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HOMOCLINIC BOUNDARY-SADDLE BIFURCATIONS IN NONSMOOTH VECTOR FIELDS KAMILADAS.ANDRADE,MIKER.JEFFREY,RICARDOM.MARTINS, 7 ANDMARCOA.TEIXEIRA 1 0 2 n a Abstract. Inasmoothdynamicalsystem,ahomoclinicconnectionisaclosed J orbit returning to a saddle equilibrium. Under perturbation, homoclinics are 0 associated with bifurcations of periodic orbits, and with chaos in higher di- 2 mensions. Homoclinic connections in nonsmooth systems are complicated by theirinteractionwithdiscontinuitiesintheirvectorfields. Aconnectionmay ] involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a S discontinuity set, with segments of the connection allowed to cross or slide D along the discontinuity. Even the simplest case, that of connection to a regular saddle that hits a discontinuity as a parameter is varied, is surprisingly . h complex. Bifurcationdiagramsarepresentedherefornon-resonantsaddlesin t theplane,includinganexampleinaforcedpendulum. a m [ 1. Introduction 1 In smooth dynamical systems, a homoclinic orbit is a closed trajectory connect- v 7 ing a saddle equilibrium to itself. Under perturbation the homoclinic orbit can 5 create a limit cycle or, in more than two dimensions, chaos. Homoclinic orbits in 8 nonsmooth systems come in multiple different forms, only the simplest of which 5 have so far been studied. For example, regular saddles with homoclinic orbits that 0 involve segments of sliding along a line of discontinuity, or homoclinics to so-called . 1 pseudo-saddlesintheslidingdynamicsitself, arestudiedasoneparameterbifurca- 0 tions in [1]. 7 A boundary homoclinic orbit, which involves a regular saddle lying on the dis- 1 : continuity set of a nonsmooth system, is novel so far as classification, because it v involves both a local bifurcation (a saddle lying on the discontinuity set, or a so- i X called boundary equilibrium bifurcation [2]), and a global bifurcation in the form r of the homoclinic connection. Its unfolding then involves the transition of the sad- a dle into a pseudo-saddle, and the appearance of not one limit cycle, but multiple. Thepreciseunfoldingissurprisinglycomplexevenintwodimensions. Thebifurca- tion diagrams for non-resonant saddles in the plane are derived here, including an example in a pendulum with a discontinuous forcing. Our interest is in systems of the form (cid:26) X(x) h(x)≥0, (1) x˙ =Z(x)= Y(x) h(x)≤0, where h:R2 →R is a smooth function having 0 as a regular value, and X,Y ∈χr, whereχr isthesetofallCr vectorfieldsdefinedinR2, andχr isendowedwiththe Cr topology. 1 2KAMILADAS.ANDRADE,MIKER.JEFFREY,RICARDOM.MARTINS,ANDMARCOA.TEIXEIRA TherighthandsideZ isanonsmoothvectorfield,whichforbrevitywemaywrite as Z = (X,Y). Denote by Ωr = χr ×χr the set of all nonsmooth vector fields Z endowed with the product topology. The set Σ = {p ∈ R2; h(p) = 0} is called the switching surface and the definition of trajectories follows Filippov’s convention, see [3]. Our objective is to study bifurcations of a degenerate cycle passing through a saddlepointofX lyingonΣ, calledahyperbolicsaddle-regularpointofZ. Weare thusconcernedwithvectorfieldsZ =(X,Y)∈Ωr whereX hasahyperbolicsaddle pointS ∈ΣwhichisaregularpointforY (meaningY isnon-zeroandtransverse X to Σ at S ). The stable and unstable manifolds of S are transverse to Σ in S X X X andtheunstablemanifoldintersectsΣtransverselyatapointP ∈Σ\{S }. The X X trajectory of Y passing through P intersects Σ transversely at S and P . An X X X example of this kind of cycle is illustrated in figure 1. Z =(X,Y) S X P X Σ Σs Σc undefined region Figure1. Adegeneratecyclepassingthroughasaddle-regularpoint. Σs,Σc are, respectively,theslidingandcrossingregions. AnoverviewofconceptsanddefinitionsaregiveninSection2. Thesettingofthe problem and the study of the first return map for the degenerate cycle are given in Section3. InSection4themainresultsandthecorrespondingbifurcationdiagrams arepresented. Section5presentsmodelshavingadegeneratecyclethroughresonant saddle-regular point. In Section 6 a model presenting a degenerate cycle is given. 2. Preliminaries The switching manifold Σ = {p ∈ R2; h(p) = 0} is the hypersurface boundary between the regions Σ+ = {p ∈ R2; h(p) > 0} and Σ− = {p ∈ R2; h(p) < 0}. It is partitioned into the following regions depending on the directions of X and Y: - the crossing region, where Σc ={p∈Σ; Xh·Yh(p)>0}; - the sliding region, where Σs ={p∈Σ; Xh(p)<0 and Yh(p)>0}; - the escaping region, where Σe ={p∈Σ; Xh(p)>0 and Yh(p)>0}. For all X ∈ χr, the scalar Xh(p) = (cid:104)X,∇h(cid:105)(p) is the Lie derivative of h with respect to X at p. The regions Σc,Σs,Σe, are open in Σ and their complement in Σ is the set of all points satisfying Xh(p)·Yh(p) = 0, called tangency points. A smooth vector field X is transversal to Σ at p∈Σ if Xf(p)(cid:54)=0. HOMOCLINIC BOUNDARY-SADDLE BIFURCATIONS 3 Taking Z = (X,Y) ∈ Ωr, if p ∈ Σ+ (resp. p ∈ Σ−) then the trajectory of Z through p is the local trajectory of X (resp. Y) through this point. If p ∈ Σc the trajectory of Z through p is the concatenation of the respective trajectories of X in Σ− and of Y in Σ−. If p∈Σs∪Σe then the trajectory of Z through this point is the trajectory of the sliding vector field Zs through p. The vector field Zs is defined as the unique convex combination of X and Y that is tangent to Σ, given by 1 (2) Zs(p)= (Yf(p)X(p)−Xf(p)Y(p)), Yf(p)−Xf(p) for p ∈ Σs ∪ Σe. It is useful to define also the normalized sliding vector field Zs (p)=Yf(p)X(p)−Xf(p)Y(p). N The singular points of Zs in Σs ∪Σe are called pseudo-singular points. Those singular points of X (resp. Y) that lie on Σ+ (resp. Σ−) are called real singular points, and those singular points of X (resp. Y) that lie on Σ− (resp. Σ+) are called virtual singular points. The singularities of Z = (X,Y) are: real singular points,pseudo-equilibriaandtangencypoints. Thepointsthatarenotsingularities are called regular points. Definition 1. A smooth vector field X has a fold singularity or a quadratic tan- gencyatp∈ΣifXh(p)=0andX2h(p)(cid:54)=0. AnonsmoothvectorfieldZ =(X,Y) has a fold singularity at p∈Σ if p is a fold for X or Y. A fold point p for X (resp. Y) is visible if X2h(p) > 0 (resp. Y2h(p) < 0) and invisible if X2h(p) < 0 (resp. Y2h(p) > 0). If p is a fold for X and a regular point for Y (or vice-versa) then p is a fold-regular point for Z. Definition 2. A nonsmooth vector field Z has a saddle-regular point at p∈Σ if p is a saddle point for X (resp. Y), and Y (resp. X) is transversal to Σ at p. Denote a trajectory of x˙ = Z = (X,Y) by ϕ (t,q) where ϕ (0,q) = q. Taking Z Z q ∈Σ+∪Σ−, a point p∈Σ, p is said to be a departing point (resp. arriving point) ofϕ (t,q)ifthereexistst <0(resp. t >0)suchthatlim ϕ (t,q)=p(resp. Z 0 0 t→t+ X 0 lim ϕ (t,q)=p). With these definitions if p∈Σc, then it is a departing point t→t− Z 0 (resp. arriving point) of ϕ (t,q) for any q ∈γ+(p) (resp. q ∈γ−(p)), where X γ+(p)={ϕ (t,p);t∈I∩{t≥0}} and γ−(p)={ϕ (t,p);t∈I∩{t≤0}}. Z Z Todistinguishbetweenthemaintypesoforbitsandcycleswehavethefollowing. Definition 3. A continuous closed curve Γ is said to be a cycle of the vector field Z if it is composed by a finite union of segments of regular orbits and singularities, γ ,γ ,...,γ , of Z. There are different types of cycle Γ: 1 2 n - Γ is a simple cycle if none of the γ ’s are singular points and the set γ ∩Σ i i is either empty or composed only by points of Σc, ∀i=1,...,n. If such a cycle is isolated in the set of all simple cycles of Z then it is called a limit cycle. See figure 2(a); - Γ is a regular polycycle if either, for all i=1,...,n, the set γ ∩Σ is empty i and at least one of γ(cid:48)s is a singular point or, for some i=1,...,n, γ ∩Σ i i is nonempty but only contains points of Σc that are not tangent points of Z. See figures 2(b) and 2(c); 4KAMILADAS.ANDRADE,MIKER.JEFFREY,RICARDOM.MARTINS,ANDMARCOA.TEIXEIRA - Γ is a sliding cycle if there exists i ∈ {1,2,...,n} such that γ is a seg- i ment of sliding orbit and, for any two consecutive curves, the departing or arriving points in Σ are not the same. See figure 2(d); - Γ is a pseudo-cycle if for some i ∈ {1,2,...,n}, the arriving points (or departing points) of γ and γ are the same. See figures 2(e) and 2(f). i i+1 (a) (b) (c) (d) (e) (f) Figure2. Illustrationofdifferenttypesofcycles: (a)simplecycle;(b),(c)regular polycycles;(d)slidingcycle;(e)pseudo-cycleand(f)slidingpseudo-cycle. Definition 4. (See [4]) An unstable (resp. a stable) separatrix is either: - a regular orbit Γ that is the unstable (resp. stable) invariant manifold of a saddle point p ∈ Σ+ of X or p ∈ Σ− of Y, denoted by Wu(p) (resp. Ws(p)); or - a regular orbit that has a distinguished singularity p∈Σ as departing (resp. arriving) point. It is denoted by Wu(p) (resp. Ws(p)) and ± means that ± ± it leaves (resp. arrives) from Σ±. Where necessary, we use the notation Ws,u(X,p) to indicate which vector field ± is being considered. If a separatrix is at the same time unstable and stable then it is a separatrix connection. A orbit Γ that connects two singularities, p and q, of Z, will be called either a homoclinic connection if p=q or a heteroclinic connection if p(cid:54)=q. Definition 5. A hyperbolic pseudo-equilibrium point p is said to be a - pseudonode if p∈Σs (resp. p∈Σe) and it is an attractor (resp. a repeller) for the sliding vector field; - pseudosaddle if p ∈ Σs (resp. p ∈ Σe) and it is a repeller (resp. an attractor) for the sliding vector field. 3. Degenerate cycle passing through a saddle-regular point We start by establishing the necessary generic conditions to obtain a degenerate cycle with lowest possible codimension. Consider a nonsmooth vector field Z = 0 (X ,Y )∈Ωr satisfying the following conditions: 0 0 HOMOCLINIC BOUNDARY-SADDLE BIFURCATIONS 5 - BS(1) : X has a hyperbolic saddle at S ∈ Σ and the invariant mani- 0 X0 folds of X at the saddle point S , Wu(X ,S ) and Ws(X ,S ), are 0 X0 0 X0 0 X0 transversal to Σ at S ; X0 - BS(2): Y is transversal to Σ, Wu(X ,S ), and Ws(X ,S ) at S ; 0 0 X0 0 X0 X0 - BS(3) : the normalized sliding vector field has S as a hyperbolic singu- X0 larity, by taking x as a local chart on Σ at S , Zs (x)=µx+O(x2) with X0 N µ(cid:54)=0; - BSC(1):theunstablemanifoldofthesaddlethatliesinΣ+,Wu(X ,S ), + 0 X0 is transversal to Σ at P (cid:54)=S . We have ϕ (t,P )∈Σ+ for all t<0; X0 X0 X0 X0 - BSC(2): Y is transversal to Σ at P and there exists t > 0 such that 0 X0 0 ϕ (t ,P )=S and ϕ (t,P )∈Σ− for all 0<t<t . Y0 0 X0 X0 Y0 X0 0 Remark1. Undertheconditionsabove,thesaddle-regularpointisontheboundary of a crossing region and an escaping or sliding region, i.e, S ∈ ∂Σe ∪∂Σc or X0 S ∈∂Σs∪∂Σc. X0 There are two different topological types of cycles satisfying BS(1)-BS(3) and BSC(1)-BSC(2), see Figure 3. undefined region Σc Σs S X0 Σ SX0 Σ P PX0 X0 (a) (b) Figure 3. A degenerate cycle passing through a saddle-regular point: (a) W+s(X0,SX0)containedintheunboundedregionand(b)W+s(X0,SX0)contained intheboundedregion. Despite the fact that both cases shown in Figure 3 are topologically distinct, theiranalysisissimilar,sowefocusoncase(a). Wheneverwerefertoadegenerate cycle through a saddle point, we refer to a cycle as given in case (a) of Figure 3. To study the unfolding of the cycle, we look carefully at the local saddle-regular bifurcation, after that we perform a study on the first return map defined near the cycle. 3.1. Bifurcationofasaddle-regularsingularity. Thesimplestcaseisthecodi- mension 1 bifurcation studied in [4, 1], which we review briefly here for complete- ness. Consider a nonsmooth vector field Z = (X ,Y ) ∈ Ωr satisfying conditions 0 0 0 BS(1)-BS(3). To study bifurcations of Z near S , the following result from [5] 0 X0 is important, describing bifurcations of a hyperbolic saddle point on the boundary Σ of the manifold with boundary Σ+ =Σ∪Σ+. Lemma 1. Let p ∈ Σ be a hyperbolic saddle point of X | , where X ∈ χr. 0 Σ+ 0 Then there exist neighborhoods B of p in R2 and V of X in χr, and a Crmap 0 0 0 β :V →R, such that: 0 6KAMILADAS.ANDRADE,MIKER.JEFFREY,RICARDOM.MARTINS,ANDMARCOA.TEIXEIRA (a) β(X)=0 if and only if X has a unique equilibrium p ∈Σ∩B that is a X 0 hyperbolic saddle point; (b) if β(X) > 0, X has a unique equilibrium p ∈ B ∩ int(Σ+) that is a X 0 hyperbolic saddle point; (c) if β(X)<0, X has no equilibria in B ∩Σ+. 0 Since Y is transversal to Σ at S , there exist neighborhoods B , of S in Σ, 0 X0 1 X0 and V , of Y in χr, such that for any Y ∈ V and p ∈ B , Y is transversal to Σ 1 0 1 1 at p. Taking B as given in Lemma 1 there is no loss of generality in supposing 0 B ∩Σ=B and then restrict ourselves to the neighborhood V =V ×V of Z 0 1 Z0 0 1 0 in Ωr. If β(X) (cid:54)= 0, for X ∈ V , there exists a fold point of X in B . The fold point 0 1 is located between the points where the invariant manifolds of the saddle cross Σ. The map s : V → Σ that associates each X ∈ V to a tangent point F ∈ Σ is 0 0 X of class Cr, F is visible fold point if β(X)<0, F is a hyperbolic saddle point if X X β(X)=0, F is an invisible point if β(X)>0. X EachZ =(X,Y)∈V canbeassociatedwithtwocurves,T andPE ,defined Z0 X Z as follows: (i) T is the curve given implicitly by the equation Xh(p) = 0, i.e., T is X X formed by the point where X is parallel to Σ. Therefore, the intersection of T with Σ gives the fold point F . X X (ii) PE is composed by those points where X and Y are parallel. So, when Z the intersection of PE with Σ is in Σs ∪Σe, this intersection gives the Z position of the pseudo-equilibrium point. ThemapsX (cid:55)→T andZ (cid:55)→PE areofclassCr. ItfollowsfromconditionsBS(1)- X Z BS(3) that the curves T , PE , Wu(X ,S ) and Ws(X ,S ) have empty in- X0 Z0 0 X0 0 X0 tersection in B up to the saddle point S . In fact, all these curves contain the 0 X0 singular point S . Condition BS(2) guarantees that PE , Wu(X ,S ) and X0 Z0 0 X0 Ws(X ,S ) do not coincide in B up to the saddle point. Condition BS(3) also 0 X0 0 ensures that T and PE are different in B except at S . X0 Z0 0 X0 ThecontinuousdependenceofthecurvesT , PE , Wu(X ,S )andWs(X ,S ), X0 Z0 0 X0 0 X0 on the vector field ensures that V can be taken in such a way that the relative Z0 position of these curves do not change for all Z ∈ V . The curve T is located Z0 X0 between Ws(X ,S ) and Wu(X ,S ), see Figure 4. + 0 X0 + 0 X0 Since S ∈∂Σs∪∂Σc there are three different cases to consider depending on X0 the position of PE in relation to T , Ws(X ,S ), and Wu(X ,S ), see 4. Z0 X0 + 0 X0 + 0 X0 These cases are named BS , BS and BS as in [1]. 1 2 3 HOMOCLINIC BOUNDARY-SADDLE BIFURCATIONS 7 T T T T X0 X0 PEZ0 PEZ0 X0 X0 PE PEZ0 Z0 Σ Σ Σ Σ (a) (b) (c) (d) Figure 4. Relative position of the curves TX0, PEZ0, W+s(X0,SX0), and W+u(X0,SX0). (a) corresponds to case BS1, (b) corresponds to case BS2 and (c)−(d)bothcorrespondtocaseBS3. TounderstandthedifferencebetweenBS , BS andBS ,forZ =(X,Y)∈V 1 2 3 Z0 let β =β(X) be given by Lemma 1. Then S is a real saddle if β >0, a boundary X saddle if β =0, or a virtual saddle if β <0. 1. Case BS : this happens when Wu(S ,X ) is between T and PE in 1 + X0 0 X0 Z0 Σ+,seeFigure4(a). Ifthesaddleisvirtual(β <0),thesaddle-regularpoint turns into a visible fold-regular point and there is no pseudo-equilibrium. The fold-regular point is an attractor for the sliding vector field. Also, when the saddle is real (β >0), an invisible fold-regular point emerges and thereexistsanattractingpseudo-node. ThepointinΣs wheretheunstable manifold of the saddle crosses Σ is located between the pseudo-equilibrium and the fold-regular point, see Figure 5. β <0 β =0 β >0 Figure5. Bifurcationofasaddle-regularpoint: caseBS1. 2. CaseBS : thiscasehappenswhenPE isbetweenT andWu(S ,X ) 2 Z0 X0 + X0 0 in Σ+, see Figure 4(b). There is a visible fold-regular point and there is no pseudo-equilibrium when the saddle is virtual (β < 0). The fold- regularpointisanattractorfortheslidingvectorfield. Whenthesaddleis real (β >0), an invisible fold-regular point and an attracting pseudo-node coexist. Thepseudo-equilibriumislocatedbetweenthepoint(inΣs)where theunstablemanifoldofthesaddlemeetsΣandthefold-regularpoint. See Figure 6. 8KAMILADAS.ANDRADE,MIKER.JEFFREY,RICARDOM.MARTINS,ANDMARCOA.TEIXEIRA β <0 β =0 β >0 Figure6. Bifurcationofasaddle-regularpoint: caseBS2. 3. CaseBS : thiscasehappenswhenT isbetweenWu(S ,X )andPE 3 X0 + X0 0 Z0 in Σ+, see Figures 4(c)-(d). When the saddle is virtual (β < 0), a visible fold-point coexists with a pseudo-saddle. There exists no pseudo- equilibriumwhenthesaddleisreal(β >0). Inthiscase,thesaddle-regular point turns into an invisible fold-regular point that is a repeller for the sliding vector field. See Figure 7. β <0 β =0 β >0 Figure7. Bifurcationofasaddle-regularpoint: caseBS3. 3.2. Structure of the first return map. Let Γ be the degenerate cycle of Z . 0 0 WeshowthatV canbechoseninsuchawaythat,foreachZ ∈V ,afirstreturn Z0 Z0 map is defined in a half-open interval, near the cycle Γ . 0 Proposition 1. Let Z be a nonsmooth vector field satisfying conditions BS(1)- 0 BS(3) and BSC(1)-BSC(2). In addition, suppose S is located at the origin. X0 Then there exists a neighborhood V of Z in Ωr such that, for all Z ∈V , there Z0 0 Z0 is a well defined first return map in a half-open interval [a ,a +δ ) with δ >0 Z Z Z Z and a ≈0. The first return map of Z can be written as Z π (x)=ρ ◦ρ ◦ρ (x), Z 3 2 1 where ρ and ρ are orientation reversing diffeomorphisms and ρ is a transition 2 3 1 map near a saddle or a fold point. Proof. We have already determined a neighborhood V = V ×V where, for all Z0 0 1 Z = (X,Y) ∈ V , Lemma 1 holds for X and transversality conditions hold for Y Z0 in B = B ∩Σ. The claimed existence follows directly by means of continuous 1 0 dependence results and properties of transversal sets. HOMOCLINIC BOUNDARY-SADDLE BIFURCATIONS 9 Nowwedeterminea foreachZ ∈V . WhenthesaddleofX isnotinΣ,there Z Z0 are at least three different points in which the invariant manifolds Wu,s(X,S ) X meet Σ. Denote these points by Pi , i = 1,2,3, where P1, P2 ∈ B and P3 ∈ X X X 1 X I (I is a neighborhood of P in Σ). Assume P1,P3 ∈ Wu(S ,X)∩Σ and 1 1 X0 X X X P2 ∈ Ws(S ,X)∩Σ. We keep this notation if the saddle is on Σ, in this case X X P1 =P2 =S . BytakingxasalocalchartforΣnear0(withx<0corresponding X X X to sliding region and x > 0 corresponding to crossing region) and by denoting Pi = x , i = 1,2, we have x ≤ x . If S ∈/ Σ then there exists a tangency point X i 1 2 X F ∈ B , also S = F when the saddle is on the boundary. Considering the X 1 X X previous chart for Σ, denote F = x , then x ≤ x ≤ x , see Figure 8. Observe X f 1 f 2 that lim x = 0 for i = 1,2,f. For Z ∈ V let a be defined as a = x if Z→Z0 i Z0 Z Z f β(X) < 0, a = x = x if β(X) = 0, or a = x if β(X) > 0. Thus, by choosing Z 1 2 Z 2 δ >0smallenoughtheexistenceofthefirstreturnmapin[a ,a +δ ]isensured Z Z Z Z by continuity. To study the structure of this first return map for Z ∈ V , we analyze it near Z0 the saddle point. Without loss of generality, suppose that Σ is transversal to the y axis at the origin. Consider a sufficiently small ε>0 and let σ denote a section transversal to the flow of X, for x > 0, through the point (0,ε). There are three options for the position of S in relation to Σ, in each case the first return map is X analysed differently. As above, consider β =β(X) then: • if β < 0, S is virtual, then the first return map is limited by the visible X fold point, F . So we have a transition map, ρ , from Σ to σ near a fold X 1 point. See Figure 8(a); • if β = 0, S ∈ Σ, then we have a transition map, ρ , from Σ to σ near a X 1 boundary saddle point. See Figure 8(b); • ifβ >0, S isreal, thenthelimitpointofthefirstreturnmapisthepoint X P2 where the stable manifold Ws(X,S ) intersects Σ near 0. There is a X + X transition map, ρ , from Σ to σ, near a real saddle point. See Figure 8(c). 1 Aftercrossingthroughσ,theorbitswillcrossΣnearP3. Sinceσisatransversal X section, the transition from σ to Σ is performed by means of a diffeomorphism ρ . 2 The flow of Y makes the transition from Σ (near P3) to Σ (near 0), then the X transversal conditions satisfied by Y give another diffeomorphism, ρ , performing 3 thistransition. Thediffeomorphismsρ andρ areorientationreversing, sowecan 2 3 write π (x)=ρ ◦ρ ◦ρ (x) for x∈[a ,a +δ ). (cid:3) Z 3 2 1 Z Z Z 1K0AMILADAS.ANDRADE,MIKER.JEFFREY,RICARDOM.MARTINS,ANDMARCOA.TEIXEIRA σ σ ρ2 ρ ρ ρ 2 1 1 P3 X F P3 Σ P1 F P2 Σ X ρ X X X X ρ 3 3 (a) (c) σ ρ 2 ρ 1 S P3 Σ X ρ X 3 (b) Figure8. Illustrationofthefirstreturnmapwithtransversalsectionτ: (a)β<0, (b)β=0and(c)β>0. Since ρ and ρ given in Proposition 1 are diffeomorphisms, the difficult part in 2 3 understanding the first return map π is the structure of ρ . From now on we as- Z 1 sume σ ⊂{(x,1)∈R2;x∈R} and Z =(X ,Y )∈Ω∞, since high differentiability 0 0 0 classes are required. We restrict the analysis to nonresonant saddles, meaning we consider Z = (X,Y) ∈ V such that the hyperbolicity ratio of S , r, is an irra- Z0 X tional number, (r = −λ /λ where λ < 0 < λ are the eigenvalues of DX(S )). 2 1 2 1 X The point S is assumed to lie at the origin. X According to [6], for each l, X is Cl-conjugated, around S , to the normal form X ∂ ∂ (3) X˜(x,y)=−rx +y . ∂x ∂y Let us assume that Σ=h−1(0) where h (x,y)=y−x+k. Then Σ intersects the k k axes at (0,−k) and (k,0), implying that if k > 0 then the saddle is real, if k = 0 then the saddle is on the boundary, and if k <0 the saddle is virtual. Denote the flow of X˜ by ϕ (t,x,y)=(ϕ (t,x,y),ϕ (t,x,y))T. In this case, the X˜ 1 2 transitiontimefromΣtoσiseasilycalculatedanditisgivenbyt (x)=−ln(x−k) 1 for each (x,x−k) ∈ Σ. Therefore, the transition map ρ˜, from Σ to σ, is given by ρ˜(x)=ϕ2(t1(x),x,x−k)=e−rt1(x)x=x(x−k)r =k(x−k)r+(x−k)r+1. Since X and X˜ are conjugated, there must exist diffeomorphisms φ and ψ, defined in a neighborhood of the origin, such that ρ = φ◦ρ˜◦ψ and ψ(0) = φ(0) = 0. We can also assume that a˜ = ψ(a ) is equivalent to a , i.e., the transition map ρõf X˜ is Z Z defined in [a˜,a˜+δ˜), δ˜>0. Then: • if k ≥0, then a˜ =k. This means P =(k,0) for k >0 and S =(0,0) for 2 X k =0; see Figures 9(a) and 9(b). (cid:18) (cid:19) k k −kr • if k <0, then k <a˜ = <0. This means that F = , ; 1+r X 1+r 1+r see Figure 9(c).