%\magnification=1200
\parskip=2pt
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\G{{\cal G}}
\def\R{{\bf R}}
\def\I{{\cal I}}
\def \d{{\rm d}}
\def\pd{\partial}
\def\sp{\sum\nolimits^*}
\def \a{\alpha}
\def \b{\beta}
\def \om{\omega}
\def \ka{\kappa}
\def \z{\zeta}
\def \l{\lambda}
\def \phi{\varphi}
\def \th{\theta}
\def \ep{\epsilon}
\def \s{\sigma}
\def \tr {transformation}
\def \com {constant of motion}
\def \coms {constants of motion}
\def \sy {symmetry}
\def \sys {symmetries}
\def \an {analytic}
\def \co {convergen}
\def \evl {eigenvalue}
\def \bif{bifurcat}
\def \sol{solution}
\def \e{{\rm e}}
\def \qq {\qquad}
\def \q {\quad}
\def \en {\eqno}
\def \cd {\cdot}
\def \grad {\nabla}
\def \Ker {{\rm Ker}}
\def \sse {\subseteq}
\def \dst {\displaystyle}
\def \pn{\par\noindent}
\def \bs{\bigskip}
\def \en{\eqno}
\def \({\big(}
\def \){\big)}
\def \Ref{\bs\bs\pn{\bf References}\parskip=5 pt\parindent=0 pt}
\def \section#1{\bs\bs \pn {\bf #1} \bigskip}
\def\~#1{\widetilde #1}
\def\.#1{\dot #1}
\def\^#1{\widehat #1}
{\nopagenumbers
~ \vskip 3 truecm
\baselineskip .8cm
{\bf \centerline {Resonant Bifurcations}}
\vskip 2 truecm
\centerline {{\bf G. Cicogna}\footnote{$^{(*)}$}{E-mail :
cicogna@difi.unipi.it} }
\centerline{Dipartimento di Fisica, Universit\`a di Pisa, }
\centerline{Via Buonarroti 2, Ed. B}
\centerline{I-56127, Pisa, Italy}
\vskip 2 truecm
\pn
{\it Abstract.}
\font\pet = cmr8 {\pet
\medskip\pn
We consider dynamical systems
depending on one or more real parameters, and assuming that,
for some ``critical'' value of the parameters, the eigenvalues of the
linear part are resonant, we discuss the
existence -- under suitable hypotheses -- of a general class of
bifurcating solutions in correspondence to this
resonance. These bifurcating solutions include, as particular cases, the
usual stationary and Hopf bifurcations. The main idea
is to transform the given dynamical system into normal form (in the
sense of Poincar\'e-Dulac), and to impose that the normalizing
transformation is convergent, using the convergence conditions in the
form given by A. Bruno. Some specially interesting situations, including
the cases of multiple-periodic solutions, and of degenerate eigenvalues in
the presence of symmetry, are also discussed with some detail.}
\bigskip
{\hfill (to appear in J. Math. Anal. Appl.)}
\vfill\eject
}
\pageno=2
~ \vskip 1 truecm
\section{1. Introduction}
In this paper we consider dynamical systems of the form
$$\.u=f(u,\lambda)\equiv A(\lambda)u+F(u,\lambda)\qq u(t)\in\R^n\qq
\l\in\R^p\eqno(1)$$
depending on one or more real parameters $\lambda$, and, assuming that,
for some ``critical'' value $\lambda=\lambda_0$ of the parameters, the
matrix $A(\lambda_0)$ admits resonant eigenvalues, we want to discuss the
existence -- under suitable hypotheses -- of a general class of
``bifurcating solutions'' $u=u_{\lambda}(t)$ in correspondence to this
resonance. These bifurcating solutions include, as particular cases, the
usual stationary bifurcation and the Hopf bifurcation as well (see, e.g.
[1]). The main idea
is that to transform the given dynamical system into normal form (in the
sense of Poincar\'e-Dulac [2-9]), and try to impose that the normalizing
transformation is convergent. The imposition of convergence is
essentially based on the application of the conditions given by Bruno [4-5],
and leads to some prescriptions which may be fulfilled thanks to the presence
of the parameters $\lambda$; in this way, the appearance of these ``resonant
bifurcating'' solutions can be automatically deduced.
Among these solutions, some attention is paid to discuss some situations
with special interest (also from the physical point of view), including the
cases of multiple-frequency periodic solutions, and of degenerate \evl s
in the presence of symmetry, giving examples for each situation.
%\vfill\eject
\section{2. Basic assumptions and preliminaries}
We need some preliminary notions and results.
Let $u=u(t)\in\R^n,\ \l\in\R^p$ and $f=f(u,\l)$ be a vector-valued
analytic function in a neighbourhood of some $u_0$ and $\l_0$ (it is not
restrictive to choose $u_0=0$ and $\l_0=0$), such that
$f(\l,0)=0$ for each $\l$ in a neighbourhood of $\l_0$. Let us denote
the linear part of $f$ by $A(\l)u$ where
$$A(\l)=\nabla_u f(\l,0)\en(2)$$
and assume that, for some ``critical'' value of the parameters (we can
assume that this value is just $\l_0=0$) the matrix
$$A_0=A(0)$$
is semisimple, i.e. diagonalizable.
Let us notice that in the case $A_0$ is not semisimple, then it could
be uniquely decomposed into a sum of a semisimple and nilpotent part:
$A_0=A^{(s)}+A^{(n)}$, and the foregoing discussion could be equally well
applied to the semisimple part $A^{(s)}$, considering in particular normal
forms with respect only to $A^{(s)}$. The introduction of normal
forms with respect to a non-semisimple matrix requires a more difficult
procedure and will not be considered here (cf. [10-11]).
Up to a linear change of coordinates (possibly after complexification of
the space), we will then assume for convenience that the matrix
$A_0$ is diagonal, with \evl s $\s_1,\ldots,\s_n$. The first important
assumption is that, for the value $\l_0=0$, the \evl s
exhibit a resonance, i.e. there are some non-negative integers $m_i$ such
that
$$\sum_{i=1}^n m_i\s_i=\s_j\qq ; \qq \sum_{i=1}^n m_i\ge 2\en(3)$$
for some index $j\in[1,\ldots,n]$.
Together with the given dynamical system (DS) (1),
we need also to consider its normal form (NF) (in the sense of
Poincar\'e-Dulac [2-9]), in a neighbourhood of $u_0=0,\ \l_0=0$. As well
known, the idea is that of performing a near-identity coordinate \tr
$$u\to v=u+\phi(u)\en(4)$$
in such a way that in the new coordinates $v$ the given DS takes
its ``simplest'' form. To define this, consider
the linear operator $\A$ \big(the ``homological operator'' [2-9]\big)
defined on the space of vector-valued functions $h(v)$ by
$$\A(h)=A_0v\cdot\grad h-A_0h \en(5)$$
Writing the NF in the form
$$\.v= g(v,\l)=A_0v+G(v,\l) \en(6)$$
the nonlinear terms $G(v,\l)$ are then defined by the property [2-9,11-14]
$$G(v,\l)\in\Ker(\A) \en(7)$$
Actually, to be more precise, one should also consider, together with (1),
the $p$ equations
$$\.\l=0$$
(as in the usual suspension procedure), and extend the homological
operator $\A$ adding to the matrix $A_0$ the last $p$ columns and $p$ rows
equal to zero, and similarly extend $G(v,\l)$ as a $(n+p)-$dimensional
vector-valued function with the last $p$ components equal to $0$. It is
important indeed to notice that it is essential here to consider the $\l$ as
independent variables; in this way, in particular, one has that
the bilinear terms in $v$ and $\l$ are included in $G(v,\l)$; see also
Remark 1 below.
Let us now briefly recall the following important results. The proof of
these can be found e.g. in [8,11-14].
\smallskip\pn
{\bf Lemma 1.} {\it Given the matrix $A_0$, the most general NF is given
by (6) with
$$G(v,\l)=\sum_i\b_i\(\l,\rho(v)\)B_iv \qq {\rm where}\qq
B_i\in\C(A_0),\ \rho=\rho(v)\in\I_{A_0}\en(8)$$
where $\C(A_0)$ is the set of the matrices commuting with $A_0$, and
$\I_{A_0}$ is the set of the \coms\ of the linear system
$$\.v=A_0v\en(9)$$
The sum in (8) is extended to a set of independent matrices $B_i$,
the \coms\ $\rho(v)$ can be chosen in form of monomials (possibly fractional)
and the functions $\b_i$ are series or
rational functions of the $v_i$ (see [13] for a detailed statement,
and below, for the cases of interest for our discussion). }
\medskip
It is now clear that the assumption (3) on the existence of some
resonance among the \evl s of $A_0$ ensures that there are nontrivial
\coms\ of the linear problem (9), and then nontrivial terms in the NF (8).
It is also well known that the coordinate \tr s taking the given DS into
NF is usually performed by means of recursive techniques, and that in
general the sequence of these \tr s is purely formal: indeed, only very
special conditions can assure the \co ce of the normalizing \tr\ (NT).
Let us now recall the basic conditions, in the form given by Bruno,
and called respectively Condition $\om$ and Condition A, which ensure
this \co ce. The first condition is (see [4-5] for details)
\pn
{\sl Condition $\om$}: {\it let $\om_k=\min|(q,\s)-\s_j|$ for all
$j=1,\ldots,n$ and all $n-$uple
of nonnegative integers $q_i$ such that $1<\sum_{i=1}^n q_i<2^k$ and
$(q,\s)=\sum_iq_i\s_i\ne\s_j$: then we require }
$$\sum_{k=1}^\infty 2^{-k}\ln \big(\om_k^{-1}\big)<\infty$$
\pn
This is a actually very weak condition, devised to control the appearance
of small
divisors in the series of NT, and generalizes the Siegel-type condition:
$$|(q,\s)-\s_j|>\ep\ \Big(\sum_{i=1}^n q_i \Big)^{-\nu}$$
for some $\ep,\nu>0$, or the much simpler condition $|(q,\s)-\s_j|>\ep>0$,
for all $n-$uple $q_i$ such that $(q,\s)\ne\s_j$ (see [2-5]).
We explicitly assume from now on that this condition is always satisfied.
The other one, instead,
is a quite strong restriction on the form of the NF. To state this
condition in its simplest form, let us assume that there is
a straight line through the origin in the complex plane which contains
all the eigenvalues $\s_i$ of $A_0$
%, and that there are eigenvalues lying
% on both parts of this line with respect to the origin
. Then the condition reads
\pn
{\sl Condition A}: {\it there is a coordinate \tr\ $u\to v$ changing $f=A_0u+F$
to a NF $g=g(v)$ having the form
$$g(v)=A_0v+\a(v)A_0v$$
where $\a(v)$ is some scalar-valued power series \big(with $\a(0)=0$\big)}.
\smallskip\pn
In the case there is no line in the complex plane which satisfies the
above property, then Condition A should be modified [4-5] (or even
weakened: for instance, if there is a straight line through the origin
such that all the $\s_i$ lie on the same side in the complex plane with
respect to this line, then the
eigenvalues belong to a Poincar\'e domain [2-5] and the \co ce is
guaranteed without any other condition); but in all our applications
below we shall assume for the sake of definiteness that the \evl s are
either all real
%(and in this case not all with the same sign)
or purely
imaginary; therefore, the above formulation of Condition A is enough to cover
all the cases to be considered.
\smallskip\pn
{\it Remark 1.} It can be useful to point out that it is essential in the
present approach to use the suspension procedure for the parameters $\l$,
i.e. to consider $\l$ as additional variables. Indeed, let us consider,
for instance, a simple standard Hopf-type $2-$dimensional \bif ion
problem with $p=1$ parameter:
$$\.u=A_0u+\l I u+{\rm higher\ order\ terms}\qq {\rm where}\qq
A_0=\pmatrix{0 & 1 \cr -1 & 0} $$
and $I$ is the identity matrix. If $\l$ is kept fixed $\not=0$, the
\evl s of the linear part $A_0+\l I$ are $\s=\l\pm i$ and, as a
consequence, there are no (analytic nor fractional) \coms\ of the
linearized problem, and the NF is trivially linear. Also, the NT would be
convergent, as a consequence of Condition A (or -- more simply -- of the
Poincar\'e criterion [2-5]). But no \bif ion can be found in this way.
Instead, considering $\l$ as an independent variable, the linear part of the
problem is $\.u=A_0u$ and now there are nontrivial \coms\ in the NF, i.e.
the functions of $r^2=u_1^2+u_2^2$ and $\l$. \hfill$\diamondsuit$
\smallskip
Given the DS (1), it will be useful to rewrite its NF, according to
Lemma 1, observing that obviously $A_0\in\C(A_0)$ and in view of the
above Condition A, in the following ``splitted'' form
$$\.v=g(v,\l)=A_0v+\a\big(\l,\rho(v)\big) A_0v+
\sp_j\b_j\big(\l,\rho(v)\big)B_jv \qq\qq \rho(v)\in\I_{A_0}\en(10)$$
where (hereafter) $\sp_j$ is the sum extended to the matrices $B_j\not=A_0$
Following Bruno [4-5], we can then say that the \co ce
of the NT is granted if $\sp_j\b_j\big(\l,\rho(v)\big)B_jv=0$. Clearly,
``\co ce'' stands for ``\co ce in some neighbourhood of $u_0=0, \ \l_0=0$''.
Let us remark, incidentally, that an algorithmic implementation of the
procedure for obtaining step by step the NF is possible (cf. e.g. [15,16]);
we stress however that actually, in this paper, we shall not need any
explicit calculation of NF's.
\hfill\eject
\section{3. The 2--dimensional case.}
For the sake of simplicity we consider first of all the case of
2-dimensional DS, with some other simplifying assumptions; more general
cases will be considered in subsequent sections.
\pn
{\bf Theorem 1.} {\it
Given a 2-dimensional DS
$$\.u=A(\l)u+F(u,\l)\qq u\in\R^2,\ \l\in\R \en(11) $$
(i.e. $n=2,p=1)$, assume that for $\l_0=0$ the
two \evl s $\s_1,\s_2$ of $A_0$ are nonzero, of opposite sign if real,
and satisfy a resonance relation (3), which in this case can be more
conveniently written in the form of commensurability of the two \evl s;
$${\s_1\over{s_1}}=-{\s_2\over{s_2}}=\th_0\en(12)$$
where $s_1,s_2$ are two positive, relatively prime, integers. Assume also
that
$$s_2{\d\over{\d\l}}A_{11}(\l)+s_1{\d\over{\d\l}}A_{22}(\l)
\Big|_{\l=0}\not=0\en(13)$$
then there is a ``resonant \bif ing'' solution $\^u=\^u_{\l}(t)$
of the form
$$\eqalign{\^u_1= & c^{(1)}\exp\big(s_1\th(\l)t\big)+
\sum_{\matrix{n_1=-\infty\cr n_1\not=s_1}}^{+\infty}
c_{n_1}\exp\big(n_1\th(\l)t\big) \cr
\^u_2= & c^{(2)}\exp\big(-s_2\th(\l)t\big)+
\sum_{\matrix{n_2=-\infty\cr n_2\not=-s_2}}^{+\infty}
c_{n_2}\exp\big(n_2\th(\l)t\big)}\en(14)$$
where for $\l\to 0$
$$s_1\th(\l)\to \s_1\q ;\q -s_2\th(\l)\to \s_2\q ;\q c^{(1)},c^{(2)}\to 0 $$
and all terms in the two series are ``higher-order terms'', i.e. terms
vanishing more rapidly than the two leading terms, and the series are
convergent in some time interval. There is also an \an\ \com\ along this
solution, which, for small $\l$, has the form}
$$\rho(\^u)=\big(c^{(1)}\big)^{s_2}\big(c^{(2)}\big)^{s_1}+{\rm h.o.t.}$$
{\it Proof.} Consider the (possibly non-convergent) coordinate \tr\ $u\to
v$ (4) which takes the given DS into NF (10). The assumption
on the resonance of the \evl s $\s_i$ and Lemma 1 imply that the DS in NF is
expected to have the following form
$$\.v=g(\l,v)=A_0v+\a(\l,\rho)A_0v+\b(\l,\rho)Bv=(1+\a)A_0v+\b Bv\en(15)$$
where $\rho=\rho(v)=v_1^{s_2}v_2^{s_1}$ is a (monomial) \com\ of the linear
problem $\.v=A_0v$, $B$ is any diagonal matrix independent of $A_0$
(i.e. such that $s_2B_{11}+s_1B_{22}\not=0$), and $\l$ can be viewed as a
\com\ of the enlarged system including $\.\l=0$. We now impose the Bruno
Condition A (in the form given above, indeed $\s_i$ are either
real or purely imaginary) in order to ensure the convergence of the NT: this
amounts to imposing
$$\b(\l,\rho)=0\en(16)$$
The expression of this function is clearly not known (unless the NF
itself is known), but the manifold defined by $\b=0$ is \an\ [4], and the
relevant behaviour of the first-order terms (in $\l$) of $\b(\l,\rho)$ can be
inferred from the original DS $\.u=A(\l)u+\ldots$ , indeed the first-order
terms (in $\l$) of $A(\l)u$, i.e.
$A_0u+\l \big(\d A(\l)/\d\l\big)\big|_{\l=0}u$ are not changed by the
NT; therefore
$$\a(0,0)=\b(0,0)=0$$
and
$$s_2{\d\over{\d\l}}A_{11}(\l)+s_1{\d\over{\d\l}}A_{22}(\l)\Big|_{\l=0}=
\big(s_2B_{11}+s_1B_{22}\big){\pd\b\over{\pd\l}}\Big|_{\l=0,\rho=0}$$
having also used $s_2\s_1+s_1\s_2=0$. Assumption (13) then shows
that one can satisfy the condition $\b(\l,\rho)=0$, thanks to the
implicit-function theorem, if $\l$ and $\rho$ are
related by a function
$$\l=\l(\rho)\qq {\rm with}\qq \l(0)=0\en(17)$$
With $\b=0$, a solution of (15) is then
$$\^v(t)=\exp\big((1+\a)A_0t\big)\^v_0$$
or, putting $\th(\l)=\th_0(1+\a)$,
$$\eqalign{\^v_1(t)=&\^v_{10}\exp(s_1\th(\l)t)\cr
\^v_2(t)=&\^v_{20}\exp(-s_2\th(\l)t)} \en(18)$$
with the constraints
$$\big(\^v_1(t)\big)^{s_2}\big(\^v_2(t)\big)^{s_1}=
\big(\^v_{10}\big)^{s_2}\big(\^v_{20}\big)^{s_1}=\rho\in\I_{A_0}\qq
{\rm and}\qq \l=\l(\rho) \en(19)$$
and where
$\a=\a\big(\l(\rho)\big)\to 0$ , $\th(\l)\to \th_0$ for $\l,v,\rho\to 0$.
On the other hand, in the \an\ manifold defined by
$\b(\l,\rho)=0$ the NT is \co t and this \bif
ing \sol\ corresponds to an \an\ \sol\ $\^u=\^u_\l(t)$ of the initial
problem. The original coordinates $u$ are in fact related
to the new ones $v$ by an \an\ \tr\ \big(the inverse of (4)\big)
$$u=v+\psi(v)$$
where $\psi$ is a power series in the $v_1,\ v_2$, and the
\co ce is granted on some neighbourhood of zero, say
$|v_1|2$: reduction to a lower dimensional
problem}
In this section, we will consider a general situation in which, given a
$n-$dimensional DS with $n>2$, it is possible to reduce the problem to a
lower dimensional case.
A quite simple but useful result is the following.
\smallskip\pn
{\bf Lemma 2.} {\it Consider a $n-$dimensional DS ($n>2$), and assume that for
$\l_0=0$ there are $r2$: a general result.}
We now consider a $n-$dimensional DS: according to the above Section, we
can assume, for concreteness, that there is a resonance involving all the
$n$ \evl s (see also Remark 4 below).
Before giving the main result of this Section, the following property may
be useful (the proof is straightforward).
\smallskip\pn
{\bf Lemma 3.} {\it Given a DS in NF $\.v=g(v)=A_0v+G(v)$, the \coms\
$\rho\in\I_{A_0}$ of the linear part are in general not \coms\ of the
full DS, but their time dependence can be expressed as a function only
of the $\rho$ themselves: $(\d \rho/\d t)_g=\Phi(\rho)$, where
$(\d /\d t)_g$ is the Lie derivative along the DS. If the DS satisfies
Condition A, then these \coms\ $\rho$ are also \coms\ of the full DS
in NF $\.v=g(v)$.}
\smallskip\pn
{\bf Theorem 2.} {\it Consider the DS (1) and assume that for the value
$\l_0=0$ the \evl s $\s_i$ of $A_0$ are distinct, real
%(not all with the same sign)
or purely imaginary, and satisfy a resonance
relation (3). Assume also that $p=n-1$, i.e. that there are $n-1$
real parameters $\l\equiv(\l_1,\ldots,\l_{n-1})$, and finally that putting
$$a^{(i)}_k={\pd A_{ii}(\l)\over{\pd \l_k}}\Big|_{\l=0}
\qq\qq i=1,\ldots,n\ ;\qq k=1,\ldots,n-1 \en(24)$$
the $n\times n$ matrix $D$ constructed according to the following
definition (notice that only the diagonal terms $A_{ii}(\l)$ of $A(\l)$ are
involved) is not singular, i.e.:
$$\det D\equiv\det \pmatrix
{ \s_1 & a^{(1)}_1 & a^{(1)}_2 & \ldots & a^{(1)}_{n-1} \cr
\s_2 & a^{(2)}_1 & \ldots & & \cr
\ldots\cr
\s_n & a^{(n)}_1 & \ldots & & a^{(n)}_{n-1} }\not=0 \en(25)$$
Then, there is, in a neighbourhood of $u_0=0,\ \l_0=0,\ t=0$, a \bif ing
solution of the form
$$ \^u_i(t)=\big(\exp(\^\a(\l)A_0t)\big) \^u_{0i}(\l)+
{\rm h.o.t.} \qq \ i=1,\ldots,n \en(26) $$
where $\^\a(\l)$ is some function of the $\l$'s such that
$\^\a(\l)\to 1$ for $\l\to 0$.}
\smallskip\pn
{\it Proof.} As in the particular cases examined in the previous
Sections, let us consider the given problem transformed into NF:
$\.v=A_0v+G(v,\l)$ in the new coordinates $v$. Let us write $G(v,\l)$ in
the following, more convenient form:
$$G(v,\l)=\sum_{i=1}^n \ka_i\big(\l,\rho(v)\big)K_iv$$
where $K_i\equiv{\rm diag}(0,\ldots,1,\ldots,0)$ is the diagonal matrix with
$1$ at the $i-$th position, or also
$$\.v_i=\s_iv_i+\ka_i\big(\l,\rho(v)\big)v_i \qq{\rm (no\ sum\ over\ }
i=1,\ldots,n) \qq \rho(v)\in\I_{A_0} \en(27)$$
Now recall that the functions $\rho(v)$ and $\ka(\l,\rho)$ can be
fractional in the components $v_i$, but in such a way that each term
$\ka_iK_iv$ is a polynomial, so that the only admitted fractional terms
have necessarily the form, {\it e.g.},
$\dst{{{v_2^{s_2}\cdot\ldots\cdot v_n^{s_n}}\over{v_1}}}$, and so on; the
assumption that the $\s_i$ are distinct ensures that the functions $\ka$
cannot be of zero degree in the $v_i$ (i.e. of the form $v_2/v_1$,
{\it e.g.}), then when $v\to 0$ all terms
$\ka_i v_i$ vanish more rapidly than $v$ and one finds
$${\pd\over{\pd v_i}}\Big(\ka_i\big(\l,\rho(v)\big)v_i\Big)\Big|_{v=0}=
\ka_i(\l,0)\qq{\rm (no\ sum\ over\ } i)$$
Then, eq. (27) can be written, at the lowest-order
$$\.v_i=\s_iv_i+\ka_i(\l,0)v_i+\ldots=\s_iv_i+\sum_{k=1}^{n-1}
q_{ik}\l_kv_i+\ldots +\ldots \en(28)$$
where $q_{ik}$ are the elements of a constant matrix with $n$ rows
and $(n-1)$ columns. On the other hand, considering the original DS
$$\.u=A(\l)u+\ldots$$
its diagonal bilinear terms (in $u$ and $\l$) $a_k^{(i)}\l_ku_i$
\big(using definition (24)\big) are just NF terms and are not changed
by the normalizing procedure; therefore, $a_k^{(i)}=q_{ik}$.
Now according to Condition A, the NF is \co t (or better: is obtained by
a \co t NT) if one can rewrite (27) in the splitted form as in (10):
$$\.v=A_0v+\a\(\l,\rho(v)\)A_0v+\sp_j\b_j\(\l,\rho(v)\)B_jv
\en(29) $$
where $\a,\b_j$ are suitable combinations of the $\ka_i$, and
can satisfy the $(n-1)$ conditions
$$\b_j(\l,\rho)=0 \en(30)$$
In fact, the hypothesis (25) ensures precisely that one is able to do this
and also to satisfy $\b_j(\l,\rho)=0$, by means of the implicit-function
theorem, giving some $(n-1)$ relations (here the $\rho$
are considered as independent variables)
$$\l_j=\l_j(\rho) \en(31)$$
Once these $n-1$ conditions are satisfied, i.e. on the manifold defined
just by (31) (which is an \an\ manifold, see [4]), the \co ce of the NT
taking the initial DS into
$$\.v=A_0v+\a(\l)A_0v=\^\a(\l)A_0v\en(32)$$
is granted. This DS can be easily solved, giving
$$\^v(t)=\exp\big((\^\a(\l)A_0t)\big)\^v_0\en(33)$$
with the $n-1$ relations
$$\l_j=\l_j\big(\rho(\^v)\big)\qq\qq
\rho=\rho\big(\^v(t)\big)=\rho(\^v_0)\in\I_{A_0}\en(34)$$
The desired result is then obtained, with similar remarks as in the proof
of Theorem 1, coming
back to the initial coordinates by means of the inverse (\co t) \tr\
$v\to u=v+\psi(v)$, where $\psi(v)$ are series of monomials of the $v_i\
(i=1,\ldots,n)$. \hfill$\bullet$
\medskip
It is immediately seen that, in particular, condition (13) of Theorem 1 is
nothing but a special case of (25). As a generalization of Corollary 1, the
case of purely imaginary \evl s is particularly interesting, because it
corresponds to the case of coupled oscillators with multiple frequencies
and gives, in the above hypotheses, the existence of multiple-periodic
\bif ing solutions. We have indeed [17]:
\smallskip\pn
{\bf Corollary 3.} {\it With the same notations as before, let $n=r=4$
and $\s_1=-\s_2=i,\ \s_3=-\s_4=mi$ (with $m=2,3,\ldots$): then, with
$\l\equiv(\l_1,\l_2,\l_3)\in\R^3$, and $\det D\not=0$, there is a
double-periodic \bif ing solution preserving the frequency resonance }
$$\om_1:\om_2=1:m$$
\smallskip\pn
{\it Example 3.} Consider a 4-dimensional DS, with $u\in\R^4$, $\l\in\R^3$,
describing two coupled oscillators with unperturbed frequencies
$\om_1=1,\ \om_2=2$:
$$\.u=A(\l)u+F(u,\l) \qq\qq {\rm where} \qq
A(\l)=\pmatrix{\l_1+\l_3 & -(1+\l_2) & \l_1 & 0 \cr
1+\l_2 & \l_1-\l_3 & 0 & \l_1 \cr
\l_2 & 0 & \l_3 & -2 \cr
0 & -\l_2 & 2 & \l_3}$$
For $\l=0$, the \evl s of $A_0$ are $\pm i,\pm 2i$,
and it is easily seen that condition (25) is satisfied (the above
procedure and notations can be extended without difficulty to the complex
space). The NF will have the form, denoting by
$w_a=v_1+iv_2,\ w_b=v_3+iv_4$ the new coordinates, in complex form
$$\eqalign
{\.w_a=&\ iw_a+\(\l_1+\b^{(1)}_1(\l,\rho)+i(\l_2+\b^{(1)}_2(\l,\rho))\)w_a
\cr
\.w_b=&\ 2iw_b+\(\l_3+\b^{(1)}_3(\l,\rho)+i\b^{(1)}_4(\l,\rho)\)w_b}$$
where we have put
$$\b_i(\l,\rho)=\b_i(\l,0)+\b_i^{(1)}(\l,\rho)$$
and the $\b_i$ are real functions of the three (functionally independent)
\coms\
$$\rho_1=|w_a|^2=v_1^2+v_2^2\equiv r^2_a \ ;\
\rho_2=|w_b|^2=v_3^2+v_4^2\equiv r^2_b \ ;\
\rho_3=(w_a^2\overline{ w_b }+{\rm c.c.})\equiv 2r^2_ar_b\cos 2\phi $$
where $\phi$ is the time phase-shift between the two components $w_a$ and
$w_b$. Notice that fractional \coms\ $\rho(w)$ may appear in this problem,
e.g. $w_a^2/w_b$ or $\overline{ w_a }w_b/w_a$, etc. (which are
functionally -- but not polynomially -- dependent on the three above),
but this would not alter the result, as shown in the proof of the theorem.
The above NF can be trasformed into the splitted form (29)
$$ \eqalign
{\.w_a=&\ iw_a+i\a\big(\l,\rho(w)\big)w_a+ \b_a\big(\l,\rho(w)\big)w_a
\cr
\.w_b=&\ 2iw_b+2i\a\big(\l,\rho(w)\big)w_b+ \b_b\big(\l,\rho(w)\big)w_b}$$
with
$$\eqalign{ \a=&-\l_2-\b^{(1)}_2+\b^{(1)}_4 \cr
\b_a=&\l_1+\b^{(1)}_1+i\(2\l_2+2\b^{(1)}_2-\b^{(1)}_4\) \ ;\
\b_b=\l_3+\b^{(1)}_3+i\(2\l_2+2\b^{(1)}_2-\b^{(1)}_4\) }$$
One can impose the \co ce of the NT solving for $\l_i=\l_i(\rho)$ the
conditions $\b_a=\b_b=0$, which actually give three real conditions,
and obtain a \bif ing double-periodic solution, with frequencies
$$\om_1=1+\a(\l) \qq {\rm and}\qq \om_2=2\big(1+\a(\l)\)$$
\pn
Just to give a concrete example, let us imagine that the NF is such that
$$\b^{(1)}_1=-\rho_1\ ;\ \b^{(1)}_2=0\ ;\ \b^{(1)}_3=-\rho_2\ ;\
\b^{(1)}_4=\rho_3$$
then the leading terms of the solution, in the original real variables
$u_i$, are
$$\^u_1=r_a \cos\om t\ ;\ \^u_2=r_a \sin\om t\ ;\
\^u_3=r_b \cos2\om(t+\phi)\ ;\ \^u_4=r_b \sin2\om(t+\phi)$$
with the constraints
$$\l_1=r_a^2\ ;\ \l_3=r^2_b\ ;\ \l_2=r_a^2r_b\cos2\phi\ ;\ \om=1+\l_2\ $$
producing (see especially the role of $\l_2$) a sort of
amplitude--phase--frequency locking in the solution. \hfill$\diamondsuit$
\smallskip\pn
{\it Remark 4.} As already remarked, it can happen that, among the $n$
resonant \evl s $\s_i$, as considered in Theorem 2, one can find some
$r