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\markboth{\hfil Dynamics at infinity \hfil EJDE--2001/22}
{EJDE--2001/22\hfil Efthimios Kappos \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 22, pp. 1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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%
Dynamics of polynomial systems at infinity
%
\thanks{ {\em Mathematics Subject Classifications:} 34C11, 34D23, 37B30, 52B12.
\hfil\break\indent
{\em Key words:} Dynamics on manifolds,
Newton polytopes, dissipative systems, peaking, \hfil\break\indent
Poincar\'e and Bendixson spheres.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted August 16, 2000. Published April 4, 2001.} }
\date{}
%
\author{ Efthimios Kappos }
\maketitle
\begin{abstract}
The behaviour of dynamics `at infinity' has not received much
attention, even though it was central to Poincar\'e's analysis
of qualitative dynamics.
Poincar\'e's `sphere' is actually a projective
plane and our treatment of dynamics at infinity in more than
two dimensions requires the use of $\mathbb{R} P^n$.
In control theory, `strange' transients have been reported
by Kokotovi\'c and Sussmann, where they
go by the name of `peaking behaviour'.
These have a simple explanation when we consider the dynamics
on the Poincar\'e compactification of state space.
In this work, we propose to give an analysis of
the issues arising in trying to examine the dynamics at infinite
radius for dynamical systems in {\em arbitrary} dimension.
Use is made of the Newton polytope and of recent results on
principal parts of vector fields.
\end{abstract}
\newtheorem{prop}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}
\newtheorem{remark}{Remark}
\section{Introduction}
\label{sec1}
The {\em `behaviour at infinity'} of a dynamical system is crucial to
an understanding of its global dynamics.
Before the development of the theory of dynamical systems, the
qualitative approach of its main pioneer, Henri Poincar\'e, involved
defining dynamics on a compact state space that is in fact the
projective plane, see~\cite{poinc}.
For a variety of reasons, the subsequent development of dynamical
systems paid little attention to the question of interesting, or
pathological dynamics {\em `far away'}
(exceptions are references~\cite{basir1, lat-sar, sarip} and a few others.)
Perhaps because many practical systems are
`dissipative,' attention has focussed on `local' problems where the
theory of normal forms plays a major role.
Still, the subject is treated in a limited way in some of the main
references, such as the book~\cite{andr} of the Andronov school, and in
Lefschetz~\cite{lef}.
Modern texts completely ignore this aspect, an exception being Perko~\cite{perko}.
Recently, in the context of nonlinear control systems having a
certain diagonal structure, the phenomenon of {\em peaking} was
observed which involves a family of trajectories originating
arbitrarily close to one of the invariant manifolds of a stable
equilibrium point that have arbitrarily large transients (see
Section~\ref{sect-peak}.) It therefore seems appropriate to
re-examine techniques for systematically analyzing trajectories
far away and to re-visit the classical subject of the Poincar\'e
and Bendixson spheres.
In this paper we set up a general method for obtaining dynamics on
compact manifolds whose trajectories are almost everywhere
in one-to-one correspondence
with the trajectories of a flow in Euclidean space.
We make an effort to update the classical treatments in~\cite{lef}
and \cite{andr} and to go beyond them in several respects.
% ------------------- Setting ---------------------
We mainly consider dynamical systems arising from a vector field defined
in euclidean $n$-dimensional space $\mathbb{R}^n$:
\begin{equation}
\dot x = F ( x) ,
\label{nl1}
\end{equation}
where $x = {( x_{1} , \ldots ,x_{n} )}^{T}$ and
$F = {( F_{1} , \ldots ,F_{n} )}^{T}$.
{\em We do not assume that the vector field is} {\bf complete}.
The main class of vector fields we shall consider is the finitely generated module
of {\bf polynomial vector fields} over $\mathbb{R} [ x_{1} , \ldots ,
x_{n} ]$, the ring of polynomials in $n$ variables.
We denote by $\deg F_{i}$ the {\em total degree} of $F_i$ and use the {\em
multi-index notation}
\[ x ^{\alpha} = x_{1}^{\alpha _{1}} \cdots x_{n}^{\alpha _{n}} , \quad
\text{so that} \quad \deg x^{\alpha} = | \alpha | = \sum _{i} \alpha _{i} . \]
The standard basis in $T _x \mathbb{R} ^n$ will be denoted by
${\bf e} _i , i=1, \ldots , n$ (rather than $\frac{\partial}{\partial x_i}$.)
We then use the notation
\[ x^{\alpha + a {\bf e}_i} = x_{1}^{\alpha _{1}} \cdots x_{i}^{\alpha _{i} + a}
\cdots x_{n}^{\alpha _{n}} . \]
% Poinc draft
The notation $( x_{1} , \ldots , \widehat{x_{i}}, \ldots , x_{n}
)$ will denote the $(n-1)$-dimensional array with the $i$th
element $x_{i}$ omitted.
% modelling issues; polynomial vfs vs transcendental; infinty of # eq pts
\section{Bendixson one-point compactification}
The obvious way to attempt to define dynamics on a compact state
space is to use stereographic projection to define a vector field on
the one-point compactification of $\mathbb{R}^n$, namely the sphere $S^{n}$.
Let us assume that the two $n$-dimensional manifolds $\mathbb{R} ^n$ and
$S^n$ are embedded in $\mathbb{R} ^{n+1}$ in such a way as to have $\mathbb{R}^n$
be the tangent plane to the sphere $S^{n}$ at the {\em `north
pole'} $\{ Z=1 \}$; the case $n=2$ helps in the visualization, see
Figure~\ref{bend-sph}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{bend-sph.eps}
\caption{The Bendixson sphere compactification.}
\label{bend-sph}
\end{figure}
We use the coordinates $X_{1}, \ldots , X_{n} , Z$ in $\mathbb{R} ^{n+1}$
and hence for the sphere $S^{n}= \{ \sum X_{i}^{2} + Z^{2} = 1
\}$. Stereographic projection from the south pole sends points at
infinity to the south pole. Now if we give the tangent plane $\{ Z
= -1 \}$ the coordinates $\xi _{1} , \ldots , \xi _{n} , \zeta$
and use projection from the north pole, we get the change of
coordinates
\begin{equation}
\xi_{i} = \frac{4 x _{i}}{ \sum _{j=1}^{n} x _{j}^{2}}
\quad \text{and conversely} \quad
x_{i} = \frac{4 \xi _{i}}{ \sum _{j=1}^{n} \xi _{j}^{2}}
\label{bend1}
\end{equation}
The elementary proof of this is given in the Appendix.
Taking the derivative with respect to time of equation~\ref{bend1}
gives
\begin{equation}
\dot \xi _{i} = \frac{1}{4}( \sum _{j \ne i} \xi
_{j}^{2} - \xi _{i}^{2} ) F_{i}
- \frac{1}{2} \xi _{i} \sum _{j \ne i} \xi _{j} F_{j}
\label{bend2}
\end{equation}
where
\[ F_{i} = F_{i} ( \frac{4 \xi}{\sum _{k} \xi
_{k}^{2}} ) . \]
This gives a vector field {\em away from the point} $\xi = 0$; we
shall denote it by $G$.
We then have that the above transformation gives a one-to-one map
between trajectories
of the system~\ref{nl1} and the trajectories of the vector field
in the complement of the origin in both spaces.
In order to obtain a well-defined system on the sphere, we need
to scale the vector field in eq.~\ref{bend2} so that is is defined
at the origin $\xi = 0$.
\begin{definition}
The class $\mathcal{N} \subset \mathcal{X} ( \mathbb{R}^n )$ of {\bf normalizable} dynamics is the
subset of the set of vector fields $F$ in $\mathbb{R}^n$ for which a function
$\rho : \mathbb{R}^n \to \mathbb{R}$ exists such that, for the transformed vector field $G$,
the limit
\[ \lim _{\xi \to 0} \rho ( \xi ) G ( \xi ) \]
is defined and the direction fields of $\rho G$ and $G$ coincide,
where $G$ is defined.
\end{definition}
The class of {\em polynomial vector fields} is normalizable.
The normalizing function can be taken to be $\rho ( \xi ) = R^{2N}$,
where $R$ is the norm of the vector $\xi$, $R^{2} = \sum _{k} \xi _{k}^{2}$
and for some positive $N$, possibly smaller than $M$,
where
\[M = \max _{i} \{ \deg F_{i} \} . \]
Since the two coordinate patches $U_{1} = \{ Z > - 1 \}$ and $U_{2} =
\{ Z < 1 \}$ cover the sphere, we have shown that
\begin{prop}
For any normalizable vector field $F$ in $\mathbb{R}^n$, there is defined a
direction field in the sphere $S^{n}$ topologically orbitally
equivalent to $F$ on the open subset $\{ Z > -1 \}$ of $S^{n}$.
\end{prop}
% ---------- Example: ----
\begin{example}
An elementary example is the non-complete vector field
\[ \dot x = x^{2} \]
in $\mathbb{R} ^{1}$ with a degenerate equilibrium at the origin.
The dynamics on $U_{2}$ are given by
\[ \dot \xi = - \xi ^{2} ( \frac{16}{\xi ^{2}} ) = - 16 \]
which means a vector field on the sphere $S^{1}$ with a {\em single}
equilibrium point.
\end{example}
\subsection*{Dissipativeness and Lyapunov functions}
The main class of dynamics in $\mathbb{R}^n$ of practical interest is the class
of {\bf dissipative dynamics}, i.e. those with a globally
asymptotically stable compact attracting set.
\begin{prop}
The system~(\ref{nl1}) is dissipative iff the point at infinity on
the sphere $S^{n}$ is a repeller.
\end{prop}
It is sometimes (but certainly not always) easier to check the
local stability of an equilibrium point rather than to come up with a
global Lyapunov function.
Thus, the above Proposition can be of practical use.
Quite often, though, the point at infinity is a highly degenerate
equilibrium, whose stability is hard to establish.
{\em Dissipativeness} can be defined by the existence of a global,
compact attractor $\mathcal{A}$ or by the existence of a {\em proper
Lyapunov function} strictly decreasing towards the value at the
compact set. The quotient flow obtained by collapsing the
attractor $\mathcal{A}$ to a point is a gradient-like flow with a single
attracting equilibrium; the repeller at infinity is the {\em
complementary repeller} of $\mathcal{A}$ in the terminology of the Conley
index (see~\cite{cigb}.)
\begin{proof}
By basic Conley index theory, the complementary attractor of a
repelling equilibrium at the North pole on the sphere is a compact
set. Thus the {\em `if'} direction follows.
Next note that, in the complement of the two poles, the change of
coordinates of equation~(\ref{bend1}) is a diffeomorphism. Thus,
the derivative of a Lyapunov function $V$ along the trajectory is
the result of evaluating an exact one-form along a vector field,
$\frac{d V}{dt}= dV ( F (x))$, which is clearly independent of the
coordinates chosen. (Note that we are considering the {\em
unscaled} version of the vector field in the patch $U_{2}$.) Since
away from the compact attractor, we have
\[ \frac{dV}{dt} < 0 \]
and since $V$ is proper, we get that the south pole is a repeller.
\end{proof}
% ------- the Lorenz equations ------------------
Let us look at a familiar example.
\begin{example}[The Lorenz dynamics]
\begin{equation}
\label{lorenz}
\begin{array}{rcl}
\dot x & = & \sigma ( y - x ) \\
\dot y & = & \rho x - y - x z \\
\dot z & = & - \beta z + x y
\end{array}
\end{equation}
where the parameters $\sigma , \rho , \beta$.
Since the divergence
$ \mathop{\rm div} F = - \sigma - 1 - \beta < 0$,
the attracting set cannot be of dimension three.
\begin{prop}
There is an increasing sequence of compact sets $K_i$ (so $K_{i+1} \supset
K_i$) such that $\lim K_i = \mathbb{R}^n$ and each $K_i$ is positively invariant for
the flow of the Lorenz system.
\end{prop}
\begin{proof}
Lorenz, see~\cite{sparrow}, Appendix C, uses the function
\[ V(x,y,z)=\frac{1}{2} ( \rho x^{2} + \sigma y^{2} + \sigma {(z - 2
\rho )}^{2}) \]
which gives
\begin{eqnarray*}
\frac{dV}{dt} = \sigma ( - \rho x^{2} - y^{2} - \beta z^{2} + 2
\rho \beta z ) \\
= \sigma ( - \rho x^{2} - y^{2} - \beta {( z - \rho )}^{2} +
\beta \rho ^{2} )
\end{eqnarray*}
which is negative as soon as the sum of squares dominates the constant
term.
The function $V$ is thus a {\em Lyapunov function} outside a compact
set and its {\em
sub-level sets} $\{ V (x ) \le k_{i} \}$ supply the desired compact
sets, for appropriate $k_{i}$.
\end{proof}
The Lorenz equations are thus dissipative for {\em all} (positive)
parameter values and thus can be defined as dynamics on the sphere $S^3$.
Also note that the levels of $V$ are clearly spheres far away.
\end{example}
\subsection*{State spaces other than euclidean ones}
% Example: the pendulum
The natural state space of a dynamical systems is often a manifold.
In cases where this manifold is a product of some euclidean space with a
compact manifold, the compactification procedure still works, by only
compactifying the euclidean summand.
The simple pendulum equations,
\begin{equation}
\begin{array}{rcl}
\dot x & = & y \\
\dot y & = & - \gamma y - \sin x ,
\end{array}
\label{pendul}
\end{equation}
for example, live in the space $\mathbb{R} \times S^{1}$.
Here, the one-point compactification gives a two-sphere, $S^{2}$.
Care must be exercised to take a Lyapunov function that is also
$x$-periodic.
\section{Poincar\'e compactification}
% ---- Rem : Genericity ---------------
The key to the Poincar\'e compactification is to consider the
state space $\mathbb{R}^n$ as the affine plane $\{ Z=1 \}$ in $\mathbb{R} ^{n+1}$
and to extend the vector field on $\mathbb{R}^n$ to a {\bf direction field}
in $\mathbb{R} P^n$ (see Figure~\ref{poinc-f1}.)
\begin{figure}[ht]
\centering
\includegraphics[width=0.6\textwidth]{poin1.eps}
\caption{The Poincar\'e compactification.}
\label{poinc-f1}
\end{figure}
Since a whole $(n-1)$-dimensional space of infinities is used, the
dynamics at infinity tend to be considerably simpler that for the
Bendixson one-point compactification.
\subsection*{Compactifying Dynamics to the Projective space}
The affine space $\mathbb{R}^n$ gives a coordinate patch
\[ \{ Z = 1 \} \]
of the projective space $\mathbb{R} P^{n}$, whose homogeneous coordinates will
be written
\[ [ X_{1} ; \ldots ; X_{n} ; Z ] . \]
The space
\[ \{ Z = 0 \} \]
provides a collection of {\em `lines at infinity'} equivalent to $\mathbb{R}
P^{n-1}$.
The other $n$ coordinate patches correspond to $\{ X_{i} \ne 0 \}$.
Let us present the
case of the $i$th coordinate patch, $\{ X_{i} = 1 \}$.
From the equality of homogeneous coordinates in the overlap,
\[ [ x_{1}; \ldots ; x_{n} ; 1 ] = [ X_1 ; \ldots ; X_{i-1} ; Z ; X_{i+1} ; \ldots ; X_{n} ] \]
we obtain by differentiation the vector field
\begin{equation}
\begin{array}{lcl}
\dot X_1 & = & Z ( F_1 - X_1 F_i ) \\
\cdots & & \cdots \\
\dot X_{i-1} & = & Z ( F_{i-1} - X_{i-1} F_{i} ) \\
\dot Z & = & - Z^{2} F_{i} \\
\dot X_{i+1} & = & Z ( F_{i+1} - X_{i+1} F_{i} ) \\
\cdots & & \cdots \\
\dot X_{n} & = & Z ( F_{n} - X_{n} F_{i} )
\end{array}
\label{poin-eq1}
\end{equation}
where each vector field component is expressed in the new coordinates
\[ \tilde F_{i} ( X_1 , \ldots , Z, \ldots X_n ) =
F_{i} ( \frac{X_{1}}{Z}, \ldots , \frac{X_{i-1}}{Z} , \frac{1}{Z},
\frac{X_{i+1}}{Z} , \ldots \frac{X_{n}}{Z} ) \]
and is hence a Laurent polynomial.
The above equations establish the equivalence of the
dynamical systems on the overlap $\{ X_{i} \ne 0, Z \ne 0 \}$ of the
two coordinate patches in $\mathbb{R} P^{n}$.
As it stands, the dynamical system above is not defined for $Z=0$.
The next step is thus to obtain if possible a well-defined {\em vector field} in
$\mathbb{R} P^{n}$ from $F$ by some kind of scaling or normalization.
In the case of polynomial vector fields, the obvious (and familiar) solution
is to multiply the right-hand sides of equation~\ref{poin-eq1}
by an appropriate power of $Z$ to obtain a polynomial vector field, call it $G_i$ (see, for
example,~\cite{perko} or \cite{andr}.)
If we scale by an {\em even} power, $Z^{2k}$, we say that {\em the scaling is even}
and we can define a vector field in $\mathbb{R} P^n$ by patching the vector fields defined in
the $(n+1)$ patches along small neighbourhoods of the codimension-two sets
$\{ x_i =1 , Z=1 \}$, where the vector fields coincide.
Let us examine the process of transforming the vector field in more detail,
with the aim of obtaining information about the {\em global dynamics} on $\mathbb{R} P^n$ and
to point out an important modelling issue motivated by the notion of {\em
genericity} in dynamical systems.
\subsection*{Newton Polytopes and Normalization}
We assume $F_{i} \in \mathbb{R} [ x_{1} , \ldots , x_{n} ]$, $1 \le i \le n$.
We work in the $i$th coordinate patch $\{ X_{i} = 1 \}$.
Let us define the following map for monomials:
\begin{equation}
c_\alpha x^{\alpha} \mapsto ( \alpha , c_{\alpha} ) \in \mathbb{Z} ^n \times \mathbb{R} ^n .
\end{equation}
We shall think of the image as a point $\alpha$ on the integer lattice of the
first quadrant of $\mathbb{R} ^n$, with the coefficient $c_{\alpha}$ as a label affixed at
the point.
The map $c_\alpha x^{\alpha} \mapsto \alpha \in \mathbb{Z} ^n$ is the {\bf exponent map}.
Now the change of coordinates between the different affine charts gives
an {\bf involution} (a linear transformation $A$ such that $A^2 = I$)
in the exponent map domain, given by the matrices
\begin{equation}
A^i = \left( \begin{array}{ccccc}
1 & 0 & \cdots & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
& \cdots & & \cdots \\
-1 & -1 & -1 & -1 & -1 \\
& \cdots & & \cdots \\
0 & \cdots & \cdots & 0 & 1
\end{array} \right) ,
\end{equation}
where $ x= ( x_1 , \ldots , x_{i-1}, x_i , x_{i+1}, \ldots x_n ) $ is meant to transform to
\[ X = ( X_1 , \ldots , X_{i-1} , Z , X_{i+1} \ldots , X_n ) . \]
Thus the monomial $c_{\alpha} x^{\alpha}$ is mapped to $c_{\alpha} X^{A \alpha}$.
For example, $3 x_1^2 x_2^4 x_3$ maps, in the $\{ X_2 = 1 \}$ patch, to
$ 3 X_1^2 Z^{-7} X_3$.
It is clear that only $Z$ appears with a non-positive exponent, namely $- | \alpha |$.
Applying the exponent map to each of the monomials of
\[ x_1 \cdots \widehat x_j
\cdots x_n F_j ( x_1 , \ldots , x_n ) \] ($F_j$ polynomial) we get
the {\em support} of $F_j$, $\mathop{\rm supp} F_j$, of the non-zero alphas.
The {\bf Newton polytope} $\Gamma$ {\bf of the polynomial vector
field} $F$ is the convex hull of $\cup _j \mathop{\rm supp} F_j$. Clearly,
$\Gamma$ is a compact convex subset of the {\em first quadrant}
$\{ x_i \ge 0 \; ; \; \forall i \}$. The shifting involved in this
definition (see~\cite{brun}) is special to vector fields; for a
polynomial, one uses the exponent map directly;
Koushnirenko~\cite{koushn} has given definitions of Newton
polytopes for power series and for Laurent polynomials as well;
these are not needed here. Even though it clearly depends on the
chosen coordinates, the Newton polytope of a polynomial $p$
contains a surprising amount of information about the
singularities of $p$ (see Arnol'd et.al.~\cite{via:dsv}.)
A {\em support hyperplane} of $\Gamma$ is a hyperplane maximizing
the value of some one-form $\beta$ on $\Gamma$.
The {\em facets} $\gamma$ of the boundary of the Newton polytope
of a vector field $F$ are intersections of $\Gamma$ with a
supporting hyperplane; they are compact, convex polytopes of
dimension at most $n-1$. The union of the facets whose support
hyperplane co-vectors have negative entries form the {\bf Newton
diagram} $\mathcal{N}$ of the vector field $F$. The restrictions
$F_{\gamma} = \sum _{\alpha \in \gamma} c_{\alpha} x^{\alpha}$ are called
the {\bf quasi-homogeneous components} of $F$.
%The restrictions $F_{\gamma}$ for $\gamma$ in the Newton polyhedron are
%called {\bf dominant}
Lastly, we use the fact that a linear transformation $A$ of vectors gives a
transformation by the inverse $A^{-1}$ for {\em co-vectors} and the fact that
the matrices $A$ above are involutions to obtain the transformation $\beta A$
for the co-vectors of the supporting hyperplanes.
%Define the $i$th {\bf shift} $\sigma _i x^{\alpha} = x^{\alpha + 1 {\bf e} _i}$.
\begin{prop}
Let $\Gamma$ be the Newton polytope of the vector field $F$.
In the $i$th patch, the Newton polytope of the transformed vector field of
equation~\ref{poin-eq1} is exactly equal to the affine transformation of $\Gamma$ by
$A^i$, followed by a shift in the $i$th direction (found from the maximal degree of
the monomials in $F$.)
Hence, the {\bf Newton diagram} of the transformed vector field is the transform of the
union of facets of $F$ with support covectors $\beta$ such that $\beta A^i < 0$.
\end{prop}
Note the convenience of the above Proposition in being able to check the single
Newton polytope of $F$, instead of computing all the transformed ones.
The notion of a {\bf principal part} of a vector field at an equilibrium
(the terms of the vector field mapping to the Newton diagram)
is crucial to the generalization
of the classical Grobman-Hartman Theorem by Brunella and Miari~\cite{brun}.
Vector fields with the same principal parts have locally equivalent dynamics.
A condition that makes the principal part concept useful is the absence of dynamics
of the centre-focus type (roughly, in the plane, we need a trajectory tending to the
equilibrium at a well-defined angle.)
We are interested in finding the {\em principal parts at infinity} of the vector field $F$.
We assume the origin is an equilibrium of the transformed vector field.
\begin{cor}
In dimension two, assume that, in the $i$th patch, the origin is a nondegenerate equilibrium,
in the sense of~\cite{brun}.
Then the vector field $G_i$ is topologically equivalent
to its $A^i$-transformed (and appropriately shifted) principal part modulo centre-focus.
\end{cor}
\begin{remark}
Computing convex hulls is a classical problem in Computational Geometry (\cite{prepa},
\cite{deberg}.)
In dimension two, it is even implemented in software such as {\tt maple} and {\tt matlab}.
\end{remark}
% Define centre-focus
% Do an example.
% ---------recall Localization-----------------
\begin{proof}
Let $\mathbf{1}$ be a vector of ones and use $( \alpha , k )$ for the exponent of the monomial
$x^{\alpha} \mathbf{e} _k$ in the $k$th component of a vector field.
Let MNP stand for the modified Newton polytope map which, using this notation, is defined by
\[ x^{\alpha} \mathbf{e} _k \mapsto x^{\alpha + \mathbf{1} - \mathbf{e} _k} e_k \]
or
\[ ( \alpha , k ) \mapsto ( \alpha + \mathbf{1} - \mathbf{e} _k , k ) . \]
Now the vector field in the $i$th patch defined in equation~\ref{poin-eq1} maps
$( \alpha , k )$ to $( A \alpha , k )$; to get the MNP, we distinguish the two
cases: $k=i$ and $k \ne i$.
Since we shall later normalize by a power of $Z$, we ignore the factor $Z$ common to
all components.
In the former case, we get
\[ ( \alpha , k ) \mapsto ( A \alpha + \mathbf{1} - \mathbf{e} _k , k ) . \]
For $k=i$, we get the monomial $( \alpha , i )$ contributing to both the $i$th and
the $k$th component of the vector field, in the first case giving
\[ ( \alpha , i ) \mapsto ( A \alpha + \mathbf{1} - \mathbf{e} _i + \mathbf{e} _i , i ) = ( A \alpha + \mathbf{1} , i ) \]
(because we multiply by $Z$) and, in the second case
\[ ( \alpha , i ) \mapsto ( A \alpha + \mathbf{1} - \mathbf{e} _k + \mathbf{e} _k , k ) = ( A \alpha + \mathbf{1} , k ) \]
(because of the $X_k$ multiplying $F_i$ in equation~\ref{poin-eq1}.)
It is easy to check that $A \mathbf{1} = \mathbf{1} - (n+1) \mathbf{e} _i $, $A \mathbf{e}_k = \mathbf{e} _k - \mathbf{e} _i$
for $k \ne i$ and $A \mathbf{e} _i = - \mathbf{e} _i$.
We now have that
\[ A ( \alpha + \mathbf{1} - \mathbf{e} _k ) = A \alpha + \mathbf{1} - \mathbf{e} _k - n \mathbf{e} _i , k \]
and
\[ A ( \alpha + \mathbf{1} ) = A \alpha + \mathbf{1} - n \mathbf{e} _i \]
and hence the involution $A$ maps the MNP of $F$ to the MNP of the transformed
vector field, except for the shift by $n \mathbf{e} _i$, which is immaterial, since we are going
to scale anyway by a power of $Z$.
The proposition now follows from the transformation rule for covectors,
under the stated conditions.
\end{proof}
The Corollary is immediate from the results of Brunella
and Miari.
%---------------------
Just as it has now become common to expect {\em local dynamics} to be of {\em low
codimension}, we can require the dynamics {\bf at infinity} to be of low
codimension as well.
The results of Brunella et.al. can be combined with the above setting
to examine when the principal parts
of vector fields at infinity are generic.
The details are left to an extended version of this work.
\section{Examples}
\subsection*{Gradient dynamics with two finite minima}
The examination of relations between properties of a polynomial, such as its
degree, and the number and nature of its critical points is an interesting and
non-trivial problem.
It turns out that to do the counting properly, one needs a definition of
{\bf critical points at infinity} for functions $f : \mathbb{R} ^n \to \mathbb{R}$ (\cite{durfee}.)
Durfee gives {\em five} different definitions, which he then shows to be equivalent.
Through our dynamical viewpoint, we approach this question via the {\em gradient vector
field} obtained from the given function.
Let us take a concrete example (adapted from~\cite{davies}.)
It is the polynomial
\[ f(x,y) = {( x^2 y - x -1)}^2 +{( x^2 -1)}^2 \]
which is easily seen to have just two (local) {\bf minima}, at
$(-1,0)$ and $(1,2)$, and {\bf no other (finite) critical points}!
In terms of the gradient flow
\[ - \nabla f (x,y) , \]
the gradient dynamics has two attractors and no other equilibria.
We shall examine the {\bf global phase portrait} of this system obtained from the
Poincar\'e compactification we have described.
Clearly, on the compact state space $\mathbb{R} P^2$, we must have more equilibrium points,
by basic Morse theory.
The phase portrait of the system dynamics $\dot x = - \nabla f(x)$ is shown in
Figure~\ref{phfin}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.6\textwidth]{phf2.eps}
\caption{Phase portrait of the two-minimum system.}
\label{phfin}
\end{figure}
The Newton polytope of the gradient vector field is shown in Figure~\ref{fig1}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{np1.eps}
\caption{Newton polygon of the two-minimum example.}
\label{fig1}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\textwidth]{cvh5.eps}
\caption{Transformed Newton polygon centred at $X=1$.}
\label{fig2}
\end{figure}
In the $y$-patch, $(x,y) \mapsto ( \frac{X}{Z} , \frac{1}{Z} )$,
we find that we have three additional equilibria, at $( \pm
\sqrt{2}, 0)$ and $(0,0)$. The pair of equilibria at $( \pm
\sqrt{2}, 0)$ are repellers, while the one at the origin is
degenerate (so that the Corollary is not applicable to it.) The
instability can be checked by effecting the shift $(X,Z) \mapsto
(X \pm 1 , Z)$ on the Newton polygon, obtaining the polygon shown
in Figure~\ref{fig2}, and checking that the equilibrium is
nondegenerate with unstable linear principal part. The three
`asymptotic' curves visible in the phase portrait of
Figure~\ref{phfin} become {\em unstable} (for the two outer
curves) and {\em stable} (for the middle one) {\em manifolds} of
the degenerate equilibrium (on the positive-$Z$ side.) On the
other side (as $y \to - \infty$) there is a single unstable curve.
It appears from the simulations that there is then a connecting
orbit (homoclinic in projective space) from the origin to itself
(in the $\{ y = 1 \}$-patch.) Its existence has not been shown
here, however.
The $X$-axis is invariant, with dynamics
\[ \dot X = X^2 ( X^2 -1 ) . \]
Notice that the two systems can be `sewn together' along the line
$\{ y = 1 \} = \{ Z = 1 \} $.
The scaling we use is even, so the phase portrait at the $y$-infinity
is patched to the finite phase portrait without a sign change.
In the $x$-patch, $(x,y) \mapsto ( \frac{1}{Z}, \frac{Y}{Z} )$, there are three
additional equilibrium points, one at the origin and two at
$( 0, \pm \frac{1}{\sqrt{2}} )$.
%Using {\tt maple}, one can shift to the non-zero equilibria and check that they are
%nondegenerate, in fact they are attractors.
\subsection*{Peaking behaviour}\label{sect-peak}
The following example demonstrates {\bf peaking behaviour} in a
so-called upper
triangular system, where the diagonal systems are both linear and
asymptotically stable.
\begin{equation}
\label{peak}
\begin{array}{rcl}
\dot x & = & - x + x^2 y \\
\dot y & = & - k y, \; k > 1
\end{array}
\end{equation}
The origin is linearly stable and hence locally stable.
The problem is that the quadratic term $x^{2} y$ prevents some
trajectories from converging to zero fast enough.
In fact, for any bound $K$, there is a trajectory whose $\omega$-limit
set is $0$ and whose distance from the $y$ axis tends to zero as $t
\to - \infty$, but such that its $x$ coordinate exceeds $K$ for some
intermediate time.
Now the dynamics on the $\{ X = 1 \}$ plane, after scaling by $Z^{3}$,
are given by
\begin{equation}
\begin{array}{rcl}
\dot Z & = & Z ( Z^{2} - Y ) \\
\dot Y & = & - Y ( (k-1) Z^{2} + Y ) ,
\end{array}
\label{peak-eq}
\end{equation}
giving a degenerate equilibrium at zero.
It is easily checked that the $Z$-axis is invariant and unstable and
that the $Y$-axis is also invariant, with dynamics $\dot Y = - Y^{2}$.
In fact, the parabola
\[ Y = (k+1) Z^{2} \]
is also invariant and stable and thus the equilibrium point
exhibits a mixed saddle-stable-unstable dynamical be\-ha\-vi\-our.
This parabola is of course the image of an invariant hyperbola in
the original plane $\{ Z = 1 \}$, see Figure~\ref{peak-f2}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{peak1.eps}
\caption{Phase portrait in the $\{ x=1 \}$ plane}
\label{peak-f2}
\end{figure}
The saddle-like {\em `sector'} is thus responsible for the peaking
be\-ha\-vi\-our. The full details of the phase portrait on $\mathbb{R}
P^{2}$ are not difficult to obtain, but we omit them here. It is
also possible to generalize this peaking example by taking {\em
cross-terms} more general than $x^2 y$. Details will be given
elsewhere.
\section{Conclusion}
We have presented but the bare elements of a theory of global (polynomial) dynamics,
combining a generalization of the classical Poincar\'e compactification with the
powerful Newton polytope method, so useful in singularity theory and algebraic geometry.
We have not touched on the topological information provided by the Whitney-Morse theory
of relations between the topology of the state manifold and the indices of the
equilibria of the vector field on it.
As the peaking example shows, a study of the compactified dynamics
is sometimes necessary to clarify apparently strange transient
dynamical behaviour. The two-minimum example shows that, even
within the class of polynomial systems, expectations on the
dynamics based on the intuition derived from compact state
manifolds are occasionally wrong (two minima and no saddles).
Compactification can resolve these ambiguities. It is clear that
more examples need to be studied and that the genericity aspects
must be more extensively addressed.
\section{Appendix}
Here we derive Equation~(\ref{bend1}).
In ${\mathbb R}^{n+1}$, write $\mathbf{v} =( \mathbf{x} ,Z)$, with $\mathbf{x} \in \mathbb{R}^n$
and $Z \in \mathbb{R}$. The unit sphere is
\[ S^n = \{ \mathbf{v} ; | \mathbf{v} | = \sqrt{| \mathbf{x} | + Z^2} = 1 \} \]
and the hyperplanes tangent to the North and South poles are
\[ \mathcal{P} _N = \{ \mathbf{v} ; Z=1 \} \; \text{and} \; \mathcal{P} _S = \{ \mathbf{v} ;
Z=-1 \} . \] The stereographic projection from the South pole
sends the point $\mathbf{v}$ to the point, $\mathbf{p} _N$, of intersection of
$S^n$ with the line
\[ \ell _N = \{ t \mathbf{v} + (1-t) (- \mathbf{e} _{n+1} ) , t \in \mathbb{R} \}
= \{ t ( \mathbf{x} ,0) + (2t-1) \mathbf{e} _{n+1} , t \in \mathbb{R} \} ,\] where
$\mathbf{e}_{n+1}$ is the unit vector in the $Z$-direction. We thus have
\begin{equation*}
{| t ( \mathbf{x} , 0) + (2t-1) \mathbf{e} _{n+1} |}^2 = t^2 {| \mathbf{x} |}^2 +
{(2t-1)}^2 | \mathbf{e} _{n+1} |^2 = 1 ,
\end{equation*}
which has the non-trivial solution
\[ t = \frac{4}{4+ \mathbf{r} ^2}, \; \mathbf{r} = | \mathbf{x} |. \]
Repeating for the projection from the North pole, we get a point
$\mathbf{p} _S$ satisfying
\begin{equation*}
{| s ( \xi , 0) - (2s-1) \mathbf{e} _{n+1} |}^2 = s^2 {| \xi |}^2 +
{(2s-1)}^2 | \mathbf{e} _{n+1} |^2 = 1 ,
\end{equation*}
giving
\[ s= \frac{4}{4+ \mathbf{r} '^2}, \; \mathbf{r} ' = | \xi | . \]
The change of coordinates means that
\begin{multline*}
\mathbf{p} _n = \frac{4}{4+ \mathbf{r}^2} ( \mathbf{x} ,0) + ( \frac{8}{4 + \mathbf{r} ^2} -1 )
\mathbf{e}_{n+1} = \\
= \frac{4}{4+ \mathbf{r}'^2} ( \xi ,0) - ( \frac{8}{4 + {\mathbf{r} '} ^2} -1 )
\mathbf{e} _{n+1} = \mathbf{p} _S
\end{multline*}
and, equating the $Z$-components, we check that
$\mathbf{r} \mathbf{r} ' = 4 $
and therefore
\[ \mathbf{x} = \frac{4+ \mathbf{r}^2}{4+ {\mathbf{r} '} ^2} \xi = \frac{4}{{\mathbf{r} '} ^2} \xi .
\]
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\noindent{\sc Efthimios Kappos } \\
Department of Applied Mathematics \\
University of Sheffield, Sheffield, S3 7RH, U.K. \\
e-mail: E.Kappos@shef.ac.uk
\end{document}