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=0pt}
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{\nopagenumbers
\titlea{ Nonlinear Lie symmetries in bifurcation
theory}{}
\smallskip
\centerline{G. Cicogna}
\centerline{\it Dipartimento di Fisica, Universita' di Pisa}
\centerline{\it Piazza Torricelli 2, I-56126 Pisa (Italy)}
\medskip
\centerline{G. Gaeta\footnote{$^*$}{Supported in part by C.N.R.
grant 203.01.48, present address: Math. Inst. Univ. of Utrecht (Netherlands
)}}
\centerline{\it Centre de Physique Theorique, Ecole Polytechnique}
\centerline{\it F-91128 Palaiseau (France)}
%\vfill\eject
\smallskip
\pn
{\bf Abstract}
\pn
We examine the presence of general (nonlinear) time-independent Lie point
\ss
in dynamical systems, and especially in bifurcation problems. A crucial
result
is that center manifolds are invariant under these symmetries: this fact,
which
may be also useful for explicitly finding the center manifold, implies that
Lie
point \ss are inherited by the "reduced" bifurcation equation (a result
which
extends a known property of {\it linear} symmetries). An interesting
situation
occurs when a nonlinear \sy of the original equation results in a linear
one
(e.g. a rotation - typically related to a Hopf bifurcation) of the reduced
problem. We provide a class of explicit examples admitting nonlinear
symmetries,
which clearly illustrate all these points. }
%\vfill\eject
\bigskip\bigskip
{\centerline {\tt Phys. Lett. A {\bf 172} (1993), 361}}
%\bigskip\bigskip
\vfill\eject
\pn
Bifurcation theory is by now one of the most useful and
used tools in attacking nonlinear problems; here we are in particular
concerned
with equivariant bifurcation theory, i.e. bifurcation in the presence of
some
\sy properties [1-7].
In general, all the developments of equivariant bifurcation theory, and its
fundamental theorems, deal with {\it linear} \ss (or, to be more
precise, with linear group actions). Quite recently, the notion of
Lie-point
(LP)
\sy received diffuse attention [8-12], not only thanks to the theoretical
appeal of LP symmetries, but also to their power in the study of concrete
nonlinear equations, often in coincidence with symbolic manipulation
computer programs [see, e.g., 13].
It is therefore very natural to try to build an
equivariant theory for general LP symmetries: in fact, some
results in this direction have been already obtained, showing that some
typical results of linear equivariant bifurcation theory (e.g. the
Sattinger
theorem on the \ss of the bifurcation \eqs obtained
via Lyapunov-Schmidt projection [2,4], and the equivariant bifurcation
lemma
[14,3]) can be suitably extended to nonlinear LP \ss [15,16]. Other
applications can be found in [8-12, 17-18] and Ref. therein.
We shall provide, in the second part of this paper, a class of explicit
examples of dynamical
systems exhibiting nonlinear LP symmetries.
Let us recall that the generic dynamical problem
$$\. u_i = F_i(\l,u) \qquad\qquad u=u(t) ;\qquad u\in R^n ;\qquad
i=1,\ldots,n
\eqno(1)$$
(where $F_i$ are smooth, e.g. $C^\infty$, functions, possibly also
depending
on one
or more real parameters $\l$) admits a time-independent LP \sy generated by
the
$C^\infty$ vector field operator (here and in the following,
$\pd_{u_i}$ stands for $\displaystyle {\pd\over{\pd u_i}}$, etc.)
$$\eta =\sum_i \theta_i(\l, u)\ \pd_{u_i} \eqno(2)$$
if and only if [8,17]
$$\big[\eta, \eta_F\big] = 0 \eqno(3)$$
where $\eta_F$ is the operator (generating the time evolution of the
dynamical
flow)
$$\eta_F =\sum_i F_i\ \pd_{u_i} \eqno(4)$$
\medskip
Given the generic dynamical problem (1), we may state the following main
result.
\medskip\pn
{\bf Theorem}.
{\it Let the linearization of the problem {\rm (1)} around the
equilibrium point $u=u_0$ (and $\l=\l_0$; we can choose $u_0=0, \l_0=0$)
possess some eigenvalues with vanishing real part, the remaining
eigenvalues
having strictly negative real part. Assume here that the (attractive)
center
manifold $M=W_c$ is uniquely defined. If the problem {\rm
(1)} admits a time-independent LP symmetry generated by an
operator $\eta$ {\rm (2)} defined in a neighbourhood of
the point $u_0$, then the center manifold is left invariant by this
symmetry,
and the equations obtained by restriction to the center manifold $M$
admit
the LP \sy $\eta \vert_M$}.
\bigskip
In short, this follows from (3) and
from the fact that LP \ss transform solutions into
neighbouring ones; if $\=u(t)$ is a solution belonging to the center
manifold
$M=W_c$, this implies that $\eta$ transforms it into a solution $u'(t)$
belonging
to a neighbourhood $U$ of $M$; but the only possibility to have a
solution $u'(t)\in U$, for all $t\in R$, is precisely that $u'(t)\in
M$ [19-23]. In the case of bifurcation problems, the above
argument can be extended to a "neighbourhood of criticality", i.e. in an
interval about $\l_0$, by a standard procedure [24, 19-23]. Let us remark
also
that the nontrivial part of the above result can be written in the form
$\eta : M \to TM$.
Notice that our approach is essentially "local", being based on Lie {\it
algebraic} method, and on local properties of manifolds as well: then, our
arguments do not involve properties, e.g. the compactness, of the full \sy
group action.
\smallskip
There are some important aspects of this result. First, one has that
the center manifold (CM) $W_c$ is an invariant manifold under the LP \sy
$\eta$
(notice that this is a peculiar property of center manifolds, and more in
general of transversally hyperbolic manifolds: in fact, a generic manifold
invariant under
the LP \sy is in general {\it not} invariant under the dynamical flow).
For sake of simplicity, we have stated our result under the assumption
that the CM is unique; it is well known, however, that in general this
condition
is not met: we refer to [19-23] for detailed and careful discussions
ensuring
the existence of CM and concerning its properties. In the case of nonunique
CM,
our result above says immediately that a LP \sy may trasform a CM into
another
CM; but we can also prove that, given a LP \sy $\eta$, it is possible to
choose
in general one CM which is invariant under this $\eta$, and/or also to
modify
the \sy $\eta$ in such a way that the given CM turns out to be \sy
invariant.
Let us
recall, incidentally, that the nonuniqueness of the CM does not affect
practical
calculations based, as usually happens, on power series expansions of the
CM
[20-23].
The property of the CM of being invariant under the LP \sy
may be useful even for explicitly finding the CM itself: indeed, the
expression describing $W_c$ must be a function of the quantities
$k_\a=k_\a(u)$
which are left invariant by the \sy $\eta$, i.e.
$$\eta \cdot k_\a = 0 \eqno(5)$$
(cf. also the method of introducing "adapted coordinates" [11], and/or
Poincar\'e - Birkhoff normal forms [25,26]).
\medskip
Another general and important point of the above theorem concerns the \sy
properties of the "reduced" bifurcation problem, i.e. of the \eqs
restricted
to the
variables $v$ spanning the center eigenspace $E_c$. Precisely, putting
$u\equiv
(v,z)$, where $v \in E_c$, $z\in E_c^\perp$, we can write locally the
CM $ W_c$ as identified by $z = \g(\l , v)$ for some function $\g$;
then the reduced \eqs [20-23] obtained from the original problem (1) are
$$\.v=\~F(\l,v)\equiv F\big(\l,v,\g(\l,v)\big) \eqno(6)$$
Now, if the LP \sy generator $\eta$ (2) is written as
$$ \eta = \phi (\l , v,z) \pa_v + \psi (\l , v,z) \pa_z \eqno(7) $$
we then have
$$\eta\vert_M = \phi \big(\l , v, \g(\l ,v)\big) \pa_v + \psi \big(\l , v,
\g(\l
,v)\big) \pa_z \eqno(8) $$
and the condition to have $\eta\vert_M : M \to T M$ is simply $\delta z
= ( \pa \g / \pa v) \delta v$, i.e.
$$ \psi \big(\l , v, \g(\l ,v)\big) = {\pa \g \over \pa v} \phi \big(\l ,
v,
\g(\l , v)\big) \eqno(9) $$
Therefore, the symmetry $\~\eta$ inherited by the reduced bifurcation
equation
(6) is given by
$$\~\eta = {\~\phi} (\l , v) \pa_v \equiv \phi \big(\l , v, \g (\l
,v)\big)\pa_v
\eqno(10) $$
This result extends to nonlinear LP \ss a well known property of
linear ones [1,23].
Now, it may also happen that general nonlinear \ss
of the original problem result in {\it linear}
\ss of the reduced bifurcation equation (6). This result clearly
cannot be predicted by standard (linear) equivariant bifurcation theory;
in particular, if the
original \eqs (1) have {\it no} linear symmetry, then linear equivariant
bifurcation theory is not able to give any \sy constraint on the form of
functions $\~F(\l,v)$, i.e. on the form of the reduced bifurcation
equations.
\bigskip
Let us provide now a family of dynamical problems admitting a
nonlinear LP symmetry; they will be useful also to illustrate all the
above discussion. Let us write, for notational
convenience, $$u\equiv (x,y,z_1,\ldots,z_{n-2})\ \in R^n \eqno(11)$$
Let
$$w_a=w_a(y,z_a) \qquad\qquad (a=1,\ldots,k=n-2) \eqno(12)$$
be arbitrary smooth functions, satisfying the conditions
$${\pd w_a\over{\pd z_a}}\ne 0 \qquad ({\rm no\ sum \ over \ } a)
\eqno(13)$$
and put
$$\chi_a=\chi_a(y,z_a)={\pd_yw_a\over{\pd_{z_a}w_a}}\qquad ({\rm no\ sum \
over \ } a) \eqno(14)$$
Consider systems of the following form
$$\eqalign
{\. x=&\ x\ f - y\ g \cr
\. y=&\ y\ f + x\ g \cr
\. z_1=&\ -y\ f\ \chi_1 - x\ g\ \chi_1 + {h_1\over{\pd_{z_1}w_1}} \cr
\vdots \cr
\. z_k=&\ -y\ f\ \chi_k - x\ g\ \chi_k + {h_k\over{\pd_{z_k}w_k}} \cr }
\eqno(15) $$
where $f,g,h_1,\ldots,h_k$ are $n$ arbitrary smooth functions of the
quantities
$$r^2=x^2+y^2 \qquad {\rm and\ }\qquad w_1,\ldots,w_k\eqno(16)$$
(and possibly also of one or more real parameters $\l$, to include the case
of
bifurcation problems). Then, using (3),
it is easy (even if tedious) to verify that the
above system (15) admits the LP \sy generated by
$$\eta=y\pd_x-x\pd_y+x\chi_1 \pd_{z_1}+\ldots
+x\chi_k \pd_{z_k} \eqno(17)$$
In particular (with $n=3$, the extension to $n>3$ is
straightforward; we write here $z, w$, etc. instead of $z_1, w_1$, etc.),
if
$w=w(y,z)$ and $h=h(r^2,w)$ in (15) satisfy
$$\lim_{y,z\to 0}\pd_y w = 0 \qquad
{\rm and} \qquad \lim_{y,z\to 0}{h(r^2,w)\over{z\ \pd_z w}}=\ell < 0
\eqno(18)$$
then the origin $u=u_0=0$ is an equilibrium point for the problem (15),
with an
attractive 2-dimensional center manifold $W_c$ tangent at $u_0=0$ to
the center eigenspace $E_c$ ($z=0$). If
the functions $f(\l,r^2,w), \ g(\l,r^2,w)$ (with $\l \in R)$ satisfy also
$$f(0,0,0)=0,\quad \pd_\l f(0,0,0)\ne 0, \quad g(0,0,0)\ne 0 \eqno(19)$$
then the standard conditions for the appearance of Hopf
bifurcation are met.
The invariance under the \sy (17) implies that the CM must be a function of
the quantities $k_\a(u)$ defined in (5), which are now just $r^2$ and
$w_a$.
E.g.,
let us choose in (12,15), with $n=3$, $$w=z\ {\rm e}^{-y}$$
which gives the nonlinear LP \sy
$$\eta= y\pd_x-x\pd_y-xz\pd_z\ ,$$
and also choose
$$f=h=-w+r^2$$
leaving arbitrary $g$. Then, it is easy to
explicitly obtain the CM, which is given by the surface
$$z={\rm e}^yr^2\ . $$
Also, more in general, if in (15), again with $n=3$, one has $f=h$
\big(independently of the choice of $w=w(y,z)$\big), then the equation
$$h(r^2,w)=0\eqno (20)$$
gives the implicit expression of the center manifold. Another simple
situation
occurs if in (15) there is some $w=w_0=$ const such that $h(r^2,w_0)=0$;
then,
for any choice of $f, g$, the equation
$$w(y,z)=w_0\eqno(21)$$
gives the CM, as easily verified by direct calculation.
\bigskip
The possibility that the reduced bifurcation \eqs (6) exhibit a linear
symmetry, even in the absence of linear \ss of the original equations, is
well
illustrated by the following simple problem. Let us choose in (15) $\
w=z-y^2,
\ \ h=-w, \ \ f=\l-r^2\ $, with arbitrary $g(\l,r^2,w)$ \big(satisfying in
particular all conditions (18,19)\big): the resulting problem
$$\eqalign
{\. x=&\ x\ (\l -r^2) - y\ g(\l,r^2,w) \cr
\. y=&\ y\ (\l-r^2) + x\ g(\l,r^2,w) \cr
\. z=&-z+y^2\big(1+2(\l-r^2)\big)+2xy\ g(\l,r^2,w)\cr }
\eqno(22) $$
admits the LP symmetry generated by
$$\eta=y\pd_x-x\pd_y-2xy\pd_z \eqno(23)$$
It is easy to verify that $z=y^2$ is the CM \big(in agreement with
(21)\big), and that the reduced \eqs for the variables $v\equiv (x,y)$
exhibit a
(linear) rotation \sy generated by $$\eta_0=y\pd_x-x\pd_y \ . $$
Notice that a stable Hopf bifurcation for (22) appears at $\l_0=0$,
described by
$$r^2=x^2+y^2=\l, \qquad z=y^2$$
with frequency $\om(\l)=g(\l,\l,0)$.
%\bigskip\bigskip
\vfill\eject
\parindent 0pt
{\bf References}
\bigskip
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Boston, 1982)
[4] D.H. Sattinger, "Branching in the presence of symmetry"
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\bye