Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R 3

Transcription

1 Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fiels in R 3 Jeroen S.W. Lamb, 1 Marco-Antonio Teixeira, 2 an Kevin N. Webster 1 1 Department of Mathematics, Imperial College, Lonon SW7 2AZ, Unite Kingom 2 IMECC, University of Campinas, CP 6065, Campinas SP, Brazil August 3, 2004 Abstract We stuy the ynamics near a symmetric Hopf-zero bifurcation in a Z 2R)-reversible vector fiel in R 3, with reversing symmetry R satisfying R 2 = I an im FixR) = 1. We focus on the case in which the normal form for this bifurcation isplays a egenerate family of heteroclinics between two asymmetric sale-foci. We stuy local perturbations of this egenerate family of heteroclinics within the class of reversible vector fiels an establish the generic existence of hyperbolic basic sets horseshoes), inepenent of the eigenvalues of the sale-foci, as well as cascaes of bifurcations of perioic, heteroclinic an homoclinic orbits. Finally, we iscuss the application of our results to the Michelson system, escribing stationary states an travelling waves of the Kuramoto-Sivashinsky PDE. Contents 1 Introuction 2 2 Elementary stationary an perioic solutions 9 3 Normal form approximation 10 4 Proof of Theorem Heteroclinic cycle bifurcation 14 a) Sections an return maps b) Dynamics of the first hit maps c) Symmetric an asymmetric 2D heteroclinic orbits ) Perioic solutions e) 1D heteroclinic orbits an homoclinic orbits f) Horseshoes Consequences for the Michelson system 26 1

2 1 Introuction In this paper we stuy the generic ynamics near a Hopf-zero bifurcation steay-state/hopf interaction) at a symmetric equilibrium solution of a Z 2 R)-reversible vector fiel in R 3, where the reversing symmetry R is such that im FixR) = 1. Such a Hopf-zero bifurcation arises in the Michelson system [33] ẋ = y, ẏ = z, ż = c x2 y. 1.1) which is reversible with respect to the time-reversal symmetry Rx, y, z) = x, y, z). 1.2) Recall that reversibility means that xt) is a solution of 1.1) if an only if Rx t) is a solution, or equivalently that the vector fiel anticommutes with R. The Hopf-zero bifurcation occurs precisely when c = 0. Namely, at c = 0, 0, 0, 0) is an equilibrium at which the erivative has eigenvalues {0, ±i}. The Michelson system is obtaine from a stationary-state an/or travelling wave reuction from the Kuramoto-Sivashinsky KS) PDE u t + uu x + u xx + u xxxx = ) The reversibility of the Michelson system is a consequence of the Z 2 -invariance of the KS system with respect to the transformation u, x, t) u, x, t). The KS system has been intensively stuie as a moel PDE with complex behaviour. Equilibria an perioic solutions of the Michelson system correspon to spatially constant an spatially perioic stationary an travelling solutions of the KS equation. Likewise, homoclinic an heteroclinic solutions between equilibria of the Michelson system represent stationary an travelling solutions that converge to spatially constant solutions as x ±. Our stuy of the reversible Hopf-zero bifurcation is partially motivate by the Michelson system, but also forms part of a program aressing the systematic stuy of the ynamics near local bifurcations in reversible an reversible-equivariant) vector fiels. Many ynamical systems that arise in the context of applications possess robust structural properties, such as for instance symmetries or Hamiltonian structure. In orer to unerstan the typical ynamics of such systems, their structure nee to be taken into account, leaing one to stuy phenomena that are generic among ynamical systems with the same structure. Recently, there has been a surging interest in the stuy of systems with time-reversal symmetries, in particular since the group theoretical classification of linear reversible equivariant systems by Lamb & Roberts [29]. Since then, the linear normal form an unfoling theory of reversible equivariant linear systems has been evelope by Hoveijn, Lamb & Roberts [22, 23]. Steay state bifurcation has been stuie recently by Buono, Lamb & Roberts [6] an Hopf bifurcation by Buzzi & Lamb [7]. Such bifurcations are characterize by the appearance of a zero eigenvalue or a egenerate pair of purely imaginary eigenvalues of the linear part of the vector fiel at an equilibrium point. Whereas [6, 7] restrict to the escription of elementary equilibria an perioic solutions, in this paper we escribe more complicate locally recurrent ynamics. We consier a one-parameter family of Z 2 R)-reversible ynamical systems ẋ = F x, μ), x = x, y, z) R 3, 1.4) where F R 3 R R 3, μ R, with symmetric R-invariant) equilibrium 0 at μ = 0, where im FixR) = 1. By Bochner s Theorem [34], we may assume without loss of generality that R acts locally) linearly, eg as the twofol rotation aroun the y-axis in 1.2). R-Reversibility means that It also arises in a 2-parameter moel of a feeback system in [26]. F R = R F. 1.5) 2

3 A first important observation is that R-reversible vector fiels of the type escribe above o not typically have symmetric equilibria. Namely, by virtue of the reversibility, F maps FixR) into Fix R). Since, im FixR) = 1 an im Fix R) = 2 it follows that F FixR),0) cannot hit 0 transversally. On the other han, if we consier a one-parameter family of reversible vector fiels, we fin that if F 0, 0) = 0, it generically hits 0 transversally so that the equilibrium point is typically persistent an isolate in FixR) R where R enotes the parameter space). We are therefore le to consier the one-parameter family F with symmetric equilibrium 0, 0). The eigenvalues of D x F 0, 0) are either {0, ±α} or {0, ±αi}, with α R. Varying the parameter μ reveals that the isolate symmetric equilibrium is typically as long as α 0) a fol point, where two branches of asymmetric equilibria meet see Section 2). In case the eigenvalues of D x F 0, 0) are real, 0, ±α) with α R +, the recurrent ynamics is restricte to a one-imensional centre manifol, on which the equations of motion are reversible an of the form ż = μ + az 2 +, with Rz) = z. the corresponing local ynamics is rather simple, featuring persistent heteroclinic connections between the two branches of asymmetric equilibria. We are thus le to focus on the Hopf-zero case where the nonzero eigenvalues are purely imaginary ±αi. Without loss of generality, in this case we may take the linear part of the R-reversible vector fiel as D x F 0, 0) = 0 α 0 α ) Associate with the pair of purely imaginary eigenvalues, varying μ, one fins in aition to the branches of asymmetric equilibria, a one-parameter family of symmetric R-invariant) perioic solutions with perio near 2π/α branching off the equilibrium point see Section 2). Since symmetric equilibria with the attache branches of equilibria an perioic solutions escribe above arise persistently in one-parameter families, it is natural to consier the Hopf-zero bifurcation as a coimension-one local bifurcation. However, as we shall see later, many aspects of the local ynamics epen sensitively on the nonlinear terms an are actually not finitely etermine. In this situation, rather than aiming to escribe the local ynamics in all its etails, it is necessary to be less ambitious, an we corresponingly focus on certain generically robust aspects of the ynamics of one-parameter families of reversible vector fiels in R 3 passing at μ = 0 through a Hopf-zero bifurcation point. Our strategy is to stuy the local ynamics in first appoximation by a normal form approach. It is a stanar result of Birkhoff normal form theory [15] that by an R-equivariant coorinate transformation preserving the R-reversibility), the vector fiel can be mae to be S 1 -equivariant up to arbitrary high orer in its Taylor expansion, where S 1 = {expd x 0, 0)s) s R}. Such S 1 -equivariant vector fiels can be stuie from the symmetry reuce vector fiel on R 3 /S 1 = {θ = 0}, taking r cos θ, r sin θ, z) as cylinrical coorinates for R 3. It turns out that the phase portraits of the reuce vector fiels are finitely etermine [36]. There are six ifferent cases, most of them yieling relatively uncomplicate ynamics near the bifurcation point. However, in one of the cases incientally corresponing to the situation in the Michelson system) an elliptic point an heteroclinic cycle are simultaneously born an the situation is more complicate. The situation is illustrate in Figure 1, with the phase portraits of the reuce normal form vector fiel in R 3 /S 1 epicte when μ < 0, μ = 0 an μ > 0. When μ > 0 there is a heteroclinic cycle between the two asymmetric equilibria on the z-axis, an at the same time also a symmetric elliptic point on the r-axis. In R 3 we fin corresponingly a heteroclinic cycle between two asymmetric equilibria of salefocus type, whose one- an two-imensional stable an unstable manifols exactly coincie. The two one-imensional manifols coinciing in a straight line an the two-imensional manifols coinciing on a two-sphere. Also, in R 3 we have an elliptic perioic solution, an the phase space between the heteroclinic cycle an the perioic solution is foliate by invariant two-tori. It is essential to notice that the flow in the latter normal form approximation is very egenerate in the context of reversible flows in R 3. The reason for the egeneracy is of course the S 1 -equivariance it 3

4 z z z r r r μ < 0 μ = 0 μ > 0 Figure 1: The reuce normal form vector fiel on R 3 /S 1 = {θ = 0} near the Hopf-zero bifurcation at μ = 0, for the case a > 0, b = 1 referring to the coefficients in 3.3). gaine in the normal form approximation. The task is to unerstan the ynamics when the S 1 normal form symmetry is broken by flat perturbations, ie perturbations that are small beyon any algebraic orer). Our first result concerns the generic occurrence of an infinite number of homoclinic an heteroclinic bifurcations in the neighbourhoo of the reversible Hopf-zero bifurcation: Theorem 1.1 Denote by X μ R the space of one parameter families of R-reversible vector fiels 1.4) exhibiting the Hopf-zero bifurcation as above at μ = 0, enowe with the C topology. There exists an open subset U X μ R containing the origin, which is etermine by the 2-jet of the vector fiels at 0, 0) R 3 R, such that the set of vector fiels for which in a neighbourhoo of the origin in R 3 R there exists a countable infinity of homoclinic orbits an heteroclinic cycles between the two sale-focus fixe points, is resiual in X μ R U. The proof of this result is obtaine by an explicit perturbation argument an presente in Section 4. In fact, the homoclinic an heteroclinic orbits referre to in Theorem 1.1 are in fact homoclinic an heteroclinic orbits that lie close to the initial egenerate) heteroclinic cycle, making no more than one revolution in its neighbourhoo. The asymmetric) homoclinic cycles mentione in Theorem 1.1 generically unfol following the classical treatment by Shilnikov, see [21, 35]. In particular, this means that one fins generically nontrivial hyperbolic basic sets horseshoes) if the moulus of real part of the complex eigenvalues of the sale-foci are smaller than the moulus of the real eigenvalues. This is etermine by the 2-jet of the normal form: 0 < a < 2 in terms of coefficient of 3.3). In this paper, however, we focus on the ynamics inuce by the unfolings of the heteroclinic cycles mentione in Theorem 1.1. Their occurrence relies on the reversibility of the vector fiel. Apart from the fact that it supplies us with a lot of information on the local ynamics near the zero-hopf bifurcation, we woul like to emphasize that the stuy of the unfoling of such cycles is also of inepenent interest, for instance with relevance to the Michelson system, where at a special value of c an explicit expression for a 1D-heteroclinic solution has been obtaine in [24], see also Section 6. The starting point of the analysis of the heteroclinic cycle bifurcations is sketche in Figure 2. We have two asymmetric sale-foci p 0 an p 1 that are R-images of each other. The two-imensional stable manifol of p 0 transversally intersects the two-imensional unstable manifol of p 1 to yiel a robust heteroclinic connection. At parameter value λ = 0, the one-imensional unstable manifol of p 0 intersects FixR), giving rise to a coincience with the one-imensional stable manifol of p 1, resulting in a heteroclinic cycle. Clearly, the occurrence of such a heteroclinic cycle is generically persistent an isolate in one-parameter families of reversible vector fiels of the type consiere in this paper. We further assume that the one-imensional heteroclinic connection unfols generically along the λ- parameter family. A more technical escription of the generic hypotheses is formulate in Section 5, 4

5 Fix R Σ 1 ψ 2 ψ 2 φ σ 1 σ 1 Σ 0 p 0 p 1 φ σ 0 ψ 1 ht) ψ 1 σ 0 Figure 2: The heteroclinic cycle at λ = 0 with section planes an first hit maps. where the ynamics near the cycle is stuie using Poincaré maps φ, ψ 1, ψ 2, φ, ψ 1, ψ 2 between the sections σ 0, σ 1, σ 0, σ 1, Σ 0, Σ 1 as inicate in Figure 2. The analysis of the unfoling of the symmetric heteroclinic cycle yiels many aitional heteroclinic an homoclinic cycles, as well as perioic an aperioic solutions. In orer to facilitate the iscussion, we introuce some terminology in orer to istinguish ifferent types of these orbits. A heteroclinic orbit connecting p 0 to p 1 will be calle a 1D heteroclinic orbit, an is always symmetric R-invariant). A 1D heteroclinic orbit that intersects the section Σ 1 see Figure 2) n times is calle an n-1d heteroclinic cycle. A symmetric heteroclinic orbit intersects FixR) exactly once. Note that if n is o it intersect FixR) in Σ 1, an if n is even in Σ 0. Corresponingly we refer to such symmetric heteroclinic orbits also as upper an lower 1D heteroclinic orbits. The 1D heteroclinic orbit we start from is a 1-1D heteroclinic cycle. Heteroclinic orbits connecting p 1 to p 0 are corresponingly calle 2D heteroclinic orbits. We similarly call such connections which intersect Σ 0 n times n-2d heteroclinics. Symmetric 2D heteroclinics are upper if n is even an lower if is n is o. Note that 2D heteroclinic orbits nee not always be symmetric. Likewise, one can characterize symmetric perioic orbits which always intersect FixR) precisely twice by the number of times n that Σ 1 or Σ 0 ) is intersecte. If n is even then the intersections of perioic solutions with FixR) are either upper or lower, whereas if n is o the intersections are always mixe one upper an one lower). There also may arise homoclinic solutions connecting p 0 to p 0 or p 1 to p 1. By symmetry, whenever there exists a homoclinic solution to p 0 then such a solution also arises to p 1. We call a homoclinic solution n-homoclinic if it intersects Σ 0 an Σ 1 ) n times. Analogously we call a heteroclinic cycle n-heteroclinic if it intersects Σ 0 an Σ 1 ) n times. The main results of the heteroclinic cycle bifurcation analysis in this paper are summarise in the following theorem. Its proof is iscusse in Section 5. Theorem 1.2 Consier a one-parameter family of R-reversible vector fiels F x, λ) in R 3, with at λ = 0 a symmetric heteroclinic cycle between two asymmetric sale-foci. Then, generically, the following statements hol: A. At λ = 0 for all n > 1 there is a countably infinite number of upper an lower) symmetric, an asymmetric n > 2), transverse n-2d heteroclinic orbits accumulating to the symmetric hetero- 5

6 clinic cycle. Note that in combination with the 1-1D heteroclinic, each n-2d heteroclinic solution constitutes a heteroclinic cycle. B. For each heteroclinic cycle there exists a countable infinity of perioic solutions converging to the heteroclinic cycle as their perio goes to infinity. For small λ, these perioic solutions form a one parameter family in R 3 R, parameterise by λ as an oscillating function of the perio. The type of the perioic orbits is in corresponence to that of the heteroclinic cycle that is accumulate: If the heteroclinic cycle intersects Σ 1 n times then so o the perioic solutions. If the heteroclinic cycle is symmetric, then so are the perioic solutions, an the perioic solutions intersect FixR) is in the same sections) Σ i, as the heteroclinic cycle. A similar set of perioic solutions acummulates each homoclinic solution, if the eigenvalues of the sale-foci satisfy Shilnikov s conition that the moulus of the real eigenvalue is larger than the moulus of the real part of the complex eigenvalue) [19, 35]. C. For each n > 1 there exists a countably infinite set of parameters {λ n) k }, converging exponentially to zero as k such that at λ = λ n) k there exists a symmetric) n-1d heteroclinic orbit. These orbits converge to the initial 1-1D heteroclinic orbit as k. Consequently, the set of parameter values for which there exists a 1D-heteroclinic orbit forms an accumulation set. D. For each n > 1 there exists a countably infinite set of parameters {λ n) j }, converging exponentially to zero as j such that at λ = λ n) j, p 0 an p 1 have asymmetric) homoclinic solutions. These orbits converge to the initial heteroclinic cycle as j. E. At λ = 0 there exists an inecomposable R-invariant nonuniformly hyperbolic invariant set, containing a countable infinity of nontrivial hyperbolic basic sets horseshoes), whose ynamics is topologically conjugate to a full shift on an infinite number of symbols. For small nonzero λ, an R-invariant uniformly hyperbolic basic set remains whose ynamics is topologically conjugate to a full shift on a finite number of symbols tening to infinity as λ 0). We woul like to highlight the result uner E in the above theorem, as it establishes the existence of nontrivial basic sets horseshoes) in the unfoling of the symmetric heteroclinic cycle, inepenent of the eigenvalues at the sale-foci. This is in sharp contrast to the Shilnikov conition for the existence of horseshoes in the unfoling of a homoclinic orbit to a sale-focus which is incientally automatically satisfie in the case of volume preserving vector fiels, cf [5]). Many of the results in Theorem 1.2 on the ynamics near the heteroclinic cycle o not crucially rely on the reversibility of the system. In general issipative non-reversible) systems heteroclinic cycles between two sale-foci of the type introuce above have coimension two. Apart from the fact that in such systems 1D heteroclinic connections can only be expecte to be seen in two-parameter unfolings, our results on the existence of horseshoes inepenent of any Shilinikov conition) remain vali as their proof relies only on the geometry of the flow near the cycle. We note that Bykov [9, 10] also stuie the unfoling of such cycles in two-parameter families of non-reversible vector fiels. He obtaine results on the existence of certain perioic, homoclinic an heteroclinic solutions but none on the existence of horseshoes. The construction of the R-invariant non-uniformly hyperbolic invariant set at λ = 0 mentione in Theorem 1.2.E, by the intersection of strips is sketche in Figure 3. By combining Theorem 1.1 an Theorem 1.2 above, we obtain the following conclusion concerning the generic ynamics near the Hopf-zero bifurcation. See Figure 6 for a sketch. 6

7 Fix R 0 Fix R 1 Σ 0 Σ 1 ψ 2 φ ψ 1 B 0 B 1 H i W s p 0 ) H i+1 W u p 1 ) V i V i+1 M i S 0 S 1 ψ 1 φ ψ 2 M i+1 Figure 3: Sketch of the construction of the invariant set conjugate to a full shift on an infinite number of symbols with horizontal strips H i an vertical strips V i. The construction starts with consiering intersecting strips B 0 an B 1 = R 0 B 0 ) in Σ 0 ajacent to the traces of the two-imensional stable an unstable manifols of the sale-foci. The images S 0 := ψ 2 φ ψ 1 B 0 ) an S 1 := ψ 1 φ ψ 2) 1 B 1 ) in Σ 1 are logarithmic spiralling strips that are exactly each other s R 1 -image. Let {M i } enote the countable set of intersections of S 0 an S 1 containing part of the FixR 1 -axis. The images H i := ψ 1 φ ψ 2M i ) an V i := ψ 2 φ ψ 1 ) 1 M i ) form sets of horizontal an vertical strips. 7

8 Theorem 1.3 Denote by X μ R the space of one parameter families of R-reversible vector fiels 1.4) exhibiting the Hopf-zero bifurcation as above at μ = 0, enowe with the C topology. There exists an open subset U X μ R containing the origin, which is etermine by the 2-jet of the vector fiels at 0, 0) R 3 R, such that the set of vector fiels for which in a neighbourhoo of the origin in R 3 R there exists for each n Z + a countable infinity of n-homoclinic orbits, a countable infinity of symmetric n-heteroclinic cycles, a countable infinity of asymmetric n-heteroclinic cycles n > 2, a countable infinity of n-perioic orbits accumulating to n-heteroclinic cycles), a countable infinity of hyperbolic basic sets horseshoes), is resiual in X μ R U. The subset of X μ R U for which which in a neighbourhoo of the origin in R3 R U there exists a hyperbolic basic set horshoe) is open an ense. Our work has parallels with that of Broer & Vegter [5], Delshams & Martinez-Seara [11] an Gaspar [17], who stuie the Hopf-zero bifurcation in the context of issipative an volume preserving vector fiels, isplaying a similar family of egenerate heteroclinic cycles in normal form approximation. The main ifference between the issipative/volume preserving an reversible contexts is that in the latter context one fins coimension one heteroclinic cycles, whereas in the former context such heteroclinic cycles have coimension two. From our results we see that the heteroclinic cycles arising ue to the reversibility have important implications for the local ynamics. Our work also forms part of a systematic effort to stuy local bifurcations in reversible equivariant systems. The Hopf-zero bifurcation is one of the simplest examples of a coimension one local bifurcation in reversible vector fiels. The behaviour of the equilibria in this example follows that preicte by the more general treatment of reversible equivariant steay-state bifurcations of Buono, Lamb an Roberts [6]. The branch of perioic solutions is reminiscent of a Liapunov Centre family of perioic solutions embee in R 3 R. This case stuy illustrates how some results that are well-known to hol in reversible systems in R 2n with im FixR) = n, o not hol without moification in o imenions. We note above the Liapunov centre family of perioic solutions embee in R 3 R. Another striking example encountere in this stuy is the absense of a one-parameter family of perioic solutions accumulating to symmetric heteroclinic cycles. In even imensions, it is well known that such families exist in phase space, converging to a persistent heteroclinic [12]. Here, however, the heteroclinic cycle is only persistent in one-parameter families, an there exists a one-parameter family of perioic solutions converging to the heteroclinic in R 3 R. In fact, varying the parameter towars the heteroclinic cycle bifurcation point, there appears an increasing number of isolate perioic solutions approximating the heteroclinic cycle, with at the bifurcation point an infinite iscrete set of isolate perioic solutions with growing perio, approaching the heteroclinic cycle as the perio goes to infinity see Theorem 1.2 B). As mentione before, this paper is partially motivate by the Michelson system, which arises as a reuction to travelling wave an steay-state solutions in the Kuramoto-Sivashinsky partial ifferential equation. The moel is an example of a reversible vector fiel in R 3. However, at the same time the Michelson system has more structure: it is for instance, analytic even quaratic), volume preserving an also it has the property that it can be written as a thir orer ODE in one variable. It is thus natural to ask what our analysis of the generic reversible Hopf-zero bifurcation can tell us about the Michelson system. The Michelson system exhibits a Hopf-zero bifurcation of the type iscusse in this paper. Importantly, Theorem 1.1 an Theorem 1.3 also hol if the vector fiel is not only reversible but also volume preserving. The following theorem illustrates that we can establish the valiity of the most important conclusions of Theorem 1.1 an Theorem 1.3 also for the Michelson system. 8

9 Theorem 1.4 Consier the Michelson system 1.1) with parameter c. Then, in every parameter interval 0, δ] with δ > 0, there exists for each n Z + a countable infinity of n-homoclinic orbits, a countable infinity of n-heteroclinic cycles n > 1, a countable infinity of n-perioic orbits accumulating to n-heteroclinic an n-homoclinic cycles), a countable infinity of hyperbolic basic sets horseshoes), The proof of this result is iscusse in Section 6. We note that Aams et al. [1] also prove the existence of heteroclinic cycles in this system arbitrarily close to the singularity in parameter space. We here show that this fact inee coincies with the behaviour one woul expect near reversible Hopf-zero bifurcation points. 2 Elementary stationary an perioic solutions In this section we examine the simplest solutions emerging in the unfoling of the Hopf-zero equilibrium, ie those that relate to solutions of the linear approximation: stationary solutions an perioic solutions with perio approximately equal to 2π/α. The existence of such solutions can be proven using Liapunov- Schmit reuction. We briefly set out the Liapunov-Schmit reuction technique, along the lines of e.g. [20]. The first t step is to introuce new functions ut) = x ), so that 1.4) transforms into α1+τ) N u, μ, τ) = 1 + τ) u t 1 F u, μ) = ) α Restricting ourselves to 2π-perioic solutions, we can view N as a ifferential operator N : C 1 2π R R C 2π, where C 2π an C 1 2π are Banach spaces of continuous, respectively continuously ifferentiable 2π-perioic functions into R 3. By varying the newly introuce small variable τ, one keeps track not only of solutions of 1.4) with perio 2π/α but also of solutions with nearby perio. N inherits the R-reversibility of F, ie RN ut), μ, τ) = N Ru t), μ, τ). Moreover, ue to the fact that F is autonomous, N is also S 1 -equivariant: φn u, μ, τ) = N φu, μ, τ), with φ [0, 2π = S 1 acting on C 2π an C 1 2π as φut) = ut φ). The next step is to consier the erivative D u N 0, 0, 0) of N : It follows that im ker D u N 0, 0, 0) = 3 since D u N 0, 0, 0) = t 1 α D uf 0, 0, 0). ker D u N 0, 0, 0) = {exp 1 α D uf 0, 0, 0)t)u 0 u 0 R 3 }, = {Ime it v), Ree it v), z) v C, z R}, with v = y + ix C an z R. Using the fact that D u N 0, 0, 0) is a Freholm operator of inex zero [13, 20] we may write C 2π = ker D u N 0, 0, 0) range D u N 0, 0, 0), C2π 1 = ker D u N 0, 0, 0) M, 2.2) 9

10 where M = ranged u N 0, 0, 0) C 1 2π. We now may employ the Implicit Function Theorem to solve N M = 0, leaving to solve a reuce equation fv, μ, τ) = 0, where f : ker D u N 0, 0, 0) R R ker D u N 0, 0, 0), where we recall that ker D u N 0, 0, 0) = R 3 = C R. As the splitting 2.2) is Z 2 R) S 1 invariant, the reuce equation inherits the R-reversibility an S 1 -equivariance of N, where the inuce action of Z 2 R) S 1 on C R is given by Rv, z) = ˉv, z) an φv, z) = e iφ v, z). Consequently, the reuce bifurcation equations take the form ) vzp v fv, z, μ, τ) = 2, z 2, μ, τ) + ivq v 2, z 2, μ, τ) f z v 2, z 2 = 0, 2.3), μ, τ) where p, q an f z are real functions, an f z 0, 0, 0, 0) = 0. A subset of the solutions is obtaine by choosing v FixS 1 ), ie v = 0, z). In that case we are left to solve f z 0, z 2, μ, τ) = 0. As all solutions in FixS 1 ) are invariant with respect to time shifts, an are therefore equilibria, this equation must be invariant with respect to parameter translations of the form τ ε, an thus in fact be inepenent of τ. It can be checke that with the generic conitions F 3 μ 0, 0), 2 F 3 z 0, 0) 0 where F 2 3 enotes the thir component of the vector F ) we arrive at f z 0, z 2, μ, τ) = f z 0, z 2, μ, 0) = μ + cz 2 +, with c 0. The equilibria thus typically form a one-parameter family consisting of two asymmetric branches an a symmetric R-invariant) fol point z = 0). We note that this equation can be obtaine also irectly from applying the Liapunov-Schmit reuction to F x, μ) = 0, which invariably applies in case there is one zero eigenvalue, an hence is inepenent of the nature of the nonzero eigenvalues as mentione in Section 1). It remains to look for solutions outsie FixS 1 ), which correspon to perioic solutions. Detaile analysis of the application of the Implicit Function Theorem in orer to erive the reuce equations show, in analogy to the treatment of the Hopf bifurcation in [20], that q0, 0, 0, 0) = 0 an q τ 0, 0, 0, τ) 0. Hence, we may apply the Implicit Function Theorem to fin τ v 2, z, μ) so that q v 2, z, μ, τ v 2, z, μ)) = 0. Focusing furthermore on reversible solutions in FixZ 2 R)) = {v = ˉv, z = 0}, it remains to solve gv 2, μ) = f z v 2, 0, μ, τv 2, 0, μ)) = 0. Uner the generic hypotheses F 3 μ 0, 0), 2 F 3 r 0, 0) 0 where r = x 2 + y 2 ) we have D 2 μ g0, 0) 0, an we can use the Implicit Function Theorem to fin μv 2 ) so that gv 2, μv 2 )) = 0 an μ v 0. This solution correspons to a 2 one parameter family of perioic solutions with perio converging to 2π/α as v Normal form approximation In this section we erive a Birkhoff normal form for the reversible Hopf-zero bifurcation an its unfoling. This normal form or its truncations) provies an approximation of the ynamics in the neighbourhoo of the Hopf-zero bifurcation point. We consier a symmetric equilibrium point of an R-reversible vector fiel 1.4) in R 3 with Hopf-zero linear part given by 1.6). It is a stanar result of Birkhoff normal form theory [15] that one can fin coorinate transformations that rener the vector fiel S 1 -equivariant up to arbitrarily high orer, where S 1 = {expd x F 0, 0)s) s [0, 2π/α }. Moreover, the corresponing coorinate transformations can be taken to be R-equivariant, preserving the R-reversibility of the vector fiel, see e.g. [25]. Employing cylinrical coorinates x = r cos θ, y = r sin θ the R-reversible formal) S 1 -equivariant Birkhoff normal form takes the form θ = fr 2, z 2 ) ṙ = rzgr 2, z 2 ) 3.1) ż = hr 2, z 2 ) 10

11 where h0, 0) = h z 0, 0) = 0 an f0, 0) = α. In fact, using a reparametrization of the vector fiel we may take without loss of generality fr 2, z 2 ) = α to be constant in a neighbourhoo of 0, 0). We shall enote the S 1 -equivariant normal form vector fiel by X μ. Note that the original vector fiel F is not conjugate to X μ, but conjugate to X μ = X μ + Y μ, where Y μ x, y, z) : R 3 R 3 is small beyon all algebraic orers flat) in x, y, z). Due to the S 1 -equivariance the vector fiel is θ-inepenent, an thus it is possible to consier the normal form vector fiel on the reuce phase space R 3 /S 1 = {θ = 0}, yieling up to thir orer: ṙ = a 1 rz, ż = b 1 r 2 + b 2 z 2 3.2) where a 1, b 1, b 2 are constants. Takens [36] showe that the ifferential equation 3.2) is 2-etermine up to C 0 orbital equivalence, uner the generic conitions that a 1, b 1, b 2 0 an b 2 a 1 0. Note that the S 1 -equivariant 2-jets are the same in the generic coimension 2) an the reversible coimension 1) case. There are six topologically ifferent phase portraits for the truncate vector fiels an they can be foun for instance in [21]. The accoring one-parameter reversible versal unfoling follows irectly by restriction from the general two-parameter versal unfoling in [21, 36], cf [22]. After applying some aitional rescalings, one obtains the normal form ṙ = arz, ż = μ + br 2 z 2, 3.3) where a R an b ±1 are constants an μ is the unfoling parameter. Phase portraits for this normal form can be foun in [21, 36]. In Figure 1 the phase portraits before, at an after the Hopf-zero bifurcation are epicte in the case that a > 0, b = 1, which is the case arising in the Michelson system an the one we focus on in this paper. We emphasize again that the normal form vector fiel, although a goo approximation, is highly egenerate because of the S 1 normal form symmetry. Theorem 1.1 iscusses some of the consequences of small S 1 -symmetry breaking perturbations. Its proof is iscusse in the following section. 4 Proof of Theorem 1.1 We are intereste in the reversible Hopf-zero bifurcation, when the 2-jet normal form coefficients in 3.3) satisfy a > 0, b = 1. These conitions form the open conitions mentione in Theorem 1.1. In this case, ue to the S 1 normal form symmetry, the flow of the normal form when μ > 0 gives rise to a highly egenerate approximation in which the stable an unstable manifols of the sale-foci coincie in a line an a 2-sphere. A generic R-reversible but S 1 -symmetry breaking perturbation woul remove such egeneracies, an the question is what remains. Our first observation is that ue to the fact that Fix R intersects the heteroclinic 2-sphere of the normal form flow transversely in two points, in the break up of such a 2-sphere by some small perturbation at least two symmetric ie setwise R- invariant) 2D heteroclinic orbits will remain to exist. The situation is analogous to the illustration of the perturbe globe in [5]. Our proof of Theorem 1.1 is constructive an relies on the following lemma. Lemma 4.1 Let Xμ be an S 1 -symmetric vector fiel in R 3 with a egenerate heteroclinic cycle as arising in the normal form 3.3) with a > 0 b = 1 an μ > 0, cf Figure 1. There exists a flat perturbation Y μ such that the perturbe vector fiel X μ + Y μ has a sequence of Shilnikov homoclinic bifurcations at a iscrete set of parameter values μ i which accumulate at μ = 0. There also is a sequence of parameter values μ j accumulating at μ = 0 for which the two sale-foci are connecte by a heteroclinic cycle. 11

12 Recall that the Shilnikov homoclinic bifurcation is a generic coimension one homoclinic bifurcation of a homoclinic orbit to a sale-focus in R 3 ). The proof closely follows the constructions by Broer & Vegter [5] of an infinite sequence of Shilnikov homoclinic bifurcations near Hopf-zero bifurcation in volume-preserving vector fiels, involving an explicit escription of flat perturbations Y μ. We consier flat perturbations Y μ that are the compositions of two flat perturbations Y μ = Y μ Y μ 2, such that the perturbation Y μ 1 manifols an the perturbation Y μ 2 an unstable manifols. We consier first the perturbation Y μ controls the position of the one-imensional stable an unstable yiels a transversal intersection between the two-imensional stable μ. We set Y2 = δμ)p μ 2, where δμ) is a flat function in μ at μ = 0. This perturbation is esigne to cause a transversal intersection between the two imensional manifols W s p 0 μ)) an W u p 1 μ)). First note that the invariant 2-sphere for the normal form X μ, forme by W s p 0 μ)) an W u p 1 μ)), is given by the equation [21] r 1 + a + z2 = μ. We choose the support of P μ 2 to be a torus centre on the circle r2 = 1 + a)μ, in R 3. The following lemma implies that we can construct a perturbation Y μ 2 = δ 2μ)P μ 2 such that X μ + Y μ 2 has transverse heteroclinic connections lying in W s p 0 μ)) W u p 1 μ)). We refer to [38] for the proof. Lemma 4.2 [38]) Let X R be the space of R-reversible vector fiels in R 3 enowe with the C topology, where Rx, y, z) = x, y, z), an let S R X R be the subset for which all fixe points have transversally intersecting invariant manifols. Then S R is resiual in X R. The flat perturbation Y μ 1 is constructe in orer to manipulate the one-imensional stable an unstable manifols. We write Y μ 1 x, y, z) = δ 1μ)P μ 1 x, y, z), where δ 1 μ) is some appropriate flat function in μ at μ = 0 appropriate in a sense to be specifie later) an P μ 1 x, y, z) = y yβ μξ)), ) x yβ μξ)), 0 4.1) where β μ : R 3 R is given by β μ x, y, z) = γ 1 μ r).γ 1 z), μ > 0 small. 4.2) μ2 with x = r cos θ, y = r sin θ, an γ : R R is an even bump function with support suppγ) = [ 2, 2] an γs) 1 for s [ 1, 1], cf [5]. Observe that β μ is thus constructe to be R-invariant, where Rx, y, z) = x, y, z). The support of P μ 1 in R3 is a cyliner η μ, given by We efine two subsets ν 1 μ, ν 2 μ η μ by η μ = {x, y, z) R 3 : x 2 + y 2 = r 2μ, z 2μ 2 }. ν 1 μ = η μ {x, y, z) R 3 : x 2 + y 2 = r μ}, an ν 2 μ = ν 1 μ {x, y, z) R 3 : z μ 2 }. See Figure 4 for a sketch. It follows from 4.1) an 4.2) that for x, y, z) νμ, 1 Y μ 1 x, y, z) = δ 1μ).γ 1 μ z) 2 x an for x, y, z) νμ, 2 Y μ 1 x, y, z) = δ 1μ) μ x. Note that Y1 is a C reversible ivergence free flat perturbation. The flat perturbation Y μ 1 escribe above has the following consequences for the positions of the one-imensional stable an unstable manifols of the sale-foci p 0 μ) an p 1 μ) [4, 5]: Lemma 4.3 [4]) Consier vector fiel X μ + Y μ 1 as consiere above where X μ is the S 1 -equivariant normal form vector fiel with for μ > 0 a heteroclinic cycle between sale-foci p 0 μ) an p 1 μ), an 12

13 z μ μ ν 1 μ μ 2 0 νμ 2 νμ 1 νμ 2 νμ 1 μ 2 μ 2 r ν 1 μ μ 2 Figure 4: The support box η μ for the perturbation P μ 1. Y μ 1 is the flat perturbation introuce above. Let r μ be the r-coorinate of the 1-imensional unstable manifol W u p 0 ) when it exits η μ, with μ > 0 sufficiently small. Then r μ μδ 1 μ) 4.3) For the proof of this lemma, we refer to [4]. Lemma 4.3 emonstrates that we can control the orer of magnitue of the r-coorinate of the oneimensional unstable manifol of p 0 μ) by using a flat perturbation. Note that we also have θ μ = o1) as μ 0 +, where θ μ is the θ-coorinate of the 1-imensional unstable manifol W u p 0 ) when it exits η μ. Proof of Lemma 4.1 The perturbe vector fiel X μ +Y μ 1 +Y μ 2 has transverse heteroclinic connections for all μ. We follow the behaviour of the two-imensional manifol W s p 0 μ)) in negative time, as it gets close to the one-imensional manifol W s p 1 μ))). By the λ-lemma, W s p 0 μ)) wraps itself tightly aroun W s p 1 μ)) in a logarithmic spiral. Consier the top of the support box η μ, that is η μ {z = 2μ 2 }. We assume here that the angle θ is lifte to R. Then W s p 1 ) intersects η μ {z = 2μ 2 } along the z-axis the perturbations Y μ 1 an Y μ 2 o not affect the relevant part of W s p 1 )), an W s p 0 ) will trace out a 1-imensional curve in this section, with equation rθ, μ) δ 3 μ)e aθ μ, where δ 3 μ) is a flat function epening on Y μ μ 2. Note that the perturbation Y1 oes not affect this logarithmic spiral. Since θμ = o1) an from 4.3), if we set, for example μδ 1 μ) = δ 3 μ)e 1/μ we obtain a sequence of Shilnikov homoclinic bifurcations. Finally, choosing Y μ 1 = sin 1 μ )δ 1μ)P μ 1 will create a sequence of 1-D heteroclinic connections ientical to the connection in the S 1 -equivariant normal form) whenever sin 1 μ ) = 0. When sin 1 μ ) = 1 we have r μ μδ 1 μ) as before. This completes the proof of Lemma 4.1. Proof of Theorem 1.1 It remains toverify the generic occurrence of sequences of global bifurcations as escribe in Lemma 4.1. From Section 3 we know that { X μ } the set of S 1 -symmetric vector fiels is ense in X R in the C topology. Since the perturbations we have use to prove existence are flat, this proves that our bifurcation sequences are ense in X R. Let Bhom/het k be the set of vector fiels with k homoclinic resp. heteroclinic) bifurcations close to the Hopf-zero bifurcation point. Given any element X μ + Y μ in the ense set which has an infinite sequence of homoclinic an heteroclinic bifurcations accumulating at the bifurcation point, this family will have k such bifurcations for all μ > 0 sufficiently small. Each such bifurcation in persistent in the C 1 topology, thus Bhom/het k is open in the More precisely, the relation 4.3) means there exist constants C 1, C 2 such that r μ C 1μδμ) an μδμ) C 2 r μ. 13

14 C 1 topology. Note that this set says nothing about what happens in a neighbourhoo of the bifurcation point itself as any finite number of bifurcations is boune away from zero. Hence, for each vector fiel in the set k Bk hom/het, which is resiual in the C1 -topology, we have a countable infinity of homoclinic an heteroclinic bifurcations. This completes the proof of Theorem 1.1. Remark 4.4 Theorem 1.1 also hols in the case of volume-preserving vector fiels [5], an in the case of reversible volume-preserving vector fiels, since the flat perturbation use in the argument may be chosen to be reversible an volume preserving at the same time. In the next section, we show that typically, in the neighborhoo of the heteroclinic bifurcations escribe above, there exist countably many other homoclinic an heteroclinic bifurcations, all accumulating on the heteroclinic bifurcation. 5 Heteroclinic cycle bifurcation In this section we stuy the ynamics near a heteroclinic cycle bifurcation. We consier a one-parameter family of R-reversible vector fiels F : R 3 R R 3, with R as before, satisfying the following hypotheses: [H1] F has two fixe points p 0 an p 1, such that Rp 0 ) = p 1. [H2] Dfp 0 ) has one real eigenvalue μ > 0 an a complex pair of eigenvalues ρ ± i with ρ, > 0. [H3] There exists an isolate symmetric heteroclinic solution ht) containe in the transversal) intersection of the twoimensional stable manifol of p 0 an the twoimensional unstable manifol of p 1, ie ht) W s p 0 ) W u p 1 ) for all λ [ ε, ε] for ε sufficiently small. [H4] at λ = 0 the unstable manifol of p 0 coincies with the stable manifol of p 1, ie W u p 0 ) = W s p 1 ), so that F, 0) has a symmetric heteroclinic loop. Aitionally W u p 0 ) passes with positive spee) through FixR) at λ = 0. Hypothesis [H1-H4] are robust, ie satisfie in an open subset of one-parameter families of smooth R-reversible vector fiels in R 3. See Figure 2 for a sketch of the situation. In this section we etail some important aspects of the ynamics near such a heteroclinic cycle an its unfoling, as formulate in Theorem 1.2. Our results inclue the existence of certain symmetric heteroclinic an perioic solutions close to the original heteroclinic cycle. This section is organize as follows. In Subsection a) we introuce first hit maps between surfaces of sections an return maps. In Subsection b) we iscuss the some of the tranformation properties of these return maps, which are use in Subsection c), ), e), f) to iscuss the existence of respectively 2D-heteroclinic orbits, symmetric perioic solutions, 1D-heteroclinic an homoclinic solutions an horseshoes, at an near the bifurcation point λ = 0. a) Sections an return maps In this section we efine the surfaces of sections that we employ to efine return maps to stuy the ynamics near the heteroclinic cycle. We efine two main local sections, Σ 0 an Σ 1, satisfying: The sections are setwise invariant uner R: RΣ 0 ) = Σ 0 an RΣ 1 ) = Σ 1. Consequently, FixR) bisects Σ 0 an Σ 1. We istinguish between the local actions of the time-reversal symmetry R: R 0 = R Σ0 an R 1 = R Σ1. The sections are locally transverse to the the flow of F at λ = 0 an hence also at sufficiently small values of λ). 14

15 The section Σ 0 is locally transverse to W u p 1 ) an W s p 0 ). Similarly the section Σ 1 is locally transverse to W u p 0 ) an W s p 1 ). We efine some more sections close to the sale-foci p 0 an p 1 : σ 0 is a local section transversal to W s p 0 ) an W u p 1 ), σ 1 is a local section transversal to W u p 0 ), σ 0 = Rσ 0 ), an σ 1 = Rσ 1 ). A sketch of the total configuration an the position of the surfaces of section is given in Figure 2. The next step is to construct a return map that is built from the composition of first hit maps between the surfaces of section. Such first hit maps are locally well efine for small λ) since at λ = 0 the surfaces of section intersect the heteroclinic cycle transversally. We thus efine the Poincaré return map F 0 : Σ 0 Σ 0 : F 0 = ψ 1 φ ψ 2 ψ 2 φ ψ 1, 5.1) where the maps ψ 1 : Σ 0 σ 0, φ : σ 0 σ 1, ψ 2 : σ 1 Σ 1, ψ 2 : Σ 1 σ 1,φ : σ 1 σ 0, an ψ 1 : σ 0 Σ 0 are first hit maps see the illustration in Figure 2). We may use the reversibility of the vector fiel to express the maps ψ 1, ψ 2 an φ in terms of ψ 1, ψ 2, φ, R, R 0 an R 1. Namely: ψ 1 = R 0 ψ 1 1 R, ψ 2 = R ψ 1 2 R 1, φ = Rφ 1 R. Consequently, we have: F 0 = R 0 ψ1 1 φ 1 ψ2 1 R 1 ψ 2 φ ψ ) an it is reaily verifie that F 0 is a R 0 -reversible map, ie F 1 0 = R 0 F 0 R ) Similarly it is easy to show that the Poincaré return map F 1 : Σ 1 Σ 1 satisfies an that hence F 1 is a R 1 -reversible map, ie F 1 = ψ 2 φ ψ 1 R 0 ψ 1 1 φ 1 ψ 1 2 R ) F 1 1 = R 1 F 1 R ) We now focus on the local return maps. We consier the local map φ about the sale point p 0, the corresponing properties for the map φ can be euce from the form of φ an the fact that φ = Rφ 1 R. This local map will provie the key to unerstaning the features of the ynamics we are intereste in. Since the sale fixe point p 0 is hyperbolic, there is a unique hyperbolic sale point p λ 0 for each λ sufficiently small, an without loss of generality we may assume that the equilibrium p 0 is at p, 0, 0) for all λ small. Similarly we may choose R to be spanne by 0, 1, 0). We now choose local coorinates aroun p 0 such that p 0 is at the origin. Recall the eigenvalues of the fixe point p 0 are ρλ)±iλ), μλ), with ρλ), μλ) > 0. From now on we suppress the argument λ. It can be shown [3] that there exists a local C 1 change of coorinates an a reparametrization of time), such that in these coorinates the flow in an ɛ-neighbourhoo near the sale point p 0 is linear, so that the local stable an unstable manifols are straightene: x L = ρ/μ)x L + /μ)y L y L = /μ)x L ρ/μ)y L 5.6) z L = z L We efine the local section σ 0 as follows: σ 0 = {x L, y L, z L ) R 3 x L = 0, y L = y ± δ}, where the point of first intersection of ht) an σ 0 is 0, y, 0), an δ is sufficiently small so that 0, y, 0) is the only intersection of ht) with σ 0. Note that {0, y, 0)} σ 0 is the trace of the two imensional stable manifol W s p 0 ) in σ 0. We subsequently choose σ 1 = {x, y, z) R 3 z = }. 15

16 Note that {0, 0, )} σ 1 is the trace of the one imensional unstable manifol W u p 0 ). The flow of 5.6) can be integrate explicitly to give: x L t) = x L 0) exp ρt μ ) cos μ t) + y L0) exp ρt μ ) sin μ t) y L t) = y L 0) exp ρt μ ) cos μ t) x L0) exp ρt μ ) sin μ t) z L t) = z L 0) expt), 5.7) from which the time of flight from σ 0 to σ 1 can be calculate to be t = lnz L 0)/). We thus fin the following expression for the first hit map φ : σ 0 σ 1 : zl ) ρ/μ φy L, z L ) = y L sin μ ln zl ) ) zl ) ρ/μ, y L cos μ ln zl ) )). 5.8) The analysis near the equilibrium point p 1 is analogous, yieling the φ = φ 1 in terms of local coorinates x L, y L, z L ) near p 1. Throughout the remainer we use x L, y L, z L ) to enote local coorinates near p 0 an x L, y L, z L ) to enote local coorinates near p 1. To complete the constuction, we choose the remaining surfaces of section in terms of local coorinates Σ 0,1 = {x i, y i, z i ) R 3 x = 0}, 5.9) where y i, z i ) are chosen such that 0, 0) is a point of intersection of the heteroclinic cycle an the sections Σ i at λ = 0. For the construction of the global maps, we may write, for example ψ 1 : Σ 0 σ 0 as ψ 1 y0 z 0 ) = yl ) y = z L 0 ) ) y0 + A ) z 0 where the ots enote terms of higher orer. Since the map ψ 1 is a iffeomorphism, A is a nonsingular matrix. We can similarly write ) ) ) xl y1 xl ψ 2 = = B ) y L z 1 y L ) ) ) ψ 2 y1 x = L z 1 y = RB 1 y1 R ) L z 1 ) ) ) ψ 1 y L y0 z L = = R z 0 A 1 y R L y 0 z L ) It shoul be note that with the above choices the compositions ψ 2 ψ 2 an ψ 2 ψ 2 appear to be orientation reversing if we ientify the local coorinates x L, y L, z L ) near p 0 with the local coorinates x L, y L, z L ) near p 1, but this is just ue to the choices of local coorinates. Finally, the local reversing symmetries R 0,1 act on the sections Σ 0,1 as: b) R 0,1 : 0 y z Dynamics of the first hit maps 0 y z, 5.14) In this section we establish three lemmas that are central to the proof of Theorem 1.2. Throughout this section we assume that the hypotheses [H1]-[H4] are satisfie. The first result concerns the fact that line segments transversal to the local stable manifol in σ 0 get mappe by φ to a logartihmic spiral in σ 1. Its proof is immeiate from 5.8). 16

17 Lemma 5.1 Let y L s), z L s)) be a line segment in σ 0, parameterise by s, such that y L 0) is close to y, an y L s), z L s)) transversely intersects the local stable manifol W s p 0 ), ie z L 0) = 0 an zs) s s=0 0. Then the image of y L s), z L s)) uner the local map φ is a logarithmic spiral in σ 1. That is, in polar coorinates x L = r sin θ, y L = r cos θ, the image of y L s), z L s)) takes the form zl s) r, θ) = y L s) ) ρ/μ, μ ln ) ) zl s). 5.15) The secon result iscusses how the image of the above logartihmic spiral uner ψ 2 ψ 2 in which is still a logarthmic spiral) is subsequently mappe to σ 0 by the first hit map φ. Lemma 5.2 Consier a logarithmic spiral Γ = x L s), y L s)) in σ 1 with x L s) = ay Ls) y L s) = cy Ls) zls) zl s) ) ρ/μ sin )) μ ln zls) + by L s) )) + y L s) ) ρ/μ sin μ ln zl s) zls) zl s) ) ρ/μ cos )) μ ln zls) +... )) +... ) ρ/μ cos μ ln zl s) 5.16) where a, b, c, are constants, a bc = 1, an the remainers... enote terms of higher orer in z L s). The coorinates y L s), z L s)) refer to the coorinates of the logarithmic spiral in Lemma 5.1. Then the image of Γ uner φ in σ 0 consists of a countably infinite set of lines, accumulating exponentially fast in the C 1 -topology) to the trace of the unstable manifol W u p 1 ) σ 0 = {z L = 0}, with exponent μ π. Proof Recall that φ 1 = RφR, so the pre-image of a point y L, z L ) σ 0 is, in polar coorinates x L, y L ) = r sin θ, r cos θ ): ) z r, θ ) = y L ρ/μ L, ) ) z μ ln L. 5.17) [ ] [ ] a b 1 0 We first consier the case where the linear part C = of ψ c 2 ψ 2 equals C = an 0 1 neglect the higher orer terms. We note here that ue to the reversibility an our choice of coorinates which accounts for the minus sign) we always have etc) = 1. Then, in r, θ ) coorinates Γ has the form ) ρ/μ r, θ zl s) ) = y L s), ) ) μ ln zl s). 5.18) We note that the equations for θ are moulo 2π, an that the θ equation in 5.17) is vali for θ sufficiently large, an the equation for θ in 5.18) is vali for θ sufficiently large. We first consier y L σ 0 fixe, an search for values of z L σ 0 that are in the image of Γ uner φ. By equating the raius coorinates of 5.17), 5.18), we obtain ) μ/ρ z L yl s) = z L s). 5.19) y L Note that 5.19) gives z L as a function of s. y Ls) is O1) in s as s 0, so z L s) an z Ls) are of the same orer as s 0. Now recall that the equations for the arguments in 5.17), 5.18) are moulo 2π. Hence, with z L s) an z L s) sufficiently small, equating the angle equations in 5.17) an 5.18) yiels ) μ/2ρ z Ls) yl s) = exp μ ) nπ, 5.20) y L 17