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{\nopagenumbers ~\vfill
\titlea{Conditional symmetries and conditional constants of motion
for dynamical systems}\footnote{}{\giorno }
\vfill
\centerline{Giuseppe Gaeta}
\centerline{{\it Centre de Physique Theorique, Ecole Polytechnique}}
\centerline{{\it F-91128 Palaiseau (France)}}
\vfill
{\bf Summary.} {\small We apply conditional Lie-point symmetries to
the study of dynamical systems, discussing in particular their
relation to conditional constants of motion; we also briefly discuss
the relation between conditional symmetries, conditional constants of
motion, and invariant manifolds.}
\vfill ~
\vfill \eject
~~ Introduction \dotfill 1
1. Lie-point symmetries for dynamical systems \dotfill 2
2. Module structure of the symmetry algebra \dotfill 4
3. Symmetry reduction for ODEs and symmetry adapted coordinates
\dotfill 5
4. Topology of trajectories and Lie-point symmetries
\dotfill 7
5. Smooth structures and Lie-point symmetries \dotfill 7
6. Conditional symmetries, general setting \dotfill 9
7. Conditional symmetries and conditional constants of motion for DS
\dotfill 11
8. CS, CCM, invariant manifolds \dotfill 15
9. Module structure of CS \dotfill 17
10. Hyperbolic fixed points, stable and unstable manifolds \dotfill 18
11. Nonhyperbolic fixed points, center manifolds \dotfill 20
12. Bifurcation of fixed points \dotfill 21
~~ References \dotfill 22
\vfill \eject
} \pageno=1
\titleb{Introduction}
Symmetry methods are among the most effective tools in studying
nonlinear differential equations, both ODEs and PDEs [1-4]. Here we
are specially interested in ODEs, which we can write in full
generality as a first order system; for ease of notation we will
assume the system is autonomous. (This is no restriction since if we
have a nonautonomous system, we can always add one variable $x_0$ and
one equation $ \xd_0 = 1$, changing it into an equivalent autonomous
system).
We will write $\xi = ( x_1 , \dots , x_N ) \in M \sse \R^N $ and
write the ODE $\De$ as
$$ \De ~:~~~~ {\dot \xi} = f (\xi ) ~~~~~~~~ f : M \to TM \eqno(1) $$
If we know the algebra $\G_\De$ of exact symmetries of (1), we can
choose symmetry adapted variables $w = w (\xi )$ on $M$, such that
$$ {\dot w}^i = 0 ~~~~~~i = 1 , ... , m ~,~ m \le N \eqno(2) $$
where $m \le N$ depends on $\G_\De$ and, more precisely, on the
dimension of the maximal solvable subalgebra of $\G_\De$ (see [1,2] for exact conditions and results);
symmetry reduction of ODEs is also described and discussed in [5,6],
and e.g. in [7-9] in the context of Hamiltonian mechanics.
In the application of symmetry methods to PDEs, a prominent role is
played also by "non classical" [10], or "conditional" [11]
symmetries. These are also related to compatibility conditions of
the original PDE and a differential constraint, or "side condition"
[12,13] imposed on it, see [14,15]; a relation also exists to the so
called "direct reduction" method [16], see [11, 14,15,17]; see also
[18,19] for related discussion.
Quite surprisingly, the application of this method to ODEs and
dynamical systems (DS) - by which we mean ODEs in the form (1) -
seems not to have been considered in the Lie-theoretic literature;
our goal here is to study this question.
It should be mentioned that the application of symmetry methods to
equations of the simple form (1) can run into unexpected troubles -
not met for PDEs - precisely because of the semplicity of the
equations [20], which introduces degeneracies. A typical problem is
constituted by the fact that $\G_\De$ turns out to be an
infinite-dimensional Lie algebra [4]; this fact is better understood
by remarking that, if $I_\De$ is the algebra of
conserved quantities for the equation $\De$, then $\G_\De$ is a
module over $I_\De$ [20-23], as discussed below (see section 2); in the
presence of nontrivial conserved quantities, $\G_\De$ will be
infinite dimensional as a Lie algebra; this does not forbid that it
is finite dimensional as a module over $I_\De$.
Considering conditional symmetries (CS) will naturally lead us to
consider "conditional constants of motion" (CCM); it will moreover
turn out that certain subsets of CS for $\De$ have a structure of
module over an associated subset of CCM.
Pretty much as in the PDE case, use of CS will permit us to determine
a set of special solutions to the nonlinear DS under study. Actually,
CS and CCM will be related to {\it invariant manifolds} of the DS,
and the methods developed in the PDE frame can help to identify
these. We will discuss this issue, and in particular the case in
which the DS admits a fixed point $\xibar$, so that it is natural to
consider the stable, unstable and center manifolds [24-27] for this
fixed point; similar considerations would apply if considering a
periodic solution rather than a stationary one, or even more general
invariant sets. We will then be specially interested in the case the
DS (1) depends on external control parameter and the fixed point
$\xibar$ undergoes a bifurcation; this will also permit to make
contact with recent works of ours [21,28-30].
We should point out that our results will be in the direction of a
pioneering work by Wulfman [31]. Also, after the completion of this
work we came across a paper by Sarlet, Leach and Cantrijn [32], who
consider CCM in the context of hamiltonian systems; they call them
"configurational invariants", and discuss their relation to a weak
form of complete integrability (which in our present vocabulary would
be "conditional complete integrability"). Although the paper [32]
does not mention conditional symmetries - among other good reasons,
because they had not been introduced yet - the reader will have no
difficulties in placing it into the present framework; this will also
exempt us from dealing specifically with hamiltonian dynamical
systems.
Formulas are numbered consecutively in each section; obviously
$(i.j)$ will refer to formula $j$ of section $i$, and we will omit
the section index when referring to formulas in the same section.
Finally, we would like to thank proff. D. Levi and F. Verhulst for
interesting discussion connected with this work; and more
specifically proff. L. Michel and P. Winternitz for a lunch which is
at the origin of the present paper.
%\vfill \eject
\titleb{1. Lie-point symmetries for dynamical systems}
Let $M$ be a smooth manifold, which we think as embedded in $\R^N$;
a {\it Dynamical System} (DS) on $M$ is a system of autonomous first
order ODEs
$$ \xd = f(x) \eqno(1) $$
$$ x = ( x^1 , \dots , x^N ) \in M ~~,~~~~ f : M \to TM \eqno(2) $$
where $f$ is a smooth function. If $f$ is smooth of class ${\cal C}^k$
(analytic), we talk of a ${\cal C}^k$ (analytic) DS; unless otherways
specified, in the following we consider smooth, i.e. ${\cal
C}^\infty$, DS.
The DS (1) corresponds to the flow on $M$ under the action of the
{\it vector field} (VF)
$$ \^f = f(x) \pa_x \equiv f^i (x) { \pa \over \pa x^i} \eqno(3) $$
The set of tangent (smooth) VF on $M$ equipped with the Lie
commutator $[.,.]$ form a Lie algebra, which we denote by $\M$.
Clearly, to any VF $\^f \in \M$ we can associate a smooth function $f
: \R^N \to \R^N$, $f: x\in M \to T_x M$ (actually, a section of the
tangent bundle of $M$, $TM$), which gives its component expansion in
the $x$ coordinates, as in (3). The space of such smooth functions $M
\to TM$ will be denoted by $\V$; this is naturally equipped with a
Lie-Poisson bracket $\{ .,. \}$ induced by $[.,.]$, which is given by
$$ \{ f,g \} = ( f \c \grad ) g - (g \c \grad ) f \eqno(4) $$
A VF $\^s \in \M$ is a {\it Lie-point} (LP)
symmetry\footnote{$^1$}{More precisely, a VF would be the generator of
a LP symmetry; we will use this innocent abuse of language, which
should not cause any confusion, for ease of discussion.} of the DS (1)
if under its flow solutions of (1) are transformed into (generally,
different) solutions of (1). In the present setting, this is equivalent
to the condition $[\^f , \^s ] =0$ or, in terms of (4),
$$ \{ f , s \} = 0 \eqno(5) $$
{\small
{\bf Remark 1.} Notice that, although in the present setting both $f$
and $s$ do not involve neither depend on time, so that they give {\it
Lie-point time-independent} (LPTI) vector fields, the case of
nonautonomous VF is readily recovered by adding a new coordinate
$x_0$, whose evolution is given by $\xd_0 = 1$. If $s$ does
depend on time but $f$ does not, and we keep the time variable
distinct from the space ones, eq.(5) would be replaced by
$$ s_t + \{ f , s \} = 0 \eqno(5') $$
More in general [1-4], LP symmetries would be VF on $\R \x M$, of the
form $$ \eta = \phi^i (x,t) {\pa \over \pa x^i} + \tau (x,t) { \pa
\over \pa t} \eqno(6) $$
Under this, i.e. under $e^{\eps \eta}$ with $\eps \to 0^+$,
$$ t \to t + \d t ~,~~ x^i \to x^i + \d x^i ~,~~ \xd^i \to \xd^i + \d
\xd^i \eqno(7) $$
$$ \eqalign{ \d t = \eps \tau (x,t) ~~~,& ~~~ \d x^i = \eps \phi^i
(x,t) \cr
\d \xd^i = \eps [ D_t ( \phi^i - \tau \xd^i ) + \tau {\ddot x}^i ] =&
\eps [ \pa_t \phi^i + \xd^j \pa_j \phi^i ] \cr} \eqno(8) $$
where $\pa_i \equiv \pa / \pa x^i$, and we have used the
prolongation formula [1-4].
Applying this to (1), we get on solutions (i.e. for $\xd^i = f^i
(x)$)
$$ \pa_t \phi^i + f^j \pa_j \phi^i - f^i \pa_t \tau - f^i f^j \pa_j
\tau = \phi^j \pa_j f^i $$
which, if $\eta$ does not involve a reparametrization of time (if
not possibly a trivial shift), so that $\tau = const.$, reads
$$ \pa_t \phi^i + \{ f , \phi \} = 0 $$
and we recover (5'), or (5) if $\phi$ does not depend on time.}
{\small
{\bf Remark 2.} Notice that if $\tau$ is not identically zero, we can
divide (6) by $\tau$, reaching the form $\pa_t + \phi \pa_x$; anyway,
this could introduce singularities, corresponding to zeroes of
$\tau$, and take us out of the class of smooth VF.}
{\small
{\bf Remark 3.} It is particularly convenient, in the study of
autonomous DS, to consider VF for which $\tau = 0$, $\phi = \phi
(u)$; some reasons for this will be recalled below, for other see
e.g. [20-23].}
It is easy to see, e.g. by Jacobi identity, that the VF satisfying
(5) - or even (5') - form a Lie algebra under $[.,.]$ or, which is
the same, under $\{ .,. \}$. This algebra will be denoted by $\G_f$.
\titleb{2. Module structure of the symmetry algebra}
Let us now consider the case (1.1) admits a {\it constant of motion}
(CM); i.e., there is a function $P(x) : M \to \R$ such that
$$ {d \over dt } P(x) \equiv ( \xd \c \grad ) P = ( f(x) \c \grad ) P
(x) = 0 \eqno(1) $$
In this case, it is easy to see that if $\{ f ,s \} = 0$, then also
$\{ f , g(P(x)) s \} = 0$ for smooth $g$. Indeed,
$$ \{ f , g (P(x)) s \} = g' (P(x)) s(x) (f(x) \c \grad ) P(x) +
g(P(x)) \{ f, s \} = g(P(x)) ~ \{ f,s \} \eqno(2) $$
The constants of motion form an algebra under addition (i.e. if $P_1
(x)$, $P_2 (x)$ are CM, then also $P_+ (x) = P_1 (x) + P_2 (x)$ is a
CM); this algebra will be denoted by $I_f$.
Suppose now that $ s_1 , s_2 \in \G_f $, $P_1 , P_2 \in I_f$; the
above (2) and bilinearity of $\{ .,. \}$ show that then
$$ P_1 (x) s_1 (x) + P_2 (x) s_2 (x) \in \G_f ~~~~~~ \all P_1 , P_2
\in I_f ~,~ \all s_1 , s_2 \in \G_f \eqno(3) $$
This shows that $\G_f$ is an infinite dimensional algebra for
nontrivial\footnote{$^2$}{$I_f$ does always contains the functions
which are constant on the whole $M$, which are trivially also CM.}
$I_f$; More in general it proves that
{\bf Lemma 1.} {\it $\G_f$ is a module over the algebra $I_f$.}
Notice that although $\G_f$ is infinite dimensional as a Lie algebra,
it is in general, for $f \not\equiv 0$, finite dimensional as a
module. For definition and properties of modules, see any text on
algebra, or e.g. [33].
{\small
{\bf Remark 4.} If we see (1) as an equation determining the $P(x)$
which are constants of motions for $f$, this is a first order linear
PDE and its solutions are therefore detrmined by solving the
associate characteristic system [34,35]
$$ {d x^1 \over f^1 (x) } = \dots = {d x^N \over f^N (x) } $$
}
\titleb{3. Symmetry reduction for ODEs and symmetry adapted
coordinates}
Let us briefly recall how the knowledge of $\G_f$, or part thereof,
can be used to simplify the DS (1.1). If we know a single VF $\^s =
s(x) \pa_x \in \G_f$, we can change coordinates and pass to
coordinates $w(x)$ such that, say,
$$ \^s = {\pa \over \pa w^N } \eqno(1) $$
i.e. such that $\^s$ is along one of the coordinates, which are
therefore called {\it symmetry adapted} [2].
The commutation relation $[ \^f , \^s ] = 0$ being satisfied
indipendently of the coordinate choice, in the $w$ coordinates $\^f $
will be independent of $w^N$. This means that the system (1.1) will
be transformed, in the coordinates $w$, into
$$ {\dot w} = g (w) ~~~~;~~~~ \pa g(w) / \pa w^N = 0 \eqno(2) $$
so that we are reduced to a system of $(N-1)$ equations for the first
$(N-1)$ variables, plus an equation giving the evolution of the
variable $w^N$ in terms of the others, i.e. of solutions of the
$(N-1)$ dimensional system.
{\bf Example 1.} Let us consider the system
$$ \eqalign{
\xd =& \a (r^2 ) x - \b (r^2 ) y \cr
\yd =& \b (r^2 ) x + \a (r^2 ) y \cr } \eqno(3) $$
where $r^2 = x^2 + y^2$, and $\a , \b $ are smooth functions. This is
symmetric under rotations, generated by the VF $\^s = -y\pa_x + x
\pa_y $. Passing to polar coordinates $(r , \th )$, we have $\^s =
\pa_\th$, and the time evolution is given by
$$\eqalign{ {\dot r} =& \a (r^2 ) \cr {\dot \th} =& \b (r^2 ) \cr}
\eqno(4) $$
{\small
{\bf Remark 5.} As shown concretely by this example, the passage to
symmetry adapted coordinates can require a singular change of
coordinates.}
In the case we know a subalgebra $\G_0 \sse \G_f$, the procedure can
be iterated $d$ times, where $d$ is the dimension of the maximal
solvable subalgebra in $\G_0$ [1,2]; it should be stressed
that the VF in $\G_0$ to be successively rectified should be chosen in
the appropriate order, see [1]. In this way (1.1) is reduced to an
$(N-d)$ dimensional system plus $d$ "quadrature" equations.
If we know a constant of motion $P(x)$, we can also take advantage of
this by passing to coordinates $z(x)$, chosen in such a way that $z^N
= P(x)$; in these new coordinates, (1.1) will read
$$ \eqalign{ {\dot z}^i =& h^i (z) ~~~~~ i = 1 , ... , N-1 \cr
{\dot z}^N =& 0 \cr } \eqno(5) $$
Notice that the $h^i$ will depend on all the $z$ coordinates,
including $z^N$.
{\bf Example 2.} If $\a = 0 $ in (3), $r$ is a CM; indeed with $\a =
0$ the polar coordinate equations (4) are just
$$ {\dot r} = 0 ~~~~;~~~~{\dot \th } = \b (r^2 ) \eqno(6) $$
More in general, if we know $p$ functionally independent CM, $P_1 (x)
, ... , P_p (x)$, we can pass to coordinates $z_1 , ... , z_{N-p} ,
\th_1 , ... , \th_p $, with $\th_i \equiv P_i (x) $; in these
coordinates,
$$ \eqalign{ {\dot z}^i =& h^i (z, \th ) \cr
{\dot \th}^j =& 0 \cr } \eqno(7) $$
{\small
{\bf Remark 6.} The CM can be seen as invariant functions for the
one-parameter group $e^{\lambda \^f }$. Then a celebrated theorem of
Hilbert (for polynomial functions; extended by Schwarz [36,37] to the
smooth case) ensures that [38-41] there exists a basis $\Th_1 , ... ,
\Th_d $ of invariant functions (respectively, polynomials), such that
any $P(x) \in I_f$ (polynomial $P(x) \in I_f$) can be written as a
function (as a polynomial) of the $\Th$'s; i.e. as $P(x) = {\widetilde
P} ( \Th (x))$.}
Notice that the $\th$'s enter in (7) only as parameters; the problem
is therefore reduced to an $(N-p)$ dimensional one with $p$
parameters. Fixing the values of the $\th$'s identifies a submanifold
$M_\th \ss M$, by definition invariant under the flow of (1); one is
therefore legitimate to consider the reduction of (1.1) to this
$$ \xd = f_\th (x) ~~~~~~ x \in M_\th ~,~ f_\th \equiv f \vert_\th :
M_\th \to T M_\th \eqno(8) $$
{\small
{\bf Remark 7.} Notice that not necessarily $\dim ( M_\th ) = [ \dim
(M) - p ]$: e.g., in the example 2 above, for $r=0$ we have a zero
dimensional $M_\th = \{ 0 \}$.}
In the following, we will assume that the reduction corresponding to
global CM and/or symmetries has already been performed.
{\small
{\bf Remark 8.} It is clear from (7) that $\^s \in \G_f$ will be, in
the $(z,\th )$ coordinates, of the form
$$ \^s = r(z, \th ) \pa_z + t (\th ) \pa_\th \eqno(9) $$
as can be easily verified by direct computation.}
{\small
{\bf Remark 9.} It should be stressed that $( f \c \grad ) P = 0$
does {\it not} imply $ ( s \c \grad ) P = 0$ for $\{ f , s \} = 0 $,
as the following example shows.}
{\bf Example 3.} Let us consider again (3), with $\a = 0 $, $\b ( r^2
) = 1$. In this case $r$ is a CM, but the system is symmetric under
$\^s = x \pa_x + y \pa_y $, which generates scaling transformations
and does {\it not} leave $r$ invariant.
\titleb{4. Topology of trajectories and LP symmetries}
It should be briefly recalled that LP transformations, being smooth
and locally invertible\footnote{$^3$}{The invertibility is global if we
consider groups of LP transformations.}, cannot transform a subset $A
\ss M$ into a topologically different one.
This applies in particular to the trajectory of a point $x \in M$
under the flow of (1), so that trajectories of topologically
"special" types, e.g. fixed points, periodic orbits, or orbits
filling densely a (topological) $k$-dimensional torus, cannot be
transformed to orbits of different types. Isolated fixed points,
periodic orbits, $k$-dimensional invariant tori, are therefore
invariant under LP transformations [20,23].
{\small
{\bf Remark 10.} Notice that if we consider full solutions instead
than trajectories, i.e. the graph $(t,x(t))$ instead than the set $\{
x(t) ~,~t \in \R_+ \}$, we do not have topological difference between
graphs of solutions corresponding to topologically different
trajectories.}
\titleb{5. Smooth structures and Lie-point symmetries}
It should be understood that preservation of topology of trajectories
is not the only limit on the type of solutions that can be connected
by a LP transformation. Roughly speaking, preservation of topology
just follows from the fact that LP transformations are one to one
(invertible) and continuous; the fact that they are not only ${\cal
C}^0$ but ${\cal C}^\infty$, or analytic, poses further constraints,
as we are going to discuss in this section [20,23,42].
A first obvious remark is that a polynomial (in general, a ${\cal
C}^\infty$) LP transformation cannot transform a ${\cal C}^\infty$
solutions into a ${\cal C}^k$ one, $k < \infty$ (this will play a role
in the discussion of center manifolds).
Another useful remark is that a polynomial LP transformation cannot
connect two trajectories that diverge exponentially. More in general,
let $x(t)$ be the solution to (1.1) with initial datum $x(0) = x_0$;
this identifies an invariant curve under $\^f$ in $M$, the
trajectory $\ga_0$ of $x(t)$. Suppose now that the linearization of $\^f$ at $\ga_0$
has positive eigenvalues in some direction $\xi \in T_{x_0} M$ (transversal to $\ga_1$); let $\^s$ be a VF such that $s(x_0 ) = \xi$, and let $x_1 = e^{\eps \^s } x_0$ (with $\eps$ small). Let us consider an interval of $\ga_0$ of
lenght $\ell > 0$, and a tubular neighbourhood $u_\d$ of $\ga_0$ of
radius $\d$. Then the trajectory $\ga_1$ issued from $x_1$ will get
out of $u_\d$ for $\ell$ long enough (the trajectories $\ga_0 , \ga_1$ diverge with positive Lyapounov exponent [43]). By the uniform continuity of LP
transformations, $\ga_1$ and $\ga_0$ cannot be connected by a LP
transformation.
There is a point in the above reasoning that should be emphasized: the
separation is exponential in the curvilinear coordinate along $\ga_0$
(by taking $\d$ small enough, we can always choose a system of
coordinates in $u_\d$ such that $\ga_0$ corresponds to, say, $x_2 =
x_3 = ... = x_N = 0$, so the exponential in $x_1$ is well defined),
but not necessarily exponential in time.
For given $\d_0$ and $\ell$, by taking $\eps < \d_0$ small enough, we
can guarantee that $\vert e^{\eps \^s } x - x \vert < \d_0$ $\all x \in
\ga_0 (\ell )$; but for any $\d_0$ , $\ell$ we can find a point $x_1
= e^{\eps \^s } x_0$, $\eps < \d_0$, such that $\vert \ga_1
(\lambda ) - \ga_0 (\lambda ) \vert > \d_0$ for $\lambda_0 < \lambda
< \ell$.
Notice that exponential separation of solutions in time does not
forbid that their two trajectories are connected by a LP
transformation, as the example 4 below shows; also, the trajectories
can separate exponentially in the curvilinear coordinate even if the
solutions do not separate exponentially in time, see example 5.
{\bf Example 4.} Let us consider the simple system
$$ \eqalign{ \xd =& x \cr \yd =& y \cr } $$
which has solutions $(x(t),y(t)) = ( x_0 e^t , y_0 e^t )$, so that
the trajectories are just straight lines $y=cx$, percurred with
exponentially increasing speed. Let us consider the motions with
initial data
$$ p_0 = (a , 0) ~~~;~~~p_1 = (a , \eps ) $$
Then, $\^s = {\rm arctg}(\eps / a) ( x \pa_y - y \pa_x ) $ transforms
the trajectory $y=0$ issuing from $p_0$ into the trajectory $y = (\eps
/ a ) x$ issuing from $p_1$.
{\bf Example 5.} Let us consider the system
$$ \eqalign{ \xd =& f(x) \cr \yd =& f(x) \c y \cr } $$
for which $dy / dx = y$, i.e. the trajectories are given by
$$ y = c e^x $$
with $c$ a constant. The trajectories $y=0$ issuing from $p_0 = (a ,
0)$ and $y = \eps e^x$ issuing from $(a , \eps e^a )$
diverge exponentially in $t$. If $f(x)$ is such that $e^{x(t)}$ is not
exponential in $t$, e.g. $f(x) = {\rm log}(x)$, the solutions do not
diverge exponentially in time.
\titleb{6. Conditional symmetries. General setting}
Let us briefly recall the setting for the determination (and use) of
conditional symmetries, following [11] (see [12-19] for further
details and related topics); the specifities of the DS case will be
discussed in the next section.
Let us consider a differential equation $\De$ and its symmetry
algebra $\G_\De$; we will denote by $y$ the independent variables and
by $u$ the dependent ones. Denoting a VF $\eta$ as
$$ \eta = \phi^\a (y,u) {\pa \over \pa u^\a} + \xi^i (y,u) {\pa \over
\pa y^i} \eqno(1) $$
if $\De$ is of order $n$, the VF $\eta \in \G_\De$ are those for which
$$ \cases{ \eta^{(n)} \c \De = 0 & \cr \De = 0 & \cr} \eqno(2) $$
where $\eta^{(n)}$ is the $n$-th prolongation of $\eta$ [1-4].
As already recalled, knowledge of $\G_\De$ allows for a reduction of
the equation $\De$; in particular, one can look for solutions
$u=f(y)$ that are invariant under a subgroup $\G_0 \sse \G_\De$. With
this invariance ansatz, eq.(2) reduces to simpler ones: indeed, now
we can express $u(y)$ in terms of the (differential) invariants of
$\G_0$.
It should be noted that, given a VF $\eta_0 = \phi_0 \pa_u + \xi_0
\pa_y $, the condition of invariance of $u = f(y)$ under this reads
$$ \De_0 \equiv \phi_0^\a + \xi_0^i {\pa u^\a \over \pa y^i} = 0
\eqno(3) $$
so that if we know a priori that $\eta_0 \in \G_\De$, and therefore
that (2) is satisfied, the solutions invariant under $\eta_0$ can be
seen as solutions of the system made of (2) and (3), and in
particular of
$$ \cases{ \De = 0 & \cr \De_0 = 0 & \cr} \eqno(4) $$
Notice that it can happen that a solution $u = f(y)$ of $\De$ is also
invariant under VF which are {\it not} in $\G_\De$.
{\bf Example 6.} Deferring to the next section the consideration of
examples in the frame of DS, consider the PDE
$$ (\pa_1^2 + \pa_2^2 ) u = - 2 \a u + \a (y_1^2 + y_2^2 ) u + \b
y_2^2 u + (\b / \a ) y_2 \pa_2 u \eqno(5) $$
where $\a , \b$ are real constants, $u \in \R$, $y \in \R^2$, and
$\pa_k = \pa / \pa y^k$. This is easily seen to be not invariant for
rotations in the $(y_1 , y_2 )$ plane, i.e. under the VF $\eta_0 =
(y_2 \pa_1 - y_1 \pa_2 )$. On the other side, the function
$$ f(y) = {\rm exp} \left[ - \unm (y_1^2 + y_2^2 ) \right] \eqno(6) $$
is obviously rotationally invariant, and gives a solution to (5).
This suggests that symmetry reduction could be possible, and useful,
also considering symmetries (of solutions) which are {\it not} in
$\G_\De$.
Solving the original equation $\De = 0$ with the ansatz of invariance
under $\eta_0$ amounts to solution of (4); notice that there we do
not care about the first of (2). If we give an arbitrary $\eta_0$,
anyway, the system (4) will in general not have solutions, so that
this method is useful only if we are also able to determine the
$\De_0$ compatible with the original $\De$, i.e. the $\eta_0$ that
can be symmetries of solutions of $\De$. A method for determining
these, and the corresponding invariant solutions, does indeed exist,
and we do now briefly illustrate it.
Before doing this, it is maybe worth remarking explicitely that
although we always discuss invariance under a single VF for ease of
notation, the whole discussion is immediately extended to an algebra
of VF by considering invariance under all its generators at the same
time.
Let us now consider $\eta_0$ to be a not specified VF: i.e., the
functions $\phi_0 , \xi_0$ are not known; the solutions of $\De$
which are also invariant under $\eta_0$ will again be given by
solutions to (4), but this should now be seen not as an equation for
$u=f(y)$, but as an equations for the set of unknown functions
$\phi_0 (y,u)$, $\xi_0 (y,u)$ and $f(y)$.
We can apply to (4) the known methods for solving a system of PDEs
(notice that even if $\De$ was an ODE, $\De_0$ will be a PDE), among
which symmetry methods.
Let us indeed determine the symmetry of (4); the determining
equations for this amount to
$$ \cases{ \eta^{(n)} \c \De = 0 & \cr \eta^{(1)} \c \De_0 = 0 & \cr
\De = 0 = \De_0 & \cr} \eqno(7) $$
Notice that if we choose to consider $\eta_0 = \eta$, which is surely
legitimate as $\eta_0$ is completely generic, the second of these is
automatically satisfied, since $\De_0 = 0$. This means that (7) is
equivalent to (2) and (3), i.e. it poses no restriction on $\eta_0$.
Therefore, we can apply to (4) the usual symmetry reduction methods,
and obtain solutions in this way. Such solutions of (4) give us a VF
$\eta_0$ and a function $u=f(y)$, invariant under $\eta_0$, which is
also a solution to the original equation $\De = 0$. The VF $\eta_0$
is also a symmetry of (4), and is therefore obtained by the
corresponding determining equations (7).
Notice that we can solve (7) by the well known algorithms for
determining equations [1-4], also by means of computer algebra if
desired [44,45]; once this is done, we can choose a specific
$\eta_0$ solution of (7) and pass to consider (4), i.e. determine the
invariant solution $u=f(y)$ to the original equation.
The VFs $\eta_0$ solution of (7) are symmetries of (4) but not of
$\De$ alone, except of course for the trivial case $\eta_0 \in
\G_\De$. They are therefore called {\it conditional symmetries} (CS) of
$\De$, as they are symmetries of $\De$ once this is subject to the
additional condition (3).
Notice that this additional (or side) condition, i.e. $\De_0$,
depends on $\eta_0$ itself. Therefore, the conditional symmetries of
a given equation $\De$ do not in general form an algebra, as they are
ordinary symmetries of different systems.
Notice also that the space of CS of $\De$ does naturally carry an
action of $\G_\De$, and the CS can be divided into conjugacy
classes under this action, leading to a classification.
It should be stressed that, although the introduction of conditional
symmetries was motivated by the search for invariant solutions, we
can very well have solutions to (7) under which {\it no} solutions of
$\De$ - neither therefore of (4) - are invariant. Although these VF
do not lead to symmetry reduction of $\De$ as above, they are
nevertheless CS; indeed, they exchange among themselves a set of
solutions to $\De$, those which are also solutions of the
corresponding $\De_0$ (see the discussion and definitions 1,1' and 4
below).
As already recalled, in this section we have followed [11], to which
we also refer for further details and examples; see also [12-19] for
related approaches and questions.
\titleb{7. Conditional symmetries and conditional constants of motion
for dynamical systems}
Let us now consider how the general discussion on CS given in the
previous section does apply to dynamical systems. In this case, the
$\De$ of previous section will be a system of first order, autonomous
ODEs as in (1.1), i.e.
$$ \De \equiv \xd - f(x) = 0 \eqno(1) $$
where as usual $x \in M \sse \R^N$, $f : M \to TM$. Correspondingly,
we write generic LP VF on $M \x \R$ as
$$ \eta = \s (x,t) \pa_x + \tau (x,t ) \pa_t \eqno(2) $$
where $\s \pa_x \equiv \sum_i \s^i \pa / \pa x^i$.
The equation $\De_0$, see (6.3), is in this case
$$ \s (x,t) + \tau (x,t) \xd = 0 \eqno(3) $$
which using (1) reads simply
$$ \s (x,t) + \tau (x,t) f(x) = 0 \eqno(4) $$
{\small
{\bf Remark 11} As already discussed, we can consider $M' = \R \x M
\sse \R^{N+1}$ instead than $M$, with $\xd_0 = 1$, in order to have a
time-independent formalism. With this, (1) remains unchanged but $x
\in M'$ and $f_0 (x) = 1$, while $\eta = \s^0 \pa / \pa x^0 + \sum_i
\s^i \pa / \pa x^i$ and (4) reads
$$ \s^i (x) = - \s^0 (x) f^i (x) \eqno(5) $$
We will anyway not use this setting.}
We want to concentrate our attention on autonomous, time-independent
VFs, and consider CS of this form; this means that the considered VF
will be of the form
$$ \eta = \^s = s(x) \pa_x \equiv s^i (x) { \pa \over \pa x^i}
\eqno(6) $$
rather than (2). (The above remark 11 would then permit to
reinterprete our results in a more general setting, which we will not
do explicitely.)
Now $\eta^{(1)} \c \De = 0$ will give, before restriction to solution
of (1),
$$ \xd^j \pa_j s^i - s^j \pa_j f^i = 0 \eqno(7) $$
while a solution $x = \xi (t) $ is invariant under (6) if
$$ \De_0 \equiv s( \xi(t) ) = 0 \eqno(8) $$
Notice that if we do not ask invariance under (6) of the full
solution $x = \xi (t)$, but only of its trajectory $ \ga = \{ \xi (t)
$$ \vert f (\xi (t) ) \vert \c \vert s (\xi (t)) \vert = \pm \left( f
(\xi (t)) , s (\xi (t)) \right) \eqno(9) $$
where $(.,.)$ denotes scalar product in $T_x M$; in geometrical
terms, this is just the requirement that
$$ \^s : \ga \to T \ga \eqno(10) $$
We could therefore repeat the discussion of sect.6, which would just
require to consider (1),(7) and (8) (or (10) if we are considering
invariance of trajectories) instead than (6.1) and (6.3). We will not
bother the reader with such a repetition, and just state some
definitions which are natural in view of it.
{\bf Definition 1.} {\it The VF (6) is a (LPTI) conditional symmetry of
the DS (1) if and only if there are solutions $x = \xi (t)$ to this,
such that (8) is satisfied for all $t$.}
Similarly, we say that $\^s$ is a {\it configuration symmetry} of a
solution $x = \xi (t)$ (respectively, of the DS $\De$) if the
trajectory $\ga = \ga_\xi$ is invariant under $\^s$, i.e. if (10) is
verified for $\ga = \ga_\xi$ (respectively, if all trajectories are
invariant under $\^s$, i.e. if (10) is verified for any solution
trajectory $\ga$). With this, we have the
{\bf Definition 2.} {\it The VF (6) is a (LPTI) conditional
configuration symmetry of the DS (1) if and only if there are
solutions $x=\xi (t)$ to this, such that (10) is satisfied for $\ga =
\ga_\xi$ the corresponding trajectory.}
The discussion of the previous section shows that (LPTI) conditional
symmetries are also obtained as ordinary (LPTI) symmetries of the
system
$$ \cases{ \De = 0 & \cr \De_0 = 0 & \cr } \eqno(11) $$
where $\De , \De_0$ are given by (1),(8); the same holds for
conditional configuration symmetries, provided $\De_0$ is now given
instead by (9).
This would suggest an alternative definition of CS, i.e. the
{\bf Definition 1'.} {\it The VF (6) is a (LPTI) conditional symmetry
of the DS (1) if and only if it is an ordinary (LPTI) symmetry of the
system (11).}
It should be stressed that Definition 1 and Definition 1' are {\it
not equivalent}, as example 7 below does clearly show.
In the same spirit, it is entirely natural to consider, beside
conditional symmetries, also {\it conditional constants of motion}
(CCM); as already recalled, these were introduced ante litteram by
Sarlet, Leach and Cantrijn [32]; they are also implicitely considered
in [11] (see (2.9) of that paper), provided one specializes the
discussion of Levi and Winternitz to the case of DS instead than
considering the general PDE case.
Indeed, in the same way as a conditional symmetry is a symmetry of
(11), i.e. of $\De$ subject to the additional condition $\De = 0$, we
have that
{\bf Definition 3.} {\it A function $P(x)$, $P : M \to \R$, is a
conditional constant of motion of $\De$ if and only if it is a
constant of motion for the system (11).}
Notice that, as for ordinary constants of motion, every DS admits
CCM, given by constant functions.
It should be mentioned that symmetries of $\De$ can be also CS for
it, and in the same way CM are also CCM; such CS and CCM are not
interesting in this context, since they are not really "conditional",
and are dealt with - and utilized - by standard methods; they will be
seen here as trivial. In the following, it will be implicit that we
consider nontrivial CS and CCM.
Also, it should be stressed that since by definition (11) admits
solutions only for $\De_0$ corresponding to a CS, definition 3 is
meaningful only for this case; i.e., CCM are associated with CS, as
we are going to discuss in greater detail in the following.
Before doing this, it is worth considering again, from the present
point of view, simple examples given in previous sections.
{\bf Example 7.} Let us consider the simple system of examples 1-3,
namely (say in $\R^2 \backslash \{ 0 \}$ to eliminate the trivial
solution)
$$ \eqalign{ \xd =& \a (r^2 ) x - \b (r^2 ) y \cr
\yd =& \b (r^2 ) x + \a (r^2 ) y \cr }J\eqno(12) $$
which in polar coordinates reads simply
$$ \eqalign{ {\dot r} =& \a (r^2 )
\cr {\dot \th} =& \b (r^2 ) \cr } \eqno(13) $$
If $\a (r^2 ) $ has some nontrivial zero, say $r = r_0$, then the
solutions on the invariant circle $r_0$, i.e. the solution of the
system (11) where $\De$ is given by (13) and $\De_0$ by $r = r_0$,
are exchanged among themselves by rotations, i.e. by action of the VF
$\^s = x \pa_y - y \pa_x = \pa_\th$. It should be stressed that while
this shows that $r^2 = x^2 + y^2$ is a CCM in the sense of definition
3, and that $\pa_\th$ is a conditional configuration symmetry in the
sense of definition 2, some extra care should be taken before
concluding that $\pa_\th$ is also a CS. Indeed, $\pa_\th$ will {\it
not} be a symmetry of (11) in the sense of definition 1 with this
choice of $\De$ and $\De_0$, neither any solution of (12) (other than
the trivial one $x=y=0$ that we have excluded before) can be
invariant as a full solution\footnote{$^4$}{It should be stressed that
$\pa_\th$ is, anyway, a symmetry of (11); i.e. it is a CS in the
sense of definition 1'.}. One gets easily convinced that
$$ \eta = \b (r_0^2 ) \pa_t + [ x \pa_y - y \pa_x ] = \b (r_0^2 )
\pa_t + \pa_\th \eqno(14) $$
is a conditional symmetry for (12) (if, as before, $\a (r_0^2 ) = 0$).
The above example shows that requiring the CS of a DS to be
time-independent can be a too restrictive condition, especially in
view of definition 1, so that one should consider CS of general form.
It is maybe also worth giving an example of a case in which there are
CS which are time-independent.
{\bf Example 8.} Let us consider the system ($r^2 = x^2 + y^2$)
$$ \eqalign{ \xd =& \a (r^2 ) x - \b (r^2 ) y \cr
\yd =& \b (r^2 ) x + \a (r^2 ) y \cr
{\dot z} =& f(z) + g (x,y) r^2 \cr } \eqno(15) $$
It is immediate to check that $\pa_\th = x \pa_y - y \pa_x$ is now a
CS, corresponding indeed to solutions on the $z$ axis, i.e. with
$r=0$.
Notice that unless $g(x,y) = g(r)$, $\pa_\th$ is not an ordinary
symmetry for (15); notice also that it is indeed an ordinary symmetry
for (11), and therefore a CS in the sense of definition 1', as it
clearly exchanges solutions into solutions. It is the fact that it
does not leave any nontrivial solution invariant that makes that it
does not qualify as a CS in the sense of definition 1 as well.
The above discussion clearly explains why ordinary symmetries {\it
can} be also CS, but are not necessarily such in the sense of
definition 1 (while they are necessarily such in the sense of
definition 1'). In particular, it shows that necessarily if a VF $\eta
$ is a symmetry but not a CS for $\De$, then there is no solution of
$\De$ which is invariant under $\eta$.
Notice that if we do not focus on invariant solution, the definition
1 is not so natural; it would indeed be also useful to consider as CS
those VF which, without necessarily leaving invariant the set ${\cal
S}$ of all solutions to $\De$, leave invariant some set ${\cal S}_0
\sse {\cal S}$; if ${\cal S}_0$ amounts to a single solution, we
recover definition 1. At the risk of adding some confusion to the
already intricated nomenclature for conditional or similar
symmetries, we will call these {\it weak conditional symmetries}
(WCS); i.e., we have the
{\bf Definition 4.} {\it A VF $\eta$ is a weak conditional symmetry for
the DS $\De$ if and only if it leaves invariant (not necessarily
pointwise) a set ${\cal S}_0$ of solutions to $\De$.}
Notice that with this, a CS is also necessarily a WCS; notice also
that CS in the sense of definition 1' are necessarily WCS.
Also, in this way conditional configuration symmetries appear as a
special class of WCS, the invariant set ${\cal S}_0$ being that of
solutions with different initial data $x(0)$ along the same
trajectory, i.e. the set of solutions having the same trajectory.
In conformity with the general spirit of the present paper, we will
be specially interested in time-independent WCS; notice that these
will also be "weak conditional configuration symmetries", in the
sense that they will leave invariant (not necessarily pointwise) a
subset ${\cal T}_0$ of the set ${\cal T}$ of trajectories of
solutions to $\De$.
\titleb{8. Conditional symmetries, conditional constants of motion,
and invariant manifolds}
It is clear from the discussion of sect.7, as the reader may have
already remarked, that a close connection does also exists between CS
and CCM of a DS on one side, and its {\it invariant manifolds} on the
other; we are now going to discuss this.
Let us first consider LPTI CS; in this case the additional condition
$\De_0$ in (7.11) is given by (7.8). In this way, any VF $\^s$
identifies a manifold $S_0 \sse M$, and solutions to (7.11) are
nothing else than the solutions to $\De$ which lie entirely in $S_0$.
These are associated to the set $\Xi_0 \sse S_0$ of points $x$ for
which $\xd$ lies in $T S_0$, i.e.
$$ \Xi_s = \{ x \in S_0 ~/~ \xd \equiv f(x) \in T_x S_0 \} \eqno(1) $$
It is clear that $\Xi_s$ is an invariant manifold for the DS (7.1),
i.e. (by definition)
$$ f : \Xi_s \to T \Xi_s \eqno(2) $$
Notice that a different CS $\^s' $ with $S_0' = \Xi_s$, would yield
the same invariant solutions as $\^s$; this fact, ease of notation, and
the autonomous interest of this case, suggest to consider in
particular the case $\Xi_s = S_0$, i.e. that $S_0$ is itself an
invariant manifold for the dynamical system,
$$ f : S_0 \to T S_0 \eqno(3) $$
In both cases, (2) and (3), the determination of CS greatly
simplifies or solves the problem of determining invariant manifolds
for the DS under study. It should be stressed that this simple remark
does at once put at our disposition the powerful - and, what's more,
{\it completely algorithmical} - methods developed for the study of
symmetries and CS of differential equations, for studying invariant
manifolds of dynamical systems.
Let us now concentrate on the case (3): the invariant manifold being
determined by (7.8), it follows at once that $s^i (x)$ are CCM for
the DS $\xd = f(x)$ under study. More in general, any function $P(x)$
such that $S_0$ (or $\Xi_s$ in case (2)) is a level set of $P$ will
be a CCM for this DS.
Similarly, in case (2) any function $P(x)$ such that $\Xi_s$ is a
level set of $P$ will be a CCM for the DS. In this case we can first
restrict $P(x)$ to $S_0$, and consider more simply level sets of this
restriction.
{\small
{\bf Remark 12.} Clearly, if $P(x)$ is a CCM, also $P_c (x) = P(x) +
c$ is a CCM, for $c$ a constant. Denoting by $I_0$ the algebra of
constant functions on $M$, this and previous remarks on trivial CS
and CCM suggest that - in order to avoid unneeded redundancies - one
should actually consider constants of motion, and CCM, modulo $I_0$.}
Notice now that $S_0$ is a level set of $P(x)$ if and only if
$$ ( v(x) \c \grad ) P(x) = 0 \eqno(4) $$
for all $\^v : S_0 \to T S_0$; this last condition just means that
$$ ( v(x) \c \grad ) s(x) = 0 ~~~~{\rm on} ~ s(x) = 0 \eqno(5) $$
We have therefore a simple method to determine the $P(x)$ associated
to a given CS $\^s$. Indeed, once $s(x)$ is known, (5) is a linear
equation for $v$,
$$ A(x) v(x) = 0 \eqno(6) $$
where
$$ A_{ij} (x) = \left[ {\pa s^i (x) \over \pa x^j } \right]_{S_0}
\eqno(7) $$
Once the general solution $v(x)$ of this is known, we can pass to
determine $P(x)$ from (4), i.e. from the characteristic equation
$$ {\pa x^1 \over v^1 (x) } = ... = {d x^N \over v^N (x) } \eqno(8) $$
A close connection also exists between conditional configuration
symmetries and CCM. Indeed, if $\^s$ is a conditional configuration
symmetry, the trajectories invariant under it provide invariant
manifolds; such trajectories correspond, once $s(x)$ is fixed, to the
manifold $\chi_s$ of points satisfying (7.9), i.e.
$$ \chi_s = \{ x \in M ~/~ \vert f(x) \vert \c \vert s(x) \vert = (
f(x) , s(x) ) \} \eqno(9) $$
Notice that necessarily $S_0 \sse \chi_s$, which corresponds to the
fact that any CS is a fortiori a conditional configuration symmetry.
We can then proceed as in (4)-(8), with the role of $S_0$ played by
$\chi_s$.
It is also interesting to consider weak CS, and correspondingly weak
configuration symmetries. In this case again we have a connection
with CCM.
Indeed, for $\^s$ a WCS, let us consider the invariant set ${\cal
S}_0$, see definition 4; the trajectories of these form a manifold
(recall $f$ and $s$ are smooth functions), which we call again $S_0$
and which is by construction invariant under the flow of $\^f$ (and
also of $\^s$, obviously); we can then apply again the discussion
(4)-(8).
Notice that in this way a WCS is also characterized as a VF $\^s$
such that it exists a manifold $S_0 \sse M$ for which
$$ \eqalign{ & \{ f,s \} \vert_{S_0} = 0 \cr & \^f : S_0 \to T S_0
~;~~\^s : S_0 \to T S_0 \cr } \eqno(10) $$
Finally, let us consider a weak configuration symmetry, as defined in
sect.7; again the invariant set of trajectories ${\cal T}_0$ defines
a manifold in $M$ (the union of points $x \in \ga$ for $\ga \in {\cal
T}_0$) which is invariant under both $\^f$ and $\^s$, and the same
considerations apply.
\titleb{9. Module structure of conditional symmetries}
The connection between CS and CCM discussed in the previous section
also makes that there is a module structure of CS over the
corresponding CCM, pretty much analoguous to the model structure of
$\G_\De$ over $I_\De$ discussed in sect.2.
Indeed, let us reverse the point of view of the previous section, and
consider manifolds $S_0 \ss M$ identified by
$$ s(x) = 0 \eqno(1) $$
Suppose to have chosen $S_0$ invariant under $\^f$, so that any VF
$\eta : S_0 \to T S_0$ is a WCS for $\De$; moreover $\^s$ is then a CS
for $\De$. More in general, we could have several VF which are CS for
$\De$ and which leave invariant one (or more, or all) solution lying in
$S_0$.
It should be stressed that, once $S_0$ has been fixed, the associated
WCS form a Lie algebra, which we will denote by $\G_S^W$; this
follows from the definition of WCS or equivalently from (8.10).
As for CS, those which leave invariant {\it some} solutions lying in
$S_0$ (not necessarily the same solutions for different CS) do {\it
not} form an algebra; on the other side, if we fix a given set of
solutions lying in $S_0$, possibly all of them, the CS leaving these
invariant {\it do} form an algebra. To be specific, let us consider
the CS leaving invariant all the solutions in $S_0$, and let us call
their algebra $\G_S^C$; clearly,
$$ \^s \in \G_S^C \sse \G_S^W \eqno(2) $$
Let us also consider the CCM associated to $S_0$, i.e. the funstions
$P : M \to \R$ for which $S_0$ is a level set (their relation with
$S_0$ is discussed in sect.8); these do clearly form an algebra,
which we denote as $I_S$. It is then immediate to remark, as in sect.2,
that $\G_S^C$ and $\G_S^W$ are not only algebras, but do also have
the structure of modules over $I_S$.
Similar remarks, and corresponding results, hold when considering
conditional, and weak conditional, configuration symmetries.
\titleb{10. Hyperbolic fixed points, stable and unstable manifolds}
In the last two sections we have considered the relation between CS
(and CCM) and invariant manifolds, under several points of view which
were natural for the investigation of symmetry properties. We want
now to reverse our point of view, i.e. consider this relation, and
symmetry properties, from the point of view of dynamical system
theory. This discussion, developed in this and the next two sections,
is strongly related to [29], to which we refer for further details in
the context of ordinary symmetries.
It should be mentioned that a number of results are already known in
this direction, mainly concerned with ordinary Lie-point symmetries
rather than conditional ones [20,21,28,29]; we will briefly recall
them, referring the interested reader to original papers for details
and proofs.
First of all, as already remarked in sect.4, if the smooth DS
$$ \xd = f(x) \eqno(1) $$
admits an isolated fixed point $x_0$,
$$ f(x_0 ) =0 \eqno(2) $$
then any LPTI (ordinary) symmetry $\^s = s(x) \pa_x$ must satisfy
$$ s(x_0 ) = 0 \eqno(3) $$
Notice that (3) is {\it not} a necessary condition for $\^s$ to be a
CS, but is sufficient for this ($x(t) = x_0$ being the invariant
solution).
Let us now suppose $x_0$ to be hyperbolic [24-27], i.e. let us
consider the linear operator
$$ A \equiv (Df) (x_0 ) \eqno(4) $$
and suppose that all the eigenvalues of $A$ have nonzero real part.
In this case, to $x_0$ are associated local unique {\it stable} and
{\it unstable manifolds} [24-27], $W_s$ and $W_u$, which are in
particular invariant manifolds under the flow of $\^f$ (see [24-27]
for the role they play in DS theory).
It is remarkable that the situation described above for the fixed
point case is, mutatis mutandis, met again here: namely, it can be
proved [21,28,29] that a necessary (but clearly not sufficient)
condition for $\^s = s(x) \pa_x$ to be an ordinary LP symmetry of (1)
is that $$ \^s : W_s \to T W_s ~~~;~~~ \^s : W_u \to T W_u \eqno(5)
$$
This implies that the restrictions of $\^s , \^f $ to $W_s$ and $W_u$
are well defined, and indeed it is immediate to see that a condition
(again, necessary but not sufficient) for $\^s$ to be an ordinary
symmetry is that
$$ [ \^s , \^f ] \vert_{W_s} = 0 ~~~;~~~ [ \^s , \^f ] \vert_{W_u} = 0
\eqno(6) $$ Conversely, each of these is clearly a sufficient - but not
necessary - condition for $\^s$ to be a CS\footnote{$^5$}{At first
sight, one could think this just guarantees to have a WCS, but (6) also
guarantees (3).} of (1).
Also, the invariant manifolds $W_s , W_u$ will be identified by
equations
$$ w_s (x) = 0 ~~~;~~~ w_u (x) = 0 \eqno(7) $$
Notice that these are not unique, although $W_s$ and $W_u$ are: any
function with the same zero level set would do; clearly, functions
vanishing on $W_s$ (on $W_u$) form an algebra under sum, which we
call $I_W^s$ ($I_W^u$).
Clearly, the $w_s (x) , w_u (x)$ are automatically CCM for (1); as
discussed above, the functions satisfying the first (the second) of
(6) will form a module over the algebra $I_W^s$ ($I_W^u$).
The above considerations extend, a fortiori, to conditional
configuration symmetries.
It should also be mentioned that the property of being invariant
under any LPTI (ordinary) symmetry, is not peculiar to stable and
unstable manifolds, but does actually extend to any transversally
hyperbolic manifold (more in general, set) [29].
\titleb{11. Nonhyperbolic fixed points, center manifolds}
The case in which the fixed point $x_0$ of the DS (10.1) is not
hyperbolic, i.e. the linear operator $A$ given by (10.4) has some
eigenvalue with zero real part, is slightly more complicated. In this
case (see [29] for precise conditions and statement of results),
together with stable and unstable manifolds, one has to consider
center manifolds $W_c$ [24-27].
It should be stressed that the center manifold is {\it not} uniquely
defined, and that even if the DS is ${\cal C}^\infty$ the center
manifold could happen to be nonsmooth, or smooth only of class ${\cal
C}^k$ [6].
Anyway, each of these center manifolds is still an invariant manifold
under the flow of $\^f$; nevertheless, the result of the hyperbolic
case do not extend immediately to this setting.
The obstacle to such an extension is intimately related to the
non-unicity of the center manifold, i.e. to terms beyond all orders
in the perturbative expansion of the center manifold (these are in
turn related to resurgent functions [6,46]); a
discussion of these issues is given in [29], and it would be well
out of place to reproduce it here.
We will therefore consider only the {\it Poincar\'e- Dulac center
manifold} $W_c^0$, which we define to be the (analytic) center manifold
constructed perturbatively or, equivalently, the (infinite order) jet
of center manifolds [6,35]. This is uniquely defined, as all the center
manifolds differ indeed for non-perturbative terms and they all share
the same jet.
{\bf Example 9.} Let us consider the simple DS
$$ \eqalign{ \xd =& - x^3 \cr \yd =& - y \cr } \eqno(1) $$
It is immediate to see that all the curves
$$ y = \a e^{-1/x^2} \eqno(2) $$
are center manifolds, for $\a \in \R$. the perturbative (Taylor)
expansion of all these is anyway the same, and just gives $y=0$.
Notice that although all the center manifolds are ${\cal C}^\infty$,
the one with $\a = 0$ is the only one to be analytic.
If we consider the Poincar\'e- Dulac center manifold, the same
results given in the previous section apply [28,29], in particular
for what concerns (10.5), (10.6), (10.7), up to a simple rephrasing
amounting to write $W_c$ instead than $W_s$ or $W_u$.
In general, i.e. for generic center manifolds $W_c$, even $\^s : W_c
\to T W_c$ is not true, and we can only be sure that a LPTI symmetry
transforms a solution lying on a center manifold $W_c^a$ into a
solution lying on a center manifold $W_c^b$ (possibly, but in
general not, the same), see [28,29].
\titleb{12. Bifurcation of fixed points}
In the case the DS (10.1) depends on a real control parameter $\la$,
so that we actually deal with
$$ \xd = f (\la ; x ) \eqno(1) $$
and $x_0$ is a fixed point for all values of $\la$,
$$ f( \la ; x_0 ) = 0 ~~~~~~\all \la \eqno(2) $$
which is undergoing a simple bifurcation [25,26,47,48] at $\la = \la_0$, the
setting and results of the previous two sections combine nicely.
Indeed, let us suppose $x_0$ is stable for $\la < \la_0$ and only one
real eigenvalue (or a pair of complex conjugate ones) crosses the
imaginary axis with positive speed at $\la = \la_0$ (we suppose here
the reader to have some familiarity with bifurcation theory; see e.g.
[25,26,47,48] for introduction and discussion). Then, beside stable
manifolds, for fixed $\la$ the system (1) has a center manifold $W_c$
for $\la = \la_0$, and an unstable manifold $W_u (\la )$ for $\la >
\la_0$. The theorem on persistence of transversally hyperbolic
manifolds [25] ensures that the center manifold can be uniquely
defined by requiring it to be the limit of $W_u (\la ) $ for $\la \to
0^+$.
In this case again we do not meet the problems related to
non-uniqueness of the center manifold; moreover, since $W_u (\la) $
corresponds to its perturbative expansion, this limit is precisely
$W_c^0$, the Poincar\'e- Dulac center manifold.
Once again, we refer to [29] for details.
\vfill \eject
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\bye