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{ \nopagenumbers
\titlea{Approximate symmetries in dynamical systems}
\centerline{Giampaolo Cicogna}
\centerline{\it Dipartimento di Fisica, Universit\`a di Pisa}
\centerline{\it P.za Torricelli 2, I-56126 Pisa (Italy)}
\bigskip \bigskip
\centerline{Giuseppe Gaeta}
\centerline{\it Centre de Physique Th\'eorique, Ecole Polytechnique}
\centerline{\it F-91128 Palaiseau (France)}
\centerline{\it and}
\centerline{\it Departamento de Fisica Teorica II, Universidad
Complutense}
\centerline{\it Avenida Complutense, E-28040 Madrid (Spain)}
\footnote{}{
\pn
The work of GG is partially supported by C.N.R. grant 203-01-62}
\vskip 2.5 truecm
\pn
{\bf Summary.}
\pn
The knowledge of exact symmetries of a differential
problem allows to reduce it, and sometimes to completely solve it;
it also allows to obtain exact solutions. If we are looking for
approximate rather than exact solutions, then approximate symmetries are
as good as exact ones; moreover, they can be determined
perturbatively. In this paper we introduce and study approximate
symmetries, together with some applications to the determination of
approximate solutions of dynamical systems.
\vskip 1.5 cm
\pn
{\tt PACS n. 03 20, 02 20 }
\vfill \eject }
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\pageno = 1
\titleb{1. Symmetries and approximate symmetries for dynamical systems}
Let us consider an autonomous dynamical system (DS) of the form
$$ \xd = f(x) \eqno(1) $$
with $x \in M \sse \R^n$, $f: M \to TM$. Under a coordinate
transformation generated by $s: M \to TM$, i.e.
$$ x \to \~x = x + \eps s(x) \eqno(2) $$
this is transformed, in accordance with prolongation formula [1-3], to
a new DS $\dot{\~x}=\~f(\~x)$ where
$$ \~f = f + \eps \{ f , s \} \eqno(3) $$
and where we have introduced the Lie-Poisson bracket
$$ \{ \phi , \psi \} := (\phi \cdot \grad ) \psi - ( \psi \cdot \grad
) \phi\ . \eqno(4) $$
The Lie-point time-independent (LPTI) symmetries of (1) are therefore
given by vector fields (VF) of the form
$$ \sigma := s(x) \cdot \grad \eqno(5) $$
with $s$ solution of
$$ \{ f , s \} = 0\ . \eqno(6) $$
If $\phi:= f(x) \cdot \grad$ is the VF corresponding to (1), i.e. the
generator of the dynamical flow, then (6) is just the commutation condition
$$ [ \phi , \sigma ] = 0\ . \eqno(7) $$
In order to determine explicitely $\sigma$, we should solve (6), i.e. a
set of linear non-homogeneous PDEs for $s(x)$ (the {\it determining
equations}),
$$ f^j (x) { \pa s^i \over \pa x^j } = {\pa f^i \over \pa x^j }
s^j \eqno(8) $$
Such equations can be studied by the method of characteristics [4,5],
or by means of other procedures (see e.g. [2,6,7]); suppose anyway
we want to solve them perturbatively. In particular,
assume that we have an isolated fixed point $x_0$ (we can assume $x_0=0$)
for (1), and expand
$f$ and $s$ around this (notice that necessarily $s(0 )=0$ as
well); i.e. let us write
$$ \eqalignno{
f(x) =& \sum_{m=0}^\infty f_m (x) & (9) \cr
s(x) =& \sum_{m=0}^\infty s_m (x) & (10) \cr } $$
where $f_m$, $s_m$ are homogeneous polynomials of degree $m+1$
(issued by Taylor expansion in our case). Writing (6) in terms of
these, we get a series of equations
$$ \sum_{j=0}^k \{ f_j , s_{k-j} \} = 0 ~~~~ k=0,1,2,... \eqno(11) $$
Notice that if we have solved the equations in this sequence for $k \le
k_0- 1$, i.e. have determined $s_0 , ... , s_{k_0 - 1}$, the $k_0$
equation reads
$$ \{ f_0 , s_{k_0} \} = - \sum_{j=1}^{k_0} \{ f_j , s_{k_0 - j} \}
:= F_{k_0 } (x) \eqno(12) $$
where the r.h.s. is a known function (cf. [8]).
It is maybe worth remarking explicitely that $\{ f_0 , . \}$ is known
in Normal Form (NF) theory as the {\it homological operator}
associated to (the linear part of) $f$ [5,9,10]. We will write, for
$f_0 (x) = Ax$,
$$ \{ f_0 , . \} := \L_A (.) \eqno(13) $$
so that (12) is an equation of the form
$$ \L_A (s_k ) = F_k (x) \eqno(14) $$
which can be solved if and only if $F_k (x) \in \Ran ( \L_A )$;
moreover $s_k$ is determined up to a function in $\Ker (\L_A )$. For
the relation of this approach to the Poincar\'e-Dulac procedure for
transforming a DS (equivalently, a VF) into its NF, see [6,
9-14]; this relation will also be discussed in the following.
A little thinking, or some attempts to consider concrete cases, shows
that if we choose a $s_0 \in \Ker (\L_A )$ (notice the first of (11)
is just $\L_A (s_0 ) = 0$ or, with $s_0(x)=Sx$, simply $[A,S]=0$),
in general we can solve the equations
(11), (14) only up to some order $k$; i.e., at some $k+1$ the
$F_{k+1}$ will not belong to $\Ran (\L_A )$ and (14) will not admit a
solution. In this case, we have an {\bf approximate symmetry} of
order $k$.
Let us write, for given $f$ and {\it general} $s$,
$$ \{ f,s \} := r ~~;~~ r(x) = \sum_{m=0}^\infty r_m (x) \eqno(15) $$
With this notation, we have the following
\pn
{\bf Definition.} {\it If in (15) we have $r_m = 0$ $\forall m \le
k$, then $\sigma = s(x) \cdot \grad$ is an approximate symmetry of
order $k$ for (1).}
The purpose of this note is indeed to discuss approximate symmetries,
and their use in the study of DS.
Approximate symmetries were introduced and studied in [15] (see also [16]),
but here we will proceed in an indipendent way and with different goals. Our
work will be quite related to previous work on NF theory [8,14,17], and
we have indeed tried to keep to a notation similar to the one used there.
\pn
\titleb{2. The set of approximate symmetries of a dynamical system}
Let $f$ be given, and let $\sigma_1 = s_1(x) \c \grad$ and $\sigma_2 =s_2(x)
\c \grad$ be approximate symmetries of the same order $k$ for $\phi
:= f(x) \c \grad$. We consider
$$ [\sigma_1 , \sigma_2 ] \equiv \s := s(x) \c \grad ~~;~~ s(x) =
\sum_{m=0}^\infty s_m (x) \eqno(16) $$
and would like to know if $\s$ is also an approximate symmetry.
The commutator
$$ [ \phi , \s ] \equiv \rho = r(x) \c \grad ~~;~~ r(x) =
\sum_{m=0}^\infty r_m (x) \eqno(17) $$
is also written, using Jacobi identity,
$$ \rho = [ \sigma_1 , [\phi , \sigma_2 ]] - [\sigma_2 , [\phi , \sigma_1 ]]
\equiv [\sigma_1 , \chi_2 ] - [\sigma_2 , \chi_1 ] \eqno(18) $$
Now, both $\chi_1$ and $\chi_2$ have only terms of order greater than
$k$, so that $r_m (x) = 0$ $\forall m \le k$. We have therefore
proved the following lemma I (lemma II is obvious from the definition)
\pn
{\bf Lemma I.} The set $\G_\phi^{(k)}$ of approximate symmetries of
order $k$ for a VF $\phi$ has the structure of a Lie algebra.
\pn
{\bf Lemma II.} The chain of Lie algebras $\G_\phi^{(k)}$ obeys
$\G_\phi^{(k+1)} \sse \G_\phi^{(k)}$.
Let $I_\phi$ be the algebra of invariant functions under $\phi$, i.e.
of functions $\z$ such that $\phi \c \z = 0$, or
$$ f^j (x) {\pa \z \over \pa x^j } = 0 \eqno(19) $$
>From the point of view of the DS (1), $I_\phi$ is just the algebra of
constants of motion.
Let $\G_\phi$ be the algebra of exact symmetries for $\phi$; it is
well known that [18]:
\pn
{\bf Lemma III.} The set $\G_\phi$ has, beyond the structure of Lie
algebra, the structure of a {\it module} over the algebra $I_\phi$.
We can define {\bf approximate constants of motion} by considering
functions $\z : M \to \R$ and their expansion around $x=0$ (up to an
additional constant $\zeta(0)$)
$$ \z (x) = \sum_{m=0}^\infty z_m (x) \eqno(20) $$
The derivative of $\z$ along the flow of $\phi$ is given by
$$ (f \c \grad) \z = \sum_{m=0}^\infty \sum_{k=0}^\infty (f_k \c
\grad ) z_{m-k} (x) \equiv \sum_{m=0}^\infty w_m (x) \eqno(21) $$
Let us now consider $\sigma = ( s(x) \c \grad )$, and a function $\z :
M \to \R$; consider $\~\sigma = (\z (x) s(x) \c \grad )$. We have
$$ \{ f , \z s \} = \[ ( f \c \grad ) \z \] s + \z ( f \c \grad ) s -
\z ( s \c \grad ) f = \[ ( f \c \grad ) \z \] s + \z \{ f , s \}
\eqno(22) $$
This suggests the following definition:
\pn
{\bf Definition.} {\it If in (21) we have $w_m (x) \equiv 0$ $\forall
m \le k$, we say that $\z$ is an approximate constant of motion of
order k for (1).}
Let us denote by $I_\phi^{(k)}$ the set of approximate constants of
motion of order $k$ for $f$, $\phi := (f \c \grad )$. Clearly, we
have that the product of $\z_1 , \z_2$ in $I_\phi^{(k)}$ is still in
$I_\phi^{(k)}$, i.e. we have
\pn
{\bf Lemma IV.} The set $I_\phi^{(k)}$ is an (abelian) algebra, and
the chain of algebras $I_\phi^{(k)}$ obeys
$I_\phi^{(k+1)} \sse I_\phi^{(k)}$.
The following lemma is an immediate consequence of (21):
\pn
{\bf Lemma V.} The set $\G_\phi^{(k)}$ has, beyond the structure of
Lie algebra, the structure of a {\it module} over the algebra
$I_\phi^{(k)}$.
Let us also remark that if a VF $\s$ has $s_m=0\ \forall m\le k$, then it is
automatically a (trivial) approximate symmetry of order $k$ for any DS.
Similarly, any function $\zeta$ such that $\zeta(0)=$const (possibly zero) and
$z_m=0$ for $m\le k$ will automatically be a (trivial) constant of motion of
order $k$. Thus, approximate symmetries (constant of motions) are only
determined up to such trivial ones. In the following we will only consider
nontrivial approximate symmetries and constants of motions.
\pn
\titleb{3. Approximate symmetries and normal forms}
As recalled in sect.1, the construction of approximate symmetries
bears some resemblance with the construction of normal forms (NF); in
this section we will discuss the relations between the two
procedures (for details on NF theory, we refer to [5,9,10], see also
Sect. 6).
In order to fix notation, let us sketchily recall the Poincar\'e-
Dulac procedure to transform the DS (1) into NF. For the sake of
simplicity, we will make the assumption - common in NF theory - that
$$ [A , A^+ ] = 0 \eqno(23) $$
where $A$ is the matrix defined by $(Df)(0) \equiv A$, as in
sect.1 (i.e., $f_0 (x) = Ax$). Most of our results hold even without
(23), but would then require an heavier notation; see [17] for details.
Let us explicitly remark that, as usual in NF theory, the series of
transformations we are dealing with are in general purely {\it formal}
series, i.e. no assumption is made on their convergence. A complete
discussion and a list of results concerning the convergence and analyticity
properties of these series can be found in [10]. When the NF transformation
is not convergent, we have to operate a partial NF transformation, i.e.
transform the DS~into
NF only up to some finite order\footnote{$^{(1)}$}
{More generally, in any concrete computation we can implement the
Poincar\'e- Dulac algorithm only up to finite orders.}
$k$; it should not be surprising to remark that approximate
symmetries of order $k$ are in this case essentially equivalent to exact
symmetries, as far as the NF is concerned; see also the remark at the
end of the present section.
It should be stressed that the relation between symmetry and convergence of
the NF transformation has attracted attention in recent times [19,20]; in
particular, Bruno and Walcher [20] have proved for an ample class of DS in
$R^2$ the remarkable result that the system admits a convergent transformation
into NF if and only if it admits a local one-parameter group of symmetries
(the results can presumably be generalized to higher dimensions).
Writing $f(x)$ in the form (9), we look for a series of near-identity
coordinate transformations
$$ x \to \~x := x + h_k (x) \eqno(24) $$
with $h_k$ homogeneous of order $(k+1)$; transformations (24) should
be performed successively for $k=2,3,...$ (we do not discuss linear
changes of coordinates).
Under (24), $f$ is transformed to a new $\~f = \sum {\~f}_m $,
and we have in particular
$$ \eqalign{ \~f_m = & f_m ~~~~~ {\rm for ~} m < k \cr
\~f_k = & f_k + \L_A (h_k ) \cr } \eqno(25) $$
while the $\~f_m$ with $m>k$ are changed in a complicate way.
Assumption (23) guarantees $[ \Ran (\L_A ) ]^c = \Ker ( \L_A ) $, so
that by repeatedly applying (24) we can transform (1) into a DS
$$ \xd = g(x) = \sum_{m=0}^\infty g_m (x) \eqno(26) $$
in such a way that $g_0 (x) = Ax$ and all the $g_m$ satisfy
$$ \L_A (g_m ) = 0\quad {\rm or\ equivalently}\quad
\{ Ax , g_m (x) \} = 0 \eqno(27) $$
Obviously, the same as (27) also holds for $g$ {\it tout court},
$$ \L_A (g ) = 0 \quad {\rm or\ equivalently}\quad
\{ Ax , g (x) \} = 0 \eqno(28) $$
If $A$ admits {\it no} resonant relations among its eigenvalues [5,9,10]
(see also [8,17]),
then $\Ker (\L_A ) $ is trivial, and (26) is a {\it linear system}, i.e.
$g(x) \equiv Ax$.
Notice that if $\phi = (f \c \grad )$ admits some symmetry $\sigma = (
s \c \grad )$, the geometric relation $[ \phi , \sigma ] = 0$ is
obviously invariant under changes of coordinates, and therefore under
(24), but the expression of $s$ in coordinates will change, similarly
to what happens for $f$. Indeed, under (24), we have $\~s_m = s_m$
for $m < k$, and
$$ \~s_k = s_k + \L_S (h_k ) \eqno(29) $$
where $S=(Ds)(0)$, or $s_0(x)=Sx$, as already defined in Sect.1, and
$\L_S$ is the homological operator associated to $S$: i.e.
$\L_S(.)=\{s_0,.\}$ \big(cf. (13)\big). Nevertheless we stress that,
if we write $\psi = (g \c \grad )$, then the algebras $\G_\phi$ and
$\G_\psi$ of symmetries of (1) and of (26) are just the same algebra.
If (26) reduces to the linear form (independently of any
assumption on the presence of resonances for the linear part $A$),
i.e. it is just
$$ \xd = A x \eqno(30) $$
then the algebra $N(A)$ of matrices commuting with $A$ is a symmetry
algebra for (30): indeed
any matrix $S\in N(A)$ defines a linear VF $\s=(Sx\c \grad)$ which is
a symmetry for the DS. We can easily extract from this an abelian
algebra of $n$ independent symmetries: e.g. the algebra of matrices
$A^m, m=1,...,n$ if they
are linearly independent, so that the $A^m x$ span a basis of
$\R^n$ for any $x\not= 0$ \footnote{$^{(2)}$}{Notice that, if
the $A^m , m=1,...,n$ are not independent, the system (30)
is reducible, thanks to (23), i.e. splits into independent DS of smaller
dimension.}.
Another (isomorphic) $n-$dimensional abelian algebra of symmetries is
provided, if the eigenvalues of $A$ are real, by the dilations
$\rho_j=y_j(\pa/\pa y_j)$ (no sum over $j$) along each eigenvector $y_j,\
j=1,\ldots,n$. We can partially reverse this result, precisely we get
(see also [17]):
\pn
{\bf Proposition 1.} {\it If a DS can be linearized, then it admits $n$
independent commuting symmetries, which can be simultaneously taken into
linear form by a coordinate transformation. Conversely, if there is a
coordinate system in which the DS admits $n$ linearly independent {\it linear
commuting} symmetries $\s_j=(S_jx\c\grad)$ such that all the matrices $S_j$
are diagonalizable, then the DS can be linearized.}
\pn
{\bf Proof.} In the coordinates in which the system is linear, it is easy
to construct $n$ linear independent commuting symmetries. Conversely,
given $n$
commuting symmetries $\s_j=(S_jx\c\grad)$, the $n$ commuting matrices $S_j$
can be simultaneously diagonalized: $S_j\to{\rm
diag}(\b_1^{(j)},\ldots,\b_n^{(j)})$; now, with respect to the basis
spanned by the $n$ (independent) vectors $\b^{(j)}\equiv
(\b_1^{(j)},\ldots,\b_n^{(j)})$, the symmetries $\s_j=(S_jx\c\grad)$ become
just the $n$
independent dilations along the directions $\b^{(j)}$. A DS admitting
these $n$ symmetries is necessarily linear.
If (the linear part $A$ of) the DS (1) is resonant, we cannot transform,
in general,
(1) to the linear NF (30), but only to the general NF (26),(28).
However, if the lowest order of resonances in $A$ is $(k+1)$, then
(26) will be written as
$$ \xd = A x + \sum_{m=k+1}^\infty g_m (x) \eqno(31) $$
so that the NF is linear up to order $k$.
In this case, any matrix $S \in N(A)$ will define a VF $\sigma
= (Sx \c \grad )$ which is an {\it approximate} linear symmetry of order $k$
for (31). This corresponds to the fact that (31) is linear, and therefore
integrable, up to order $k$.
We have therefore proved the:
\pn
{\bf Proposition 2.} {\it If there is a coordinate system in which
(1) admits $n$ linearly independent {\it linear
abelian} approximate symmetries of order $k$,
$\s_j=(S_jx\c\grad)$ such that the matrices $S_j$ are
diagonalizable, then the NF of the DS is linear up to order $k$.
In particular, if $A^m$, $m=1,...,n$ are linearly
independent, then the NF for (1) is linear up to order $k$ if and only
if there is a coordinate system in which the $(A^m x \c \grad )$ are
approximate symmetries of order $k$ for (1).}
Notice that in this statement, no mention is made of the orders at
which $A$ admits resonances. Indeed, if $r$ is the lowest order at
which $A$ has resonances, the above proposition is trivial for $k \le
r$, but it does also hold for $k > r$. Actually, a specific DS will
have a specific NF, and some of the resonance terms allowed by
(26)-(28) could be absent.
In general, we could say that while exact symmetries determine the
structure (26)-(28) of general NF expansion, approximate symmetries
give a more precise determination of lowest order terms in this
expansion.
In particular, once we have determined the exact symmetries of (1)
and therefore the general structure of the NF admitted by such
symmetries - which is in general more restrictive than (26)-(28), see
[8,17] - we can further restrict the {\it a priori} NF (i.e. without
actually performing the NF reduction) by considering approximate
symmetries.
The following proposition is a trivial extension of a result in [8]
and can be proved along the lines of the original result. Similar
extensions could be obtained for all the results contained in [8] and
[12,17], again with proofs equivalent to the ones for exact symmetries.
\pn
{\bf Proposition 3.} {\it If the DS $\xd = f(x)$ admits the approximate
symmetry $\sigma = ( s \c \grad ) $ of order $k$, and $f(0) = s(0) = 0$;
$(Df) (0 ) = A$, $(Ds)(0 ) = S$, and $S$ is assumed to satisfy
the condition $[S,S^+]=0$ \footnote{$^{(3)}$}{{\rm As for the analogous
condition (23) for $A$, we introduce here this condition just for
simplicity. All the results hold, suitably modified, even for generic $A$
and $S$; see [17].}}, then $\phi = (f \c
\grad ) $ and $\sigma$ can be put in {\bf Joint Normal Form} up to
order $k$. That is, there is a series of Poincar\'e- Dulac changes of
coordinates such that in the final coordinates $\phi = ( g(x) \c
\grad )$, $\sigma = (v(x) \c \grad )$, and for $m \le k$
$$ \eqalign{ g_m \in & \Ker ( \L_A ) \cap \Ker ( \L_S ) \cr
v_m \in & \Ker ( \L_A ) \cap \Ker ( \L_S ) \cr } $$ }
This result - and extensions of other results to approximate
symmetries - tell something quite intuitive: {\it if we want to
determine the truncation of the NF to order $k$, then approximate
symmetries of order $k$ are as good and useful as exact symmetries.}
In this respect, it should be observed that in actual computations
one does always consider truncations of the NF: indeed, even if (1)
presents only finitely many terms, in general its NF (26) is an
infinite series, so that we have to truncate it if we want to perform
computations for practical purposes. Beside this, (26) is in general
only a formal series; one is then particularly interested in its
asymptoticity [5,9,10], which also leads to consider truncations of (26)
rather than the full (infinite) formal series.
As mentioned in the beginning of the present section, in these cases we
have to consider only a partial NF transform, so that the constraints
posed on the NF by an exact symmetry and by an approximate symmetry of
sufficiently high order (at least of the same order as the truncation
of the NF) are exactly equivalent.
Approximate symmetries of lower order will also impose extra constraint
on the NF unfolding, and this {\it a priori} constraint on the explicit
form of the DS once transformed into NF can simplify the implementation
of the Poincar\'e- Dulac algorithm.
{\it This shows that approximate symmetries should be given (at least) the
same relevance as exact symmetries in NF theory.}
\pn
\titleb{4. Approximate symmetries and approximate solutions to
dynamical systems}
The knowledge of symmetries of a DS allows to obtain new solutions
from known ones, by just applying symmetry transformations; this fact
can also be taken as a definition of symmetry (i.e., a symmetry is a
transformation which takes solutions into solutions).
We are considering in this paper LPTI symmetries, i.e. symmetries of the
form (5), which act on the $x$ alone and do not transform - neither depend
on - the time coordinate.
Let the solution to (1) with initial datum $x (t_0 ) = x_0$ be given
by $x(t) = \Phi (t; x_0 )$, and let $T_\phi^{(\tau)}$ be the operator
of advance in time along (1), i.e.
$$ T_\phi^{(\tau)} \Phi (t; x_0 ) = \Phi (t + \tau; x_0 ) \eqno(32) $$
If $\sigma = (s (x) \c \grad ) $ is a symmetry of (1), $\sigma \in
\G_\phi$, and $\La^{(a)}$ the transformation corresponding to
$$ \La^{(a)} x = e^{a \sigma } x ~~~~ a \in \R \eqno(33) $$
then, on solutions, this means that if $x(t) = \Phi (t;x_0 )$ is a solution
to (1), also $y(t) = \La x(t)$ is a solution (we write $\La$ for
$\La^{(a)}$, for ease of notation), i.e.:
$$ y(t) := \La \Phi (t;x_0 ) = \Phi (t;\La x_0 ) \equiv \Phi(t;y_0 )
\eqno(34) $$
where
$$y_0=\La x_0 \ . \eqno(35)$$
Let us now consider $\sigma \in \G_\phi^{(k)}$, i.e. an {\it approximate}
symmetry of order $k$ for (1). Let again consider a solution $x(t) =
\Phi (t; x_0 )$, and let $\La = e^{a \sigma}$ denote a
transformation on $M$. We denote by $\Psi$ the $\La$ transform of
$\Phi$, i.e.
$$ y(t) = \La x(t) = \La \Phi (t; x_0 ) = \Psi (t; y_0 ) \eqno(36) $$
Since $\sigma$ is {\it not } an exact symmetry of $\phi$, we have in
general
$$ \Psi (t ; y_0 ) \not= \Phi (t; y_0 ) \eqno(37) $$
On the other side, since $\sigma$ is an approximate symmetry, we expect
that $\Psi (t; y_0 ) $ is an approximation of $\Phi (t; y_0 )$. If
this is the case, knowledge of the solution issuing from $x_0$ and of
an approximate symmetry, gives an approximation to other solutions.
Let us now write, for ease of notation, $\Phi_t (.)$ for $\Phi
(t,.)$, and similarly $\Psi_t (.) $ for $\Psi (t,.)$. We consider,
for an initial point $y_0 = \La x_0$, the exact solution $y(t)$, and
the curve $\~y (t)$ obtained by applying the transformation $\La$
(which will be thought as an approximate symmetry) to the exact
solution $x(t) = \Phi_t (x_0 )$; these curves are
$$ y(t) = \Phi_t (y_0 ) ~~,~~ \~y (t) = \Psi_t (y_0 ) \equiv \La
\Phi_t \La^{-1} (y_0 ) \eqno(38) $$
We are interested in the error we have if considering $\~y (t) $
instead than $y(t)$, i.e. we want to know in which sense $\~y (t)$ is
an approximate solution. We have therefore to consider the operator
$$ \Theta_t \equiv \Phi_t - \Psi_t = \Phi_t - \La \Phi_t \La^{-1}
\equiv \[ \Phi_t , \La \] ~ \La^{-1} \eqno(39) $$
The commutator appearing here can be written as
$$ \[ \Phi_t , \La \] = \[ e^{a \s } , e^{t \phi } \] =
\sum_{m,n} {t^m a^n \over m! n! } \[ \phi^m , \s^n \] \eqno(40) $$
and if $[ \phi , \s ] \simeq O (x^k )$, each of these commutators
contributes with $(m \cdot n )$ terms $ O (x^k )$, so we get as a
rough estimate
$$ \[ \Phi_t , \La \] \simeq \sum_{m,n} {t^m a^n \over m! n! } m n O(
x^k ) \simeq e^{at} O (a t x^k ) \eqno(41) $$
Thus, for small $t$, the error $w(t)=\vert \~y(t) - y(t) \vert$ grows
with a speed of order $| \La x_0 |^k = |y_0|^k$.
This could be better illustrated by some explicit computation on a concrete
example. Let us consider the very simple system
$$ \eqalign{
{\dot x_1} = & -x_1 + x_2^k \cr {\dot x_2} = & -x_2 \cr} \eqno(42) $$
this admits rotations as an approximate symmetry of order $(k-1)$: we
compute e.g. a solution $x(t)$ with initial datum $x(0) = (1,0)$; we then
consider an initial condition $y(0)$ obtained by rotating $x(0)$
through a (small) angle $\theta$; we then compute the solution $y(t)$
with initial datum $y(0)$, and the curve $\~y (t)$ obtained by
rotating through the same angle $\theta$ the curve $x(t)$. It is easily
seen that in this case the error $w(t)$ is (uniformly, with respect to $t$)
of order $O(|\th|^k)$.
It can be directly verified (possibly by means of numerical computations)
that, in many cases, the solution $\~y(t)$ obtained in this way provides
in fact a quite good approximation of the exact solution $y(t)$: this is
particularly true especially if the DS admits some (approximate)
constants of motion and/or it has a Hamiltonian structure. In this case,
actually, due to the presence of the conserved
quantity and the oscillating nature of solutions, we should rather
apply averaging and Nekhoroshev-like theorems [5,9,21] to
provide estimates, but this is beyond the scope of the present paper.
\pn
\titleb{5. Approximate symmetries and perturbation of dynamical systems}
Considerations similar to the ones presented above also apply if we
have a different kind of "perturbation" problem: suppose that (1) is a
known "integrable" system, and $\~f$ is another problem, which we could
consider as a perturbation (not necessarily "small") of (1), i.e.
$$ \xd = \~f (x) = f(x) + a r(x) \eqno(43) $$
If we are able to solve the equation for $s$
$$ \{ f(x) , s(x) \} = r(x) \eqno(44) $$
then (approximate) solutions of (43) can be obtained by applying the
appropriate symmetry operator $\La = e^{a \sigma}$, $\sigma = (s \c \grad)$,
to solutions of (1).
In the notation of the previous section, $\Phi (t;x_0 )$ will now be
the solution to $\dot x = f(x)$, i.e. to (43) for $a=0$, and
$\Psi (t;x_0 )$ will be solution to $\dot x = \widetilde f (x)$.
The idea is then that if we have an approximate symmetry of the
system $\dot x = f(x)$ which satisfies (44), once applied
to $\Phi (t;x_0 )$ this will generate a curve $\Psi (t;y_0 )$ which
is an approximation of the curve $\Phi (t;y_0 )$, but which provides
a solution to (43).
Notice that if $s$ satisfies (44) exactly, we have in this way an exact
solution to (43), while if (44) is satisfied only up to some finite order
(as it will generally be the case) we have in this way an approximate
solution to (43).
In other words, operating with a transformation $\La^{-1}$, studying time
evolution, and then applying again $\La$,
is equivalent to apply a coordinate transformation $\La^{-1}$, study time
evolution in the new coordinates, and then passing back to the old ones
by the transformation $\La$: this is precisely the approach used in NF
theory: in fact, as we have already stressed, approximate symmetry
theory (and computations) bears a strong resemblance with NF theory
(and computations); see also Sect. 6 for further clarifications on this
point. Indeed, (44) is just the homological
equation we should solve to eliminate the terms $r(x)$.
Clearly, a special but important
case is obtained when (1) is a linear system, see (30),
corresponding to the linear part of (43); this corresponds to the
discussion in Sect. 3. Indeed, such a $\sigma$ would be an approximate
symmetry for $\~f$, allowing for elimination of low-order nonlinear
terms.
Applying an approximate symmetry $\s=(s\cdot\grad)$ of order $k$ to a
DS (VF) $f$,
we obtain a new DS (VF) $\~f = f + a r$ as in (1)-(3). We wonder if
$\s$ is also an approximate symmetry for $\~f$: precisely,
let $r_m =0$ $\forall m \le k$, so that a
$s(x)$ solution to (44) is an approximate symmetry of order $k$
for (1), $\sigma \in \G_\phi^{(k)}$; consider
$$ \{ f + a r , s \} \equiv v (x) = \sum_{m=0}^\infty v_m (x)
\eqno(45) $$
recalling that $s$ is a solution to (44), we have
$$ v(x) = r(x) + a \{ r , s \} \eqno(46) $$
and, since $r_m =0 $ $\forall m \le k$, we also have
$$ v_m (x) = 0 ~~~~ \forall m \le k \eqno(47) $$
In this way we have also proved the following general result:
\pn
{\bf Lemma VI.} Consider a VF $\phi$ and the algebra
$\G_\phi^{(k)}$; let $G^{(k)} (\phi )$ be the set of all the VF
obtained by applying $\La=e^{a \sigma}$ to $\phi$, with $a \in \R$, $\sigma
\in \G_\phi^{(k)}$. Then, for any $\psi = e^{a \sigma} \phi \in
G^{(k)} (\phi )$, we have $\G_\phi^{(k)} \sse \G_\psi^{(k)}$.
The corollary below follows then from applying the lemma to $\psi$
rather than to $\phi$, or from the algebra structure of
$\G_\phi^{(k)}$:
\pn
{\bf Corollary.} In lemma VI we actually have $\G_\phi^{(k)} =
\G_\psi^{(k)}$.
Therefore, if $\~f$ is obtained from $f$ by the action of $\sigma = (s
\c \grad )$
and the first nonzero nonlinear terms in $\~f$ are of
order $k$, then $\sigma$ is an approximate symmetry of order $k$ for
both $f$ and $\~f$. Notice that $\~f$ could have approximate
symmetries of higher order, allowing for linearization up to higher
orders according to sect.3.
\pn
\titleb{6. Determination of approximate symmetries and reduction to
normal form}
We would like, also in view of the considerations presented in the
previous section, to come back to the relations existing between
approximate symmetries and reduction to normal forms.
We recall that, under the Poincar\'e-Dulac transformation (24) of order
$k$, terms $f_m$ with $m < k$ are not changed, while $f_k$ is changed
according to (25), i.e.
$$ f \to \~f = f + \L_A (h_k ) + O (k+1) \eqno(48) $$
On the other side, under an approximate symmetry transformation $ x
\to e^\sigma x$, $\sigma \in \G_\phi^{(k-1)}$, we have
$$ f \to \^f = f + r_k + O (k+1) \eqno(49) $$
$$ r_k = \sum_{j=0}^k \{ f_j , s_{k-j} \} \eqno(50) $$
If we choose $s_m = 0$ $\forall m \le k-1$, which surely gives a
(trivial) approximate symmetry of order $k-1$, then
$$ r_k = \{ f_0 , s_k \} = \L_A (s_k ) \eqno(51) $$
This can also be read as saying that: {\it solving the homological
equation at order $k$ is equivalent to determining an approximate
symmetry of order $k$ with trivial $(k-1)$ first terms.}
Let us now consider the case $f$ is in linear NF up to order $(k-1)$;
in this case,
$$ r_k = \{ f_0 , s_k \} + \{ f_k , s_0 \} \equiv \L_A (s_k ) - \L_S
(f_k ) \eqno(52) $$
Also, in this case $s(x) = Sx$, with $[A,S]=0$, gives an approximate
symmetry of order at least $(k-1)$, by definition. We can therefore
consider $\sigma \in \G_\phi^{(k-1)}$ with generator
$$ s(x) = Sx + s_k (x) + O (k+1) \eqno(53) $$
As shown above, we can choose $s_k$ to be precisely the solution of
the homological equation; this permits to eliminate the part of $f_k$
lying in $\Ran (\L_A )$ or, with assumption (23), in $[ \Ker (\L_A )
]^c$. It is natural to wonder if (52) could not be used to eliminate,
or reduce, also the part of $f$ lying in $\Ker (\L_A )$. Indeed, due
to
$$ \[ \L_A , \L_S \] = \L_{[A,S]} \eqno(54) $$
(see [17] for a proof), we have that $[A,S]=0$ implies
$$ \L_S : \Ker (\L_A ) \to \Ker ( \L_A ) \eqno(55) $$
Let us decompose $f_k$ as
$$ f_k = v_k + t_k ~~~~; ~~ v_k \in \Ran (\L_A ) ~,~ t_k \in \Ker
(\L_A ) \eqno(56) $$
Then, by a Poincar\'e-Dulac transformation, or by a transformation of
the form (53) with $S=0$, we can eliminate $v_k$. Let us now try to
eliminate $t_k$; we look therefore for a $S$ in $N(A)$ such that
$$ \L_S (t_k ) = t_k \eqno(57) $$
In particular, consider $[S,S^+ ]=0$, so that $\Ker (\L_S )$ and
$\Ran (\L_S )$ are complementary subspaces. Notice that we can limit
ourselves to considering the action of $\L_S$ on homogeneous
polynomials of order $k$, i.e. we deal with finite-dimensional spaces.
The $t_k$ is now given, and we should look for a $S \in N(A)$ such
that (57) is verified; again, we can choose a basis $\{ S_1 ,\ldots,
S_r \} $ in $N(A)$, with $r \le n^2$, and consider a finite
dimensional operator $\A_f$ acting on matrices and valued in the
space of vector functions, defined by
$$ \[ \A_f (S) \] = \{ f (x) , Sx \} + f(x) \eqno(58) $$
If $\Ker (\A_{t_k} )\cap N(A) $ is not empty, then by (57) we can
eliminate $t_k$. Otherways, we can simplify it by choosing an
appropriate $S$; this must be such that $t_k$ is not in $\Ker (\L_A )
\cap \Ker (\L_S )$, or we would have no simplification. Notice that
if $f$ is in joint normal form with all of its symmetries [8], no
further simplification is possible.
\pn
\titleb{7. Examples}
\pn
{\it Example 1.}
We discuss first a very simple example, in order to fix
the ideas developed in the previous general discussion, by means of a
concrete case.
We consider two-dimensional DS ${\dot x} = f(x)$ of the form
$$ \eqalign{
{\dot x_1} = & \a (r^2 ) x_1 - \b (r^2 ) x_2 + h_{[k]} (x_1 ,x_2 ) \cr
{\dot x_2} = & \b (r^2 ) x_1 + \a (r^2 ) x_2 + g_{[k]} (x_1 ,x_2 )\cr }
\eqno(59) $$
where $r^2 = x_1^2 + x_2^2$, $\a$ and $\b$ are polynomials or,
more in general, analytic functions of $r^2$
$$ \a (r^2 ) = \sum_{m=0}^\infty \a_{2m} r^{2m} ~;~
\b (r^2 ) = \sum_{m=0}^\infty \b_{2m} r^{2m} \eqno(60) $$
and $h_{[k]},\ g_{[k]}$ are polynomials (or analytic functions) such that
all terms of degree $