SWITCHING NEAR A NETWORK OF ROTATING NODES

Transcription

1 SWITHING NEAR A NETWORK OF ROTATING NODES MANUELA A.D. AGUIAR, ISABEL S. LABOURIAU, AND ALEXANDRE A.P. RODRIGUES Abstract. We study the dynamics of a Z 2 Z 2 -equiariant ector field in the neighbourhood of a heteroclinic netork ith a periodic trajectory and symmetric equilibria. We assume that around each equilibrium the linearisation of the ector field has non-real eigenalues. Trajectories starting near each node of the netork turn around in space either folloing the periodic trajectory or due to the complex eigenalues near the equilibria. Thus, a netork ith rotating nodes. The rotations combine ith transerse intersections of to-dimensional inariant manifolds to create sitching near the netork: close to the netork there are trajectories that isit neighbourhoods of the saddles folloing all the heteroclinic connections of the netork in any gien order. Our results are motiated by an example here sitching as obsered numerically, by forced symmetry breaking of an asymptotically stable netork ith O(2) symmetry. 1. Introduction Heteroclinic connections and netorks are a common feature of symmetric differential equations, and persist under perturbations that presere the symmetry. Start ith an asymptotically stable netork ith rotational symmetry. A perturbation that breaks part of the symmetry splits a to-dimensional connection into a pair of one-dimensional ones. The ne netork is no longer asymptotically stable, nearby trajectories follo the netork around in a complex ay that e call sitching. By a heteroclinic netork e mean a connected flo-inariant set that is the union of heteroclinic cycles. In the present case it is the orbit under the symmetry group Z 2 Z 2 of a heteroclinic cycle. These 2000 Mathematics Subject lassification. Primary: 37G30; Secondary: 3710, 3437, 3729, 3428, Key ords and phrases. heteroclinic netork, sitching, ector fields, symmetry breaking, shadoing. The research of all authors at entro de Matemática da Uniersidade do Porto (MUP) had financial support from Fundação para a iência e a Tecnologia (FT), Portugal, through the programs POTI and POSI ith European Union and national funding. A.A.P. Rodrigues as supported by the grant SFRH/BD/28936/2006 of FT. 1

2 2 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES netorks are often called heteroclinic cycles in the literature. Heteroclinic cycles and netorks are knon to occur persistently in the settings of symmetry [8], [16], coupled cell systems (ith and ithout symmetry) [6], [2] and population dynamics [12], [13], [11], [7]. They are induced by the existence of flo inariant subpaces that correspond, respectiely, to fixed point subspaces, synchrony subspaces and coordinate axes and hyperplanes. It is orthhile to describe general properties that entail sitching, so the results may be applied to examples in other contexts. We study the dynamics near a netork here all cycles hae a common node that is a closed trajectory. We proe that there are trajectories near the netork that follo its cycles in any desired order. Trajectories that go near the periodic orbit may sitch to any heteroclinic cycle, return and sitch again. There exist in the literature seeral numerical reports on complicated dynamics near heteroclinic netorks of equilibria and of equilibria and periodic trajectories, that include random isits to the nodes of the netork in any possible order [10], [8], [5], [23]. This type of behaiour is not possible around asymptotically stable heteroclinic netorks hose connections are contained in inariant subspaces. Each cycle in the netork cannot be asymptotically stable but it may hae strong attractiity properties [19], [24] so that each nearby trajectory outside the inariant subspaces ill tend to one of the cycles in the netork. Different forms of sitching hae been described in seeral contexts. Netorks here all the nodes are equilibria, hae been studied by Postlethaite and Daes [21] ho found trajectories that follo three cycles in a netork sequentially, both regularly and irregularly; by Kirk and Silber [15] near a netork ith to cycles ho found nearby trajectories that sitch in one direction. Persistent random sitching is found by Guckenheimer and Worfolk [10] and Aguiar et al [4]; noise induced sitching in Armbruster et al [5]. A problem similar to ours, here a netork inoles equilibria and a periodic trajectory, appears in the heteroclinic model of the geodynamo deried in Melbourne et al [20]. Starting ith a model ith Z 2 Z 2 SO(2) symmetry, they perturb the model so the only remaining symmetry is Id. For the perturbed model they establish sitching numerically in terms of reersals and excursions. This has motiated Kirk and Rucklidge [14] to ask hether sitching ould be obsered hen all the symmetries are broken. First they analyse partially broken symmetries in to different ays: hen only the SO(2) symmetry remains they find a eak form of sitching, here trajectories starting near one equilibrium may isit the neighbourhood of another but not return to the first one; for the Z 2 Z 2 -symmetric case they find attracting periodic trajectories and no sitching. Then

3 SWITHING NEAR A NETWORK 3 they argue that hen all symmetries are broken and the netork is destroyed, sitching ill not take place arbitrarily close to Z 2 Z 2 - symmetric problems because of barriers formed by inariant manifolds. They describe a scenario here sitching may arise, if the symmetrybreaking terms are larger than a threshold alue. They propose a mechanism for sitching arising from the right combination of homoclinic tangencies beteen the stable and unstable manifolds of a periodic orbit and specific heteroclinic tangencies beteen stable and unstable manifolds of the equilibria. Here e analyse equations ith a symmetry group Z 2 Z 2, a subgroup of that considered by Melbourne et al [20] but not acting in the same ay as in Kirk and Rucklidge [14]: each Z 2 in our setting contains a rotation by π that fixes a plane. A discussion of ho our results compare ith those of [14] and [20] appears at the end of this paper in section 9. Under generic hypotheses for this symmetry, e proe a strong form of sitching: the existence of trajectories that isit neighbourhoods of any sequence of nodes of the netork in any order that is compatible ith the netork connections. The conditions e need for sitching are stated in section 3 preceded by definitions and preliminary results in section 2. In section 4 e present an example of a Z 2 Z 2 -equiariant family of ordinary differential equations haing a netork of rotating nodes and some simulations (see figures 3 and 4). When one of the parameters is set to zero, the equations are Z 2 Z 2 SO(2)-symmetric and the netork is asymptotically stable. Sitching occurs for all small nonzero alues of this symmetry-breaking parameter. We linearise the flo around the inariant saddles in section 5, obtaining isolating blocks around each node of the netork. This section is mostly concerned ith introducing the notation for the proof of sitching that occupies the rest of the paper. The goal of this paper is to proe sitching in the neighbourhood of a heteroclinic netork that consists of four symmetric copies of a heteroclinic cycle here is a closed trajectory inariant under the symmetries and and are equilibria. The connection is one-dimensional and takes place inside a fixed-point subspace, the other connections are transerse intersections of 2-dimensional inariant manifolds. The trajectory has real Floquet multipliers and 2-dimensional stable and unstable manifolds; the linearisation of the flo near has a pair of complex eigenalues ith negatie real part and one real positie eigenalue; the linearisation of the flo near has one real negatie eigenalue and a pair of complex eigenalues ith positie real part (see figures 1 and 2).

4 4 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES In section 6 e obtain a geometrical description of the ay the flo transforms a cure of initial conditions lying across the stable manifold of each node. The cure is rapped around the isolating block of the next node, accumulating on its unstable manifold and in particular on the next connection. Thus, points on a line across the stable manifold of ill be mapped into a helix accumulating on the unstable manifold of that ill cross the transerse stable manifold of infinitely many times. Similarly, points on a line across the stable manifold of ill be mapped into a helix accumulating on its unstable manifold and thus ill cross the transerse stable manifold of infinitely many times. The geometrical setting is explored in section 7 to obtain interals on the cure of initial conditions that are mapped by the flo into cures near the next node in a position similar to the first one. This allos us to establish the recurrence needed for sitching in section 8: for any sequence of nodes like e find trajectories that isits neighbourhoods of these nodes in the same sequence. We end the paper ith a discussion (section 9) of the results obtained and of related results in the literature. 2. Preliminaries Let f be a smooth ector field on R n ith flo gien by the unique solution x(t) = ϕ(t,x 0 ) R n of (1) ẋ = f(x) x(0) = x 0. If A is a compact inariant set for the flo of f, e say, folloing Field [8], that A is an inariant saddle if A W s (A)\A and A W u (A)\A, here A is the closure of A. In this paper all the saddles are hyperbolic. Gien to inariant saddles A and B, an m-dimensional heteroclinic connection from A to B, denoted [A B], is an m-dimensional connected flo-inariant manifold contained in W u (A) W s (B). There may be more than one connection from A to B. Let S ={A j : j {1,...,k}} be a finite ordered set of mutually disjoint inariant saddles. Folloing Field [8], e say that there is a heteroclinic cycle associated to S if j {1,...,k},W u (A j ) W s (A j+1 ) (mod k). We refer to the saddles defining the heteroclinic cycle as nodes. A heteroclinic netork is a finite connected union of heteroclinic cycles. Let Γ be a compact Lie group acting linearly on R n. The ector field f is Γ equiariant if for all γ Γ and x R n, e hae f(γx) =

5 SWITHING NEAR A NETWORK 5 γf(x). In this case γ Γ is said to be a symmetry of f. We refer the reader to Golubitsky, Steart and Schaeffer [9] for more information on differential equations ith symmetry. The Γ-orbit of x 0 R n is the set Γ(x 0 ) = {γx 0,γ Γ} that is inariant under the flo of Γ-equiariant ector fields f. In particular, if x 0 is an equilibrium of (1), so are the elements in its Γ-orbit. The isotropy subgroup of x 0 R n is Γ x0 = {γ Γ, γx 0 = x 0 }. For an isotropy subgroup Σ of Γ, its fixed-point subspace is Fix(Σ) = {x R n : γ Σ,γx = x}. Fixed-point subspaces are the reason for the robustness of heteroclinic cycles and netorks in symmetric dynamics: if f is Γ-equiariant then F ix(σ) is flo-inariant, thus connections occurring inside these spaces persist under perturbations that presere the symmetry. For a heteroclinic netork Σ ith node set A, a path of order k, on Σ is a finite sequence s k = (c j ) j {1,...,k} of connections c j = [A j B j ] in Σ such that A j,b j A and B j = A j+1 i.e. c j = [A j A j+1 ]. For an infinite path, take j N. Let N Σ be a neighbourhood of the netork Σ and let U A be a neighbourhood of each node A in Σ. For each heteroclinic connection in Σ, consider a point p on it and a small neighbourhood V of p. We are assuming that the neighbourhoods of the nodes are pairise disjoint, as ell for those of points in connections. Gien neighbourhoods as aboe, the point q, or its trajectory ϕ(t), follos the finite path s k = (c j ) j {1,...,k} of order k, if there exist to monotonically increasing sequences of times (t i ) i {1,...,k+1} and (z i ) i {1,...,k} such that for all i {1,...,k}, e hae t i < z i < t i+1 and: (1) ϕ(t) N Σ for all t (t 1,t k+1 ); (2) ϕ(t i ) U Ai and ϕ(z i ) V i and (3) for all t (z i,z i+1 ), ϕ(t) does not isit the neighbourhood of any other node except that of A i+1. There is finite sitching near Σ if for each finite path there is a trajectory that follos it. Analogously, e define infinite sitching near Σ by requiring that each infinite path is folloed by a trajectory. An infinite path on Σ may also be seen as a pseudo-orbit of ẋ = f(x), ith infinitely many discontinuities. Sitching near Σ means that these infinite pseudo-orbits can be shadoed. 3. A netork of rotating nodes Our object of study is the dynamics around a special type of heteroclinic netork (see figure 2) for hich e gie a rigorous description here. The netork lies in a topological three-sphere and one of its nodes is a closed trajectory ith real Floquet multipliers and 2-dimensional stable and unstable manifolds. Near this node, the flo rotates folloing the closed trajectory. The other nodes are equilibria ith a pair

6 6 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES (P3) + (P4) - O + Fix(Z ( )) 2 γ 1 - Fix(Z ( )) 2 γ 2 Figure 1. Dynamics near a heteroclinic netork of rotating nodes. Left: dynamics around the plane Fix(Z 2 (γ 1 )) illustrating property (P3); Right: dynamics in the plane Fix(Z 2 (γ 2 )) illustrating property (P4). of non-real eigenalues. Thus the local dynamics rotates around each node. Specifically, e study a smooth ector field f on R 4 ith the folloing properties: (P1) The ector field f is equiariant under Γ = Z 2 Z 2 acting linearly on R 4 ith generators γ 1 and γ 2 and ith to transerse todimensional fixed-point subspaces. In particular, the origin is an equilibrium. (P2) There is a three-dimensional flo-inariant manifold S 3 diffeomorphic to a sphere that attracts all the trajectories except the origin. For simplicity, e assume this manifold to be the unit sphere. By (P1 P2) there are to flo-inariant circles 1 = S 3 Fix(Z 2 (γ 1 )) and = S 3 Fix(Z 2 (γ 2 )). (P3) On 1 there are exactly four equilibria that ill be denoted by +, +, = γ 2, = γ 2. Moreoer, the eigenalues of df restricted to TS 3 are: ± i and E ith E > 0, at ±; E ± i and ith E > 0, at ±. In Fix(Z 2 (γ 1 )) the equilibria ± are saddles and ± are sinks ith connections in 1 from ± to ±. These connections are persistent under perturbations that presere the γ 1 symmetry (see figure 1).

7 SWITHING NEAR A NETWORK 7 (P4) In the inariant plane Fix(Z 2 (γ 2 )) the only equilibrium is the origin and it is an unstable focus. Thus is a closed trajectory and, from (P2), this trajectory is a sink in Fix(Z 2 (γ 2 )) (see figure 1). Since is contained in the flo-inariant plane Fix(Z 2 (γ 2 )), its Floquet multipliers are real. (P5) The periodic trajectory is hyperbolic and, in S 3, dimw u () = dimw s () = 2. Moreoer, there are connections [ +] and [+ ] satisfying: W u () W s (+) and W u (+) W s (). These intersections are one-dimensional and consist of one pair of γ 1 - related trajectories. From the γ 2 symmetry, e obtain a pair of γ 1 -related one-dimensional connections in W u () W s ( ) and in W u ( ) W s (). It follos that there is a heteroclinic netork Σ inoling the saddles ±, ± and (figure 2). Such a netork Σ is hat e call a netork of rotating nodes. This paper shos sitching near this netork. The symmetry γ 1 and its flo inariant fixed point subspace ensure the persistence of the connections [± ±]. The other symmetry γ 2 is not essential for the existence of a robust netork ith these properties but it makes them more natural as illustrated by the example in section 4. The same is true for the existence of the inariant 3-sphere. In section 9 e discuss ariants of these hypotheses for hich sitching may be proed in the same ay. 4. Example Our study as initially motiated by the folloing example constructed in Aguiar [1], using the methods of [3]. This is an ordinary differential equation in R 4 gien by: ẋ 1 = x 1 (α 1 + α 2 r α 3 x α 4 x α 5 (x 4 3 r 2 1x 2 4)) x 2 ẋ 2 = x 2 (α 1 + α 2 r α 3 x α 4 x α 5 (x 4 3 r 2 1x 2 4)) + x 1 ẋ 3 = x 3 (α 1 + α 2 x α 3 x α 4 r α 5 (x 4 4 r 2 1x 2 3)) + ξh 1 (x) ẋ 4 = x 4 (α 1 + α 2 x α 3 r α 4 x α 5 (r 4 1 x 2 3x 2 4)) ξh 2 (x) here x = (x 1,x 2,x 3,x 4 ) R 4, r 2 1 = x x 2 2 and h 1 (x) = [α 1 +3α 2 (x 2 1+x 2 2)]x 1 x 2 x 4 and h 2 (x) = [α 1 +3α 2 (x 2 1+x 2 2)]x 1 x 2 x 3. The symmetries of the equation are changes of sign of pairs of coordinates: γ 1 (x) = ( x 1, x 2,x 3,x 4 ) γ 2 (x) = (x 1,x 2, x 3, x 4 ). Figures 3 and 4 sho a trajectory of this equation. In [1] it is proed that, for parameter alues such that α 1 > 0 α 3 + α 4 = 2α 2 α 3 < α 2 < α 4 < 0

8 8 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES Figure 2. Schematic description of a heteroclinic netork of rotating nodes satisfying (P1-P5). Each arro represents a 1-dimensional heteroclinic connection. There is one 1-dimensional heteroclinic connection from each equilibrium ± to each equilibrium ± (P3). There are to 1-dimensional heteroclinic connections inoling each equilibrium and the periodic trajectory (P5). x 4 x 3 Figure 3. Sitching trajectory on the example in section 4: projection into the (x 3,x 4 )-plane of the trajectory ith initial condition (0.001, 0.001, 0.001, 1), ith α 1 = , α 2 = , α 3 = 0.5, α 4 = , α 5 = 0.05 and ξ = 1.

9 SWITHING NEAR A NETWORK 9 x 1 x 3 x 4 0 time Figure 4. Sitching trajectory on the example in section 4: time series for x 1, x 3 and x 4 for a trajectory ith initial condition (0.001, 0.001, 0.001, 1), ith α 1 = , α 2 = , α 3 = 0.5, α 4 = , α 5 = 0.05 (same as figure 3). The ariable x 1 shos isits to the limit cycle, x 3 shos alternate isits to ± and x 4 shos isits to ±. and if ξ 0 is such that α 2 (α 3 α 4 ) + α 1 α 5 > 0 ξ < α 2α 3 + α 2 α 4 + α 1 α 5 2α 1 α 2 and ξ 2 < (α 4 α 3 )(2α 1 α 5 α 2 α 3 + α 2 α 4 ) 4α 2 1α 2,

10 10 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES then its dynamics satisfies (P1 P4) as e proceed to explain. For ξ = 0 the equation has more symmetry, like the model in Melbourne et al [20]: it is equiariant under the group Z 2 Z 2 SO(2), generated by κ ϕ (x) = (x 1 cos(ϕ) x 2 sin(ϕ),x 1 sin(ϕ) + x 2 cos(ϕ),x 3,x 4 ) κ 2 (x) = (x 1,x 2, x 3,x 4 ) κ 3 (x) = (x 1,x 2,x 3, x 4 ). When ξ = 0, the three-dimensional sphere S 3 r of radius r = α 1 α 2 flo inariant and globally attracting. The equilibria ± (resp. ±) lie at the intersection of S 3 r ith the x 3 -axis (resp. x 4 -axis) fixed by the subgroup generated by κ ϕ and κ 3 (resp. κ ϕ and κ 2 ). The closed trajectory is the intersection of S 3 r ith the plane fixed by the subgroup generated by κ 2 and κ 3. A direct computation of the eigenalues and eigenectors shos that the closed trajectory and the equilibria form a netork here the to-dimensional unstable (resp. stable) manifold of coincides ith the to-dimensional stable (resp. unstable) manifold of ± (resp. ±). Since all the heteroclinic connections are contained in fixed point subspaces, there is no sicthing. Moreoer, it can be shon (see [1]) that the netork is asymptotically stable by the criteria of Krupa and Melbourne [17], [18]. For ξ > 0 the SO(2) symmetry is broken and so are the todimensional connections. The only symmetries remaining are γ 1 = κ π and γ 2 = κ 2 κ 3. The symmetry-breaking term (0, 0,h 1 (x),h 2 (x)) is tangent to the inariant sphere so it is still flo inariant and properties (P1) and (P2) hold. The perturbation mantains the flo-inariance of the line that is fixed by κ π and κ 2 (resp. κ π and κ 3 ), the equilibria and the periodic trajectory are presered together ith their stability, and properties (P3-P4) hold. Using Melniko s Method, Aguiar [1] proed that the to-dimensional manifolds of the periodic trajectory and of the saddle-foci intersect transersely. Hence property (P5) holds. It ill follo from Theorem 7 that there is sitching near this netork as suggested by the trajectories like that of figures 3 and Local Dynamics near the saddles Here and in the next to sections, e define the setup for the proof of sicthing near the netork. This section contains mostly notation and coordinates used in the rest of the paper. We restrict our study to S 3 since this is a compact and flo-inariant manifold that captures all the dynamics. When e refer to the stable/unstable manifold of an inariant saddle, e mean the local stable/unstable manifold of that saddle. We use Samool s theorem to linearise the flo around each saddle equilibrium or closed trajectory. We then introduce local cylindrical coordinates and define a neighbourhood ith boundary transerse to is

11 SWITHING NEAR A NETWORK 11 2ε y x ϕ ε r (a) (b) Figure 5. oordinates on the boundaries of the neighbourhood of and : (a) cylinder all (b) top and bottom. the linearised flo. For each saddle, e obtain the expression of the local map that sends points in the boundary here the flo goes in, into points in the boundary here it goes out. These expressions ill be used in the sequel to obtain a geometrical description of the discretised flo oordinates near equilibria. Let and stand for any of the to symmetry-related equilibria ± and ±, respectiely. By Samool s theorem [22] f may be linearised around them, since nonresonance is automatic here. In cylindrical coordinates (ρ,θ,z) the linearisations take the forms: (2) ρ = ρ θ = 1 ż = E z ρ = E ρ θ = 1 ż = z. We consider cylindrical neighbourhoods of and in S 3 of radius ε > 0 and height 2ε. Their boundaries consist of three components (see figure 5): The cylinder all parametrized by x R (mod 2π) and y ε ith the usual coer (x,y) (ε,x,y) = (ρ,θ,z). To disks, the top and the bottom of the cylinder. We take polar coerings of these disks: (r,ϕ) (r,ϕ,jε) = (ρ,θ,z) here j {, +}, 0 r ε and ϕ R (mod 2π). change We use x for the angular coordinate on the cylinder all so as to aoid confusion ith the angular coordinate on the disks hen dealing ith the local maps.

12 12 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES H out,+ H in,+ H in,+ u W loc() s W loc() H out,+ s loc W () u loc W () H in,- u W loc() s W loc() H out,- H out,- H in,- Figure 6. Neighbourhoods of the saddle-foci. Left: once the flo enters the cylinder transersely across the all H in \Wloc s () it leaes it transersely across the cylinder top H out,+ and bottom H out,. Right: the flo enters the cylinder transersely across top H in,+ \Wloc s () and bottom H in, \Wloc s () and leaes it transersely across the all H out. Inside the to cylinders the ector field is linear. Note that the to flos defined by (2) hae symmetry Z 2 SO(2) gien by z z and rotation around the z axis. This is an artifact of the linearisation and has nothing to do ith the original symmetries. It ill be used implicitly in the next sections Local dynamics near. The cylinder all is denoted H in. Trajectories starting at interior points of H in go inside the cylinder in positie time and H in W s () is parametrized by y = 0. The set of points in H in ith positie (resp. negatie) second coordinate is denoted H in,+ (resp. H in, ). The top and the bottom of the cylinder are denoted H out,+ and H out,, respectiely. Trajectories starting at interior points of either H out,+ or H out, go inside the cylinder in negatie time (see figure 6). After linearisation W u () is the z-axis, intersecting H out,+ at the origin of coordinates of H out,+. Trajectories starting at H in,j, j {+, } leae the cylindrical neighbourhood at H out,j. The orientation of the z-axis may be chosen to hae [ j] meeting H out,j.

13 SWITHING NEAR A NETWORK 13 The local map near, φ : H in,+ H out,+ is gien by ( ) (3) φ (x,y) = K y δ, 1 E ln y + x + 1 E ln(ε) = (r,φ), here δ = E > 0, K = ε 1 δ > 0 and 1 E > 0. The expression for the local map from H in, to H out, is the same Local dynamics near. After linearisation, W s () is the z- axis, intersecting the top and bottom of the cylinder at the origin of the coordinates. We denote by H in,j, j {, +}, the component that [j ] H in,j. Trajectories starting at interior points of H in,± go inside the cylinder in positie time (see figure 6). Trajectories starting at interior points of the cylinder all H out go inside the cylinder in negatie time. The set of points in H out hose second coordinate is positie (resp. negatie) is denoted H out,+ (resp. H out, ) and H out W u () is parametrized by y = 0. The orientation of the z-axis may be chosen to hae trajectories that start at H in,j \W s (), j {+, } leaing the cylindrical neighbourhood at H out,j. The local map near, φ : H in,+ \W s () H out,+ is ( ) 1 φ (r,ϕ) = E ln(ε) 1 E ln r + ϕ,k r δ = (x,y), here δ = 1 > 0, K = ε 1 δ > 0 and > 0. E E The same expression holds for the local map from H in, \W s () to H out, Local dynamics near the closed trajectory. onsider a local cross section S to the flo at p. The Poincaré first return map defined on S may be linearised around the hyperbolic fixed point p using Samool s Theorem. Suspending the linear map yields, in cylindrical coordinates, the differential equations: (4) ρ = (ρ 1) θ = 1 ż = E z that are locally orbitally equialent to the original flo. In these coordinates, corresponds to the circle ρ = 1 and z = 0, W s () is the plane z = 0 and W u () is the cylinder ρ = 1. We ork ith a hollo three-dimensional cylindrical neighbourhood of ith boundary H in Hout, here trajectories starting in Hin (resp. H out ) go into the neighbourhood in positie (resp. negatie) small time. In hat follos e establish some notation for components of the boundary (see figure 7).

14 14 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES The components of H in are the to cylinder alls, Hin,+ locally separated by W u () and parametrized by the coering map: (x,y) (1 ± ε,x,y) = (ρ,θ,z), here x R (mod 2π), y < ε. We denote by H,+ in ith ρ = 1 + ε. and Hin, the component In these coordinates, H in W s () is the union of the to circles y = 0 in the to components. It diides H,+ in in to parts, Hin,+,+ and H in,,+, parametrized, respectiely, by positie and negatie y, ith a similar conention for H in,+, and Hin,,. The components H out,+ and H out, of H out separated by W s () and parametrized by the coering: (r,ϕ) (r,ϕ, ±ε) = (ρ,θ,z), are to anuli, locally for 1 ε < r < 1 + ε and ϕ R (mod 2π) and here H out,+ is the component corresponding to the + sign and H out W u () is the union of to circles parametrized by r = 1. Denote by H out,k,+ (resp. Hout,k, ), k {+, } the set parametrized by 1 < r < 1 + ε (resp. 1 ε < r < 1) in H out,k. In these coordinates the local map φ : H in,k,j H out,k,j, j,k {+, }, is gien by ( 1 φ (x,y) = jk c y δc + 1, ln(ε) 1 ) ln y + x = (r,ϕ), E c E c here δ c = c E c > 0 K c = ε 1 δ > 0 and 1 E c > Geometry near the saddles The coordinates and notation of section 5 may no be used to analyse the geometry of the local dynamics near each saddle. The manifold W s () separates the cylindrical neighbourhood of into an upper and a loer component, mapped into neighbourhoods of + and, respectiely. We sho here that initial conditions lying on a segment on the upper part of the cylindrical all around and ending at W s () are mapped into points on a spiral on the top of the cylinder H out here the flo goes out (figure 8). Initial conditions on a spiral on the top of cylindrical neighbourhood of are then shon to be mapped into points on a helix around the cylinder accumulating on W u () (figure 8). Since W s () is transerse to W u (), then the helix on H out,+ is mapped across W s () on H in infinitely many times. This ill be used in section 7 to obtain, in any segment on H in,+, infinitely many interals that are mapped into segments on H in ending at W s (). Then the image of a segment ending at W s () on one of the alls of H in is shon to be mapped into a cure accumulating on W u ()

15 SWITHING NEAR A NETWORK 15 H out,+,- u W loc() H out,+,+ H in,+,- H in,+,+ out H = H out,+ out,+ H = H out,+,+ out,- H = H out,-,+ U U U H out,- H out,+,- H out,-,- s W loc() in H = H in,+ U H in,- in H,+= in H,- = H in,+,+ H in,+,- U U H in,-,+ H in,-,- H in,-,- H in,-,+ H out,-,+ u W loc() H out,-,- Figure 7. Neighbourhood of the closed trajectory. The flo enters the hollo cylinder transersely across cylinder alls H,± in and leaes it transersely across top and bottom Hout H out,+,. H out (see figure 9). This cure meets W s () infinitely many times by transersality and it is thus mapped across W s () on H in infinitely many times. Again, e ill use this in section 7 to obtain infinitely many interals that are mapped into segments on H in ending at W s (). This structure of segments containing interals that are successiely mapped into segments ill allo us to establish a recurrence in section 8 and to construct nested sequences of interals containing the initial conditions for sitching. Definition 1. A segment β on H in (resp.: H in ) is a smooth regular parametrized cure β : [0, 1] Hp in (resp.: β : [0, 1] H in), that meets W s () (resp. W s ()) transerselly at the point β(1) only and such that, riting β(s) = (x(s),y(s)), both x and y are monotonic functions of s. The coordinates (x,y) may be chosen so as to make the angular coordinate x an increasing or decreasing function of s as conenient.

16 16 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES u W () s W () H out,+ H out,+ [ ] H in,+ H in,+ H in,- W s() [ ] [ ] H in,- u W () [ ] H out,- H out,- u W () s W () Figure 8. Local dynamics near the saddle-foci. Left: near, any segment on the cylinder all is mapped into a spiral on the top or bottom of the cylinder. Right: a spiral on the top or bottom of the cylinder near is mapped into a helix on the cylinder all accumulating on W u (). The double arros on the segment, spiral and helix indicate correspondence of orientation and not the flo. Definition 2. Let U be an open set in a plane in R n and p U. A spiral on U around p is a cure α : [0, 1) U satisfying lim α(s) = s 1 p and such that, if α(s) = (α 1 (s),α 2 (s)) are its expressions in polar coordinates (ρ,θ) around p, then α 1 and α 2 are monotonic, ith lim α 2(s) = +. s 1 It follos that α 1 is a decreasing function of s. Proposition 1. A segment β on H in,+ into a spiral on H out,+ (resp. H out, (resp. H in, ) around W u (). ) is mapped by φ Proof. Write β(s) = (x(s),y(s)) on H in,+ ith y(s) 0 monotonically decreasing and choose a parametrization of H in,+ such that x(s) is monotonically increasing. Then, riting φ (β(s)) = (r(s),θ(s)), it follos from the expression of φ in section 5.2 that r(s) is monotonically decreasing hile θ(s) is monotonically increasing. From lim s 1 y(s) = 0 and lim s 1 x(s) = x(1) the required limits lim s 1 r(s) = 0 and lim s 1 θ(s) = + follo. Definition 3. Let a,b R such that a < b and let H be a surface parametrized by a coer (θ,h) R [a,b] here θ is periodic. A helix

17 SWITHING NEAR A NETWORK 17 u W () H in,+,+ s W loc() H in,-,+ Figure 9. Local dynamics near the closed trajectory. A segment on the all ending at W s () is mapped into a cure accumulating on W u () H out. on H accumulating on the circle h = h 0 is a cure γ : [0, 1) H such that its coordinates (θ(s),h(s)) are monotonic functions of s ith lim h(s) = h 0 and lim θ(s) = +. s 1 s 1 Proposition 2. A spiral on H in,+ mapped by φ into a helix on H out,+ the circle H out W u (). (resp. H in, (resp. H out, ) around W s () is ) accumulating on Proof. Parametrize H in,+ so that a spiral σ(s) = (r(s),θ(s)) around W s () has θ(s) increasing ith s. The expression of φ of section 5.3 ensures that in φ (σ(s)) = (x(s),y(s)) e hae y decreasing ith s and x increasing ith s. The limits in the definiton of helix follo from the form of φ and from lim s 1 r(s) = 0 and lim s 1 θ(s) = +. Proposition 3. A segment β on H in,+,+ is mapped by φ into a helix on H out,+,+ accumulating on the circle W u () H out,+. The proof, as in Propositions 1 and 2, consists of using the expression of φ of section 5.4 after a suitable choice of orientation in H in,+,+. Using the symmetries of the linearised flo, it follos that Proposition 3 also holds for a segment β on H in,+ and for Hout,+, as ell as for a segment β on H in, + and for Hout, + and for β on Hin, and for Hout,, considering the circle W u () H out, (see figure 9).

18 18 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES 7. First return to Let p and q be to nodes of Σ such that there is a connection [p q]. The transition map Ψ p,q from Hp out to Hq in follos the trajectory [p q] in flo-box fashion. In this section e use this information to put together the local behaiour of trajectories that start near ±. To simplify the reading, e omit the and + signs. Let P H out be one of the points here [ ] meets H out. For small a,b > 0, the rectangle [ a, [ a 2 2] b, b 2 2] is mapped diffeomorphically into H out by the parametrization that maps the origin to P (figure 10). Its image, that e denote by R, ill be called a rectangle in H out centered at P ith height b. The ertical sides of R are the images of the segments (± a,y) ith 2 y [ b, b 2 2]. Rectangles in H out centered at a gien point are defined in the same ay; e denote them by R. By Propositions 1 and 2, the map η = φ Ψ, φ : H in H out maps a segment β on H in infinitely many times across any small rectangle in H out centered at a point in W u (). An admissible family of interals I = {[a i,b i ]} i N is one that satisfies 0 < a i < b i < a i+1 < 1 and lim i a i = 1. Proposition 4. Let R be a rectangle in H out centered at one point P of H out [ ]. For any segment β : [0, 1] H in there is an admissible family of interals {[σ i,ρ i ]} i N such that: (1) each closed interal [σ i,ρ i ] satisfies η β([σ i,ρ i ]) R ; (2) each open interal (ρ i+1,σ i ) satisfies η β((ρ i+1,σ i )) R = ; (3) the family of cures {η β([σ i,ρ i ])} accumulates uniformly on W u () R as i +. This also holds for the local map around, ith, φ and here e hae ritten, η and. Proof. Writing (x(s), y(s)) for the coordinates of the helix η β(s) on the cylinder all, e hae that y decreases ith s and that x(s) can be taken as an increasing function of s by choosing compatible orientations in H in and H out and, if necessary, by restricting the domain of β to a smaller interal (s 1, 1). In particular the helix η β may be seen as a graph (x,y(x)) here x [x(0), + ) and here y is a decreasing function of x ith lim x y(x) = 0 (figure 11). On H out the rectangle R is [n a/2,n + a/2] [ b/2,b/2] ith n N. Let σ 0 be the smallest alue of s (s 1, 1) such that (x(s),y(s)) lies on the left ertical side of R, ith y(σ 0 ) < b as in figure 11. Then 2 y(s) < b for all s [σ 2 0, 1).

19 SWITHING NEAR A NETWORK 19 [ ] Ψ, H in,+ H out,+ segment u W () P s W () R b 2 [ ] [ ] a 2 O b 2 a 2 Figure 10. Transition from to : a segment on H in is mapped into spirals on H out around W u () H out and W s () H in. The spiral is then mapped into a helix on H out accumulating on W u () and crossing infinitely many times a rectangle R centered at one of the connections starting at (see also figure 11). Double arros indicate orientation of the segment and not the flo. The sequences defining the family of interals are obtained from points here the helix meets successie copies of the ertical sides of R ith x(σ i ) = n 0 + i a/2 and x(ρ i ) = n 0 + i + a/2. The proof for φ is similar. Proposition 5. Gien a segment β : [0, 1] H in, a rectangle R of sufficiently small height and the family of interals {[σ i,ρ i ]} i N of Proposition 4. Then for sufficiently large i there are τ i ith σ i < τ i < ρ i such that Ψ, η(β(τ i )) W s () and Ψ, η β maps each one of the interals [σ i,τ i ] and [τ i,ρ i ] into a segment on one of the sets H in,+ and H in,. This also holds for Ψ, φ : H in Hin,± ith the appropriate changes. Proof. Since W s () H out meets W u () H out transerselly (property (P5)) then if the height of R is small, W s () does not meet its ertical sides. Each one of the images η β([σ i,ρ i ]) meets W s () H out transersely at a single point η β(τ i ), since they accumulate uniformly on W u () R as i +. The monotonicity of the coordinates of η β ill be presered by Ψ, close to W u (). Each component of η β([σ i,ρ i ])\ {η β(τ i )} ill be mapped into a segment, one into

20 20 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES y b 2 η(β(σ 0 )) η(β(ρ 0 )) R R R R R R R x Figure 11. A helix on a periodic coer of the cylinder all H out. After it meets the first (shaded) copy of the rectangle, the helix ill intersect the rectangle at interals hose image accumulates on the cure W u (), represented here by the x-axis. each connected component of H in,j \W s (). The proof for Ψ, φ is analogous. 8. Sitching near the Heteroclinic Netork In this section, e put together the information about the first return map to H in. In sections 6 and 7 e hae found that a segment ending at the stable manifold of one node contains interals that are mapped into segments ending at the stable manifold of the next node. Starting ith a segment on H in, here this is used recursiely to obtain sequences of nested interals containing initial conditions that follo sequences of heteroclinic connections. We say that the path s k = (c j ) j {1,...,k} of order k on the netork Σ is inside the path t k+l = (d j ) j {1,...,k+l} of order k + l (denoted s k t k+l ) if c j = d j for all j {1,...,k}. The family of closed interals I = {I i } i N is inside the family Ĩ = {Ĩi} i N (I < Ĩ) if, for all i N, I i Ĩi. If Ĩ is admissible in the sense of section 7 and I < Ĩ then I is also admissible, proided none of its interals consists of a point. Theorem 6. There is finite sitching near the netork Σ defined by a ector field satisfying (P1 P5).

21 SWITHING NEAR A NETWORK 21 segment β R R R s W loc() W s () W s() W s() W s() β } } } R R R } s W loc() R { R { s W () Figure 12. On a segment β on H in, there are infinitely many small segments that are mapped by η into R, each one containing a point mapped into W s loc (). The small segments contain smaller ones that are mapped into R and this may be continued, forming a nested sequence. Proof. Gien a path, e ant to find trajectories that follo it into the neighbourhoods of section 5 going through small disks in H out around W u (±) and through rectangles in H out and H out centered at the connections. Without loss of generality e only consider paths s k = (c j ) j {1,...,k} starting ith c 1 = [± ±]. Take a segment β on H in,± of points that follo the first connection c 1. We ill construct admissible families of interals I(c 1,...,c n ) = {I i } i N recursiely, such that points in β(i i ) follo (c 1,...,c n ) and the image of β(i i ) by the transition maps is a segment. We ill sho that s k s k+l implies I(s k ) > I(s k+l ) and thus the process ill be recursie. By Propositions 4 and 5, there is an admissible family of interals I(c 1,c 2 ) = {I i } i N such that β(i i ) is mapped by Ψ, φ Ψ, φ into a segment on H in,± ith the choice of sign appropriate for the next connection c 3. Applying the second part of Propositions 4 and 5 to this segment, e obtain an admissible family of interals I(c 1,c 2,c 3 ) < I(c 1,c 2 ) corresponding to points that follo (c 1,c 2,c 3 ) and to interals that are mapped into a segment on H in,± ith the choice of sign appropriate for folloing the connection c 4. In Proposition 5 e assume the height of the rectangle R is small and e reduce it if necessary. This is done to ensure that inside R the stable manifold W s () is the graph of a function and thus a helix only meets it once at each turn. Hoeer, as soon as the choice of height is made it may be kept throughout the proof and thus the construction of I(s k ) is recursie, proing finite sitching near Σ.

22 22 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES Theorem 7. There is infinite sitching near the netork Σ defined by a ector field satisfying (P1 P5). Proof. Fix an infinite path s = (c j ) j N on Σ. For each k N define the finite path s k of order k by s k = (c j ) j {1,...,k}, ith s k s k+1. From the proof of Theorem 6, for each k e hae an admissible family of interals I(s k ) = {J ki } i N such that I(s k ) > I(s k+1 ) and all the points in β(j ki ) follo s k. Since e hae I(s k ) > I(s k+1 ) then Λ = k=1 I(sk ) is non-empty because each set Λ i = k=1 J ki is non empty. From the definition of admissible family of interals, if e take a i Λ i then lim i a i = 1. From the construction e hae that β(a i ) follos s. Thus, e hae obtained a sequence of points β(a i ) that accumulate on Σ as i and that follo the infinite path. 9. Final remarks and discussion 9.1. Generalisation. Not all assumptions of Section 3 are essential to proe sitching, although some of them simplify the calculations. For instance, the eigenalues at ± and ± may hae any imaginary part, not necessarily 1 as in (P3). The proof also orks if at one of the pairs of nodes ± or ± the eigenalues of df are real, as long as the eigenalues at the other pair of nodes are not real it is enough to hae one pair of rotating equilibria. Any finite number of 1-dimensional connections [ ] and [ ] could hae been used instead of to for each equilibrium. The existence of sicthing near a netork may be easily generalised to a heteroclinic netork inoling any number of rotating nodes such that each heteroclinic connection inoling a periodic trajectory is transerse and there are no consecutie non-transerse heteroclinic connections on the netork. It is not essential to hae (Z 2 Z 2 )-equiariance. The symmetry γ 2 is used here to obtain the closed trajectory and the role of γ 1 is to guarantee that the one-dimensional connections [ ] are robust. Symmetries make the existence of the netork natural and ensure persistence. Sitching ill hold for any netork ithout symmetry haing the nodes and connections prescribed here, as long as the remaining assumptions are satisfied. Estimates for the transition maps may be refined as in Aguiar et al. [4] to sho that near this netork there is a suspended horseshoe ith transition map described as a full shift oer a countable set of symbols. The suspended horseshoe has the same shape as the netork. Thus, hen the symmetry of the example in section 4 is partly broken e hae instant chaos. In particular it can be shon that there are periodic orbits that follo any finite path in the netork, but this uses techniques beyond the scope of this paper.

23 SWITHING NEAR A NETWORK 23 Finite sitching is present een hen all the local maps are expanding, as in the case < E, < E, < E. This is due to the rotation around the nodes and is markedly different from the situation here all eigenalues are real. In the example of section 4 this corresponds to parameter alues here the O(2)-symmetric netork is a repeller. A path on the netork can also be shadoed by trajectories in W u () (see figure 12) because the local unstable manifold of meets H in at a segment by the transersality assumption (P5). It also follos that there are infinitely many homoclinic connections inoling the periodic trajectory, although there are no homoclinic trajectories inoling the equilibria. The geometry of W u () gets extremely complicated as e moe aay from, since it ill accumulate on the hole netork, haing, and as limit points. A complete description of the nonandering set for this type of flos is in preparation Discussion. Generic breaking of the γ 1 -symmetry destroys the netork, as in [14] and [20], by breaking the connections [ ]. If the remaining assumptions are still satisfied, a eaker form of sitching ill hold: for small symmetry-breaking terms, it may still be possible to find trajectories that isit neighbourhoods of finite sequences of nodes. This is because the spirals on top of the cylinder around of figure 8 ill be off-centered and ill turn a finite number of times around W s (). From this e may obtain points hose trajectories follo short finite paths on the netork. As W u () gets closer to W s () (as the system moes closer to symmetry) the paths that can be shadoed get longer. This is in contrast to the findings of Kirk and Rucklidge [14], ho claim that there can be no sitching in generic systems close to the symmetric case, so a comparison of the settings and results of the to papers is in order at this point. The first caeat is that the Z 2 Z 2 representations are different: in our case Fix(Z 2 Z 2 ) = {0}, and there are to transerse 2-dimensional fixed-point subspaces for the isotropy subgroups, hereas in [14], Fix(Z 2 Z 2 ) is a plane ith to 1-dimensional fixed-point subspaces for the isotropy subgroups. In their setting, our symmetries correspond to the group generated by a rotation of π in SO(2) and by the product of their Z 2 Z 2 generators. Both representations occur in the larger group Z 2 Z 2 SO(2) used in [20]. Moreoer, e are assuming the existence of an inariant 3-sphere (a natural assumption in the symmetric context, see [8]) and it is not eident that in their context such a sphere ill exist. Hoeer, the difference in the results of [14] and ours indicates that ector fields ith Z 2 Z 2 SO(2) symmetry hae codimension higher than 3 in the unierse of general ector fields. This ill mean that in general systems close to symmetry hat ill be obsered may depend

24 24 M.A.D.AGUIAR, I.S.LABOURIAU, AND A.A.P.RODRIGUES on the ay symmetries are broken and that a lot more needs to be done before sitching is ell understood. References [1] M. Aguiar, Vector Fields ith heteroclinic netorks, PhD Thesis, Departamento de Matemática Aplicada, Faculdade de iências da Uniersidade do Porto, 2003 [2] M. Aguiar, P. Ashin, A. Dias and M. Field, Robust heteroclinic cycles in coupled cell systems: Identical cells ith asymmetric inputs, MUP preprint [3] M. Aguiar, S. B. astro and I. S. Labouriau, Simple Vector Fields ith omplex Behaiour, Int. Jour. of Bifurcation and haos, Vol. 16, Nr.2, 2006 [4] M. Aguiar, S. B. astro and I. S. Labouriau, Dynamics near a heteroclinic netork, Nonlinearity 18, 2005 [5] D. Armbruster, E. Stone and V. Kirk, Noisy heteroclinic netorks, haos, Vol. 13, Nr.1, March 2003 [6] P. Ashin and M. Field, Heteroclinic Netorks in oupled ell Systems Arch. Rational Mech. Anal., Vol. 148, 1999, pages [7] W. Brannath, Heteroclinic netorks on the tetrahedron, Nonlinearity, Vol. 7, 1994, pages [8] M. Field, Lectures on bifurcations, dynamics and symmetry, Pitman Research Notes in Mathematics Series, ol. 356, Longman, 1996 [9] M. I. Golubitsky, I. Steart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II, Springer, 2000 [10] J. Guckenheimer and P. Worfolk, Instant haos, Nonlinearity, Vol. 5, 1992, pages [11] J. Hofbauer, Heteroclinic ycles in Ecological Differential Equations, Tatra Mountains Math. Publ., Vol. 4, 1994, pages , [12] J. Hofbauer, Heteroclinic ycles on the simplex, Proc. Int. onf. Nonlinear Oscillations, Janos Bolyai Math. Soc. Budapest, 1987 [13] J. Hofbauer and K. Sigmund, The Theory of Eolution and Dynamical Systems, ambridge Uniersity Press, ambridge, 1988 [14] V. Kirk and A. M. Rucklidge The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle beteen equilibria and a periodic orbit, Dynamical Systems: An International Journal, 23, 2008, pages [15] V. Kirk and M. Silber, A competition beteen heteroclinic cycles, Nonlinearity, Vol. 7, 1994, pages [16] M. Krupa, Robust Heteroclinic ycles, J. Nonlinear Sci., 7, 1997, pages [17] M. Krupa, and I. Melbourne, Asymptotic Stability of Heteroclinic ycles in Systems ith Symmetry, Ergodic Theory and Dynam. Sys., Vol. 15, 1995, pages [18] M. Krupa and I. Melbourne, Asymptotic Stability of Heteroclinic ycles in Systems ith Symmetry, II, Proc. Roy. Soc. Edinburgh, 134A, 2004, pages [19] I. Melbourne, An example of a non-asymptotically stable attractor Nonlinearity, Vol. 4, 1991, pages [20] I. Melbourne, M. R. E. Proctor and A. M. Rucklidge, A heteroclinic model of geodynamo reersals and excursions Dynamo and Dynamics, a Mathematical hallenge (eds. P. hossat, D. Armbruster and I. Oprea, Kluer: Dordrecht, 2001, pages

25 SWITHING NEAR A NETWORK 25 [21]. M. Postlethaite and J. H. P. Daes, Regular and irregular cycling near a heteroclinic netork Nonlinearity, Vol. 18, 2005, pages [22] V. S. Samool, Linearization of a system of differential equations in the neighbourhood of a singular point, So. Math. Dokl, Vol. 13, 1972, pages [23] Y. Sato, E. Akiyama and J. P. rutchfield, Stability and diersity in collectie adaptation, Physica D, Vol. 210, 2005, pages [24] T. Ura, On the flo outside a closed inariant set: stability, relatie stability and saddle sets, ontributions to Differential Equations, ol. III, No. 3, 1964, pages (M.A.D.Aguiar) entro de Matemática da Uniersidade do Porto, and Faculdade de Economia da Uniersidade do Porto, Rua Dr. Roberto Frias, Porto, Portugal address: maguiar@fep.up.pt (I.S. Labouriau and A.A.P. Rodrigues) entro de Matemática da Uniersidade do Porto, and Faculdade de iências da Uniersidade do Porto, Rua do ampo Alegre 687, Porto, Portugal address, I.S.Labouriau: islabour@fc.up.pt address, A.A.P.Rodrigues: alexandre.rodrigues@fc.up.pt