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Polygonal Approximation for Flows
A ThesisPresented to
The Faculty of the Division of Graduate Studies
by
Erik M. Boczko
In Partial Fulfillmentof the Requirements for the Degree ofDoctor of Philosophy in Mathematics
Georgia Institute of TechnologyNovember 2002
Polygonal Approximation for Flows
Approved:
Konstantin Mischaikow
Andrzej Szymczak
Thomas Morley
Andrzej Swiech
Luca Dieci
Date Approved by Chairman
Summary
The work presented in this thesis continues to emphasize a new direction in the
numerical analysis of dynamical systems. The shift in focus is accomplished by con-
sidering groups of trajectories. Briefly, using only information about the vectorfield,
a simplicial or polygonal complex is constructed that is ”aligned” with the flow. The
underlying flow induces a multivalued map or graph on the complex. The complex
and map serve to discretize the flow and essentially convert dynamics to combina-
torics. The goal of this thesis is to understand what conditions on the cells guarantee
that the induced map contains computable information about the dynamics.
Logically, this work is an extension and to some extent a completion of the work
initiated by Michael Eidenschink and Konstantin Mischaikow. In his thesis [13] Ei-
denschink introduced the idea that the Conley Index could be used to compute global
dynamical properties of flows, and developed a successfully algorithm for planar flows.
In this thesis we present results that extend the previous planar ones to arbitrary
dimensional Euclidean spaces. Also, by passing from simplicial complexes to polygo-
nal ones we circumvent the thorny issue of transversality, and isolate its consideration
to a portion of the boundary of the complex. Along these lines, we prove the result
that any strongly connected component of our polygonal complex is an isolating block.
This result ensures that we can extract Conley Index information from the induced
map.
The central result of the thesis is a statement that if each simplex in the complex
is δ-oriented with respect to the vectorfield then the complex ε approximates the
chain recurrent set. This result says that if you can lay down an oriented complex,
iii
then you can extract as much, or better said as fine, dynamical information as is
possible. Practically, this means that you can get as detailed an approximation, to
a complete Lyapunov function on the maximal invariant set under consideration, as
you are willing to pay for.
iv
Contents
Summary iii
List of Figures viii
Acknowledgements ix
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Conley Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Notation, Definitions and Preliminaries . . . . . . . . . . . . . . . . . 10
1.4 Flow Transverse Polygonal Decompositions . . . . . . . . . . . . . . . 12
2 Results 16
2.1 Strongly Connected Components are Isolating Blocks . . . . . . . . . 16
2.2 Expansion in Parallel Flow . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 δ-Oriented Simplices . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Local Expansion of the Induced Map . . . . . . . . . . . . . . 34
2.4 Global Approximation Theorem . . . . . . . . . . . . . . . . . . . . . 38
3 Computation 49
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Verifying a Given Triangulation . . . . . . . . . . . . . . . . . . . . . 51
3.3 Bistellar flips and Regular Triangulations . . . . . . . . . . . . . . . . 53
v
3.4 Voronoi Diagrams and Delaunay Triangulations . . . . . . . . . . . . 58
3.4.1 Euclidean Metric . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.1.1 The Bowyer-Watson Algorithm . . . . . . . . . . . . 60
3.4.2 Metrics Induced by a Norm . . . . . . . . . . . . . . . . . . . 61
3.4.3 Riemannian Metric . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.3.1 Anisotropic Mesh Generation . . . . . . . . . . . . . 64
3.5 Point Placement Strategies . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.1 Local Patch Scheme . . . . . . . . . . . . . . . . . . . . . . . 66
3.5.2 Global Point Placement . . . . . . . . . . . . . . . . . . . . . 69
3.6 Independent Patches Approach . . . . . . . . . . . . . . . . . . . . . 71
Bibliography 79
Vita 82
vi
List of Figures
1.1 The BIG picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 An example of a point x ∈ Rε . . . . . . . . . . . . . . . . . . . . . . 6
2.1 An internal tangency . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 An Equilateral Triangulation . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 The family of δ-oriented parallel flow complexes constructed in Example 2.23 33
2.4 Construction of an Euler path using successive cone apices. . . . . . . . 39
2.5 An E-path that is a solution up to ε . . . . . . . . . . . . . . . . . . . . 40
2.6 Estimate for Λ(λ, β, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 A triangulation that is not regular . . . . . . . . . . . . . . . . . . . . . 55
3.2 Circuits in two dimensions, and associated bistellar flips . . . . . . . . . 56
3.3 The secondary polytope of a pentagon . . . . . . . . . . . . . . . . . . 57
3.4 A (Euclidean) Voronoi Diagram and Delaunay Triangulation . . . . . . . 59
3.5 Triangle with circumcenter C, circumradius R . . . . . . . . . . . . . . 65
3.6 Quadratic approximation to affine flow . . . . . . . . . . . . . . . . . . 67
3.7 Phase portrait for the van der Pol equation . . . . . . . . . . . . . . . . 73
3.8 A strongly connected component containing a periodic orbit, computed by
the IT-algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.9 Points placed with the GZB algorithm using the curvature induced density
function, for the van der Pol oscillator . . . . . . . . . . . . . . . . . . . 75
3.10 An anisotropic Delaunay triangulation for the van der Pol oscillator, pro-
duced with the AT-algorithm and the points depicted in figure 3.9 . . . . 76
vii
3.11 Highlighted is the strongly connected component containing the period
orbit for the van der Pol oscillator, the triangulation is the same as that
shown in figure 3.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.12 The strongly connected component containing the period orbit for the van
der Pol oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
viii
Acknowledgements
Mathematics is art and beauty and, in a very real sense, it is my religion. My
understanding and appreciation of it have been facilitated by many people. Several
people have stood directly in the way. This is my one and lasting opportunity to tell
them how much they and their contributions have meant to me, and ultimately to
this document.
• Thanks GT math dept for letting me be a part of things.
• Thanks Juniors Grill: Miss Ann and Tommy for providing a homey feeling
atmosphere, and the best biscuits gravy and grits I have ever had.
• The comprehensives were an uphill struggle, but well worth it. Richard O’connell
I could not have passed them without you.
• Mona, thanks for the job and for being a friend and for the topology book I
stole from you, oops.
• Marcio Its always nice to have someone to share the pain. Thanks for all your
help and effort.
• Throughout, Russ you were always there to help me out and make me laugh,
pump me up, and listen to me complain. You are my best friend.
• Ant, you are a great mathematician and a good friend, just stop punching my
arm.
ix
• Tom, it has been a pleasure trying to figure out how you think. How does that
work, by FM of course! Thank you for your friendship.
• George Cain I could listen to your stories all day. Thanks for telling them to
me, I will never forget them or you. Your topology book is never far away.
• Bill, you are a fine mathematician. You have given me years worth of material
for therapy, for that and for your help I am truly grateful.
• Konstantin, I marvel at your intuition, I admire your dedication and vision,
and I appreciate your tutelage. I look forward, with nothing but excitement, to
working with you and learning from you in the future.
• Prof Hale, your encouragement and interest have meant the world to me.
• BFS you listened without judgment and without your help I could not have
held it together.
• Ken and Peggy, you are the best in-laws I can imagine. You have always
provided a safe comfortable retreat on the weekends or whenever we needed to
get away from it all. You have always been generous and available to help in
emergencies or whenever I needed a place to stay. I wish you had a little bigger
yard to store some sheep, but nobody’s perfect!
• Jim and Sally If I could have chosen my parents it would have been you. Your
continued friendship is among my most prized possessions.
• Sabrina you are the sun the moon and the stars.
• Debbie through thick and thin you are always there. You are my love and my
family and through good times and bad we have held it together.
x
• There is a bullpen of professors whose time I have wasted, company I have
enjoyed, advice I have squandered, and friendship I have abused. I wish to
thank you all. In roughly alphabetical order you are Professors: Chafee, Dieci,
Ghrist, Heil, Klain, Randall, Swiech, and Wang.
• Finally, I would like to thank my parents for their continued support.
xi
Chapter 1
Introduction
1.1 Motivation
There are many areas in applied science where the object of study is a large sys-
tem of ordinary differential equations. For example, in biology there is considerable
interest in understanding the dynamics of macromolecules like proteins and DNA,
at the atomic level of detail. The number of atoms in a relatively small protein in
an explicit aqueous environment is on the order of 2000. The resulting system of
differential equations would have 12,000 variables and at least a dozen parameters.
The parameters typically result from the fitting of potentials, like Leonard-Jones and
Coulomb, to spectroscopic data, and are subject to experimental error. In this area
what is typically done are numerical simulations and statistical analysis of the result-
ing time series. Inferences about the global dynamics of the system are then typically
drawn.
Because of the nonlinear reaction terms and the dimensionality, these systems are
not analytically tractable. This illustrates the point that the powerful but general
theorems in dynamical systems are often hard to apply to specific equations. From
a numerical point of view, these systems are stiff and typically time steps between
one and two pico seconds are used. The biological events of interest happen on the
order of seconds, and even with fast clusters of computers realistic simulations of
a single trajectory can take a few months. Dynamical systems theory has shown
1
that the behavior of a differential equation can only fully be understood through an
understanding of the behavior of sets of trajectories and the structure of its invariant
sets.
What is needed in applications like these is a numerical procedure whose focus is
on the structure of invariant sets rather than on individual orbits, and whose results
are robust with respect to the many layers of error and approximation involved in
the model. The Conley index is the perfect foundation on which to build such a
procedure. The focus of the Conley Index is on isolated invariant sets, the robustness
of which harmonizes perfectly with the errors introduced by numerical computation,
and Conley’s decomposition theorem makes clear the limits of what can and cannot be
realistically computed. Fast codes to compute homology and a growing list of Conley
Index theorems insure that qualitative and useful information could be extracted from
such a scheme.
The need for such a scheme, its global organization and many of its details were
provided by Eidenschink and Mischaikow [13] in their seminal work. The work pre-
sented in this thesis continues to emphasize this new direction in the numerical anal-
ysis of dynamical systems. Briefly, using only information about the vectorfield, a
simplicial or polygonal complex is constructed that is ”aligned” with the flow. The
underlying flow induces a multivalued map or graph on the complex, see figure 1.1.
The complex and map serve to discetize the flow and essentially convert dynamics to
combinatorics. The structure of the induced map recurrent sets provides information
concerning the structure of the chain recurrent set of the dynamical system. The goal
of this thesis is to understand what conditions on the cells guarantee that the induced
map contains computable information about the dynamics.
The central result of the thesis is a statement that if each simplex in the complex
is δ-oriented with respect to the vectorfield then the recurrent components of the
induced map ε-approximate the chain recurrent set. This result says that if you
2
Figure 1.1: The BIG picture
3
can lay down an oriented complex, then you can extract as much, or better said
as fine, dynamical information as is possible. Practically, this means that you can
get as detailed an approximation, to a complete Lyapunov function on the maximal
invariant set under consideration, as you are willing to pay for.
The result is a claim about sufficiency, and is probably overly restrictive, as you
can most certainly get the same result by asking far less from each simplex. However,
the result is independent of dimension and there is some hope that a general numerical
procedure can be developed that automatically produces such triangles. And most
certainly this first result will point the way to better ones.
It would have been gratifying, but too much to hope, if the algorithms and ideas
developed in this thesis would, at the time of writing, have been mature enough to
apply to the macromolecular systems that have provided the author the motivation
to pursue this thesis. Sadly they are not. There is however good reason to be
optimistic that through a better understanding of point placement algorithms and
the role of Riemannian geometry, we will be able in the near future to extend our
current algorithms efficiently to arbitrary dimensions. The reader is, of course, under
no obligation to agree!
1.2 Conley Index Theory
This thesis is not per se about the Conley Index. The results in this thesis are about
how best to discretize a flow in order to be able to extract global information about
the dynamics via the computation of Conley Indices and the application of Conley
Index theorems. Therefore it does not seem justified to provide a lengthy review of
the Conley Index theory. Detailed reviews can be found in [13, 22, 24, 23]. Here we
present those results that form the foundation for a complete exposition.
4
To be concrete, throughout, we will study a differential equation of the form
x = f(x), x ∈ Rn (1.1)
which generates a flow
ϕ : R × Rn → R
n (1.2)
We are interested in approximating the dynamics on a polyhedron X ⊂ Rn. In
applications this is typically taken to be some cube in the phase space where the
dynamics of interest, or the interesting dynamics, is known or suspected to occur.
From the dynamical systems point of view the object of interest is the structure
of the maximal invariant set in X,
Inv(X,ϕ) := x ∈ X | ϕ(R, x) ⊂ X
and our goal is to develop numerical methods to capture this information.
The first observation is that because of local and global bifurcations invariant
sets are not stable with respect to perturbations. Of course any numerical method
introduces errors and hence should be thought of as a perturbation to the system
of interest. Therefore, while it is possible to develop numerical methods to find
particular orbits, e.g. fixed points, periodic orbits, etc, or particular structures, e.g.
unstable manifolds, invariant tori, to study the general invariant set we must proceed
indirectly.
We begin with the following concept.
Definition 1.1 A compact set N ⊂ Rn is an isolating neighborhood if Inv(N,ϕ) ⊂
intN . An invariant set is an isolated invariant set if it is the maximal invariant set
of an isolating neighborhood.
The method that is being developed is intended to elucidate the structure of
isolated invariant sets. To do this it needs to be able to decompose invariant sets in
a robust manner. This leads to the following idea.
5
Definition 1.2 Let ε > 0. An ε-chain from x to y in X is a finite sequence j =
1, . . . , J of pairs
(zj, tj) ⊂ X × [1,∞) | x = z1, y = zJ , ‖ϕ(tj, zj) − zj+1‖ < ε, ϕ([0, tj], zj) ⊂ X .
The ε-chain recurrent set of X is
Rε(X) := x ∈ X | ∃ an ε− chain from x to x
The chain recurrent set is
R(X) :=⋂
ε>0
Rε(X)
ε
Figure 1.2: An example of a point x ∈ Rε
In this context Conley’s decomposition theorem [8] is as follows.
Theorem 1.3 Let Rj(X), j = 1, 2, . . ., denote the connected components of R(X).
Then there exists a continuous function V : Inv(X,ϕ) → [0, 1] such that
1. if x 6∈ R(X) and t > 0, then V (x) > V (ϕ(t, x)),
2. for each j = 1, 2, . . . there exists σj ∈ [0, 1] such that Rj ⊂ V −1(σj).
6
The chain recurrent set is not necessarily an isolated invariant set, and with nu-
merical approximations in mind, we are more interested in Rε(X) for a fixed ε > 0,
than R(X). The following is one of two important results that form the theoretical
underpinning of our approach.
Theorem 1.4 Let ε > 0 and let Rjε(X), j = 1, 2, . . . , J , denote the connected com-
ponents of Rε(X). Then,
cl(Rjε(X)
)
is an isolating neighborhood. Let M(j) := Inv(Rjε(X), ϕ). Then there exists a con-
tinuous function V : Inv(X,ϕ) → [0, 1] such that
1. if x 6∈ ∪Jj=1M(j) and t > 0, then V (x) > V (ϕ(t, x)),
2. for each j = 1, 2, . . . , J there exists σj ∈ [0, 1] such that M(j) ⊂ V −1(σj).
The sets M(j) are called Morse sets and together form a Morse decomposition of
Inv(X,ϕ). Observe that if x ∈ Inv(X,ϕ) \ ∪Jj=1M(j) then there exist two distinct
Morse sets M and M ′ such that
ω(x, ϕ) :=⋂
t≥0
cl(ϕ([t,∞), x) ⊂M
and
α(x, ϕ) :=⋂
t≥0
cl(ϕ((−∞,−t], x) ⊂M ′
With this in mind, we can also think of a Morse decomposition of Inv(X,ϕ) as a
finite collection of disjoint, compact, invariant sets indexed by a finite set J , i.e.
M(Inv(X,ϕ)) = M(j) | j ∈ J
with the added condition that one can impose a strict partial order > on the elements
of J with the property that j > k implies that there is no element x ∈ X such that
ω(x, ϕ) ⊂M(j) and α(x, ϕ) ⊂M(k)
7
Such an ordering is called an admissible order.
As will become clear in the next section, our numerical scheme is designed to
find a Morse decomposition and a discrete approximation to the associated Lyapunov
function. What remains to be discussed is how one uses the approximation to under-
stand the structure of the invariant set. This is done via the Conley index which can
be computed in terms of special isolating neighborhoods.
Definition 1.5 An isolating neighborhoodN is an isolating block if for every x ∈ ∂N ,
and every ε > 0,
ϕ((−ε, ε), x) 6⊂ N.
The immediate exit set for N is given by
N− := x ∈ N | ∀ ε > 0, ϕ((0, ε), x) 6⊂ N .
If N is an isolating block, then the Conley index of Inv(N,ϕ) is given by
CH∗(N) := H∗(N,N−).
There are a variety of results that use the Conley index to deduce information about
the structure of the dynamics of the invariant set [22, 24].
We have argued that the above approach to dynamics is well suited to numerical
computations. However a major distinction between the topological dynamics of the
Conley index theory and any computational method is that the latter is necessarily
combinatorial in nature.
With this in mind we consider the phase space to be decomposed into a finite set
of cells, P as opposed to a topological space. Furthermore, because this combinatorial
system arises via an approximation, the dynamics is generated by a multivalued map
F on this finite set. More precisely, for every cell P ∈ P, F(P ) ⊂ P. Note that we
8
allow F(P ) = ∅. To emphasize the fact that we are doing dynamics with multivalued
maps we will write
F : P−→→P
A full trajectory through P ∈ P is an bi-infinite sequence Pn | n ∈ Z satisfying
Pn+1 ∈ F(Pn) and P0 = P . The maximal invariant set of P under F is defined to be
Inv(P,F) := P ∈ P | ∃ full trajectory through P
The multivalued map acts on sets of cells in a natural way. Let A ⊂ P, then
F(A) :=⋃
A∈A
F(A).
Inductively, Fj+1(P ) := F(Fj(P )) for every P ∈ P, and we denote the total forward
image of a cell by
Fω(P ) :=
∞⋃
j=0
Fj(P ).
Turning now to the dynamics, we call P ∈ P recurrent if there exists i > 0 such
that
P ∈ Fi(P ).
We denote the set of recurrent elements by RF (P) or just simply RF when no con-
fusion can arise. We define an equivalence relation and equivalence classes on the set
of recurrent elements by
P ∼ Q ⇔ ∃ i, j > 0 such that P ∈ Fi(Q) and Q ∈ Fj(P ) (1.3)
Often we will refer to an equivalence class as an strongly connected component, this
language comes from graph theory and emphasizes the close connection between the
multivalued map and directed graphs. As will be described in the next section, in our
approximation scheme, each equivalence class will lead to an isolating neighborhood
9
of a Morse set. To obtain an approximate Lyapunov function, we can choose any
function W : P → [0, 1] which satisfies the following property.
Q ∈ F(P ) ⇒ W (P ) ≥ W (Q) and W (P ) = W (Q) ⇔ P ∼ Q. (1.4)
Because the polyhedral set we study is not necessarily nor typically an invariant
set, given a typical point x ∈ X, there is no reason to believe that its entire forward
trajectory lies in X. Therefore, define
τx := max t ≥ 0 | ϕ([0, t], x) ⊂ X
1.3 Notation, Definitions and Preliminaries
We begin some definitions concerning simplicial complexes. While most of these
definitions are standard [25] we include them for the sake of completeness.
Let a0, . . . , ak be an affinely independent set in Rn. The k-dimensional simplex
K spanned by a0, . . . , ak is
K :=
x ∈ Rn | x =
k∑
i=0
tiai, where
n∑
i=0
ti = 1 and ti ≥ 0
The interior of K is defined by
int(K) :=
x ∈ Rn | x =
k∑
i=0
tiai, where
k∑
i=0
ti = 1 and ti > 0
.
The barycenter of K is
b(K) :=1
k + 1ai
k∑
i=0
Definition 1.6 A simplicial complex K in Rn is a collection of simplicies in Rn
satisfying:
1. Every face of a simplex of K is in K.
10
2. The intersection of any two simplicies of K is a face of each of them.
Set
K(l) := K ∈ K | dimK = l
The dimension of the simplicial complex K is determined by its highest dimensional
simplex. More precisely,
dimK := maxK∈K
dimK.
We will employ the following terminology. Given a simplex, or a polytope of
dimension d, we will refer to the d− 1 dimensional faces as facets, and a proper face
refers to a nonempty face of dimension no greater than d− 2.
For the purpose of this work we are only interested the following special class of
simplicial complexes.
Definition 1.7 A simplicial complex K in Rn is full if every simplex in K is a face
of an n-dimensional simplex in K.
Given a full simplicial complex K in Rn, K ∈ K(n−1) is a free face if there exists
a unique L ∈ K(n) such that K is a face of L.
Definition 1.8 Let K be a full simplicial complex. Let |K| be the subset of Rn given
by the union of the simplicies in K. |K| is the polytope of K. A polygon is a subset of
Rn that is the polytope of a full simplicial complex.
Definition 1.9 Let P = |P| be a polygon. The boundary of P is
∂P :=K ∈ P (n−1) | K is free
and the boundary of the polygon is
∂P := |∂P|.
11
Definition 1.10 Let X ⊂ Rn be the polytope of the full finite simplicial complex K.
A polygonal decomposition of X consists of a finite collection of polygons P1, . . . PNsuch that each polygon Pi is the polytope of a simplicial complex Pi ⊂ K,
X =
N⋃
i=1
Pi,
and dim(Pi ∩ Pj) ≤ n− 1 for all i 6= j.
1.4 Flow Transverse Polygonal Decompositions
We are interested in the relationship between polygons and the vector field of our
differential equation (1.1) restricted to a polygonal region X. Throughout this section
X = |K| where K is a full finite simplicial complex. Our first step will be to use K to
construct a polygonal decomposition of X which is compatible with the vector field
f .
Given K ∈ K(n−1), let
ν(K)
denote one of the two unit normal vectors to K. To determine a unique choice of
ν(K), let L ∈ K(n) such that K is a facet of L. Then,
νL(K)
is defined to be the inward unit normal of K with respect to L.
Definition 1.11 K ∈ K(n−1) is a flow transverse facet, if
ν(K) · f(x) 6= 0
for every x ∈ K. Let P = |P| be a polygon where P ⊂ K. P is a flow transverse
polygon if every K ∈ ∂P is flow transverse.
12
Let P = |P| be a polygon. Observe that if K ∈ ∂P, then there exists a unique
KP ∈ P (n) such that K is face of KP . K is an exit face of P if
νKP(K) · f(x) < 0 ∀ x ∈ K.
Define
P− := K ∈ ∂P | K is an exit face of P (1.5)
The following result follows directly from the definition of flow transversality and
the fact that simplices are compact.
Lemma 1.12 If P is a flow transverse polygon, then f(x) 6= 0 for all x ∈ ∂P .
Definition 1.13 Let P = P1, . . . , PN be a polygonal decomposition ofX. Let Pi =
|Pi|. P is a flow transverse polygonal decomposition of X if the following condition is
satisfied. If Pi is not flow transverse and K ∈ ∂Pi such that ν(K) · f(x) = 0 for some
x ∈ K, then |K| ⊂ ∂X.
Observe that in the above definition, every polygon is flow transverse, except,
perhaps, those which share an n − 1 dimensional simplex with the boundary of X.
Furthermore, these polygons which intersect the boundary are also flow transverse
with the possible exception of their n−1 dimensional faces which lie on the boundary
of X. The motivation for this definition is that while we can control the structure of
the polygons within X, the boundary of X is fixed and therefore for such points we
have no control on the flow transversality or lack thereof.
In practice we generate polygonal decompositions in a natural way. One starts
with a simplicial complex and agglomerates together any adjacent simplices whose
common facet is not flow transverse. The end result is a polygonal decomposition
where the boundary facets of the polygons are by definition flow transverse, save
perhaps the ones on the boundary of the entire complex. The following definition
formalizes this procedure.
13
Definition 1.14 Starting with any simplicial complex K and any vector field f there
is a minimal flow transverse polygonal decomposition of X denoted by P(K, f) which
is defined as follows. Let Ki, Kj ∈ K(n) such that Ki∩Kj = L ∈ K(n−1). Set Ki ∼ Kj
if ν(L) · f(x) = 0 for some x ∈ L or if i = j. Extend this relation by transitivity.
Then ∼ is an equivalence relation on K. Define P(K, f) = P1, . . . , PN to be the
polygons defined by the equivalence classes. We shall let Pi ⊂ K be the simplicial
complex such that Pi = |Pi|.
Our goal is to use P(K, f) to approximate the dynamics of the flow ϕ restricted
to X. To do this we will use the following definitions.
Definition 1.15 Two distinct polygons Pi, Pj ∈ P(K, f) are adjacent if they share
an n− 1 dimensional simplex, i.e. if
∂Pi ∩ ∂Pj ∩ K(n−1) 6= ∅.
Definition 1.16 Let Pi, Pj ∈ P(K, f) be adjacent polygons and let K ∈ ∂Pi∩∂Pj ⊂K(n−1). Pi is in the image of Pj if
νPi(K) · f(x) > 0
for x ∈ K.
The multivalued or flow induced map is the most important object of study. In
practice and in its definition it is a combinatorial object. We define polygons that
contain equilibria to be recurrent, notice that the on the complement of these polygons
the magnitude of the vectorfield is bounded from below thus can be normalized, this
will be of importance later. Also notice that if a polygon contains an equilibrium
then because of lemma 1.12 it must be in the topological interior of the polygon.
The flow induced multivalued map Fε : P(K, f)−→→P(K, f) is defined as follows.
Pi ∈ Fε(Pi) if and only if ∃x ∈ Pi such that |f(x)| < ε
14
and
Pj ∈ Fε(Pi) if and only if Pj is in the image of Pi.
In what follows, we will assume that ε is fixed and so to simplify the notation we will
write F = Fε.
15
Chapter 2
Results
2.1 Strongly Connected Components are Isolating
Blocks
In this section we briefly describe some of the relationships between the induced map
on polygons F, and the flow ϕ generated by the differential equation (1.1). On the
interior of every facet of P(K, f) that is not at the boundary of the complex, flow
transversality completely defines the relationship between f and F. At an interior
point of a flow transverse facet the flow can not be tangent to the facet, it must
cross one way or the other. What is more subtle and potentially problematic is what
happens at points that represent lower dimensional intersections between adjacent
polygons P and Q. Flow transversality alone cannot rule out a tangency in the flow,
see for instance Figure 2.1. However, what can be shown is that if we have a strongly
connected component S of the multivalued map F on P(K, f) then there can be no
internal tangencies along the boundary ∂S.
This important result shows that strongly connected components are isolating
blocks and therefore they contain information about the dynamics that can be ex-
tracted via a homology computation. This result is important for another reason, it
implies that we need only know the behavior of the vectorfield on facets in order to
create a well defined induced map on polygons. From a computational and combina-
torial point of view this is very important. In earlier work, Eidenschink [13] defined
16
A R
PQ
Figure 2.1: An internal tangency
the multivalued map inductively, by the behavior at all faces. In higher dimensions
this would be very inefficient and computationally expensive.
Throughout this section let x0 lie at the boundary of some polygon that is not at
the boundary of the complex P(K, f). Further we assume that this point x0 does not
lie in the interior of some facet of the complex K, but rather in the interior of some
proper face K0 of dimension 0 ≤ l ≤ (n − 2). Consider a small ball Bρ(x0) where ρ
is chosen small enough so that for any K ∈ K if K ∩ Bρ(x0) 6= ∅ then x0 ∈ K.
For notational convenience, we will take x0 = 0; in the general case the hyperplanes
H discussed below should be replaced by affine spaces x0 +H.
By flow transversality, K0 and f(x0) determine an (l + 1)-dimensional space W .
Let H∗ denote the span of W⊥ and f(x0), which implies codimH∗ = l. Consider
any facet L ∈ Kn−1 that contains x0, then it necessarily contains K0, and let H
denote the codimension-1 hyperplane determined by L. Then H is transverse to H∗
since f(x0) /∈ H by flow transversality. Thus codim(H ∩ H∗) = l + 1. Also, for
Sρ := ∂Bρ(x0) define
S∗ρ := Sρ ∩H∗
which is homeomorphic to an (n − l − 1)-dimensional sphere. Therefore, slicing the
polygonal complex with the hyperplane H∗ and looking locally in Bρ(x0), we have
17
the following Lemma.
Lemma 2.1 The (n− l − 1)-dimensional cellular complex
S∗ρ :=
K ∩ S∗
ρ | K ∈ star(x0) ∩ K(n)
is a triangulation of the sphere S∗ρ which is in 1-1 correspondence with star(x0)∩K(n).
To understand the dynamics through the complex in Bρ(x0) consider the projec-
tion of f(x) onto the tangent space at x of Sρ. This is given by
fρ(x) = f(x) − (f(x) · r(x))r(x)
where r(x) = (x− x0)/ρ. Notice that ‖r(x)‖ = 1.
Observe that for ρ small enough, the flow in Bρ(x0) is nearly parallel. For the
constant (and hence parallel) flow, x = f(x0), the projected vector field f(x0) −(f(x0) · r(x))r(x) has exactly two critical points which are located at the poles x0 ∓ρf(x0)/‖f(x0)‖ where the function
V (x) = −r(x) · f(x0)
attains its maximum and minimum values of ∓‖f(x0)‖ on Sρ. Furthermore, V (x) is
a Lyapunov function on Sρ. The following Lemma formalizes how these properties
are preserved for fρ on Sρ with ρ small.
Lemma 2.2 For every δ > 0, there exists ρ0 > 0 such that for 0 < ρ < ρ0:
1. if x ∈ Sρ is a critical point of fρ, then ‖f(x0)‖ − |V (x)| < δ, and
2. if x ∈ Sρ \ x | ‖f(x0)‖ − |V (x)| < maxδ, δ∗, then
−fρ(x) · f(x0) ≤ −δ < 0
where δ∗ = 2δ/‖f(x0)‖.
18
Proof. First choose α > 0 so that ‖f(x0)‖ − |y| < δ whenever ‖f(x0)‖2 − y2 < α
and |y| ≤ ‖f(x0)‖. By continuity, ρ0 can be chosen so that
‖f(x) − f(x0)‖ < minα/2‖f(x0)‖, δ
and
|f(x) · r(x) − f(x0) · r(x)| ≤ ‖f(x) − f(x0)‖ ‖r(x)‖ < minα/2‖f(x0)‖, δ
for all ρ < ρ0. If x is a critical point of fρ, then f(x) = (f(x) ·r(x))r(x) which implies
‖(f(x0) · r(x))r(x) − f(x0)‖ ≤ ‖(f(x0) · r(x))r(x) − (f(x) · r(x))r(x)‖
+‖f(x) − f(x0)‖ < α/‖f(x0)‖.
Taking the dot product with f(x0) gives ‖f(x0)‖2 − (r(x) · f(x0))2 < α which by the
choice of α implies (i).
Moreover,
−fρ(x) · f(x0) = −f(x) · f(x0) + (f(x) · r(x))(f(x0) · r(x)).
Again ‖f(x) − f(x0)‖ < δ implies −f(x) · f(x0) < −‖f(x0)‖2 + δ, and |f(x) · r(x) −f(x0) · r(x)| < δ implies |f(x) · r(x)| < |f(x0) · r(x)| + δ. Hence
|f(x) · r(x)| · |f(x0) · r(x)| < (|f(x0) · r(x)| + δ)|f(x0) · r(x)|
< (‖f(x0)‖ − δ + δ)(‖f(x0)‖ − δ∗)
< ‖f(x0)‖2 − 2δ
where the second inequality follows from
x ∈ Sρ\x | ‖f(x0)‖ − |V (x)| < maxδ, δ∗.
Therefore −fρ(x) · f(x0) < −‖f(x0)‖2 + δ + ‖f(x0)‖2 − 2δ = −δ, which proves (ii).
The last statement says that V is a Lyapunov function for the flow x = fρ(x) on
Sρ\x | ‖f(x0)‖ − |V (x)| < maxδ, δ∗.
19
Lemma 2.3 There exists a ρ1 > 0 such that for all 0 < ρ < ρ1 we have
1. S∗ρ is transverse to the flow f ∗
ρ on S∗ρ ,
2. the flow induced multivalued map F∗ρ : S∗
ρ−→→S∗
ρ is locally equivalent to the flow
induced map F.
Proof. Let f ∗(x) = PH∗f(x) be the projection of f(x) onto the hyperplane H∗.
Since f(x0) ∈ H∗, we can apply Lemma 2.2 to the vector field f ∗ on H∗ and consider
the vector field f ∗ρ on S∗
ρ . Any (n− l−2)-dimensional face L∗ in S∗ρ is the intersection
of S∗ρ with a hyperplane H∩H∗ where H is determined by a facet as described above.
For any point x ∈ L∗, the vector field f ∗(x) /∈ H by flow transversality, and hence
f ∗ρ (x) = f ∗(x) − (f ∗(x) · r(x))r(x) /∈ H since r(x) ∈ H.
Definition 2.4 Let P ∈ P(K, f) with x ∈ ∂P and let F := K ∈ K(n−1)∩P | x ∈ Kbe the collection of facets containing x. We say that P is field enclosing for f at x if
f(x) · νP (K) > 0
for every K ∈ F .
Lemma 2.5 Let x0 ∈ intK ∩ intX where K ∈ Kl with 0 ≤ l ≤ (n− 2). Then there
exist unique elements A and R in P(K, f) such that A is field-enclosing for f at x0
and R is field enclosing for −f at x0. Moreover, for every P ∈ P(K, f) such that
P ∩Bρ(x0) 6= ∅, A ∈ Fω(P ) and P ∈ Fω(R).
Proof: The existence of the polyhedra A and R with the stated enclosure properties
follows immediately from flow transversality. The nontrivial assertion of the lemma
is that A ∈ Fω(P ) for every P ∈ P(K, f) such that P ∩ Bρ(x0) 6= ∅.Then applying that result to the backward flow, given by the vector field −f ,
and noting that K1 ∈ Fω(K2) iff K2 ∈ (−F)ω(K1), we obtain K ∈ Fω(R) for every
K ∈ star(x0) ∩ K(n).
20
First, we choose ρ > 0 small enough so that Lemma 2.3 applies. Let R∗ = R∩ S∗ρ
and A∗ = A ∩ S∗ρ . For any ρ > 0, R∗ and A∗ contain sectors around ±f(x0) of fixed
angular size, i.e. by choosing δ small enough, x ∈ S∗ρ : ‖f ∗(x0)‖ − |V (x)| > δ ⊂
int(R∗ ∪ A∗). Then, by Lemma 2.2, we have that V (x) is a Lyapunov function on
S∗ρ\ int(R∗ ∪ A∗).
Step 1: If ϕ ∈ S∗ρ\A∗, then minx∈ϕ V (x) is attained at a vertex of ϕ.
The set Hmin = x ∈ H∗ : V (x) = minx∈ϕ V (x) is a (d− l− 1)-dimensional affine
space of the form xmin + spanf(x0)⊥. Futhermore, Hmin is tangent to the sphere
S∗ρ only at the global minimum of V which is attained at the pole in A∗, and thus for
ϕ 6= A∗, we have that Hmin intersects S∗ρ transversely.
IfHmin∩int(ϕ) 6= ∅ then ϕmust contain points for which V is less than minx∈ϕ V (x),
and hence Hmin ∩ ϕ ⊂ ∂ϕ. The set ∂ϕ is composed of the intersection of S∗ρ with
(d− l− 1)-dimensional hyperplanes. If Hmin ∩ϕ does not contain a vertex, then Hmin
and ϕ intersect at a point on int(η) for some (d− l − 2)-dimensional η ∈ ∂ϕ. Then,
Hmin ∩ int(ϕ) 6= ∅, which we have already concluded is impossible.
Step 2: If ϕ ∈ S∗ρ\A∗, then there exists ψ ∈ (F∗
ρ)ω(ϕ) such that minx∈ψ V (x) <
minx∈ϕ V (x).
Let v ∈ S∗ρ be a vertex at which V (x) attains its minimum in ϕ. Since V (x)
decreases in the direction f ∗ρ (v), the element ϕ cannot be field-enclosing at v. By
flow transversality, there exists ψ1 ∈ F ∗ρ (ϕ) such that v ∈ ψ1. If minx∈ψ1
V (x) <
minx∈ϕ V (x), then we have proven the claim, otherwise minx∈ψ1V (x) = minx∈ϕ V (x),
which is again attained at v. In this case, ψ1 cannot be field-enclosing at v, and we
can repeat the previous step to obtain ψ2 ∈ F∗ρ(ψ1), which implies ψ2 ∈ (F∗
ρ)2(ϕ).
Since there are only finitely many elements in S∗ρ , this process must terminate and
yield ψ ∈ (F∗ρ)ω(ϕ) such that minx∈ψ V (x) < minx∈ϕ V (x).
21
Step 3: If ϕ ∈ S∗ρ\A∗, then A∗ ∈ (F∗
ρ)ω(ϕ).
If ϕ = A∗, then A∗ ∈ Fω(A∗), and hence we assume that ϕ 6= A∗. From Step 2
we can find ψ1 ∈ (F∗ρ)ω(ϕ) such that minx∈ψ1
V (x) < minx∈ϕ V (x). If ψ1 6= A∗, then
we can repeat the previous step to obtain ψ2 ∈ F∗ρ(ψ1), which implies ψ2 ∈ (F∗
ρ)2(ϕ),
and minx∈ψ2V (x) < minx∈ψ1
V (x). Since there are only finitely many elements in S∗ρ ,
this process must terminate, at which point A∗ = ψ ∈ (F∗ρ)ω(ϕ).
Step 3 immediately implies the desired result by the correspondences between the
maps F and F∗ρ and the sets star(x0) ∩ K(n) and S∗
ρ in Lemma 2.3.
The following theorem shows that the multivalued map provides an outer bound
for flow. How good an approximation the map is to the flow depends on how fast the
multivalued map expands in the directions orthogonal to the flow. This is the subject
of subsequent sections, where it is shown that if we impose certain natural orientation
and scale conditions on the cells of the complex then we can in fact expect that the
multivalued map expands linearly rather than exponentially fast, at least locally.
Theorem 2.6 Let x0 ∈ X. Let P ∈ P(K, f) such that x0 ∈ P . Then
ϕ(x0, (0, τx0)) ⊂ int(|Fω(P )|)
Proof. We will show that for any t ∈ [0, τx0], if ϕ(x0, t) ∈ |Fω(P )| ∩ int(X), then
there is an ε > 0 such that t + ε ∈ [0, τx0] and ϕ(x0, [t, t + ε]) ⊂ |Fω(P )|. The result
then follows by considering τ = inft ∈ [0, τx0] | ϕ(x0, t) 6∈ |Fω(P )|. If τ exists, then
ϕ(x0, τ) ∈ ∂X.
To prove the above claim, we consider three cases. First, if ϕ(x0, t) ∈ int(P ′)
for some P ′ ∈ Fω(P ), then the desired ε exists since int(P ′) is open. Second, if
ϕ(x0, t) ∈ int(K) for some facet K, then by flow transversality, there exist elements
Pi and Pj in P(K, f) such that K ∈ ∂Pi ∩ ∂Pj and Pi ∈ F(Pj). By definition, there
22
exists ε > 0 such that ϕ(x0, [t, t + ε]) ⊂ Pi. We are assuming that ϕ(x0, t) ∈ |Fω(P )|so that either Pi ∈ Fω(P ) or Pj ∈ Fω(P ), which implies Pi ∈ Fω(P ).
The remaining case is that ϕ(x0, t) ∈ int(K) for some proper face K 0 ≤ l :=
dim(K) ≤ (n − 2). We are assuming that ϕ(x0, t) ∈ |Fω(P )|, which implies that
ϕ(x0, t) ∈ P ′ for some P ′ such that K ⊂ P ′ and K ∈ star(ϕ(x0, t)) ∩ Fω(P ). Let
A ∈ P(K, f) be the element which is field-enclosing at ϕ(x0, t). Then, there exists
ε > 0 such that ϕ(x0, [t, t + ε]) ⊂ A. By Lemma 2.5, we have A ∈ Fω(P ′) ⊂ Fω(P ),
which completes the proof.
We now have all the ingredients to prove the theorem we have been aiming for,
and that is to show that strongly connected components of the multivalued map are
in fact isolating blocks. To prove this result we need to rule out internal tangencies.
Since these cannot occur in the interior of boundary facets, if they happen they must
occur along lower dimensional proper faces.
Theorem 2.7 Let S ⊂ P(K, f) ∩ int(X) be a strongly connected component of F,
then |S| is an isolating block for the flow.
Proof. Let x ∈ ∂S be such that there exists an interval of time I := (−ε, ε) such
that ϕ(I, x) ⊂ S. In other words the orbit through x is tangent to the boundary of
S and does not leave immediately in either forward or backward time, see Figure 2.1.
Let Q := star(x) and let Qout be those polygons in the star that do not belong to S.
They cannot all belong to S otherwise the point x cannot be on the boundary of S.
Let Qin be the rest. Now the polygons A and R guaranteed by Lemma 2.5 cannot
belong to the latter set.
Suppose they did. Then by definition they are both in S. Let K ∈ Qout, then by
Lemma 2.5 K ∈ Fω(R) and A ∈ Fω(K) and since A and R are in S we have that
R ∈ Fω(A). But taken together, these imply that K ∈ S, a contradiction.
23
So now suppose that A ∈ Qout. There exists a sequence ai∞1 ⊂ int(A) such that
lim ai = x. There also exist an ε > λ > 0 and δ > 0 and a polygon P ∈ Qin such that
B(ϕ(λ, x), δ) ⊂ int(P ). Because the flow is Lipschitz in X we have continuity with
respect to initial data which implies that there exists an N > 0 such that for n > N
we have |ϕ(λ, an) − ϕ(λ, x)| < δ, which implies that ϕ(λ, an) ∈ int(P ). Now, by
Theorem 2.6 we have that P ∈ Fω(A). But this implies that A ∈ S, a contradiction.
The same argument works for R by reversing the direction of time. These contra-
dictions imply that there cannot be an internal tangency at the boundary of S, and
completes the proof.
2.2 Expansion in Parallel Flow
In this section we document some results that were developed prior to developing the
notion of a δ-oriented simplex, which plays the prominent role in the local and global
approximation theorems to follow. This section deals with a simple, strictly planar
result. The ideas, while not general, pointed the way. What was clear is that in order
for the forward image complex of a given cell to ”shadow” the flow but not expand
exponentially fast in the orthogonal directions, the cells of the complex had to posses
some regularity properties.
In what follows we need the notions of conical and convex combinations. One
relationship between cones and flows is that locally around ordinary points every
continuous vectorfield must lie in a cone, this is a trivial consequence of continuity.
Definition 2.8 A set S is a cone if
x ∈ S ⇒ λx ∈ S ∀λ ≥ 0
24
A set S is convex if
x, y ∈ S and 0 ≤ λ ≤ 1 ⇒ λx + (1 − λ)y ∈ S
The example S = (x, y) : xy ≥ 0 shows that not every cone is convex.
Definition 2.9 Let S ⊂ Rn. The cone generated by S is
Cone(S) := λx ∈ Rn : x ∈ S and λ ≥ 0
Definition 2.10 Let S ⊂ Rn and let x ∈ R
n \ S. The cone generated by x and S
is
Cone(x, S) := Cone( y − x : y ∈ S )
If the set S is convex then so are the sets Cone(S) and Cone(x, S). If the set S
is taken to be a disk that lies in a hyperplane perpendicular to the vector f, and a
point is selected along a line parallel to f through the center of the disk, then one
gets the following more familiar cone.
Definition 2.11 Let x ∈ Rn, f ∈ Sn−1 and ε > 0. The right circular cone oriented
along the f direction is denoted by
C(f, ε) := x ∈ Rn : x · f ≥ ε‖x‖
The affine version is denoted by
C(x, f, ε) := x + C(f, ε)
In general a set that is closed under finite positive linear combinations is a convex
cone. Given a finite set of vectors F, one can consider the smallest convex set or
convex cone that contains F .
25
Definition 2.12 Given a finite set of vectors F = v1, v2, . . . , vk ⊂ Rn we denote
the polyhedral cone generated by F as
(v1, v2, . . . , vk) := ∑
i
λivi : λi ≥ 0
We often use the standard notation [v0, v1, . . . , vn] for the convex hull of a finite
point set.
It is convenient, although perhaps not fundamental, to define a metric whose level
sets are long and skinny in the direction of the parallel flow. The eigenvalues are
chosen so as to make an isosceles triangle in the direction of the flow equilateral,
alternatively and more to the point, it relates a line of slope ε with an angle of π/3.
While the notion of a parallel flow metric is not essential to prove the result of this
section, these metrics are an essential part of our computational algorithm.
In this short section we are considering the parallel flow in the e1 direction in the
plane.
x = 1
y = 0
Definition 2.13 Define the parallel flow metric to be
Θ(ε) :=
1 0
0 13ε2
Remark. We will immediately abuse notation and drop the ε dependence to min-
imize notational clutter. Θ(ε) = Θ. The matrix Θ induces a norm equivalent to the
standard Euclidean norm through an inner product in a natural way. The angle θ
between any two vectors u and v is defined to be
cos(θ) =< u,Θv >√
< u,Θu >< v,Θv >
For the remainder of this section, all angles are measured relative to Θ.
26
Theorem 2.14 (Equilateral Enclosure) Let K be any equilateral triangulation,
with respect to Θ(ε), with the property that all the triangles in K have diameter d 1,
then for all K ∈ K there exists a vertex x such that Fω(K) ⊂ C(x, e1, 3ε).
Proof. An equilateral triangle with respect to the metric Θ must have all angles
equal to π/3. An equilateral triangulation is uniquely determined therefore, by a set
of three lines whose angles with respect to each other are exactly π/3, a choice of
edge length, and the placement of a single vertex. A given equilateral triangulation is
constructed by parallel translation of a ”fundamental” set of three lines, and at each
vertex of such a complex this ”fundamental” set of three lines meet.
Given the slope m, of one line, we can solve for the slopes of the other two lines
that determine a ”fundamental” set.
Edgeslopes = m, mε− 3ε2
ε+m,mε+ 3ε2
ε−m (2.1)
By symmetry we need only consider m ∈ (−ε, ε]. Let us first consider m ∈ (0, ε).
Letting n := mε−3ε2
ε+m, and p := mε+3ε2
ε−m we have the following inequalities:
0 < m < ε
ε < |n| < 3ε
3ε < p
(2.2)
The slope n is negative and all the slopes are nonzero which imply that each
triangle is transverse to the parallel flow. For any given triangle σ order the vertices
by their y-coordinate. There must be a middle vertex in this ordering. Call this
vertex v0. The triangle σ must be either field enclosing at v0 or reverse field enclosing
at v0. Now we argue that in σ the edges incident to v0 have slopes m and n.
Because n is the only negative slope the only possibilities are m,n or p, n.Suppose at v0 the edges incident have slopes p, n. But at each vertex of the complex
K three lines pass through that vertex. So at v0 we have a line L of slope m that
27
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
x x
x
x
m
n
p⊕
Figure 2.2: An Equilateral Triangulation
passes through v0 and this line must have a segment that lies entirely inside the
triangle σ, since its slope is between n and p, the slopes of the edges at v0. But this
segment of L forms the edge of some simplex in K. This contradicts the fact that Kis a simplicial complex. Hence, each simplex σ ∈ K has a field enclosing vertex at
which the slopes of the incident edges are exactly m and n, both of which have slope
less than 3ε in absolute value.
Each simplex either field encloses or reverse field encloses. We can label each sim-
plex in the complex K as positively oriented, shown as ⊕ in Figure 2.2, or negatively
oriented, shown as in Figure 2.2, respectively. Furthermore positively and nega-
tively oriented simplices are adjacent to each other across an edge with slope p. Let
l be the length of the medians. Suppose that we start a trajectory at a point x0 that
sits inside a positively oriented simplex σ. Let v0 be the field enclosing vertex of σ.
The forward image Fω(σ) must be contained within the cone generated by the vectors
v0 + me2, v0 + ne2 since the flow is moving to the right across the corresponding
rays. But the above cone is certainly contained in the cone generated by the vectors
v0 + 3εe2, v0 − 3εe2.The proof for m ∈ (−ε, 0) is symmetric. The proof for m = ε and m = 0 follow
by an explicit computation and the same logic.
28
2.3 Orientation
Let σ = [v0, v1, . . . , vn] be an n-simplex in Rn. We denote by
σi := [v0, v1, . . . , vi−1, vi+1, . . . , vn] (2.3)
the facet opposite the vertex vi. Let H be the unique hyperplane containing σi, and
let νi denote the unit inward normal that has the property p ∈ H ⇒ ηi · (vi− p) > 0.
Let f : Rn → R
n.
Definition 2.15 We say that the facet σi is an infacet with respect to the vectorfield
f if there exists a point x ∈ σi such that f(x) · νi ≥ 0.
We will occasionally require the following notation. Let IF(σ) denote the set of
infacets of the simplex σ.
The important thing to notice is that an infacet may not be transverse to the flow,
that is there may exist a point in the facet at which the vectorfield lies in the facet.
However, the opposite of an infacet is a flow transverse exit face of σ.
At each vertex of a simplex the incident edges form a cone. The most natural and
intuitive notion of a simplex oriented with respect to the flow begins with the idea of
a field enclosing vertex.
Definition 2.16 Given a n-simplex σ = [x0, x1, . . . , xn]
(xi)+ := ( xj − xi | j 6= i)
and
(xi)− := ( xi − xj | j 6= i)
29
Definition 2.17 We say that a simplex σ is field enclosing at a vertex xi if
f(xi) ∈ (xi)+
and reverse field enclosing if
f(xi) ∈ (xi)−
Definition 2.18 Given a n-simplex σ let
(σ)+ =⋃
i
(xi)+
(σ)− =⋃
i
(xi)−
(σ) = (σ)+ ∪ (σ)−
In the plane, for any non degenerate simplex, σ, (σ) = R2 . In higher dimensions
however this is not true. Many of the results and certainly lots of the intuition in
Eidenschink’s thesis are based upon the validity of this claim in 2d, while most of
the hard work in generalizing those very same results was in finding a new notion to
replace field enclosing.
Proposition 2.19 If d = 2 and σ is an affinely independent triangle then (σ) = R2.
Proof. Let σ = [v1, v2, v3] . Since the vertices of σ are affinely independent, the
vectors i := (v1 − v3) and j := (v2 − v3) are linearly independent. Now the vector
k := (v2 − v1) cannot be in the positive or negative cone of v3 since k = j− i. Since
i and j span the plane, they break it into 4 quadrants or cones, let us label these
clockwise as 1 := (i, j), 2 := (j,−i), 3 := (−i,−j) and 4 := (−j, i). We saw above
that k lives in the interior of the cone 2. But then clearly the cone 2 is the union of
the cones (j, k) and (k,−i). Further the cone 4 is the union of the cones (−j,−k)and (−k, i). And this completes the proof
30
Proposition 2.20 If d > 2, (σ) 6= Rn
Proof. Consider the simplex ∆ = [0, e1, e2, . . . , en]. The vector v = (1, 1 . . . , 1,−1)
is not in F(∆). Now, consider an arbitrary non degenerate simplex σ, with ver-
tices xi : 0 ≤ i ≤ d . The affine isomorphism f , constructed by defining f(ei) = xi
with f(0) = x0, is a bijection and preserves affine and hence conical combinations.
Therefore, the vector f(v) cannot be in F(σ)
The following theorem from the combinatorial theory of polytopes is the final
word on these issues [14].
Theorem 2.21 (Gunter Ewald) Let P be a polytope then normal cones to the
proper faces of P form a complete fan.
The previous discussion is intended to show that it is not possible to define a
natural orientation condition on each simplex of a complex, in a way that makes
essential use of a field enclosing vertex. It is possible to do so in R2 but nowhere
else. This was the major conceptual stumbling block to developing a dimension
independent notion of orientation.
2.3.1 δ-Oriented Simplices
In this section we define the notion of δ-oriented simplex and describe the local
consequences of having a complex consisting of such simplices. These local results
have global consequences and allow us to prove a sufficient condition for the induced
map recurrent sets to approximate the chain recurrent set, what in the sequel is called
the Global approximation theorem.
The cone enclosure lemma is a local result that says that if the vectorfield lies in
a cone then the total forward image of an oriented simplex must also be confined to
a cone. This allows us to control the spread of the multivalued map and is the main
ingredient in the global approximation theorem.
31
Definition 2.22 An n-simplex σ = [v0, v1, . . . , vn] is called δ-oriented with respect
to a vectorfield f : Rn → Sn−1 if for each infacet σi there exists a point y in that
facet such that the following condition holds
(vi − y) · f(y) ≥ δ‖vi − y‖
Let K be a simplicial complex in Rn. The complex K is called δ-oriented with
respect to the vectorfield f : Rn → Sn−1 if every simplex σ ∈ K is.
The property of being δ-oriented is a local one, meaning that it depends only
on the vectorfield in a neighborhood of a simplex, in fact it depends only on the
vectorfield on the boundary of the simplex. An equally important feature of this
definition is that it is computable.
Several questions arise in connection with this definition. First, do such complexes
exists? This question is in part answered by Example 2.23. Second, is the condition
overly restrictive or is it in some sense tight. If a simplex has an infacet in a cone,
but the simplex itself meets the exterior of the cone, then in fact that simplex cannot
be oriented. In this sense the condition is tight.
The following example provides an illustration of the oriented concept in the
simplified case of parallel flow.
Example 2.23 Let K(s, t) be the parameterized family of simplicial complexes in
R3 described below, see Figure 2.3. The complexes are constructed from identical,
parallel layers of equilateral triangulated planes, shifted relative to one another such
that the vertices of one layer are the circumcenters of the previous layer. The param-
eter ’s’, describes the size of the equilateral triangles, while ’t’, describes the distance
along the vectorfield between layers.
In the Figure 2.3, the upper layer of vertices are solid, the bottom layer is shown
dashed. Up to symmetry there are three geometrically different simplices represented
32
B C
D E
I
K
L
J
Figure 2.3: The family of δ-oriented parallel flow complexes constructed in Example 2.23
33
by:
σ1 := F,D,B, L ’forward facing’
σ2 := F, L, J, I ’rear facing’
σ3 := C,K,B, L ’filler’
A straightforward but tedious series of calculations shows that for the parallel
flow vectorfield f = (0, 0, 1), if the following condition is met then the complexes are
δ-oriented.
τ ≥ 4sδ
3√
1 − δ2
The points are given by F = (0, 0, 0),E = (2s, 0, 0),C = (s,√
3s, 0), J = (s, s√3, t),I =
(0, 2√3s, t).
2.3.2 Local Expansion of the Induced Map
In this section we prove two lemma’s that characterize the local expansion of the
induced map, given that all the simplices are δ-oriented. These results justify the
δ-orientation condition as useful. These local estimates will then be used in the
subsequent section to prove the global approximation theorem.
The following geometrically obvious estimate is a required ingredient for the up-
coming lemma.
Proposition 2.24 Let x, g, f ∈ Sn−1. Let 1 ≥ ω > λ > 0. If 1 ≥ δ ≥ ωλ +√
(1 − λ2)(1 − ω2) and g · f ≥ ω and x · g ≥ δ , then x · f ≥ λ.
Proof. Without loss of generality, by applying a rotation, we can assume g = e1.
Then we have that
g · f = f1 = cos(θ1) ≥ ω
x · g = x1 = cos(θ3) ≥ δ
34
and define θ2 by x · f = cos(θ2). It follows that
∑ni=2 f
2i = 1 − f 2
1 = sin2(θ1) ≤ 1 − ω2
∑ni=2 x
2i = 1 − x2
1 = sin2(θ3) ≤ 1 − δ2
∑ni=1 xifi = x1f1 +
∑i = 2nxifi = cos(θ2) ≥ x1f1 − |∑n
i=2 xifi|
By Cauchy-Schwartz we get
cos(θ2) ≥ x1f1 − (n∑
i=2
x2i )
1/2(n∑
i=2
f 2i )
1/2
= cos(θ1) cos(θ3) − sin(θ1) sin(θ3)
≥ ωδ −√
1 − ω2√
1 − δ2
Now if we impose the condition
εδ −√
1 − ω2√
1 − δ2 ≥ λ (2.4)
then cos(θ2) ≥ λ as we wish. Solving the inequality 2.4 for δ leads to solving a
quadratic inequality in δ which is positive precisely when
1 ≥ δ ≥ ωλ+√
(1 − λ2)(1 − ω2) (2.5)
This geometric relationship appears repeatedly, therefore we set it off as a defini-
tion.
∆(ω, λ) := ωλ+√
(1 − λ2)(1 − ω2) (2.6)
Lemma 2.25 (Cone Enclosure Lemma) Let 1 ≥ ω > λ > 0, and let δ ≥∆(ω, λ). Let f0 ∈ Sn−1 and the vectorfield f : R
n → Sn−1∩C(f0, ε) . Let x0 ∈ R and
denote ζ := C(x0, f0, λ). Let K be a δ-oriented complex with respect to the vectorfield
f and suppose that for some σ0 ∈ K we have that |σ0| ⊂ ζ , then |Fω(σ0)| ⊂ ζ.
35
Proof. Suppose not. Then there exists a chain of simplices σ0, σ1, . . . , σp where
σi+1 ∈ F(σi) i = 0, 1, . . . , p − 1. The chain has the property that |σi| ⊂ ζ i =
0, 1, . . . , p − 1 but σp ∩ (Rn \ ζ) 6= ∅. σp is the first simplex in the map chain to
leave the cone. Now σp ∩ σp−1 is a facet of both, lets call it σ∗p, and by assumption,
it lies entirely in the cone ζ. The vertex opposite σ∗p in σp is denoted v∗p and
the unit inward normal to σ∗p in σp is denoted ν∗. Since σp ∈ F(σp−1) we know
from the definition of the multivalued map F that there exists an x in σ∗p such that
f(x) · ν∗ ≥ 0. But by definition σp is δ-oriented which implies that there exists a
point y in σ∗p such that
(v∗p − y) · f(y) ≥ δ‖v∗p − y‖
But we also have by assumption that f(y)·f0 ≥ ω and these two relations together
imply
(v∗p − y) · f0 ≥ λ‖v∗p − y‖
by the Proposition 2.24. We know that y ∈ σp−1 and hence y ∈ ζ. Since ζ is a
cone we have that the points in the half line given by
y + t(v∗p − y)
‖v∗p − y‖must lie in ζ for all t ≥ 0. Choosing t = ‖v∗p − y‖ we have that v∗p ∈ ζ. The
vertices of σp are those of σ∗p together with v∗p. All of these vertices have been shown
to lie in the cone ζ. ζ is convex, hence it contains the convex hull of σ∗p and v∗p
which is by definition the simplex σp. But this contradicts the assumption that σp
meets the complement of ζ.
For lack of a better one, we use the phrase ”linearly expanding multivalued map”
to encapsulate the conditions guaranteed by the cone enclosure lemma. We end this
36
section with the apex location lemma. The previous lemma showed that if a simplex
is oriented and is contained in a cone then so must its forward image. The apex
location lemma shows that any oriented simplex is contained in some cone whose
apex is a computable distance away. This distance depends on the orientation of the
cone and the projection of the simplex in that direction.
Definition 2.26 Let g ∈ Sn−1 and σ ∈ K. Letting h(σ, g) := Inf y · g : y ∈ σ
Hg := x ∈ Rn : x · g = h(σ, g)
is the supporting hyperplane to σ such that σ ∈ H+g , the positive closed half-space.
Let Pg(·) represent orthogonal projection onto Hg and
Rg(σ) := minσi∈IFdiam(Pg(σi))
.
Lemma 2.27 (Apex Location Lemma) Let 0 < λ < 1, g ∈ Sn−1, and the
vectorfield f : Rn → Sn−1. Let σ ∈ K be δ-oriented with respect to f . Let Hg be as
in Definition 2.26. Then there exists a point p ∈ Rn with the properties that
1. d(p,Hg) = Rgλ√1−λ2
2. |σ| ⊂ C(p, g, λ)
Proof. Consider a vertex v ∈ σ ∩Hg. v cannot be the vertex opposite and infacet,
since it is δ-oriented. Also, σ must have an in-facet, otherwise it would have to
contain a fixed point, which is ruled out by our restriction on the vectorfield. Now
let σ∗ be an infacet containing v, and let P be it’s orthogonal projection into Hg.
By assumption the polytope P can be contained in a disk of radius less than or
equal to R, in the hyperplane Hg. Therefore there exists a point p∗ ∈ Hg that has
the property ‖p∗ − y‖ ≤ R over y ∈ P. Now consider the point p = p∗ − Rλ√1−λ2
g,
37
consider the disk D := Hg∩B(p∗, R) and the cone formed by the point p and the disk
D. This cone is precisely C(p, g, λ) and contains the infacet σ∗, because it contains
its projection P ⊂ D, and the cone is oriented in the orthogonal projection direction
g. Because σ is δ-oriented the same essential argument used in lemma 2.25 implies
that |σ| ⊂ C(p, g, λ).
2.4 Global Approximation Theorem
The global approximation theorem proves any map recurrent chain is contained in
some component of the ε-chain recurrent set. We prove that any point in a map
recurrent chain is ε-chain recurrent. The proof uses the natural device of cones to
control the vectorfield locally. A given map chain is covered by overlapping cones, the
local shadowing theorems of the previous section allow us to do this. The successive
apices of the cones along the map chain are connected to produce a polygonal path.
The relationship between the cones and the vectorfield allow us to conclude that the
polygonal paths are ε-approximate trajectories. A Gronwall estimate then allows us
to conclude that it must remain close to a true orbit for at least time one. This
allows us to build ε-chains. The construction is straightforward and is illustrated in
Figure 2.4.
The following theorem is a result about Euler like paths, and is a simple conse-
quence of the Gronwall inequality, and can be found in Hurewicz [17]. It is a key
ingredient used to prove the global approximation theorem and we state it here for
completeness.
Definition 2.28 Let f(t, x) be continuous on some domain D. Let t ∈ I be an
interval. Then a function y : I → Rn is a solution of y = f(t, y) up to error ε if:
1. (t, y(t)) ∈ D for all t ∈ I
38
ε
Figure 2.4: Construction of an Euler path using successive cone apices.
39
Figure 2.5: An E-path that is a solution up to ε
2. y(t) is continuous on I
3. y(t) has a piecewise continuous derivative on I which may fail to be defined
only for a finite set of points F ⊂ I
4. ‖y′(t) − f(t, y(t))‖ ≤ ε for all t ∈ I \ F
Theorem 2.29 Let (t0, x0) be a point of a region R in which the vectorfield f is
Lipshitz with constant k. Let y1 and y2 be defined on some interval t0 ∈ I and be
solutions up to error ε1 and ε2 respectively. If p(t) := y2(t) − y1(t) and ε := ε1 + ε2
Then
‖p(t)‖ ≤ ‖p(t0)‖ek|t−t0| +ε
k(ek|t−t0| − 1)
The following several claims are simple estimates that allow us to show that the
E-path construction yields an ε-approximate trajectory for time one.
Proposition 2.30 Let 0 < λ < 1 and h ∈ Sn−1. Suppose σ ∈ C(x, h, λ) is δ-oriented,
with maximal infacet circumradius R, barycenter b, g = f(b), and supporting hyper-
plane Hg as in Definition 2.26. Let r = λR√1−λ2
be as in the apex location Lemma 2.27.
Suppose further that the distance from x to any point in σ is larger than β > 0. Then
there exists a Λ < λ such that the apex p such that σ ∈ C(p, g, λ), guaranteed by apex
40
location Lemma 2.27, satisfies p ∈ C(x, h,Λ). Moreover, Λ can be chosen to satisfy
the equation
Λ(λ, β, r) =λ√β2 − r2
β− r
√1 − λ2
β(2.7)
β
λ
Λ
r
xh
q
Figure 2.6: Estimate for Λ(λ, β, r)
Proof. Consider the diagram in Figure 2.6. The ”worst case”, that is the smallest
value for Λ < λ, happens when a point q of σ ∩ Hg is on the boundary of the cone
C(x, h, λ). We know from Lemma 2.27 that the apex p of the cone C(p, g, λ) is within
a ball of radius r = Rλ√1−λ2
. Let T be a vector through x that is tangent to B(q, r).
T makes the largest angle, cos−1(Λ), with h when they are coplanar with the vector
q − x. Letting θ represent the angle between the vectors q − x and T we have that
cos−1(Λ) = θ + cos−1(λ) (2.8)
41
From trigonometry we have the following relations
1. sin(θ) = rβ
2. cos(θ) =
√β2−r2β
3. sin(cos−1(λ) =√
1 − λ2
Applying the double angle formula for cosine to Equation 2.8 we have
Λ = λ cos(θ) − sin(θ) sin(cos−1(λ)
and the result follows.
Remark. Given that r < β < 1 we have that
λ(1 − r
β) − r
√1 − λ2
β≤ Λ ≤ λ (2.9)
Notice that in equation 2.9 in order to force Λ → 1 as λ → 1 we need to require
that r → 0. One way to accomplish this is by requiring that the maximal infacet
projection diameter R ≈ (1 − λ). Since λ = cos(θ) is directly related to the angle of
the cone, what we require is R = o(sin(θ)), i.e., that it go to zero faster than sin(θ)
as θ → 0. These observations motivate the following.
Let us choose a continuous function S(λ) with the properties that
1. 0 < S(λ) < 1 for 0 < λ < 1
2. S → 0 as λ→ 1
Given the function S, which we will refer to as a ”scale” function, since it is going to
dictate the size of the triangulation, we impose the conditions:
r(λ, β) ≤ βS(λ) (2.10)
r +D < β (2.11)
42
It is important to note that while we have stated the restriction on r it is equivalent
to a restriction on R. Under these conditions we have that equation 2.9 yields
λ− S(λ)(λ+√
1 − λ2) ≤ Λ ≤ λ (2.12)
The point is that by introducing a scale function and imposing the above restrictions
on r and D, Λ becomes independent of β and has the prescribed limiting behavior.
Proposition 2.31 Let 0 < Λ < λ < 1 and g ∈ Sn−1. Suppose that the vectorfield f
lies in the cone C(g, λ) ∩ Sn−1. Let C(g, λ) ⊂ C1(g,Λ) and let p ∈ C1 with ‖p‖ ≥ β.
Let ν be a unit vector in the p direction and P (t) = tν. Then for any time t ∈ (0, ‖p‖)we have that
‖P ′(t) − f(P (t))‖ ≤√
2(1 − Λλ+√
1 − Λ2√
1 − λ2)1/2 (2.13)
Proof. The construction is illustrated in Figure 2.4. Let θ1 = cos−1(Λ) and θ2 =
cos−1(λ). Then the largest angle between any vector in the cone C1 and the vectorfield
is θ1+θ2. Therefore ν ·f(P (t)) = cos(θ) ≥ cos(θ1+θ2) = Λλ−√
1 − Λ2√
1 − λ2 finally,
for any two unit vectors k, l it follows that k · l ≥ ξ implies that√
2√
1 − ξ ≥ ‖k− l‖.
Remark. If we wish to show that a polygonal path, constructed as is P in equa-
tion 2.13, is an ε-approximate trajectory then we must be able to bound the right
hand side of equation 2.13. We introduce the equation
εp(λ) :=√
2(1 − Λλ+√
1 − Λ2√
1 − λ2)1/2 (2.14)
Under the ”scale” conditions imposed on r in equation 2.10, it follows from equa-
tion 2.12 that
limλ→1
εp(λ) = 0 (2.15)
43
We introduce a final definition.
Definition 2.32 We say a complex K is a (λ, β)-scale complex if there exists a scale
function S(λ) such that for each cell σ ∈ K we have that r(σ) and D(σ) satisfy the
conditions imposed in Equations 2.10.
Theorem 2.33 (Global Approximation Theorem) Let X ⊂ Rn be a compact
polyhedral set, endowed with a continuous unit vectorfield f : X → Sn−1. For every
ε > 0 there is a 1 > λ > 0 such that for any ω in the interval (λ, 1) there exists a
1 > δ > 0 and a β > 0 such that if K is a δ-oriented (λ, β)-scale simplicial complex
on X with respect to f , and J = kini=1 ⊂ K is any induced map recurrent chain of
simplices, then |J | is a subset of some component of Rε, the ε-chain recurrent set of
the maximal invariant set of X .
The goal of the proof is to show that for each point x in |J | we can construct
an ε-chain from the point back to itself. The proof proceeds as follows. First we
show that starting from x we can flow forward along ϕ(t, x) for a time τ no less than
one and find that B(ϕ(τ, x), ε) ∩ |J | 6= ∅. This part of the proof follows the outline
of Figure 2.4, and uses the construction of an E-path that ”shadows” the trajectory
ϕ(t, x). An E-path is a piecewise linear curve, each line segment of which is forced by
construction to remain close to the vectorfield. The geometric device of cones, along
with the error estimates for cones that we have developed in this and the previous
section, insure that the E-path remains close to the trajectory.
The successive E-paths wind their way around the map-recurrent chain |J |, each
one adding a new point to a growing ε-chain E = x = x0, x1, x2, ...xn. We show how
to construct the first E-path and the rest follow by induction. The last step in the
proof is to show that the process terminates with a final jump back to x. This will
follow from the fact that the set of simplices in J is finite and each E-path requires a
minimum number of simplices.
44
Proof. Fix ε > 0. Let L be the Lipschitz constant of f on X . Choose a λ such that
εp(λ) < Lε2(e3L−1)
. Choose an ω in the interval (λ, 1).
A choice of ω implies a Lebesgue number Ω(ω), as described below. The function
g(x, y) = f(x)·f(y) is continuous and therefore the set G = g−1((ω, 1]) is open. About
every point (x, x) there is a rectangle U × V contained in G and the set W = U ∩ Vis a neighborhood of x with the property that for any two points x, y ∈ W implies
that g(x, y) > ω. Let W = Wx : x ∈ X be an open cover of X consisting of such
sets. Let us denote by Ω(ω) the Lebesgue number of this cover. If Ω(ω) > 1 choose
Ω = 1. Choose β < min 116, Ω
6, ε
2e3L , and δ ≥ ∆(ω, λ)
Suppose we have a complex K with the assumed properties, and J is a map
recurrent chain. Since J is finite the elements can be enumerated, and since J is a
map recurrent chain, there exists a permutation π that is a cycle of the order of the
number of simplices in J such that F|J(ki) = kπ(i).
Let x ∈ |J | be arbitrary. Then there exists some simplex k = ki0 ∈ J that
contains x. Let b0 be the barycenter of ki0 , g = f(b0) and let C0 = C(p0, g, λ) be the
cone with apex p0 containing ki0 and F0 = J ∩ Fω(ki0) ∩B(p0,Ω) guaranteed by the
cone shadowing and apex location Lemma’s.
Having defined these quantities we can now proceed to construct what we will call
an E-path, an Euler like polygonal path that is constructed so that it remains close
to the true orbit through x for at least time one. The construction is inductive. In
this section we have developed error estimates for cones that contain the vector field,
cones of this type are the principle objects used in the construction of an E-path.
Choose a simplex ki1 in F0 whose apex, p1, is farthest from p0, but is no farther
than 5β away. In fact, the scale conditions imposed on the complex require that
there must be such an apex between 4β and 5β. This is because for each simplex
r +D < β < Ω/6 and the chain in F0 can only exit the ball B(p0,Ω) out the end of
the cone C0, which is a distance greater than 6β from p0.
45
Let g1 = f(b1) and C1 = C(p1, g1, λ) be the cone containing ki1 , construct the set
F1 etc. Continuing the process with the cone C1 we can find a cone C2 and so on.
The sequence of cones covers an initial segment of the forward image of the simplex
ki0 in the map chain J . The E-path is constructed from the sequence of cones by
connecting the successive apices. The result is a polygonal path P (t).
1. t0 = 0
2. ν0 = p1−p0‖p1−p0‖ .
3. νk =pk+1−pk
‖pk+1−pk‖
4. tk+1 = tk + ‖pk+1 − pk‖
5. P (t) = pk + (t− tk)νk for t ∈ [tk, tk+1].
Because of the scale conditions and the argument in Proposition 2.30 we know
that pk+1 ∈ C(pk, g,Λ) for all k ≥ 0. Consequently, by Proposition 2.31 we know that
along our E-path we have that ‖P ′(t) − f(P (t))‖ < εp(λ), for all but a finite number
of times t, associated to the apices where the straight line edges meet.
Since 5β > tk+1 − tk > 3β the process can end with a total time τ along the path
1 ≤ τ < 2 − β. This is because we can choose the smallest positive integer n such
that 3nβ ≥ 1, and the constraint on β < 1/16 implies that 5(n+ 1)β < 2.
The path is completed by choosing the final vertex pn+1 to be the barycenter of
the simplex kin . Since the distance from pn to pn+1 is less than r +D < β we know
that the total time along the path P from p0 to pn+1 is less than two. Let J0 = J
and J1 = J0 \ [ki0 , kin), where the notation [ki0, kin) represents all the simplices in the
map chain between the endpoints.
The function P satisfies the requirements in Definition 2.28. Let H(t) := ‖P (t)−ϕ(t, x)‖. From Theorem 2.29 we know that
H(t) ≤ eLtH(0) +εp(λ)(eLt − 1)
L(2.16)
46
We have that H(0) = ‖x − p0‖ ≤ r +D < β and from the choice of β above we see
that the first term in Equation (2.16) is less than ε/2. By the same token, our choice
of λ above and the bound on εp(λ) imply that the second term in Equation (2.16) is
less than ε/2. It follows that H(t) < ε for 0 ≤ t < 3.
We have shown that it is possible to jump from ϕ(x, τ) to a point pn+1 ∈ |J | with
a distance of less than ε. Label x = x0 and x1 = pn+1, these are the points in our
growing epsilon chain E . Now starting from the point x1 we can construct another
E-path P (t) that shadows the trajectory ϕ(x1, t). By induction we can extend our
ε-chain to include a point x2 ∈ |J | and so on. The i-th E-path results in the addition
of a new point xi to the set E and the creation of the set Ji ⊂ J . The subsets Ji are
a strictly monotone decreasing sequence, hence the process must end.
We now show that this process must terminate in a finite number of steps, l, with
the inclusion in E of xl = x0. There are two cases to consider. Suppose that during
the construction of the first E-path, the sequence of cones Ci, it happened that the
simplex satisfies k ⊂ Ci ∩B(pi, 5β), and that the total time along the E-path to pi is
less than 1 − 5β. Let pi+1 = p0, now our E-path is a closed polygonal path with a
transit time along the path of less than one. We can traverse this path, in the same
direction, as many times as needed to produce a total transit time 1 ≤ τ ≤ 2. In this
case we hop from ϕ(τ, x) directly back to x.
In the other case the inductive construction of the ε-chain E = x0, x1, ... con-
tinues until the initial simplex k satisfies k ⊂ Ci ∩ B(pi, 5β) for some i. This must
happen because the set of simplices J is finite, and each ”link” pi− pi−1 in an E-path
removes at least one more simplex from the set. What is left to consider is if, as
above, the current E-path returns to k with a total transit time less than 1 − 5β. In
this case we concatenate the previous E-path with the current one. The total transit
time along the previous path was by construction less than two and the current path
is less than one. The concatenated path has length less than three and this is the
47
reason for our conservative choice of e3L in the Gronwall estimate.
Thus in every case we have shown that it is possible to construct an ε-chain from
x to itself. Since x ∈ |J | was arbitrary, the theorem is proved.
48
Chapter 3
Computation
3.1 Introduction
One of the goals of this thesis work was to extend the algorithm(s) developed by
Eidenschink and Mischaikow [13] in the plane, to arbitrary dimensions. Their algo-
rithm was based on the concept of bistellar flip which is known to be highly inefficient
and insufficient in higher dimensions. While this thesis cannot claim to have solved
the problem, hopefully it can claim to have identified the major obstacles. Below we
attempt to give a reasonable summary of that part of computational geometry that
relates to the triangulation of a finite point set and the manipulation of that triangu-
lation. This provides the background for our current computational strategy which is
described, along with several ideas and strategies with which we have experimented.
Finally, we briefly discuss the connections with Riemannian geometry and some open
research questions and directions for further study. In the literature, especially in
the applied sciences, the terms mesh and triangulation are used synonymously with
simplicial complex. We will follow suit.
There exist well established numerical methods that generate triangulations to
solve mathematical problems. These fall into roughly two categories, finite element
methods that are based on interpolation, and computational geometry, mesh opti-
mization methods that adapt triangulations, for applications like image processing,
49
via local rearrangements like bistellar flips. Neither of these appear to be imme-
diately applicable for our purposes. There is however, a very recently introduced
notion of triangulations that satisfy metric specifications [3], which as an algorithmic
approximation of the Delaunay triangulation of a Riemannian manifold. This concept
appears to be intimately related to our problem.
There are several hurdles to be overcome in discretizing the flow with polygonal
decompositions. The properties that we desire in a complex
Flow transverse facets
Locally, linearly expanding induced map(3.1)
are not of the sort, that can be accomplished by interpolating a given function, as
is done in finite element grid generation. A common approach in triangulations is
to start with an initial mesh and to iteratively refine that mesh via local or global
procedures till the resulting mesh conforms to some convergence criteria. Often such
methods have relied on bistellar flips, a generalization of the edge flipping opera-
tion. But as is discussed in greater detail below, it has been proven that there exist
triangulations that admit no flip operations.
What has always been clear, and what forms the foundation of our intuition, and
perhaps the very thing that leads us astray, is that we can solve the problem locally.
That is, the flow box theorem allows us to solve the problem locally by considering
polygonal decompositions of parallel flow. In two dimensions this is seductively easy,
as theorem 2.14 shows. In three dimensions already things become complicated, as
it is no longer possible to tile space with copies of a single regular simplex. But as is
indicated by example 2.23, it is still possible. The situation is analogous to that in
the theory manifolds, our problem has been that we cannot find a object that plays
a role equivalent to that of a partition of unity.
50
3.2 Verifying a Given Triangulation
How can we use the concepts from the local and global approximation theorems to
compute? We do not yet have the understanding to construct an algorithm that can
generate triangulations that satisfy the conditions of the theorem. There are a few
ways in which the definition of δ-oriented, and the local and global approximation
theorems can be used to help further the development of algorithms. One way is sim-
ply to compute for a given triangulation, a histogram of the number of triangles that
satisfy a δ-orientation condition for the interval [δ, δ+ h]. This provides a measure of
the quality of the triangulation. To invoke the conclusions of the approximation the-
orems, one also needs to know that a δ-oriented triangle lives within a neighborhood
on which the vectorfield lives in an appropriately narrow ”ω” cone. This latter mea-
surement appears hard to verify in practice. Regardless, the existence of a geometric
measure of quality, that has a theoretical justification, is a large step forward.
A second and more ambitious application is to extract from a given triangulation
an estimate of the ”epsilon” that it satisfies, by backtracking through the small pa-
rameters involved in the global approximation theorem. We now describe how this
might be done.
Let S be a set of polygons that is comprised of a strongly connected component and
perhaps its star. With the notation used in the global approximation theorem 2.33
make the following measurements:
Measurements
1. Compute L The Lipshitz constant of f on S
2. Compute δ := minσ∈Sδ(σ)
3. Compute D := maxσ∈Sdiam(σ)
51
4. Compute R := maxσ∈Sρg(σ)
Calculations
1. In Theorem 2.33 we have the condition that λ < ω < 1, and from Lemma 2.25
we have that λ,ω, and δ are related by Equation (2.6). If for instance we
choose, ω =√λ, and substitute that into Equation (2.6), then we can solve
δ = λ√λ+
√(1 − λ2)(1 − λ) for largest λ ≤ 1 set λ = soln
2. set r = λR√1−λ2
3. set β = D + r
4. Compute Λ := Λ(λ, β, r)
5. Compute G :=√
2(1 − Λλ+
√(1 − λ2)(1 − Λ2))1/2
6. set ε := 2(e3L−1)G
L
We briefly indicate how one might compute the orientation condition on triangles.
The first step is to determine which facets are infacets, the second step determines,
for each infacet, the largest value of δ for which it satisfies the δ-orientation condition.
Orientation Algorithm
Repeat for each facet of σ := [v0, v1, v2]
1. For facet [v0, v1] say, compute η the unit inward normal. Let λ ∈ I := [0, 1] and
define h : R → R by
h(λ) := f(λv0 + (1 − λ)v1) · η
Use Newton’s method to find
M := maxλ∈Ih(λ)
If M ≥ 0 then proceed to step 2, otherwise set G2 := 1.
52
2. Let
g(λ) :=(v2 − λv0 − (1 − λ)v1) · f(λv0 + (1 − λ)v1)√|v2 − v1|2 + λ|v1 − v0|2 + 2λ(v2 − v1) · (v1 − v0)
Set
G2 := maxλ∈Ig(λ)
Again, computed with Newton’s method or Quasi-Newton.
When this is done for all facets let δ(σ) := minG0, G1, G2. This number represents
the largest number δ for which the triangle satisfies the δ-orientation condition.
We now illustrate how to compute ρg(σ) the minimum infacet projection diameter
in triangles. Let g = f(b)||f(b)|| where b is the barycenter of the triangle σ, and g⊥ is a
unit vector perpendicular to g
1. Find the vertex v such that min0≤i≤2v · g
2. If an edge [v, w] is an infacet (determined in step 1 of above algorithm) then
compute the projection of [v, w] onto Hg, which in this case is just Pw :=
|(v − w) · g⊥|
3. ρg(σ) := min[v,w] infacetPw
3.3 Bistellar flips and Regular Triangulations
In this section we discuss the problem of triangulating the convex hull of a finite, fixed
set of points A = ain1 ⊂ Rd, with Q := Conv(A). In order to make clear that we are
considering all possible triangulations of the set Q with the vertices being drawn from
the set A we adopt the pair notation (Q,A), as in [15]. We require that the vertices
of the simplices that triangulate (Q,A), must be contained in the set A, but they
do not necessarily have to exhaust them. As was discussed in the previous section,
a successful planar algorithm was developed in [13] that used the operation of edge
53
swapping to improve the alignment of triangles with the flow. Empirically it was
observed that the same general algorithm was very inefficient in three dimensions.
This section recounts the theory that explains these observations and shows that
no general procedure following the blueprint developed in [13] can be designed to
succeed. As a consequence, we have had to turn to more geometrical and dimension
independent approaches, and these will be described in the sections that follow.
Let L := l|l : A → R+ and consider the following procedure: Compute the
convex hull P of the set of points L := (a1, l(a1)), (a2, l(a2)), ...(an, l(an)) ⊂ Rd+1
consider those facets F of P where the (d + 1)st entry of the unit outward normal
vector is negative. That is, those facets that line the bottom of the polytope P.
If we orthogonally project these facets down onto Rd we get a decomposition of Q.
This decomposition will be a simplicial complex if the lower facets of P are simplices.
In [15] an elegant and elementary proof is given that this happens for almost all
functions in L.
The argument is as follows: suppose some facet F of P is not a simplex, then
it must contain at least d + 2 vertices, each of the form (ai, l(ai)), all of which lie
in a common hyperplane. This means that the determinant whose rows are formed
by the vectors (1, ai, l(ai)), must vanish. The determinant is a linear function in the
”parameters” l(ai), and its being zero, defines a hyperplane in the space L. The
coefficients of this hyperplane are uniquely determined by the points in A. The
number of possible ”bad” hyperplanes is bounded by(d+2n
). After deleting the union
of the ”bad” hyperplanes, A, any function in l ∈ L \ A will yield a triangulation of
Q.
We call this process ”lifting” and the function l a lift. Any triangulation derived
by lifting is called regular. Possibly the best known example is the Delaunay triangu-
lation. It can be proven that the the lift l(x) := |x|2 yields the Delaunay triangulation
of (Q,A). It is also possible to prove that the triangulation presented in figure 3.1 is
54
not regular.
Figure 3.1: A triangulation that is not regular
Let us denote by Si(T ) the set of simplices in T that contain the vertex ai. To
each triangulation T of (Q,A) one can associate a function φT in L in the following
way. Choose any translation invariant volume form and define
φT (ai) :=∑
σ∈Si
V ol(σ) (3.2)
Define the secondary polytope, denoted by Σ(A), to be the convex hull, in the space
L, of all the points φT , T ranging over all triangulations of (Q,A). The importance
of this object will become apparent after a short discussion of bistellar flips.
A collection of points in Rd form a circuit if any proper subset is affinely inde-
pendent, while the entire collection are affinely dependent. Thus, circuits must be
collections of d + 2 elements. Take any non degenerate lift, the convex hull of the
lifted points be will a simplex. The result is that there are exactly two triangulations
of the convex hull of these points, projecting the lower facets, or the upper facets,
and these are the only two regardless of the lift employed. So if one has a circuit,
one can triangulate that circuit in exactly two ways, and the act of replacing the one
55
Figure 3.2: Circuits in two dimensions, and associated bistellar flips
with the other is called a bistellar flip. In the plane there are two possible bistellar
flips, an edge swap and the insertion of a vertex, see figure 3.2.
The significance of the secondary polytope Σ(A) is that its vertices correspond
to regular triangulations of (Q,A), and moving from one vertex to another along an
edge represents a modification by a bistellar flip, see figure 3.3. As a consequence one
can select optimal triangulations by linear programming over Σ(A), and it is possible
to transform any regular triangulation into another by a sequence of bistellar flips.
There exist relatively fast algorithms for enumerating all regular triangulations [21].
It is known that an arbitrary triangulation of n points in plane has at least n− 3,
possible bistellar flips [10]. It was an open problem until very recently, whether or not
two arbitrary triangulations in higher dimensions could be connected by a sequence
of bistellar flips, or stated differently, whether or not there exists a point set whose
space of triangulations is disconnected. In 2000 this problem was solved by Francisco
Santos who proved that there do exist triangulations that admit no bistellar flips [27]!
He has constructed at least two examples, one of them a 6 dimensional set of 324
points. In addition it has been shown in [10] that one can construct a family of
56
!"
#$
%%&&
'(
)*
+,
-.
//0112
3=(3,4,2,5,1)
1=(5,1,4,,4,1)2=(3,1,5,2,4)
4=(1,5,2,4,3)
5=(1,3,4,2,5)
Figure 3.3: The secondary polytope of a pentagon
triangulations in R3 that have arbitrarily large flip deficiency. These results indicate
that it is very unlikely, even in three dimensions, that a general meshing process
based on bistellar flips can be developed that converges efficiently. These results
were unknown to Eidenschink, although he observed that his algorithm was highly
inefficient in three dimensions, precisely because of the paucity of bistellar flips that
would improve a configuration.
The results of this section are not just of importance because they show what
not to do, in fact the theory of lifting may play an important role in proving the
existence of triangulations that satisfy the properties 3.1. Conley’s decomposition
theorem states that for every isolated invariant set there exists a complete Lyapunov
function. Does this function provide a lift? Do the Lyapunov functions associated
with a Morse decomposition provide lifts. Does the scalar curvature function k(x),
57
whose value ought to dictate the relative local density of vertices, provide a lift that
can be used to prove the existence of triangulations with the required properties.
Considering that an often applicable Lyapunov function for an attracting fixed point
is given by l(x) = |x|2, and that this is precisely the lift that generates a regular
triangulation, this observation if you will, offers an avenue for future investigation.
3.4 Voronoi Diagrams and Delaunay Triangulations
Triangulations, a term that often is used to denote higher dimensional simplicial
complexes as well, have a rich mathematical history and is currently a very active
area. Voronoi diagrams and Delaunay triangulations of a finite point set is a central
theme in computational geometry, whose major application is in the generation of
meshes used in the simulation of physical processes [11].
3.4.1 Euclidean Metric
Here we review some of the definitions and properties following [16] and [11]. Given an
open set Ω ⊂ Rn the set Uiki=1 is called a tessellation of Ω if Ui ⊂ Ω, and Ui∩Uj = ∅
and ∪Ui = Ω. Given a finite point set ziki=1 the Voronoi region corresponding to
the point zi is given by
Vi := x ∈ Ω : |x− zi| < |x− zj| i 6= j (3.3)
The (closed) Voronoi regions are polytopes with at most k − 1 facets, as they are
the intersections of a finite number of closed half spaces, for instance see figure 3.4.
A point x that belongs to m (closed) Voronoi regions is equally far from the m
generators, and thus the generators must lie on a common sphere. If the generators are
in general position then m ≤ n. Thus, in the generic situation, the dual of the Voronoi
diagram, yields a simplicial complex called the Delaunay complex. The vertices are
58
Figure 3.4: A (Euclidean) Voronoi Diagram and Delaunay Triangulation
the generators zi and two vertices are connected by an edge if the generators share a
common facet in the Voronoi diagram and so on, see for example [11] section 9.
A centroidal Voronoi diagram is a special instance when the generators are the
mass centroids of their respective Voronoi regions. Given a region U and a density
function ρ(x) ≥ 0 defined on U the centroid is defined by
z∗ :=
∫Uxρ(x)dx∫
Uρ(x)dx
Centroidal Voronoi diagrams are not unique, however under very general assumptions
they exist and are the minimizers of a well defined functional, see [16].
All Delaunay simplices posses regularity properties, and apparently this is a source
of their wide appeal. Any non degenerate simplex defines a unique point O, the so
called ”circumcenter” that is equidistant from all its vertices, with distance or radius
R. In a Delaunay complex each simplex has the property that its open ball B(O,R)
does not contain any vertices of the complex. In the plane, Delaunay triangulations
enjoy the property that, of all triangulations of a fixed point set, it is the one that
maximizes the minimum angle in any triangle, a proof sketch can be found in [11].
59
This property does not generalize to higher dimensional Delaunay complexes, how-
ever, there are angular properties, like the angle co associated to a facet β(f), that
satisfy the theorem that the Delaunay complex minimizes the maximum angle co
associated to its facets, details can be found in [28]. Another interesting theorem is
that any simplicial complex in which the circumcenters are contained within their
respective simplices is automatically a Delaunay complex [26].
3.4.1.1 The Bowyer-Watson Algorithm
There are several algorithms to compute the Delaunay complex of a given input set
of points. Among them is a method that is often called the incremental insertion
method, which we will refer to as the BW algorithm, developed independently by
Bowyer [6] and Watson [30]. There are three advantages to this method, it works
in arbitrary dimensions, it requires only the specification of the input points, and
it can be modified to work with non Euclidean metrics. The algorithm works by
inserting vertices zi, one at a time, in such a way that a Delaunay triangulation is
maintained at every step. The procedure is inductive and can be described by the so
called Delaunay kernel, which is comprised of two components the cavity and the ball.
Given a Delaunay triangulation of the first k points Tk, consider the set of simplices
whose circumdisks contain the point zk+1. This collection is called the cavity, Ck. For
each facet f at the boundary of the cavity Ck, construct a new simplex as the convex
hull of the point zk+1 and f . This latter set of simplices are referred to collectively
as the ball, Bk. The BW algorithm can be described by
Tk+1 = Tk − Ck +Bk (3.4)
What is proven in [6, 30] is that the cavity is star-shaped with respect to the in-
serted vertex, and that the simplices of the ball have empty circumdisks. Efficient
implementations of the BW algorithm, for use in arbitrary dimensions, exist [4].
60
An interesting computational aside is the problem of point location. Given a trian-
gulation and a point how does one find a simplex containing the point, short of brute
force search? In [11] it is proven that Delaunay triangulations are acyclic, this means
among other things, that one can design a procedure to move along the sequence of
simplices that intersect a line segment without repetition and the possibility of an
infinite loop. The triangulation in figure 3.1 is not acyclic. Thus, in Delaunay trian-
gulations it is possible to perform point location by the following procedure: given
an input point z and any vertex v of T , define the segment z − v, and follow the set
of simplices that intersect this line from the known one containing v to the unknown
one containing z. This problem helps to speed the computational construction of the
cavity. It is not necessary to check every simplex in the triangulation, one first finds
a simplex containing the point being inserted, this simplex must be in the cavity, and
then one proceeds by a local search.
3.4.2 Metrics Induced by a Norm
The case of parallel flow, and the device of introducing the parallel flow metric, makes
it clear that we wish to consider the construction of Delaunay triangulations in more
general settings. The simplest step from the Euclidean case is to alter the inner
product by introducing a constant matrix M , as was done in Definition 2.13. The
level sets of the induced metric are ellipses, instead of circles. All the definitions and
properties described above for Voronoi diagrams and Delaunay triangulations go over
to this case in a trivial way.
Clearly, the set in equation 3.3 can be generalized by replacing the Euclidean
distance with more general metrics. This generalization has been widely consid-
ered [16, 7, 19] for metrics induced by a norm, such as the Lp norms, and efficient
algorithms exists.
The BW algorithm can be easily adapted to work with the metrics described in
61
this section. The computation of the cavity is the only place where the metric plays
a role. What needs to be checked is that given a simplex it uniquely defines a level
set of the metric and that if a point is inside this level set then it is ”visible” to all
the facets. That these facts are indeed true follow from the fact that the level sets
are convex and that the Voronoi regions are star shaped.
3.4.3 Riemannian Metric
A Riemannian manifold is a smooth manifold M , together with a smooth map
<,>: X →⋃
x∈Xinner products on T(X, x) (3.5)
The Riemannian structure <,> (x) smoothly changes the inner product from point
to point in the manifold.
One of the problems in extending Voronoi diagrams and Delaunay triangulations
to the Riemannian setting is that given n+1 points there can be many spheres through
these points and they are not necessarily embedded [20]. As one might expect, and
as is pointed out in [20], an arbitrary triangulation of a Riemannian manifold will
not satisfy the empty circumdisk property because for an arbitrary set of points this
notion is not well defined. A natural approach, and that which the authors of [20]
adopt, is to show that if the set of vertices is sufficiently dense, then the Delaunay
condition is well defined and such triangulations exist and are unique for arbitrary
dimensional closed Riemannian manifolds. The authors of [20] purport to prove this
very fact. Before proceeding it seems appropriate to mention the paper [20] does not
contain any proofs, and that no subsequent paper(s) by these or any other authors
have appeared in press with such a proof. We believe the claim has been proven for
the special case of two dimensional closed compact connected Riemannian manifold
by the first author of [20], and by W. Kalies.
We now briefly outline the result described in [20].
62
Definition 3.1 The injectivity radius at a point x in M is the largest radius, r, for
which B(x, r) is an embedded ball. The injectivity radius of M , denoted inj(M), is
the infimum over those at each point.
Let κ be an upper bound on the sectional curvature of M
Definition 3.2 A subset C ⊂M is strongly convex if given any two points in C the
unique shortest geodesic connecting them has its interior in C.
Definition 3.3 The strong convexity radius at a point x of M is the largest r > 0
for which ¯B(x, r) is strongly convex. The strong convexity radius of M is the infimum
of these radii over all the points of M .
Definition 3.4 The density radius at a point x of M , denoted rad(x) is 1/5 the
strong convexity radius at x.
Definition 3.5 A finite set of points X ⊂ M is said to be sufficiently dense if for
ever y in M and z in B(y, 4rad(y)), the ball of radius rad(y) centered at z, contains
a point of X in its interior.
Let κ be an upper bound on the sectional curvature of M
Claim 3.6 (Leibon and Letscher) If X ⊂ M is a finite, generic and sufficiently
dense set of points then there exists a unique Delaunay triangulation with vertex set
X.
Previously, in definition 2.13, we introduced the notion of a parallel flow metric
The parallel flow metric, is a constant, symmetric, positive definite matrix Θ(ε) that
induces a metric via the inner product < x,Θ(ε)y >. If we were so inclined, we could
use this metric, to generate a Voronoi diagram and/or the Delaunay triangulation of
any fixed set of points in general position, as was described in the last section.
63
In the setting of non-parallel flow we can associate to each point x in the phase
space a matrix M(x, ε) := Dh∗(x)Θ(ε)Dh(x) by pulling back the parallel flow metric
by the flow box maps h. Extend the unit vector f(x) in the direction of the vectorfield
at x, to an orthonormal basis B(x). The matrix Dh(x) is a matrix that has the vectors
of B(x) as columns, ordered in such a way that the vector f(x) corresponds to the
eigenvalue 1. This collection of inner products may define a continuous Riemannian
structure over our compact polyhedral set X. Our intuition is that this Riemannian
structure encodes the geometry of the flow that we wish to capture, and provides
a theoretical framework with which to attack the triangulation problem in a well
defined and systematic way.
3.4.3.1 Anisotropic Mesh Generation
In the setting of Riemannian geometry, the authors of [3] describe the following
procedure to create a so called anisotropic Delaunay triangulation. This is a heuristic
approach to approximate the Delaunay triangulation of a Riemannian manifold. The
procedure uses the BW algorithm and differs from the Euclidean case, only in the
construction of the cavity. Before stating the procedure we need to introduce their
notation.
Given a point z they define the length of the straight line segment from point
p to point q measured from z as lz(p, q) :=< p − q,M(z)(p − q) >. Now given a
simplex σ = [v0, v1, ..., vn] and a point z it is possible to find a point O(σ, z) that is
equidistant from the vertices of σ, call the common distance R(σ, z), all lengths being
measured with lz. Given the point O(z) it is possible to decide if a point p is within
the circumdisk of σ by checking if lz(O(z), p) < R(z), see for instance figure 3.5. Now
define a measure αz(p, σ) := lz(O(z),p)R(z)
.
So given a point z and a simplex σ the authors proposed the following modification
of the cavity building procedure in BW. To decide if σ is in the cavity of z they say
64
R
C
p
q
Figure 3.5: Triangle with circumcenter C, circumradius R
yes ifn∑
i=0
αvi(z) + αz(z) < n + 2 (3.6)
The authors of [3] prove that the cavity so constructed is star shaped about z.
This follows from the fact that if z satisfies equation 3.6 then it must be inside at
least one of the circumdisks, now actually a circumellipse, which is convex, and hence
is visible to all its facets. It is no longer necessarily true that circumdisks of the new
simplices in the ball do not contain any vertex of the triangulation. However this does
not effect the generation of the mesh, only its quality. We will refer to this procedure
as anisotropic BW.
3.5 Point Placement Strategies
Both plain vanilla BW and anisotropic BW require the user to specify a set of points to
be triangulated. Consider the case of a global parallel flow, where we have equipped
the space with the parallel flow metric. It should be clear that we can generate
65
terrible triangulations of this easy flow, by choosing badly placed vertices. We can
create triangles that are not flow transverse and by stacking right triangles we can
generate complexes whose induced map has arbitrarily high expansion perpendicular
to the flow. The moral is that the metric alone will not solve the problem. If given the
proper freedom to chose, the metric chooses that triangle that is long and skinny in
the direction of the flow. Therefore, a key aspect of generating global triangulations
with anisotropic BW that satisfy equation 3.1 is to carefully select the vertices of the
triangles. The rest of this section describes two ideas that we have implemented.
3.5.1 Local Patch Scheme
Initially we developed an iterative strategy for point placement, that is how and where
to place the vertices to be triangulated with the BW algorithm using anisotropic
mesh generation. Our strategy was based on the following observation. Each simplex
σ := [v0, v1, ..., vn] provides an affine approximation to the vectorfield, that is one can
find a matrix A and vector w such that
f(vi) = Avi + w i = 0, 1, ..., n (3.7)
Since the simplices are convex, the affine vectorfield at any point inside σ is simply
a convex combination of the values at the corresponding vertices. Moreover, if we
have a simplicial complex then this approximation procedure yields a global piecewise
affine vectorfield that is Lipshitz continuous and thus defines a flow over the complex
that approximates the true flow, and the difference can be made arbitrarily small by
decreasing the diameter of the simplices. Our thought was to approximate the affine
flow on each simplex by a quadratic flow box approximation, and use these flow lines
to place a set of points locally in the neighborhood of each simplex. The following
set of equations were developed.
66
Let
h(t, λ) := p+ tq + λq⊥ + t2/2r (3.8)
be the quadratic approximation to the affine flow lines. p is some point like the
barycenter of the triangle under consideration, q = f(p)/|f(p)| is the vectorfield, q⊥
is a unit vector normal to q and r = Aq. The following estimate is based on ex-
tending a triangle forward along the flow while it remains transverse to the quadratic
approximation, see figure 3.6.
λ)
(t+s, + k)
(t+s, − k)λ
λ
h(t,
Figure 3.6: Quadratic approximation to affine flow
Notice that there is an upper bound to how far we can go, imposed by the fact
that the quadratic approximations can cross, said a different way until the Jacobian
of the transformation becomes singular, which occurs when tcrit := − q·qr·q . We solve
h(t + s, λ+ k) − h(t, λ) = γ(q + (t+ s)r) (3.9)
for s and |k|, γ is a slack variable. With the notation
J := q·rr·q⊥
T := r·q⊥q·q
Γ := q·rq·q
W := 1 + Γt
Notice that T is the curvature of the quadratic approximation to the flow, and is
constant. After dotting equation 3.9 with q and q⊥, and solving the resulting system,
we can eliminate γ and find that
s2(T/2) − s(Γk) − kW = 0 (3.10)
67
which in turn yields
s = Γ|k/T | + |1/T |√J2k2 + 2|k/T |W. (3.11)
We impose the additional cone growth condition that
|k|/s ≤ tan(θ) (3.12)
The θ is envisioned as a global parameter that either remains constant during the
iteration or that we gradually decrease during the iteration, akin to annealing.
There are essentially two cases to consider. If Γ > 0 then s ≥√J2k2 + 2|k/T |W
and hence it is sufficient to impose that the right hand side condition be met
|k|/s ≤ |k|√J2k2 + 2|k/T |W
≤ tan(θ) (3.13)
We solve
k2 ≤ J2k2 tan2(θ) + 2W |k/T | tan2(θ)
to get
|k| ≤ 2W tan2(θ)
(1 − J2 tan2(θ))|T | (3.14)
provided 1/J2 > tan2, otherwise k is free.
If Γ < 0, then in a similar way, we derive that our constraints are met when
|k| ≤ 2W tan(θ)
(1 + 2|J | tan(θ))|T |
The estimates above can used in the following way. Given a simplex, map the
vertices to the parallel flow, and find the smallest rectangle containing them. Place a
regular grid of vertices in that box, spaced according to s and k given above, for some
fixed value of θ. Using the flow box map, transform the points back to the phase
space, the image triangles satisfy the imposed conditions. The following algorithm
was developed.
68
Iterative Triangulation (IT)-Algorithm
1. Initialize with an arbitrary mesh M0 and a value of θ0.
2. Initialize a list of points Li+1 = ∅, For each simplex in the current mesh Mi,
generate a local patch of points using the above procedure and the value θi, and
add them to the list Li+1.
3. Delete the triangulation Mi and use the set of points Li+1 to generate mesh
Mi+1 with BW and anisotropic mesh generation. Update θ as desired.
The van der Pol oscillator, whose phase portrait is shown in figure 3.7, represents
a good test case because of the non constant curvature and the existence of a peri-
odic orbit. This algorithm performed in practice, figure 3.8 shows an example of a
strongly connected component containing the periodic orbit of the van der Pol equa-
tion computed by the iterative triangulation algorithm. There were, however, several
problems with this method. Points from overlapping neighborhood’s were filtered out
according to a heuristic scheme in [3]. Approximately one third of the total points at-
tempted were filtered out. We were not able to develop a better heuristic for filtering.
The iterative nature made it costly when the triangulations became fine, but this is
precisely when the quadratic approximations become close to true. For unoptimized
and poorly written code the following running time performance was observed on a
sun ultra 10 workstation: 22,069 points ∼ 1 minute, for 76,351 points ∼ 9 minutes
and for 254,662 points ∼ 139 minutes.
3.5.2 Global Point Placement
In sharp contrast to the above method is our current strategy. This strategy is based
on the following intuition. The relative density of points should be related to the
69
curvature.
k(x) =
√|f(x)|2|Df(x)f(x)|2− < f(x), Df(x)f(x) >2
|f(x)|3 (3.15)
Further, that the Delaunay triangulation of the compact polyhedral set we are inter-
ested in, equipped with the Riemannian structure,
gεx(u, v) :=< Dh(x)u,Θ(ε)Dh(x)v >
k(x) ∗ |f(x)|2 (3.16)
will satisfy equation 3.1. And finally, if the vertices are dense enough and form a
centroidal Voronoi diagram with respect to the density k(x), then the triangulation
they form will be the desired Delaunay complex.
In [16] a Monte-Carlo algorithm is described that generates an approximation to
the centroidal Voronoi diagram with prescribed density. We will refer to this algorithm
as GZB.
These ideas produce the following dimension independent procedure.
Anisotropic Triangulation (AT)-Algorithm
1. Use the GZB algorithm to globally collocate a set of n vertices according to the
density function induced by the curvature k(x).
2. Use anisotropic mesh generation to generate a triangulation T of these vertices.
This algorithm has produced good quality triangulations for a variety of flows in
the plane. The process is illustrated with the van der Pol oscillator. The figures
depict an experiment in which Θ(0.2), was used along with 10,000 vertices. The first
figure 3.9 in the series shows the resulting point distribution generated by applying
the GZB algorithm with the density function given by the curvature. The second
figure 3.10 shows a triangulation that results from applying the anisotropic BW al-
gorithm to the points displayed in the previous figure, there are 19,950 triangles.
Figure 3.11 shows the strongly connected path component of the induced map that
70
contains the periodic orbit highlighted within the full triangulation. The last figure
in the series shows the strongly connected component in isolation, which is comprised
of 3,373 triangles.
An important point to mention is that this algorithm can be used to refine an
existing triangulation. Indeed, given a strongly connected component C of triangula-
tion T , like the one shown in figure 3.12, the algorithm can be applied to the region
C alone, thus allowing a larger number of points to be used to refine the structure
of the region in C alone. This procedure may be one way of beating the curse of
dimensionality. An initial mesh is generated using as many points as can be afforded,
the strongly connected components identified and subjected to refinement, possibly
in parallel.
3.6 Independent Patches Approach
An alternative and straightforward strategy, proposed by Andrzej Szymczak, is to
abandon the desire for a global triangulation, in favor of a multivalued map, which in
the end is the object that is mathematically important. The idea is to stitch together
a global multivalued map from locally optimal overlapping polygonal patches. Ideas
along these lines have been used in other computational dynamics problems involving
triangulations, see for instance [29]. This method has been partially investigated by
the author, his advisor and Marcio Gameiro. A balanced quadtree data structure,
was used to keep track of local patches and facilitate the computation of intersections.
The computations were done in the following order. A cutoff angle was chosen. First
cut and balance the cells of a quadtree according to an angle criterion. To decide if a
cell should be quadrasected, a large number of spatially random points were chosen
and if the angle between the vectorfield at these points and the vectorfield at the
center of the box under consideration was larger than the cutoff angle, cut this cell
71
of the quadtree in four and balance. Next generate a flow box triangulation in each
cell of the quadtree, according to the procedure outlined in the section 3.5.1 above.
For each patch, trim away any simplices that are completely outside the boundary of
their cell, but so that the simplices cover their cell. Define the multivalued map in
the following way
1. For simplices inside the same patch the multivalued map remains unchanged.
2. Let s1 and s2 be simplices in adjacent overlapping patches. We say that s1 is
mapped in s2 iff s1 has an exit face and this exit face intersects s2, AND s1 is
in the boundary of its patch.
In several trials, it was found that this method required one or two orders of magnitude
more triangles than other methods with the same resulting isolating block for the
periodic orbit of the van der Pol oscillator.
72
Figure 3.7: Phase portrait for the van der Pol equation
73
Figure 3.8: A strongly connected component containing a periodic orbit, computed bythe IT-algorithm
74
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 3.9: Points placed with the GZB algorithm using the curvature induced densityfunction, for the van der Pol oscillator
75
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 3.10: An anisotropic Delaunay triangulation for the van der Pol oscillator, producedwith the AT-algorithm and the points depicted in figure 3.9
76
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 3.11: Highlighted is the strongly connected component containing the period orbitfor the van der Pol oscillator, the triangulation is the same as that shown in figure 3.10
77
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
Figure 3.12: The strongly connected component containing the period orbit for the vander Pol oscillator
78
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Vita
Erik Miklos Boczko is a graduate of Manhattenville College. After repeating college
algebra and calculus he decided that math was not all that hard if you kept up, and
actually was kind of fun. While working as a lab tech at Rockefeller University a
few years later he took math classes for fun at Hunter College at night. During his
graduate work at Carnegie Mellon University he sat in on as many math classes as
his advisor would allow and became obsessed with both the mysteries of mathematics
and the comforts of its rigour. Despite the warnings of many people, he could not
resist the temptation of remaining a graduate student for a while longer. After a brief
stint in ”paradise”, otherwise know as Cleveland, and with the help of his advisor,
he now resides in Nashville, where he practices ”informatics” and enjoys the sound
of his daughter laughing.
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