%\magnification 1200 %\baselineskip=0.7cm \def\({\left(} \def\){\right)} \def\[{\left[} \def\]{\right]} \def\R{{\bf R}} %\parindent=0pt \baselineskip=0.7cm \def \pd{\partial} \def \pa{\partial} \def \pn{\par\noindent} \def \l{\lambda} \def \ep{\varepsilon} \def \phi{\varphi} \def \g{\gamma} \def \z{\zeta} \def \a{\alpha} \def \de{\delta} \def \om{\omega} \def \th{\vartheta} \def\=#1{\bar #1} \def\~#1{\widetilde #1} \def\.#1{\dot #1} \def\^#1{\widehat #1} \def\"#1{\ddot #1} \def \ss{symmetries } \def \sy{symmetry } \def \eqs{equations } \font\tafont = cmbx10 scaled\magstep2 \font\tbfont = cmbx10 scaled\magstep1 \font\sixrm = cmr6 \font\pet = cmr9 \def\titlea#1#2{{~ \vskip 1 truecm \tafont {\centerline {#1}} \medskip {\centerline{#2}} } \vskip 2truecm } \def\titleb#1{ \bigskip \bigskip {\tbfont {#1} } \bigskip} \def\Ref{\vskip 1 truecm {\it References} \bigskip \parskip=5 pt \parindent =0pt} \hfuzz=2pt \tolerance=500 \abovedisplayskip=2 mm plus6pt minus 4pt \belowdisplayskip=2 mm plus6pt minus 4pt \abovedisplayshortskip=0mm plus6pt minus 2pt \belowdisplayshortskip=2 mm plus4pt minus 4pt \predisplaypenalty=0 \clubpenalty=10000 \predisplaypenalty=0 \clubpenalty=10000 \widowpenalty=10000 \parindent=0pt \parskip=10pt {\nopagenumbers \titlea{ Nonlinear Lie symmetries in bifurcation theory}{} \smallskip \centerline{G. Cicogna} \centerline{\it Dipartimento di Fisica, Universita' di Pisa} \centerline{\it Piazza Torricelli 2, I-56126 Pisa (Italy)} \medskip \centerline{G. Gaeta\footnote{$^*$}{Supported in part by C.N.R. grant 203.01.48, present address: Math. Inst. Univ. of Utrecht (Netherlands )}} \centerline{\it Centre de Physique Theorique, Ecole Polytechnique} \centerline{\it F-91128 Palaiseau (France)} %\vfill\eject \smallskip \pn {\bf Abstract} \pn We examine the presence of general (nonlinear) time-independent Lie point \ss in dynamical systems, and especially in bifurcation problems. A crucial result is that center manifolds are invariant under these symmetries: this fact, which may be also useful for explicitly finding the center manifold, implies that Lie point \ss are inherited by the "reduced" bifurcation equation (a result which extends a known property of {\it linear} symmetries). An interesting situation occurs when a nonlinear \sy of the original equation results in a linear one (e.g. a rotation - typically related to a Hopf bifurcation) of the reduced problem. We provide a class of explicit examples admitting nonlinear symmetries, which clearly illustrate all these points. } %\vfill\eject \bigskip\bigskip {\centerline {\tt Phys. Lett. A {\bf 172} (1993), 361}} %\bigskip\bigskip \vfill\eject \pn Bifurcation theory is by now one of the most useful and used tools in attacking nonlinear problems; here we are in particular concerned with equivariant bifurcation theory, i.e. bifurcation in the presence of some \sy properties [1-7]. In general, all the developments of equivariant bifurcation theory, and its fundamental theorems, deal with {\it linear} \ss (or, to be more precise, with linear group actions). Quite recently, the notion of Lie-point (LP) \sy received diffuse attention [8-12], not only thanks to the theoretical appeal of LP symmetries, but also to their power in the study of concrete nonlinear equations, often in coincidence with symbolic manipulation computer programs [see, e.g., 13]. It is therefore very natural to try to build an equivariant theory for general LP symmetries: in fact, some results in this direction have been already obtained, showing that some typical results of linear equivariant bifurcation theory (e.g. the Sattinger theorem on the \ss of the bifurcation \eqs obtained via Lyapunov-Schmidt projection [2,4], and the equivariant bifurcation lemma [14,3]) can be suitably extended to nonlinear LP \ss [15,16]. Other applications can be found in [8-12, 17-18] and Ref. therein. We shall provide, in the second part of this paper, a class of explicit examples of dynamical systems exhibiting nonlinear LP symmetries. Let us recall that the generic dynamical problem $$\. u_i = F_i(\l,u) \qquad\qquad u=u(t) ;\qquad u\in R^n ;\qquad i=1,\ldots,n \eqno(1)$$ (where $F_i$ are smooth, e.g. $C^\infty$, functions, possibly also depending on one or more real parameters $\l$) admits a time-independent LP \sy generated by the $C^\infty$ vector field operator (here and in the following, $\pd_{u_i}$ stands for $\displaystyle {\pd\over{\pd u_i}}$, etc.) $$\eta =\sum_i \theta_i(\l, u)\ \pd_{u_i} \eqno(2)$$ if and only if [8,17] $$\big[\eta, \eta_F\big] = 0 \eqno(3)$$ where $\eta_F$ is the operator (generating the time evolution of the dynamical flow) $$\eta_F =\sum_i F_i\ \pd_{u_i} \eqno(4)$$ \medskip Given the generic dynamical problem (1), we may state the following main result. \medskip\pn {\bf Theorem}. {\it Let the linearization of the problem {\rm (1)} around the equilibrium point $u=u_0$ (and $\l=\l_0$; we can choose $u_0=0, \l_0=0$) possess some eigenvalues with vanishing real part, the remaining eigenvalues having strictly negative real part. Assume here that the (attractive) center manifold $M=W_c$ is uniquely defined. If the problem {\rm (1)} admits a time-independent LP symmetry generated by an operator $\eta$ {\rm (2)} defined in a neighbourhood of the point $u_0$, then the center manifold is left invariant by this symmetry, and the equations obtained by restriction to the center manifold $M$ admit the LP \sy $\eta \vert_M$}. \bigskip In short, this follows from (3) and from the fact that LP \ss transform solutions into neighbouring ones; if $\=u(t)$ is a solution belonging to the center manifold $M=W_c$, this implies that $\eta$ transforms it into a solution $u'(t)$ belonging to a neighbourhood $U$ of $M$; but the only possibility to have a solution $u'(t)\in U$, for all $t\in R$, is precisely that $u'(t)\in M$ [19-23]. In the case of bifurcation problems, the above argument can be extended to a "neighbourhood of criticality", i.e. in an interval about $\l_0$, by a standard procedure [24, 19-23]. Let us remark also that the nontrivial part of the above result can be written in the form $\eta : M \to TM$. Notice that our approach is essentially "local", being based on Lie {\it algebraic} method, and on local properties of manifolds as well: then, our arguments do not involve properties, e.g. the compactness, of the full \sy group action. \smallskip There are some important aspects of this result. First, one has that the center manifold (CM) $W_c$ is an invariant manifold under the LP \sy $\eta$ (notice that this is a peculiar property of center manifolds, and more in general of transversally hyperbolic manifolds: in fact, a generic manifold invariant under the LP \sy is in general {\it not} invariant under the dynamical flow). For sake of simplicity, we have stated our result under the assumption that the CM is unique; it is well known, however, that in general this condition is not met: we refer to [19-23] for detailed and careful discussions ensuring the existence of CM and concerning its properties. In the case of nonunique CM, our result above says immediately that a LP \sy may trasform a CM into another CM; but we can also prove that, given a LP \sy $\eta$, it is possible to choose in general one CM which is invariant under this $\eta$, and/or also to modify the \sy $\eta$ in such a way that the given CM turns out to be \sy invariant. Let us recall, incidentally, that the nonuniqueness of the CM does not affect practical calculations based, as usually happens, on power series expansions of the CM [20-23]. The property of the CM of being invariant under the LP \sy may be useful even for explicitly finding the CM itself: indeed, the expression describing $W_c$ must be a function of the quantities $k_\a=k_\a(u)$ which are left invariant by the \sy $\eta$, i.e. $$\eta \cdot k_\a = 0 \eqno(5)$$ (cf. also the method of introducing "adapted coordinates" [11], and/or Poincar\'e - Birkhoff normal forms [25,26]). \medskip Another general and important point of the above theorem concerns the \sy properties of the "reduced" bifurcation problem, i.e. of the \eqs restricted to the variables $v$ spanning the center eigenspace $E_c$. Precisely, putting $u\equiv (v,z)$, where $v \in E_c$, $z\in E_c^\perp$, we can write locally the CM $ W_c$ as identified by $z = \g(\l , v)$ for some function $\g$; then the reduced \eqs [20-23] obtained from the original problem (1) are $$\.v=\~F(\l,v)\equiv F\big(\l,v,\g(\l,v)\big) \eqno(6)$$ Now, if the LP \sy generator $\eta$ (2) is written as $$ \eta = \phi (\l , v,z) \pa_v + \psi (\l , v,z) \pa_z \eqno(7) $$ we then have $$\eta\vert_M = \phi \big(\l , v, \g(\l ,v)\big) \pa_v + \psi \big(\l , v, \g(\l ,v)\big) \pa_z \eqno(8) $$ and the condition to have $\eta\vert_M : M \to T M$ is simply $\delta z = ( \pa \g / \pa v) \delta v$, i.e. $$ \psi \big(\l , v, \g(\l ,v)\big) = {\pa \g \over \pa v} \phi \big(\l , v, \g(\l , v)\big) \eqno(9) $$ Therefore, the symmetry $\~\eta$ inherited by the reduced bifurcation equation (6) is given by $$\~\eta = {\~\phi} (\l , v) \pa_v \equiv \phi \big(\l , v, \g (\l ,v)\big)\pa_v \eqno(10) $$ This result extends to nonlinear LP \ss a well known property of linear ones [1,23]. Now, it may also happen that general nonlinear \ss of the original problem result in {\it linear} \ss of the reduced bifurcation equation (6). This result clearly cannot be predicted by standard (linear) equivariant bifurcation theory; in particular, if the original \eqs (1) have {\it no} linear symmetry, then linear equivariant bifurcation theory is not able to give any \sy constraint on the form of functions $\~F(\l,v)$, i.e. on the form of the reduced bifurcation equations. \bigskip Let us provide now a family of dynamical problems admitting a nonlinear LP symmetry; they will be useful also to illustrate all the above discussion. Let us write, for notational convenience, $$u\equiv (x,y,z_1,\ldots,z_{n-2})\ \in R^n \eqno(11)$$ Let $$w_a=w_a(y,z_a) \qquad\qquad (a=1,\ldots,k=n-2) \eqno(12)$$ be arbitrary smooth functions, satisfying the conditions $${\pd w_a\over{\pd z_a}}\ne 0 \qquad ({\rm no\ sum \ over \ } a) \eqno(13)$$ and put $$\chi_a=\chi_a(y,z_a)={\pd_yw_a\over{\pd_{z_a}w_a}}\qquad ({\rm no\ sum \ over \ } a) \eqno(14)$$ Consider systems of the following form $$\eqalign {\. x=&\ x\ f - y\ g \cr \. y=&\ y\ f + x\ g \cr \. z_1=&\ -y\ f\ \chi_1 - x\ g\ \chi_1 + {h_1\over{\pd_{z_1}w_1}} \cr \vdots \cr \. z_k=&\ -y\ f\ \chi_k - x\ g\ \chi_k + {h_k\over{\pd_{z_k}w_k}} \cr } \eqno(15) $$ where $f,g,h_1,\ldots,h_k$ are $n$ arbitrary smooth functions of the quantities $$r^2=x^2+y^2 \qquad {\rm and\ }\qquad w_1,\ldots,w_k\eqno(16)$$ (and possibly also of one or more real parameters $\l$, to include the case of bifurcation problems). Then, using (3), it is easy (even if tedious) to verify that the above system (15) admits the LP \sy generated by $$\eta=y\pd_x-x\pd_y+x\chi_1 \pd_{z_1}+\ldots +x\chi_k \pd_{z_k} \eqno(17)$$ In particular (with $n=3$, the extension to $n>3$ is straightforward; we write here $z, w$, etc. instead of $z_1, w_1$, etc.), if $w=w(y,z)$ and $h=h(r^2,w)$ in (15) satisfy $$\lim_{y,z\to 0}\pd_y w = 0 \qquad {\rm and} \qquad \lim_{y,z\to 0}{h(r^2,w)\over{z\ \pd_z w}}=\ell < 0 \eqno(18)$$ then the origin $u=u_0=0$ is an equilibrium point for the problem (15), with an attractive 2-dimensional center manifold $W_c$ tangent at $u_0=0$ to the center eigenspace $E_c$ ($z=0$). If the functions $f(\l,r^2,w), \ g(\l,r^2,w)$ (with $\l \in R)$ satisfy also $$f(0,0,0)=0,\quad \pd_\l f(0,0,0)\ne 0, \quad g(0,0,0)\ne 0 \eqno(19)$$ then the standard conditions for the appearance of Hopf bifurcation are met. The invariance under the \sy (17) implies that the CM must be a function of the quantities $k_\a(u)$ defined in (5), which are now just $r^2$ and $w_a$. E.g., let us choose in (12,15), with $n=3$, $$w=z\ {\rm e}^{-y}$$ which gives the nonlinear LP \sy $$\eta= y\pd_x-x\pd_y-xz\pd_z\ ,$$ and also choose $$f=h=-w+r^2$$ leaving arbitrary $g$. Then, it is easy to explicitly obtain the CM, which is given by the surface $$z={\rm e}^yr^2\ . $$ Also, more in general, if in (15), again with $n=3$, one has $f=h$ \big(independently of the choice of $w=w(y,z)$\big), then the equation $$h(r^2,w)=0\eqno (20)$$ gives the implicit expression of the center manifold. Another simple situation occurs if in (15) there is some $w=w_0=$ const such that $h(r^2,w_0)=0$; then, for any choice of $f, g$, the equation $$w(y,z)=w_0\eqno(21)$$ gives the CM, as easily verified by direct calculation. \bigskip The possibility that the reduced bifurcation \eqs (6) exhibit a linear symmetry, even in the absence of linear \ss of the original equations, is well illustrated by the following simple problem. Let us choose in (15) $\ w=z-y^2, \ \ h=-w, \ \ f=\l-r^2\ $, with arbitrary $g(\l,r^2,w)$ \big(satisfying in particular all conditions (18,19)\big): the resulting problem $$\eqalign {\. x=&\ x\ (\l -r^2) - y\ g(\l,r^2,w) \cr \. y=&\ y\ (\l-r^2) + x\ g(\l,r^2,w) \cr \. z=&-z+y^2\big(1+2(\l-r^2)\big)+2xy\ g(\l,r^2,w)\cr } \eqno(22) $$ admits the LP symmetry generated by $$\eta=y\pd_x-x\pd_y-2xy\pd_z \eqno(23)$$ It is easy to verify that $z=y^2$ is the CM \big(in agreement with (21)\big), and that the reduced \eqs for the variables $v\equiv (x,y)$ exhibit a (linear) rotation \sy generated by $$\eta_0=y\pd_x-x\pd_y \ . $$ Notice that a stable Hopf bifurcation for (22) appears at $\l_0=0$, described by $$r^2=x^2+y^2=\l, \qquad z=y^2$$ with frequency $\om(\l)=g(\l,\l,0)$. %\bigskip\bigskip \vfill\eject \parindent 0pt {\bf References} \bigskip [1] D. Ruelle, Arch. Rat. Mech. Anal. {\bf 51} (1973) 136 [2] D.H. 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