Homotopy Invariants of Time Reversible Periodic Orbits I. Theory

journal of differential equations 126, 204223 (1996) article no. 0049 Homotopy Invariants of Time Reversible Periodic Orbits II. Towards Applications Bernold Fiedler Freie Universitat Berlin, Institut fur Mathematik I, Arnimallee 2-6, D-14195 Berlin, Germany and Steffen Heinze Universitat Heidelberg, Institut fur Angewandte Mathematik, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany Received December 10, 1993; revised March 1, 1995 For a reversible periodic orbit # we apply the sequence of homotopy invariants deg n(#), n=1, 2, 3, ..., defined in [Fiedler 6 Heinze] (1996), We use this sequence to prove a global bifurcation result for reversible periodic orbits with prescribed minimal period. This result will be applied to second order systems with Neumann boundary conditions. A discussion and remarks on the sequence of degrees concludes the paper.  1996 Academic Press, Inc. 1. Introduction We consider time reversible systems of ordinary differential equations, i.e., x* =f (x) x # R 2N, f # C 1, (1.1) x, (1.2) with f (Rx)= &f (Rx), R := I \0 for all where I denotes the N_N identity matrix. 204 0022-039696 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved. + 0 , &I 205 HOMOTOPY INVARIANTS, II For convenience we repeat some properties of reversible systems. We also give the definition and properties of the sequence of degrees associated to a reversible periodic orbit #. For further definitions and the proofs, see [Fiedler 6 Heinze] (1996). Denote by , t(x) the flow generated by (1.1). A periodic orbit # is called reversible if it intersects the fixed point space Fix(R) of R in precisely two points p and q. These occur as zeros of the shooting map ? t , ?t : R N  RN (1.3) p [ .& t ( p), = where p # Fix(R) and . & t ( p) indicates the component of . t( p) in Fix(R) . If p is a zero of ? t , then so is q=. t( p). Moreover, ? nt( p)=? nt(q)=0, n=1, 2, 3, . . .. In the nondegenerate case, we define the sequence of Brouwer degrees d n( p) and d n(q) of ? nt , at p and q, by d n( p)=det D? nt( p){0, d n(q)=det D? nt(q){0, n=1, 2, 3.... (1.4) We also define the orbit degree deg n(#)=12(d n( p)+d n(q)), n=1, 2, 3, ... . (1.5) These degrees are related via the complex Floquet multipliers on the unit circle e i?j , j=1, ..., r, and _ & , the number of real Floquet multipliers in (&, &1). This is expressed in Theorem 2.1 in [Fiedler 6 Heinze] (1996), which we recall next. 1.1. Theorem. Let # be a nondegenerate reversible periodic orbit. Let n be any positive integer. Then r d 2n&1( p)=d 1( p) } (&1) (n&1)_& } ` (&1) [(n&12) j ] , (1.6a) j=1 r d 2n( p)=d 2( p) } (&1) (n&1) _& } ` (&1) [nj ] , (1.6b) j=1 where [ } ] denotes the integer part. The same holds for q replacing p. If r=0, that is for hyperbolic #, we assign the usual value 1 to the empty products over j. The degrees at p and q are related: d n( p)=d n(q) d n( p)=(&1) _& d n(q) for odd n, (1.7a) for even n. (1.7b) 206 FIEDLER AND HEINZE In particular, r deg 2n&1(#)=deg 1(#) } (&1) (n&1) _ & ` (&1) [(n&12) j ] , (1.8a) j=1 r deg 2n(#)=deg 2(#) } (&1) (n&1) _ & ` (&1) [nj ] , (1.8b) j=1 where deg 2n(#)=0 for odd _ & and all n. These formulas will be used in Section 2 to prove a global bifurcation result. In Section 3 we apply this result to second order Neumann problems. We also recall some terminology. 1.2. Definition. Let # be a reversible periodic orbit. Assume that &1 is not a Floquet multiplier, and +1 is only a trivial multiplier: algebraically double and geometrically simple. We call # elliptic, if there are nontrivial Floquet multipliers on the unit circle. Otherwise, we call # hyperbolic. Let _ & count the real Floquet multipliers in (&, &1). We call # Mobius, if _ & is odd. Otherwise we call # non-Mobius. We call _ &(mod 2) the Mobius parity of #. 2. A Global Bifurcation Theorem In this section we will prove the following global bifurcation result for reversible periodic orbits. 2.1. Theorem. Let the following three assumptions hold for the reversible system x* =f (x) and some T>2. (i) Among the reversible equilibria ! # Fix(R) there is exactly one, say !=0, with eigenvalues on the imaginary axis. These are the simple eigenvalues \?i. (ii) The set of reversible periodic orbits with period T is bounded. (iii) All reversible periodic orbits of (not necessarily minimal) period T are nondegenerate. Then there exists a nondegenerate reversible periodic orbit # of period T such that one of the following three mutually exclusive statements (a&c) holds. (a) # is hyperbolic non-Mobius with minimal period T. (b) # is hyperbolic Mobius with minimal period T or T2. (c) # is elliptic, but T might not be its minimal period. 207 HOMOTOPY INVARIANTS, II More precisely, assume that all reversible periodic orbits of (not necessarily minimal) period T are hyperbolic. For t>0 let a t , (c t ), respectively, denote the number of non-Mobius, (Mobius) reversible periodic orbits of minimal period t. Then a T +c T +c T2 is odd. (2.1) In a way, this theorem generalizes the case N=1. Indeed, neither Mobius nor elliptic orbits can occur in two dimensions. Only case (a) remains. Unlike usual results on global Hopf bifurcation, however, our theorem does not assert the reversible orbits to form a continuum or a connected curve. 2.2. Lemma. Let x=! # Fix(R) be a reversible equilibrium of a reversible system x=f (x), that is, ? t(!)=0 for all t. Assume that the linearization, B ), is nonsingular and the block matrix M=f $(!)=( AC D y* =My (2.2) does not possess a solution of period 2t. Then the local contribution of ! to deg ? t is given by deg loc(? t , !) : =sign det ?$t(!)=sign det C } ` sign sin(|t). (2.3) i| # spec(M) |>0 In the product, multiple eigenvalues are repeated with algebraic multiplicity; an empty product equals 1. Proof. By perturbation invariance we may assume all eigenvalues of M to be simple. By reversibility, R!=! implies MR= &RM. Hence A=D=0. Moreover, } # spec(M) if and only if } 2 # spec(BC)=spec(CB). (2.4) Indeed, by some row manipulations, det \ &} C &}+} &1BC B =det &} C + \ + 0 =det (} 2 &BC), &} for the nontrivial case }{0. In particular, i| # spec(M), |>0 generate precisely the negative eigenvalues &| 2 of BC. Now exponentiate M, letting At \C t Bt =exp (Mt), Dt + (2.5) 208 FIEDLER AND HEINZE for t>0. Note that deg loc(? t , !)=sign det C t , (2.6) by definition. We claim Ct = \  t 2j+1 (CB) j C. (2j+1)! j=0 + : (2.7) Indeed, an easy induction shows that 0 B C 0 (BC) j 0 2j + \ (CB) + , (BC) B 0 B 0 \C 0 + = \(CB) C 0 + , \ = 0 j j 2j+1 j for all integer j0. Plugging this into the exponential series for exp(Mt) proves (2.7). By (2.6), (2.7), and (2.4)  deg loc(? t , !)=sign det C } sign ` }~ # spec(BC) t 2j+1 (2j+1)! \ + t \ : }~ (2j+1)!+ : }~ j j=0  =sign det C } sign ` k # spec(BC) }~ <0 =sign det C } ` i| # spec(M) |>0 =sign det C } ` sign 2j+1 j j=0 \ 1  (&1) j : (|t) 2j+1 | j=0 (2j+1)! + sign sin (|t). i| # spec(M) |>0 This proves the lemma. K We note that Lemma 2.2 could also be derived from Theorem 1.1, essentially putting p=q=!. With Lemma 2.2 and under assumptions (i)(iii) in Theorem 2.1 we can now compute the contribution of the reversible periodic orbits to the degree deg(? t ) at positive non-integer tT2. Let EFix(R) denote the set of reversible equilibria. Consider the uniformly bounded sets P t :=[x # Fix(R) | ? t(x)=0]"E (2.8) of nonstationary points with period 2t. In Lemma 2.3 below we prove that the set P t is in fact compact. To compute the contribution of P t to deg(? t ), HOMOTOPY INVARIANTS, II 209 for tT2, we choose a large ball BFix(R) containing all intersections with Fix(R) of reversible periodic orbits with period tT2, according to assumption 2.1(ii). Moreover, we assume that no reversible equilibria lie on the boundary of B. Let P 0t B"E (2.9) denote an open neighborhood of the compact set P t of reversible periodic ``orbits.'' Choose P 0t =< if P t =<. We will compute deg(? t , P 0t , 0) in two different ways. First we use homotopy invariance; see (2.10) in Lemma 2.3 below. Second we add, mod 2, the individual contributions of reversible periodic orbits of ``period'' 2t; see (2.16). Comparing will prove Theorem 2.1. Global existence of . t and ? t will be a minor technical point later on. Since the relevant reversible periodic orbits are in some large bounded set, we may multiply the vector field outside by some positive scalar function, symmetric with respect to R, such that the maximal local flow . t becomes a global flow. Orbits are unchanged by this modification. Thus we will only consider global flows below. 2.3. Lemma. Let 0<tT2 be non-integer. Then the set P t of reversible periodic orbits of period 2t defined in (2.8) is compact. The Brouwer degree of ? t in its open neighborhood P 0t is given by deg(? t , P 0t , 0)=e 0(1&sign sin(?t)), (2.10) for some constant e 0 # [ \1]. Proof. We prove compactness of the set P t first. The set P t is bounded, by assumption 2.1(ii). It is closed by assumption 2.1(i) as we now prove, indirectly. If a sequence of non-stationary periodic orbits with period 2t converges to a reversible equilibrium !, then the linearization (2.2) at ! must also possess a nontrivial periodic solution with period 2t. This fact is a consequence of the virtual period proposition; see e.g. [MalletParet 6 Yorke] (1982), Proposition 3.1. In particular, the linearization at ! must possess purely imaginary eigenvalues. In our case, !=0 by assumption 2.1(i), and therefore t # Z, again by 2.1(i). This contradicts our assumption t  Z. Therefore, the set P t is indeed compact. We compute deg(? t , P 0t , 0) next. By additivity of the Brouwer degree, deg(? t , P 0t , 0)=deg(? t , B, 0)& : ! # E&B deg loc(? t , !). (2.11) 210 FIEDLER AND HEINZE We split the sum over ! # E & B into the contribution from the reversible Hopf point !=0 and a remainder, e, from reversible hyperbolic equilibria: : deg loc(? t , !)=deg loc(? t , !=0)+e. (2.12) !#E&B Both contributions are known from Lemma 2.2. Note that e does not depend on t>0. In contrast, deg loc(? t , !=0)=e 0 } sign sin(?t), (2.13) for some fixed e 0 = \1. To simplify the right hand side of (2.12), let { min <1 be chosen small enough such that P t =<, for t{ min . (2.14) By boundedness assumption 2.1(ii) such a lower bound 2{ min on the periods exists; see [Lasota 6 Yorke] (1971). For such t, (2.11)(2.12) imply deg(? t , B, 0)=e+e 0 , (2.15) since t{ min <1 and P 0t =<. By homotopy invariance of degree, (2.15) holds for 0<tT2. Thus we may rewrite (2.11)(2.12) for general noninteger 0<tT2 as deg(? t , P 0t , 0)=e 0(1&sign sin(?t)). This proves (2.10), and the lemma. K So much for the homotopy way to compute deg(? t , P 0t , 0). Next, we compute the same quantity by adding up the local contributions of all reversible periodic orbits with period 2t=T. By compactness of P t and by our nondegeneracy assumption 2.1(iii) there are only finitely many such orbits. Recall that we assume them not to be of elliptic type. Because T=2t may not be the minimal period, being just a period, we have to account for minimal periods Tj, j=1, 2, 3, . . . and sum over their respective contributions. To do this denote a Tj : the number of hyperbolic non-Mobius reversible periodic orbits with minimal period Tj, c Tj : the number of hyperbolic Mobius reversible periodic orbits with minimal period Tj, as in Theorem 2.1. Note that a Tj , c Tj , j=1, 2, 3, . .. count all orbits with 211 HOMOTOPY INVARIANTS, II period T, in absence of elliptic orbits. Adding their mod 2 contributions to (12) deg(? t , P 0t , 0), for 0<t=T2  Z, we claim : a Tj + : c Tj #[T2](mod 2). j (2.16) j odd The sums are running over integer j1, with the specified restrictions. Note that a Tj =c Tj =0 for Tj2{ min , by the lower period bound (2.14). Therefore the sums are finite. As usual [t] denotes the largest integer t. To prove (2.16), we invoke Lemma 2.3, and Theorem 1.1. By (2.10) at t=T2, 1 2 deg(? t , P 0t , 0)= \(1&sign sin(?t))2#[t](mod 2) (2.17) equals the right hand side of (2.16). Consider a hyperbolic non-Mobius orbit # of minimal period Tj=2{ next. By Theorem 1.1 it contributes deg j (#)= \1# +1(mod 2) (2.18) to (12) deg(? t , P 0t , 0), t=2j{. Indeed, _ & is even for a non-Mobius orbit, d 1( p) and d 2( p) are \1, and r=0 for a hyperbolic orbit. This explains the sum over a Tj in (2.16). Consider a hyperbolic Mobius orbit # of minimal period Tj=2{ next. Since _ & is odd, this time, it contributes (2.18) to (12) deg(? t , P 0t , 0) if and only if j is odd. Since ``elliptic'' orbits do not occur, additivity of Brouwer degree thus proves (2.16). For later use, we need a variant of (2.16). In fact (2.16) holds for any fraction Tk, k=1, 2, 3, . . . replacing T. Thus : a Tkj + : c Tkj #[T(2k)](mod 2), j for all k1. (2.19) j odd 2.4. Lemma. Let a Tj , c Tj , j1, be any sequence of integers satisfying (2.19) for all k1. Assume a Tj =c Tj =0 for all j>j max . Then a Tk +c Tk +c T(2k) # 1 (mod 2), {0 (mod 2), for kT2 for k>T2 (2.20) holds for all integers k1. Proof. We recast (2.19) in a more convenient form, summing over all k1. Of course, all sums will in fact be finite. For integers, let j | m indicate that j divides m; by D(m) we denote the number of divisors j of m, including j=1 and j=m. Note that D(m)=(' 1 +1) } } } (' + +1), (2.21) 212 FIEDLER AND HEINZE if m=p '11 } } } p '++ is the prime factor decomposition. In particular D(m)#1 (mod 2) (2.22) if m is a square. Otherwise D(m)#0. For any integer m, let m odd denote the maximal odd factor of m. Throughout, let # denote equality mod 2. With these conventions we recast (2.19), substituting m=kj : [T(2k)]#: : a Tkj +: : c Tkj k k #: m j k \ : 1+ a j odd Tm +: j|m m \ + : 1 c Tm j | modd #: D(m) a Tm +: D(m odd ) c Tm m m #: a Tm2 + : m # : c T(2:m2 ) :0 m odd : : (a T(2:m2 ) +c T(2:m2 ) +c T(2 } 2:m2 ) ). (2.23) :0, : even m odd We have used (2.22) and replaced m by m 2. All sums run over all indices 1, unless otherwise indicated. Again (2.23) holds true for T replaced by T$=Tk$, k$=1, 2, 3, .... The form (2.23), containing expressions a T $ +c T $ +c T $2 (2.24) for certain fractions T $=Tk$ on the right, lends itself to an inductive proof over increasing T$=Tk$. To begin the induction, consider T $=Tk$<Tj max . Then the terms (2.24) are all zero. Recursively, we increase T $ but keep T$2<1. Since the left hand side of (2.23) remains zero, we conclude a Tk +c Tk +c T(2k) #0 (2.25a) Tk<2. (2.25b) as long as This proves the bottom part of (2.20). To begin the induction over increasing T $ for the top part of (2.20), let T$=Tk$ be chosen minimal such that T $21. Invoking (2.23), with T $ replacing T, and (2.25a) yields a T $ +c T $ +c T $2 #1. (2.26a) HOMOTOPY INVARIANTS, II 213 For general T $21 note that the right hand side of (2.23) contains precisely [- T2] terms (2.24) with (12) T $=(12) T(2 :m 2 )1. Therefore we will have proved (2.26a) for all T $=Tk2, (2.26b) recursively, if we only prove the following claim : [tk]#[- t] (mod 2), for all t=T2>0. (2.27) k Note that (2.26a, b) prove the top part of (2.20). It remains to prove (2.27). This relation was observed by [Dirichlet] (1887), p. 52; for a proof see also [Polya 6 Szego ] (1976), exercises VIII. 79, 80. We reproduce the short argument. In planar cartesian (x 1 , x 2 )-coordinates, the left hand side of (2.27) counts the integer lattice points in the set [(x 1 , x 2 ) | 0<x 1 , x 2 , and x 1 x 2 t], (2.28) adding up the vertical slices x 1 =k, 0<x 2 [tx 1 ]. Since the set (2.28) and the integer lattice are invariant under a reflection at the diagonal x 1 =x 2 , only the diagonal of (2.28) contributes, mod 2. On x 1 =x 2 the set (2.28) obviously contains exactly [- t] lattice points, which is the right hand side of (2.27). The lemma is therefore proved. K With Lemma 2.4 the proof of Theorem 2.1 is complete, provided t=T2>1 is not an integer. We now treat the remaining case 2t= T2 # Z. The sum a T +c T +c T2 (2.29) accounts for nondegenerate hyperbolic non-Mobius reversible periodic orbits of minimal period T4, and also for such orbits with minimal period T22 of Mobius type. By local reversible Hopf bifurcation such orbits cannot accumulate at any equilibrium !. Indeed, the minimal periods near the only candidate, the reversible Hopf point !=0, are located near the limiting value 2; see e.g. [Vanderbauwhede] (1982). Moreover, the reversible periodic orbits bifurcating from !=0 are not of Mobius type. In particular, the sum (2.29) is finite and remains invariant under small perturbations T&= of T. Since k(T&=)2  Z can be assumed, the above results imply a T +c T +c T2 =a T&= +c T&= +c (T&=)2 #1(mod 2), and Theorem 2.1 is proved. K (2.30) 214 FIEDLER AND HEINZE 3. Discussion In this section we explore some largely open neighborhood of our results. We begin with the question of homotopy invariance of the whole sequence deg n(#), n=1, 2, 3, ..., and related objects. This question is closely tied to subharmonic bifurcation. We then get entangled in the notorious similarities between Hamiltonian and reversible systems. Since our result on periodic orbits with prescribed minimal period is quite different in spirit, we also include some remarks on global Hopf bifurcation which aims at unbounded continua of periodic orbits. After a brief excursion into partial differential equations, we finally summarize our results for second order boundary value problems. To study homotopy invariance of our orbit degree deg n(#) let # 0 be a nondegenerate reversible periodic orbit, with associated p 0 , q 0 and minimal half period { 0 as in Section 1.2 of [Fiedler 6 Heinze] (1996). Since ?{0( p 0 )=0, det ?${0( p 0 ){0, (3.1) the orbit # 0 continues to a local branch #({), through p({) nearby p({ 0 )=p 0 . If we also assume # 0 to be hyperbolic, then we know that deg n(#({))=deg n(# 0 )= { (&1) (n&1) _&2 deg 1(# 0 ) deg 2(# 0 ) for odd n, for even n, holds. In particular, the deg n(#({)) are independent of { in a neighborhood of { 0 which can be chosen independently of n. In contrast, suppose next # 0 is elliptic. Then the deg n(#({)) will typically oscillate rapidly as { varies, especially for large n. Specifically, let exp (\?i j ({)) denote pairs of simple Floquet multipliers on the unit circle and suppose that one of them crosses a primitive n th root of unity as { varies. For simplicity, also assume _ & is even. Then deg kn(#({)), k=1, 2, 3, ..., change sign, accordingly, since sin((12) kn ? j ({)) does. By standard degree theory, this implies that additional zeros of ? t bifurcate from the primary branches (n{, p({)) or (n{, q({)). The associated bifurcation is a subharmonic bifurcation; see e.g. [Vanderbauwhede] (1990) and the references there. In Fig. 3.1a3.1d we schematically depict a subharmonic bifurcation, together with a few other cases. The vertical axis indicates Fix(R). The horizontal axis may be viewed as ``time'' {; but note that we identify points on the same orbit and omit the multiples of the minimal {. Also, the secondary n{-branch is drawn near the primary branch to emphasize the bifurcation aspect. In fact, Fig. 3.1d ignores certain details of the subharmonic bifurcation. For example, suppose n4 is even. Then there are in fact two secondary half-branches, one terminating at p and the other at q. HOMOTOPY INVARIANTS, II 215 Fig. 3.1. Bifurcations of periodic orbits with minimal period near T. (a) hyperbolic, no bifurcation, (b) saddle node, (c) period doubling, (d) subharmonic, n3. Since the set of n th roots of unity, n=3, 4, 5, . .. is dense on the unit circle, we also expect the set of subharmonic bifurcation points to be dense, typically, along any elliptic branch. It was observed by [Vanderbauwhede] (1990), p. 956, that also one of the secondary branches might be elliptic. And so on, for a cascade of subharmonic bifurcations. We want the alternative in Theorem 2.1, minimal period versus elliptic orbit, to be understood in this framework. By our definition, an elliptic orbit is allowed to also possess Floquet multipliers off the unit circle. Only in two degrees of freedom, x # R 2N, N=2, does ellipticity imply linearized stability. Typically, in fact, nonlinear stability also holds due to the existence of families of two-dimensional KAM tori; see [Sevryuk] (1986). For N=1, clearly, Mobius orbits and elliptic orbits are equally impossible. But that case is amenable to phase plane analysis anyhow. See e.g. [Schaaf] (1990) for a recent study. We now sketch an idea on how to drop the nondegeneracy assumption (iii) in Theorem 2.1. In fact, by genericity arguments in the spirit of [Mallet-Paret 6 Yorke] (1982), [Fiedler] (1985) and [Fiedler] (1988), it is possible to find a sequence f m  f of reversible systems for which assumptions (i)(iii) and, therefore, conclusions (a)(c) hold. (Admittedly, we omit some details here.) In the limit m  , however, minimal periods may drop by an integer factor n2, similarly as for subharmonic bifurcations. In that case, the limiting periodic orbit # must possess a Floquet multiplier / which is an n th root of unity, by the virtual period proposition; see e.g. [Chow et al.] (1983), [Fiedler] (1985). Therefore, conclusions (a)(c) remain valid if we also allow ``elliptic'' orbits to just possess Floquet multipliers &1. Our oddness claim (2.1), however, has to be dropped since finiteness of a T +c T +c T2 cannot be guaranteed in degenerate cases. 216 FIEDLER AND HEINZE It is tempting to recast the infinite sequence deg n(#) of degrees into a more analytic quantity. Zeta functions are one possibility. Given a sequence d =(d 1 , . . .) of integers, let   ` d (s) := : d n n &s. (3.3) n=1 The sum converges absolutely for Re s>;+1, provided d n Cn ; (3.4) for some constants C, ;. For example, we may consider d n = 12 deg(? nt , P nt , 0) # [&1, 0, 1], (3.5) for the set P nt of reversible periodic orbits defined as in (2.8); see also Lemma 2.3. Which reversible periodic orbits #, { appear in P nt ? Certainly { and t must be rationally related, that is l{=jt (3.6) for some relatively prime positive integers l, j. Moreover n=kj (3.7) _ k =deg kl (#) (3.8) must be a multiple of j. Let and call _ =(_ 1 , _ 2 , . . .) the type of the nondegenerate orbit #. For fixed _ let a j (_ )  denote the number of reversible periodic orbits # of type _ for   which (3.6) holds. Then (3.5)(3.8) with the notation (3.3) imply : ` a (_ ) } ` _ =` d . (3.9) _ Here a(_ )=(a 1(_ ), a 2(_ ), . . .); we assume that only a finite number of types   convergence as in (3.4) for a(_ ). To prove (3.9) just  occurs, and we assume   note that by definition : : a j (_ ) _ nj =d n .  _ j|n (3.10)  For illustration, suppose that only the constant non-Mobius hyperbolic types _ = \(1, 1, 1, 1, .. .)= \e are present. Defining a j :=a j (e )&a j (&e )     we obtain  j | n a j =d n . HOMOTOPY INVARIANTS, II 217 By the Mobius inversion theorem, a n = : +( j) d nj (3.11) j|n where +( j) is the Mobius function: +( j)=(&1) l if j is a product of l distinct primes, and +( j)=0 otherwise. See e.g. [Abramowitz 6 Stegun] (1965). Note that |a n | 2 l } sup |d n | (3.12) n if n contains l distinct prime factors. Therefore, |a n | grows very slowly for bounded sequences d n . Still, the rather restrictive growth condition (3.4) impedes our definition (3.3) of the zeta function ` d , ` a . Other definitions are possible, for example  Z d (x)= : d n x n&1, (3.13a) n=1 or more generally  Z d. (x)= : d n . n(x)  (3.13b) n=1 for a linearly independent system . n(x) of functions. Let a n denote the number of reversible periodic orbits of period n. Then the radius of convergence s of Z d satisfies log (1\)lim sup ((1n) log a n ) allowing for an exponential rather than just polynomial growth of a n . Also note that Z d is related to the logarithmic derivative of a ``false zeta function'' in the spirit of [Smale] (1967). Unfortunately, the coupling of local degrees of reversible periodic orbits with rationally related periods is not as easily expressed, in this case, as in (3.9). In Theorem 2.2 in [Fiedler 6 Heinze] (1996) we recovered information on the Floquet multipliers from the sequence of degrees. This suggests yet another approach to the problem of recasting the whole sequence deg n(#) into a manageable form. Indeed, we could associate to # a spectral degree given by explicit expressions for the Fourier coefficients a * of the series deg n(#). The map from the deg n(#) to the a * is linear and one-to-one. In particular, additivity of degree carries over to the a * . One can therefore base an analysis on the a * instead of the deg n(#). We do not pursue these questions any further, here. Instead, note the difference between our present global result and the approach which we took in Theorem 2.1. Rather than considering the infinitely many multiples nT of 218 FIEDLER AND HEINZE a period T we took effectively a finite number of integers fractions Tn, that is, all multiples of the basic ``concert pitch'' frequency 1T: harmonics instead of subharmonics, following an ancient tradition in musicology. We have lamented above about the apparent lack of continuity in the periodic solutions found. Results on global Hopf bifurcation typically address this point. See for example [Alexander 6 Yorke] (1978), [Geba 6 Marzantowicz], [Ize et al.] (1989), and the references there. One popular approach is the following. Rescaling period T to one, we may seek 1-periodic solutions x( } ) of 1 F(T, x) := & x* +f (x)=0. T (3.14) Note that F commutes with the group SO(2)=RZ acting by time shift: (%x)(t) :=x(t+%), % # RZ. (3.15) Indeed, f is autonomous. The parameter (1T) multiplies the infinitesimal generator (ddt) of the group action. If f is reversible, then O(2), generated by the above SO(2) and a reflection }, commutes with F. Here } acts by (}x)(t) : =Rx(&t). (3.16) Inserting an additional real parameter *, unbounded continua of periodic solutions can be obtained by abstract topological methods in the SO(2) case, under suitable assumptions; see the above references. The O(2) case can be treated similarly; note that reversible periodic orbits are fixed under }. In all their abstract elegance and beauty, these results tend to ignore a quite special property of periodic solutions x( } ) which we have emphasized very much: if (T, x(t)) is a 1-periodic solution then so is (nT, x(nt)), for any n1. Such a rescaling property is absent for general SO(2)- or O(2)equivariant problems. It is correspondingly difficult to control the minimal period alias, in group jargon, the isotropy of x( } ) in the resulting unbounded continua. ``Snakes'' are a remedy, at least in the generic case of non-reversible vector fields f. See the original paper [Mallet-Paret 6 Yorke] (1982), and [Fiedler] (1988) for detailed references. As we recall from the introduction, the basic tool is an orbit index ,(#) which averages the local fixed point indices i n =i(6 n ), n1, of the iterates of a Poincare map 6 of #. In fact ,(#)= 12 ((&1) _ + +(&1) _+ +_& ), (3.17) in the nondegenerate case where _ + counts the real Floquet multipliers in (1, ) and (&1) _ & is essentially the Mobius parity. We claim that (&1) _ + , (&1) _& , and therefore all i(6 n ), are determined by our orbit degree HOMOTOPY INVARIANTS, II 219 deg (#), in the reversible case, if we ignore for a moment the fact that periodic orbits are not isolated. Indeed the trivial Floquet multiplier +1 may be assumed to have algebraic multiplicity two. It is easy to see, that, if / is a Floquet multiplier, for a reversible system, then so are /, / &1 and / &1. This implies _ + +_ & +r+1#N (mod 2), where r counts the ``elliptic'' pairs. In [Fiedler 6 Heinze] (1996), Theorem 2.2, we have computed r and _ & mod 2 from deg (#). This proves our claim. An analogous generic theory for autonomous Hamiltonian systems, although sketched in its beginnings, was never quite pushed to completion. Adding some artificial dissipation, however, provides some global results; see e.g. [Fiedler] (1988), Section 8.4.2. But for reversible systems no systematic ``dissipation trick'' is known and a ``snakes theory'' is still missing entirely. Not even to speak of reversible systems which are in addition equivariant with respect to a linear group action on x # R 2N... . Along a continuous branch of reversible periodic orbits the minimal period may become unbounded in several ways. One possibility is homoclinic period blow-up: the periodic orbits approach a reversible homoclinic orbit and disappear in a ``blue sky catastrophe.'' For a recent account of this phenomenon, observed already by [Devaney] (1976), see [Vanderbauwhede 6 Fiedler] (1991). In a reversible traveling wave setting, this effect can be described as a family of spatially periodic traveling waves limiting onto a single pulse wave. In [Chow 6 Deng 6 Fiedler] (1990) it was argued that the ``snakes'' orbit index ,(#) of approximating periodic orbits #, mentioned in (3.17) above, can be used for continuation purposes of the limiting homoclinic orbit. For a detailed exposition see [Fiedler] (1992). Whether a similar scheme works in the reversible case, with deg (#) replacing ,(#) must fortunately also remain open at this time. Partial differential equations are still another open topic. For example, consider u tt = \2 x u+g(u, u t ) (3.18) where t # R, x is in a bounded domain 0, u=u(t, x) satisfies appropriate boundary conditions on 0, and g is even in u t . Depending on the sign + or & of 2 x u, the equation is semilinear hyperbolic or elliptic, respectively. Anyhow u(&t, x), t # R, is a solution if and only if u(t, x) is. In some elliptic cases equation (3.18) can be reduced to a reversible finite-dimensional system, and our degree method readily applies. See e.g. [Kirchgassner] (1982) for a local and [Mielke] (1990) for a global reduction. At present, we are not able to develop a degree theory, directly, for solutions of the infinite-dimensional problem (3.18) which are periodic in t and reversible. 220 FIEDLER AND HEINZE As a curiosity we remark that, even for (reversible) equilibria, the infinite dimensional part of the Floquet spectrum tends to be hyperbolic, for the elliptic equation, whereas it will be ``elliptic'' for the hyperbolic equation. So much for nomenclature. Summarizing, let us apply Theorem 2.1 to our original Neumann problem u # R N, u +g(u, u* )=0, 0t+, (3.19) with g even in u* . If, instead, g is odd in u then analogous results can be obtained for the Dirichlet case. But let us focus on the Neumann problem now. Assume, for simplicity u Tg(u, u* )<0, for all |u| 2 >C 1 , u* = u. (3.20a) Then |u| 2 C 1 for any solution of the Neumann boundary value problem, 0t+. Also assume at most linear growth |u* Tg(u, u* )| C 2(1+ |u* | 22 ), (3.20b) uniformly for all |u(t)| 2 C 1 and all u* . Then in addition |u* | 22 e 2C2+ &1, for 0t+. These a priori bounds show that assumption (ii) of Theorem 2.1 holds. Since g is even in u* , the linearization at any reversible equilibrium u(t)#!, u* (t)#0 is given in block matrix form by I \ &g (!, 0) 0+ ; 0 (3.21) u g u denotes the partial derivative. To guarantee assumption (i) we therefore assume if g(!, 0)=0 then g u(!, 0) does not possess real eigenvalues 0, except for a simple eigenvalue ? 2 at say !=0. (3.22) (Here we recall relation (2.4) between the spectrum of &g u and that of the block matrix (3.21).) Following our remarks above, we cheerfully ignore assumption (iii) and obtain nonconstant solutions u of our boundary value problem (3.19), for any +=T2>1. We recall that (3.19) is Hamiltonian, in addition to being reversible, if g=g(u)={ u G(u) is the gradient of a scalar function G which does not depend on u* . In particular, our theorems remain valid for such Hamiltonian systems. We sketch the existence proof using variational methods. Let I(u)= | T2 0 (u* 22&G(u))(t) dt (3.23) HOMOTOPY INVARIANTS, II 221 for u # H 1(0, T2). The condition (3.20a) on g(u) implies that G(u) is uniformly bounded from above. Thus I(u) is bounded from below. For T>2 it is easily seen, that u#0 is a saddle for the functional (3.23). This implies the existence of nontrivial minimizers, proving Theorem 2.1 in this case. For general, not necessarily reversible Hamiltonian systems there is an enormous literature concerning periodic solutions. Mostly, the crucial ingredient is a variational principle on the space of closed loops, with periodic solutions as critical points. See for example [Rabinowitz] (1983, 1986), [Zehnder] (1987), [Ekeland] (1985), [Struwe] (1990), and the references there. One such result, due to [Rabinowitz] (1983), states that for any T>0 there exists a solution of (not necessarily minimal) period T, provided the Hamiltonian grows superquadratically at infinity. Drastically oversimplified, we sketch a corresponding bifurcation in Figure 3.2a. Under additional assumptions, minimality of the period T was also investigated successfully; see e.g. [Ekeland 6 Hofer] (1985) and the references there. In Figure 3.2b, we similarly oversimplify our Theorem 2.1, for comparison. Neither result actually obtains continuous branches of periodic solutions. Note that our result does not use any variational structure. The difference in Figs. 3.2a and 3.2b is due to the different conditions at infinity, only, in our opinion. Conversely it is tempting to extend our orbit degree deg n(#) to the general Hamiltonian case, even when there is no reversibility. Indeed, we suggest to take Theorem 2.1, (2.4) in [Fiedler 6 Heinze] (1996) for a definition of an orbit degree rather than a consequence. In exchange, homotopy invariance properties then have to be proved. It seems viable to achieve this by a detailed analysis of subharmonic bifurcations for Hamiltonian systems according to the same list as given in Figure 3.1 for the reversible case, see [Meyer] (1970). This time the horizontal axis might indicate period or, alternatively, energy. We find it intriguing, in this context, that the Maslov index of a Hamiltonian periodic orbit, governing an appropriate Morse theory, can likewise be expressed in terms of certain Fig. 3.2. Global branches of periodic orbits. (a) Superquadratic Hamiltonian, (b) a priori bounded reversible case. 222 FIEDLER AND HEINZE Floquet multipliers on the unit circle; see [Zehnder 6 Salamon] (1988). The periodically forced nonautonomous case, being the main emphasis there, should be seen as a symplectic companion in conjunction with the case of reversible diffeomorphisms. Distinguishing carefully between even and odd iterates, we hope an analogous degree theory for reversible diffeomorphisms can be developedbut not in the present paper. In absence of elliptic periodic reversible orbits, the solutions in Theorem 2.1 provide injective maps (u, u* ): [0, +]  R 2N, (3.24) except for possibly identical endpoints in the case of a Mobius orbit of minimal period T2=+. This injectivity replaces, for systems, monotonicity of u which can be asserted trivially for the case N=1. In the case N=1, even the stability for the corresponding scalar parabolic equation u t =u xx +g(u, u x ), 0t+ (3.25) can be determined. 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