%\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