenvectors of the
stitched Koopman operator captures both the invariant sets of the state space.

Observe that, we have now computed two global Koopman operators, one assuming
the knowledge of the entire state space to obtain ${\cal K}$ and the other by
stitching the local invariant subspaces to obtain ${\cal K}_{\cal S}$. Fig. 10
shows the attractor sets on the state space identified using ${\cal K}_{\cal
S}$ and moreover they also approximate well the invariant sets shown in Fig.
7. This validates the proposed approach of phase space stitching to compute
the global Koopman operator from local Koopman operators.

Global Koopman Operator using Symmetry Properties: Clearly, the phase space of
the bistable toggle switch consists of two invariant subspaces, $x_{1}>x_{2}$
and $x_{1}<x_{2}$ respectively and they are symmetric to each other. In
particular, these two invariant subspaces are reflective of each other. In the
scenario where the knowledge of this symmetry is known and the time-series
from an experiment is available on any of the invariant subspaces, then phase
space stitching (Algorithm 1) is invoked to identify the stitched Koopman or
equivalently global Koopman operator.

We demonstrate the global Koopman operator computation using DMD, however
additional work is needed to show this under EDMD or deepDMD. The local
Koopman operator corresponding to the region $x_{1}>x_{2}$ using the DMD is
given by

$\displaystyle{\cal K}_{right}=$ $\displaystyle\begin{bmatrix}0.6039&0.0313\\\
-0.4784&1.0375\end{bmatrix}.$

Then the Koopman operator corresponding to the reflective space $x_{1}<x_{2}$
is obtained as

$\displaystyle{\cal K}_{left}=$ $\displaystyle\gamma^{-1}{\cal
K}_{right}\gamma=\begin{bmatrix}1.0375&-0.4784\\\ 0.0313&0.6039\end{bmatrix}.$

where $\gamma$ denotes the reflective transformation between the invariant
subspaces such that
$(x_{1},x_{2})^{\top}\xmapsto{\gamma}(x_{2},x_{1})^{\top}$. The matrix
representation for $\gamma$ corresponding to the reflective symmetry is given
by $\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$. Then the global stitched Koopman
operator is given by

$\displaystyle{\cal K}_{global}=\begin{bmatrix}{\cal K}_{right}&\\\ &{\cal
K}_{left}\end{bmatrix}.$

We next demonstrate the phase space stitching results on a nonlinear second-
order system.

###  6.2 Second Order System with Bilinear and Quadratic Terms

Consider a second-order dynamical system governed by the dynamics:

$\displaystyle\dot{x}_{1}=$ $\displaystyle x_{1}-x_{1}x_{2}$ (11)
$\displaystyle\dot{x}_{2}=$ $\displaystyle x_{1}^{2}-2x_{2}$

The system (11) has 3 equilibrium points at $(\sqrt{2},1)$, $(-\sqrt{2},1)$
and $(0,0)$. It is seen that the origin is a saddle point and the other two
equilibrium points are stable.

Data Generation: The Koopman operator is obtained by training on the complete
state space data with $81$ initial conditions and for each initial condition,
$21$ time-points and is denoted by ${\cal K}$ where $30$ Gaussian radial basis
functions with $\sigma=0.4$ are used. The observable functions used for the
bistable toggle switch are used for this system as well however the centers
for the radial basis functions are computed according to the time-series data
of (11). The phase portrait of this system is shown in Fig. 11.

Figure 11: Phase portrait corresponding to the system (11). Circles indicate
the initial conditions, green arrow indicate the direction of the vector field
at the initial condition and the red dots are the equilibrium points.

Phase Space Exploration (Global to Local): The global Koopman operator, ${\cal
K}$ is computed and the eigenvalues of ${\cal K}$ are shown in Fig. 12.
Clearly the three dominant eigenvalues of ${\cal K}$ are located at
$\lambda=1$ with geometric multiplicity ($m_{g}$) equal to 1. The
eigenfunctions corresponding to these dominant eigenvalues of ${\cal K}$ are
shown in Fig. 13.

Figure 12: Eigenvalues of the stitched Koopman operator, ${\cal K}$
corresponding to the system given in Eq. (11).

Figure 13: Eigenvectors corresponding to the dominant eigenvalues of ${\cal
K}$ on state space. (a) and (b) stable equilibrium points and (c) saddle
point.

Phase Space Exploration (Local to Global): Assume there is no knowledge of the
complete phase space and the knowledge about the local invariant sets are only
known. Then, the corresponding local Koopman operators are denoted by ${\cal
K}_{left}$ and ${\cal K}_{right}$. Furthermore, the stitched Koopman operator
with respect to these local Koopman operators is given by ${\cal K}_{\cal S}$.
The eigenvalues and the sparse structure of the stitched Koopman operator when
compared to ${\cal K}$ are shown in Fig. 14 and Fig. 15 respectively.

Figure 14: Eigenvalues of the stitched Koopman operator ${\cal K}_{\cal S}$
corresponding to the system (11).

Figure 15: (a) Sparse structure of the Koopman operator ${\cal K}$ (b) Sparse
structure of the stitched Koopman operator ${\cal K}_{\cal S}$.

Stitched Koopman Operator Validation: The eigenvector plots corresponding to
the dominant eigenvalues of ${\cal K}_{\cal S}$ are shown in Fig. 16 (a) and
(b). It can be seen that the stitched Koopman operator identifies the
attractor sets in the state space.

Figure 16: (a) and (b) Eigenvectors of ${\cal K}_{\cal S}$ associated with
dominant eigenvalues $\lambda=1$ on the state space. Eigenvectors of the
stitched Koopman operator captures both the invariant sets of the state space.

Therefore, the stitched Koopman operator describes the global behavior of the
system and identifies the attractor sets of the phase space in comparison to
${\cal K}$ (which me may not have access to in real experiments).

Global Koopman Operator using Symmetry Properties: We now discuss how the
global Koopman operator can be identified if the knowledge of time-series data
from any one invariant subspace is available and under the assumption that the
type of symmetry between the invariant subspaces of the dynamical system is
known. We demonstrate the global Koopman operator computation using DMD,
however additional work is needed to show this under EDMD or deepDMD. Let’s
say we have access to the time-series data in the region
$M_{right}=\\{x\in\mathbb{R}^{2}:x_{1}>0\\}$. The local Koopman operator
applying DMD on the data is given by

$\displaystyle{\cal K}_{right}=\begin{bmatrix}0.9782&0.0253\\\
0.7755&-0.0955\end{bmatrix}.$

Let $M_{left}$ be another invariant subspace such that,
$M_{left}=\\{x\in\mathbb{R}^{2}:x_{1}<0\\}$. Under the assumption that the
invariant subspaces $M_{right}$ and $M_{left}$ has a reflective symmetry, that
is, $(x_{1},x_{2})^{\top}\xmapsto{\gamma}(-x_{1},x_{2})^{\top}$. Then it
follows from Theorem 11 that the Koopman operator corresponding to $M_{left}$
is given by

$\displaystyle{\cal K}_{left}=\gamma^{-1}{\cal
K}_{right}\gamma=\begin{bmatrix}0.9782&-0.0253\\\
-0.7755&-0.0955\end{bmatrix},$

where $\gamma=\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix}$ is the 2-dimensional
representation of the non-identity element in the symmetry group. Moreover,
the local Koopman on $M_{left}$ matches with the local Koopman computed from
time-series data of $M_{left}$. Therefore applying the phase space stitching
result (from Section 4.3) by noticing that the observables are indeed the
states, the global Koopman is given by

$\displaystyle{\cal K}_{global}=\begin{bmatrix}{\cal K}_{right}&0\\\ 0&{\cal
K}_{left}\end{bmatrix}.$

###  6.3 Topologically Conjugate Systems

In the following study, we consider two topologically conjugate dynamical
systems and with the knowledge of the topological conjugacy and one of the
systems time-series data, the evolution of the other system is identified
applying the results developed in Section 5. Let
$T_{1}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ and
$T_{2}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be two dynamical systems that are
topologically conjugate with the homeomorphism
$h:\mathbb{R}^{2}\to\mathbb{R}^{2}$ such that

$\displaystyle\dot{x}=$ $\displaystyle T_{1}(x)$ $\displaystyle\dot{y}=$
$\displaystyle T_{2}(y)$

where

$\displaystyle x=\begin{bmatrix}x_{1}\\\ x_{2}\end{bmatrix},\quad
y=\begin{bmatrix}y_{1}\\\ y_{2}\end{bmatrix},\quad
T_{1}(x)=\begin{bmatrix}-x_{1}\\\ -x_{2}+x_{1}^{2}\end{bmatrix},\quad
T_{2}(y)=\begin{bmatrix}-y_{1}\\\ -y_{2}\end{bmatrix}.$

The phase portraits for the system $T_{1}$ and $T_{2}$ are shown in Fig. 17
(a) and (b) respectively.

Figure 17: (a) Phase portrait of $T_{1}$ and (b) Phase portrait of $T_{2}$.
The dynamical systems $T_{1}$ and $T_{2}$ are topologically conjugate via the
homeomorphism $h$ as shown in Eq. (12).

The homeomorphic function between $T_{1}$ and $T_{2}$ is given by

$\displaystyle h(y)=\begin{bmatrix}y_{1}\\\ y_{2}-y_{1}^{2}\end{bmatrix}$ (12)

To demonstrate the application of proposed Koopman operator theoretic methods
for topologically conjugate dynamical systems in Section 5, we begin with the
assumption of knowledge of system $T_{2}$ and the homeomorphism $h$ and
eventually study the dynamical system $T_{1}$.

Data Generation: Time-series data corresponding to $81$ initial conditions and
for each initial condition, $21$ time-points are collected for system $T_{2}$.
This time-series data is used to compute the finite dimensional approximate
Koopman operator, ${\cal K}_{\Theta}$ using EDMD by choosing the dictionary
function as

$\displaystyle\Theta(y)=\begin{bmatrix}y_{1}\\\ y_{2}\\\
y_{1}^{2}\end{bmatrix}$ (13)

Then the finite dimensional Koopman with respect to the chosen observables,
$\Theta$ is given by ${\cal K}_{\Theta}$ as shown in Fig. 18.

Figure 18: The matrix representation of the Koopman operator ${\cal
K}_{\Theta}$ with the choice of observables $\Theta$ defined in Eq. (13).

Koopman Operator using Topological Conjugacy: Once the dictionary functions
for $T_{2}$ is fixed, the dictionary functions corresponding to the dynamical
system $T_{1}$ is chosen as $\Psi=\Theta\circ h^{-1}$, so that we have

$\displaystyle\Psi(x)=\begin{bmatrix}x_{1}\\\ x_{2}+x_{1}^{2}\\\
x_{1}^{2}\end{bmatrix}$ (14)

The time-series data corresponding to $T_{1}$ is obtained by the action of the
homeomorphic function on the time-series data on $T_{2}$. Then the
corresponding finite dimensional Koopman operator denoted by ${\cal K}_{\Psi}$
is shown in Fig. 19.

Figure 19: The matrix representation of the Koopman operator ${\cal K}_{\Psi}$
with the choice of observables $\Psi$ defined in Eq. (14).

It is clear from Figs. 18 and 19 the Koopman operators, ${\cal K}_{\Theta}$
and ${\cal K}_{\Psi}$ are same and validates the result of Theorem 13 on
topological conjugate systems under the knowledge of the homeomorphic
function.

##  7 Conclusion

Data-driven operator theoretic methods are developed for global phase analysis
and learning of nonlinear dynamical systems. When new spatial time-series data
arrives, we proposed and proved conditions on the residual error function
under which the Koopman operator must be modified. We also proposed and proved
conditions for discovering new invariant subspaces in the state space of a
dynamical system exclusively from the time-series data, based on the spectral
properties of the Koopman operator. In contrast to phase space exploration
results, when the local behavior in each invariant subspace is known, we
developed a phase space stitching result by combining the local evolutions to
predict global system dynamics. Next, we considered the scenario where full
experimental measurements are not available, e.g., some portion of the phase
space is difficult to measure or access. However, under the knowledge of
symmetry relation between the invariant subspaces, that is the system is
equivariant, we show the global model estimation is possible using the time-
series data from only one of the invariant subspace. Finally, we show that
global-to-local decompositions of the Koopman operator for a dynamical system
can also exploit topological conjugacy, to simultaneously find model
decompositions for topologically conjugate dynamical systems. When applying
our local-to-global estimation algorithms, we found we were able to
reconstruct an approximation to the global Koopman operator for two example
systems: a genetic toggle switch and nonlinear system with quadratic and
bilinear terms. Furthermore, on topologically conjugate systems, we
demonstrate that the phase space evolution of one system can be identified
with the time-series data from the other system using the topological
conjugacy. Future work will center on the development of software and
recommender algorithms for active learning, based on the theory from this
paper.

##  8 Acknowledgments

This work was supported partially by Defense Advanced Research Projects Agency
(DARPA) Grants. DEAC0-576RL01830, HR0011-12-C-0065, and FA8750-19-2-0502,
Institute of Collaborative Biotechnologies Grant, Army Research Office Young
Investigator Program Grant W911NF-20-1-0165. Any opinions, findings and
conclusions or recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of the DARPA, the Army
Research Office, the Institute of Collaborative Biotechnologies, the
Department of Defense, or the United States Government.

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