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{\nopagenumbers
~ \vskip 2 truecm
\centerline{\bf An equivariant branching lemma for relative equilibria}
\footnote{}{{\tt \giorno }}
\vskip 2 truecm
\bigskip\bigskip\bigskip
\centerline{Giuseppe Gaeta}
\medskip
\centerline{\it Dipartimento di Fisica,}
\centerline{\it Universit\`a di Roma, 00185 Roma (Italy)}
\bigskip
\centerline{\tt giuseppe.gaeta@roma1.infn.it}
\vskip 3 truecm
{\bf Summary} The ``equivariant branching lemma'' of Cicogna and
Vanderbauwhede is extended to consider the branching of relative
equilibria via bifurcation in equivariant dynamical systems. The
results rests on projection to group-orbit space.
\vfill\eject}
\pageno=1
{\bf 1. }ÊThe Equivariant Branching Lemma (EBL) is a fundamental tool in
equivariant bifurcation theory \ref{1-5}; this was first formulated
independently by Cicogna \ref{6,7} and Vanderbauwhede \ref{8} for the
case of bifurcation of stationary solutions, and then extended in a
number of ways, in particular to include Hopf bifurcation \ref{9} and
problems whose symmetry corresponds to a nonlinear group action
\ref{10}.
The basic idea behind the EBL in its various versions is to use the symmetry
properties of the system (or class of systems) under study to guarantee the
dynamical invariance of a lower-dimensional space or manifold, and to apply on
the restriction of the initial system to this invariant space standard (usually
topological) tools to guarantee the existence of a bifurcating branch of
solutions; thus any version of the EBL is strictly connected to a suitable
version of the Reduction Lemma (RL) \ref{4}.
In the present short note we want to remark that a simple use of a rather old
result by Michel \ref{11} allows to obtain, under certain condition, am
EBL for the bifurcation of branches of {\it relative equilibria} out of
stationary solutions.
Relative equilibria -- i.e. group orbits which are invariant under the
dynamics -- received in recent years extended attention, also in the
context of bifurcation theory \ref{12-14}, leading to a good
understanding of them; I do not claim the result given here is more
powerful than those which are present in the literature, but its
derivation -- and the check that a given system satisfies the assumptions
required for its validity -- is so simple that I think it can however be
of interest to report it.
\bigskip
{\bf 2. }ÊIn his work \ref{11} motivated by the theory of strong
interactions \ref{15-17}, Michel proved a remarkable general theorem,
valid for the action of compact groups on finite dimensional manifolds.
His investigation was naturally set in the variational frame, but he
actually proved results on general equivariant vector fields, and I would
like to quote his results in this language in view of the application to
relative equilibria.
We denote by $G$ the finite dimensional, compact Lie group acting on the
manifold $M$; the quotient $M/G$ is also called the orbit space and
denoted by $\Omega$ (see e.g. \ref{4,5,18,19} for its relevance in
nonlinear dynamics and equivariant bifurcation theory). In general, $\Om$
is not a manifold.
In more abstract terms, $\Omega$ is the set of classes of equivalence on $M$
under the group action. The orbit $G(p)$ of a point in $M$ will also be
denoted as $\om (p) $ or $\om_p$.
For each point $p \in M$ we can consider the set of elements in $G$ which
leave $p$ unchanged,
$$ G_p \ := \ \{ g \in G \ | \, gp \, = \, p \, \} \ ; \eqno(1) $$
this is called the isotropy subgroup of $p$.
We introduce another equivalence relation in $M$ as follows: we consider points
with isotropy subgroups conjugated in $G$ as having the same {\it isotropy
type}, and denote the set of points having the same isotropy type as a {\it
stratum} in $M$:
$$ \s (p) \ := \ \{ q \in M \ | \ \exists g \in G \ : \ G_q = g G_p
g^{-1} \, \} \ . \eqno(2) $$
Strata are submanifolds in $M$.
Notice that $q = gp$ implies $G_q = g G_p g^{-1}$, so that necessarily
$$ \om (p) \ \sse \s (p) \eqno(3) $$
for any point $p \in M$; this means also that we can define strata in
$\Omega$; the stratum of an orbit $\om \in \Omega$ will be denoted as
$\S (\om)$.
When the group action in $M$ is regular (this is automatically true if
$G$ is compact), the topology induced by the group action is compatible
with the topology on $M$, and thus the topology in $M$ induces a natural
topology in $\Om$. In other words, it is then possible to define properly
a distance $d (\om_1 , \om_2 )$ between group orbits (e.g. as the minimal
distance in $M$ between points $p \in \om_1$ and $q \in \om_2$).
We say that an orbit $\om$ is {\it isolated in its stratum} if there is
an $\varepsilon > 0$ such that $ d (\om , \mu ) > \varepsilon $ for all
the $\mu \not= \om$ in $\S (\om ) \ss \Om $.
We refer to \ref{17,20-22} for further detail on this geometric
construction and its applications in Physics.
\bigskip
{\bf 3. }ÊLet us now consider an equivariant vector field $X$ on $M$;
that is, a field such that for all $Y \in {\cal G}$ (the Lie algebra of
$G$), satisfies $ [X,Y] = 0$.
We will think, for ease of language, $M$ as embedded in $R^N$, and use
coordinates $\{ x_1 , ... , x_N \}$ in $M$; thus $X$ can be written in
coordinates as corresponding to the Dynamical System
$$ {\dot x} \ = \ f(x) \eqno(4) $$
where $f : M \to \T M$.
The equivariance of $f$ means that for any $x \in M$ and all $g \in G$,
$$ f (gx) \ = \ \phi_g [ (D_g f) (x) ] \ . \eqno(5) $$
Here $\phi : \T M \to \T M$ is the isomorphism induced by $g$ on tangent
spaces, i.e. $\phi_g$ maps $\T_x M$ into $\T_{gx} M$.
It was proved by Michel \ref{11} that
$$ f(x) \ \in \ \T_x \s(x) \ ; \eqno(6) $$
this also means that, as it was already proved by Bierstone and Schwarz
\ref{20,23-25} the vector field $X:M \to \T M$ can be projected to a
vector field $X_\Om : \Om \to \T \Om$ defined on the orbit space (recall
each stratum is a smooth manifold).
It should be noticed that (6) also means that the closure of a stratum is an
invariant set under any equivariant dynamics, and we can consider the
restriction of (4) to it. This is a general ``reduction lemma'' in the
spirit of Golubitsky and Stewart \ref{4}.
Now, from (6) it also follows that if an orbit
$\om$ is isolated in its stratum, then for all $x \in \om$
$$ f(x) \ \in \ \T_x \om \ . \eqno(7) $$
As noticed by Michel, in the case of invariant variational problem
one can conclude that all the $x \in \om$ are critical for any
invariant potential \ref{11,17} (necessarily the gradient is orthogonal
to the group orbit, and thus must vanish).
In the case of general equivariant vector fields
(7) can be interpreted as follows:
\smallskip\noindent
{\bf Lemma 1.} {\it If a $G$-orbit $\om$ on the $G$-manifold $M$ is
isolated in its stratum, then it is a relative equilibrium for any
equivariant vector field $X$ defined on $M$.}
\bigskip
{\bf 4. } Let us now consider the case where the equivariant vector field
$X$ depends on a control parameter $\la \in \La$ ($\La$ is an interval
on the real line), so that $X^{(\la )}$ is $G$-equivariant uniformly in
$\la$; in other words, (4) is now replaced by
$$ {\dot x} \ = \ f(x; \la ) \eqno(4') $$
and (5) holds for all $\la \in \La$.
We assume moreover that $(a)$ $x \equiv x_0$ (call $\om_0$ the
corresponding $G$-orbit) is a solution for all $\la \in \La$, and that
it is stable for $\la < \la_0$, and unstable for $\la > \la_0$ (this
$\la_0$ is assumed to be in the interior of $\La$). To avoid explosive
behaviour, we assume also that $(b)$ there is an open set ${\cal B}$,
containing $x_0$ in its interior and of positive radius (in the metric
defined on $M$), such that $X^{(\la )}$ points inward on $\pa {\cal B}$,
for all values of $\la \in \La$. As standard in bifurcation theory, we
also assume that $(c)$ the real part $\sigma_i^r (\la )$ of the critical
eigenvalues of $L(\la ) := (Df)(x_0 ; \la )$ satisfies
$$ \left( {{\rm d} \s_i^r \over {\rm d} \la}
\right)_{\la_0} \ > \ 0 \ . \eqno(8) $$
These assumption guarantee in a standard wayÊthat $x_0$
bifurcates at $\la = \la_0$.
Assume now that there is a stratum $\S_0 \ss \Om$ which is
a one-dimensional manifold: as $X_\Om^{(\la)}$ is tangent to strata, it
is directed along $\S_0$ at all $\om \in \S_0$; moreover the above
assumptions guarantee that for $\la > \la_0$ there is a branch of zeroes
of $X_\Om^{(\la)}$ lying precisely in the intersection of $\S_0$ with
${\cal B}$ and distinct from $\om_0$. A zero of $X_\Om^{(\la)}$
corresponds, as already remarked, to a relative equilibrium for the
dynamics on $M$, described by $X^{(\la)}$; there is a branch of
equilibria for $X_\Om^{(\la)}$ bifurcating off $\om_0$, we can conclude
there is a branch of relative equilibria for $X^{(\la)}$ bifurcating
off $x_0$.
We can express the result of our discussion as follows; here we assume
that $x_0$ is invariant under the full $G$:
\smallskip\noindent
{\bf Lemma 2.} {\it Let us consider a one-parameter family of vector
fields $X^{(\la)}$ on $M$, and let $x_0 \in M$ with $G_{x_0} = G$ be an
isolated zero of $X^{(\la)}$ for all $\la \in \La$; let assumptions
$(a)$,$(b)$,$(c)$ above be satisfied. Then, there is a local branch of
relative equilibria bifurcating from $x_0$ at $\la = \la_0$.}
\smallskip
Notice that if $G_{x_0} \not= G$, then $\om_0$ does not reduce to $x_0$
alone, and $x_0$ is a degenerate zero of $X$; however, the discussion
in terms of $X_\Om$ does still hold, and we can again deduce the
bifurcation of a branch of relative equilibria. Also, even if $x_0$ is
not a zero of $X$, but $\om_0$ is a zero of $X_\Om$ and we focus on the
dynamics transversal to $\om_0 \ss M$, we can again conduct the same
discussion in terms of $X_\Om$, and deduce the existence of a branch of
relative equilibria bifurcating off relative equilibria (notice
relative equilibria can correspond to different subgroups of $G$, so
that we can have a symmetry breaking leading from a type of relative
equilibrium to another).
\bigskip
{\bf 5. } We can actually refine (6) -- and the above remark -- a little
further: following Golubitsky and Stewart, let us define for each $H
\sse G$ the submanifold
$$ M^{(H)} \ := \ \{ x \in M \ | \ H \sse G_x \} \ . \eqno(9) $$
Now, if in (5) we consider $g \in G_x$, we have immediately that
$$ f(x) \ = \ (D_g f) (x) \ \ \ \ \forall g \in G_x \eqno(10) $$
and thus that
$$ f(x) \ \in \ \T_x M^{(G_x)} \ . \eqno(11) $$
Combining (7) and (11) we conclude immediately that for an orbit
$\om$ isolated in its stratum we have, for all $x \in \om$,
$$ f(x) \ \in \ \T_x \om \, \cap \, \T_x M^{(G_x)} \ . \eqno(12) $$
(notice in this case we could as well consider the manifold of points
whose isotropy subgroup is exactly equal to $G_x$, as on $\om$ we cannot
have a $G_y$ which strictly contains $G_x$).
However, $T_x \om$ is by definition equivalent to the linear space
spanned by the Lie algebra $\G$ of $G$, or more precisely to $\G / \G_x$
(where $\G_x$ is the Lie algebra of $G_x$). Thus the flow $\Phi_f (x;t)$
under an equivariant vector field will take $x$ into $y \in \om (x)$ such
that $G_y = G_x$; such $y$ can be written, as we are on $\om (x)$, as $y
= h x$ for some $h \in G$, and we conclude that necessarily $h \in N_G
[G_x ]$, the normalizer of $G_x$ in $G$. Notice that, since $G_x$ acts as
the identity, we can actually consider $h \in D^0_x$, where $D \sse G$ is
defined as $$ D_x \ := \ N_G [G_x] / G_x \eqno(13) $$
and $D^0$ is the connected component of the identity in $D$. Thus,
necessarily
$$ f(x) \ \in \ \T_x [ D^0 (x) ] \ . \eqno(14) $$
The above construction and discussion are due to Golubitsky and
Stewart; when $G_x$ is a maximal isotropy subgroup of $G$, they
obtained -- as well known -- powerful general results for linear
actions: based on a homological classification of possible $D^0$, they
obtained a classification of elementary bifurcations \ref{4}.
In the present context, the above discussion shows that to investigate
the possible relative equilibria arising by the bifurcation mechanism we
are considering, we have to focus attention on the subgroups $D^0$.
In particular, if $D^0 = \{ e \}$, then the relative equilibria whose
existence is granted from Lemma 2 are actually equilibria {\it tout
court}. Similarly, if $D^0 = SO(2)$ we are guaranteed that these
relative equilibria are actually periodic orbits (possibly degenerate,
i.e. stationary).
It should also be stressed that all the relative equilibria lying on
the same $G$-orbit must be isomorphic: this poses some apriori
restrictions on the possible kind of relative equilibria. E.g., we can
have periodic orbits only if $\om$ admits a fibration in circles $S^1$,
and $n$-periodic orbits only if $\om$ admits a fibration in tori ${\bf
T}^n$.
Finally, we would like to notice that in the same way as the
``standard'' equivariant branching lemma (EBL) can be extended from
one-dimensional strata to higher dimensional cases by the use of
topological arguments guaranteeing the existence of zeroes of a
vector field on an invariant set $\S$ based on the topology of $\S$
\ref{7}, the same kind of extensions can be obtained for our
``relative equilibria EBL'' (lemma 2): extensions of
the standard EBL apply to bifurcations of stationary points for
$X_\Om^{(\lambda)}$, and these in turn correspond to relative
equilibria of $X^{(\lambda)}$.
\vfill\eject
{\bf References}
\bigskip
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\parindent=20pt
\parskip=2pt
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\bye