Homoclinic and heteroclinic bifurcations close to a twisted. heteroclinic cycle. Martn G. Zimmermann, Dept. of Quantum Chemistry. Uppsala University

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1 Homoclinic and heteroclinic bifurcations close to a twisted heteroclinic cycle. Martn G. Zimmermann, Dept. of Quantum Chemistry Uppsala University Box 58, S-75 UPPSALA, Sweden and Mario A. Natiello y Dept. of Mathematics Royal Institute of Technology S- 4 STOCKHOLM, Sweden Abstract We study the interaction of a transcritical (or saddle-node) bifurcation with a codim- /codim- heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reactiondiusion equation. The manifold organization is such that two branches of homoclinic orbits to each xed point are created when varying the two parameters controlling the codim- loop. It is shown that the homoclinic orbits may become degenerate in an orbit- ip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-ip bifurcation, where a Smale's horseshoe is found. Keywords: heteroclinic cycle; T-point; orbit-ip bifurcation; inclination-ip bifurcation. Running title: Bifurcations near a twisted heteroclinic cycle. Introduction The traveling wave ODE of the FitzHugh-Nagumo reaction-diusion (RD) equation was analyzed by several authors (Rinzel & Terman, 98). The existence of a heteroclinic cycle between equilibrium points has been shown. This cycle is composed of a pair of codimension- heteroclinic orbits between two real saddle equilibriums (Kokubu, 988; Present address: Institut Mediterrani d'estudis Avancats (IMEDEA), CSIC-UIB, E-77 Palma de Mallorca, Spain. zeta@hp.uib.es y mario@math.kth.se

2 Chow et al., 99), that is, one has to adjust a single parameter of the system for each trajectory to exist. The two-parameter unfolding of this system gives rise to two branches of homoclinic connections to each equilibrium. In (Deng, 99a) these results were extended by studying the eect of the twisting of the global connections. It is shown that if one of the connections is twisted (see (Deng, 989) and below for a characterization of twistedness) it implied the existence of heteroclinic orbits with an arbitrary number of loops, branching from the codim- point. It was also shown (Deng, 99b) that the FitzHugh-Nagumo equation satises the above conditions, and the resulting solutions correspond to fronts with an arbitrary number of \bumps". In (Zimmermann et al., 997) a pulse bifurcation in the excitable regime of a FitzHugh- Nagumo-type system was reported. It was observed that a stable pulse to a homogeneous state bifurcated leading to the onset of spatiotemporal chaos. The traveling-wave ODE of this equation was analyzed numerically, nding that a dierent kind of heteroclinic cycle between two real saddle equilibrium points was at the core of the PDE, namely one composed by a codim- and a twisted codim- heteroclinic orbit. The locus of this cycle in parameter space was shown to coincide up to numerical accuracy to the onset of spatiotemporal chaos in the excitable regime of the PDE. The study in (Zimmermann et al., 997) also revealed several stable multi-hump pulses coexisting with the main single-hump solution in the PDE. It was observed that the corresponding multi-loop homoclinic orbits also bifurcated in a multi-loop heteroclinic cycle of the same type as the single-loop one. In parameter space, these bifurcations occured in an open region of parameters, close to the original single-loop heteroclinic cycle and seemed to accumulate very close to the single-loop homoclinic branch. Numerical continuation for the single-, double-, and triple-loop heteroclinic cycles suggests that the orbits approached in phase space, as well as in parameter space, where a transcritical bifurcation of the xed points occurs. Motivated by the above numerical observations, in this article we study a model of the interaction of a transcritical (or saddle-node) bifurcation of xed points with a codim- heteroclinic cycle of the above type. The main goal in this article is to investigate what are the consequences of this interaction in terms of existence of multi-loop homoclinic and heteroclinic orbits, as well as onset of chaotic dynamics. The unfolding of codim-/codim- cycles has been studied in (Bykov, 993) where it is found that two homoclinic branches are born from this global bifurcation. This bifurcation was called T-point (Terminal-Point) (Glendinning & Sparrow, 986) because in parameter space the homoclinic branches were terminating at the codim- point. A common tool in analyzing these problems is to dene a suitable codim- transversal section to the heteroclinic cycle and further dene two mappings whose composition gives a Poincare-map. With the construction of this return map it is possible to derive information on the periodic orbit organization (saddle-node and period-doubling bifurcations) as well as on homoclinic/heteroclinic bifurcations. This approach has been applied successfully in several homoclinic and heteroclinic systems (Sil'nikov, 965; Glendinning & Sparrow, 984). The return map we construct requires 3 \unfolding" parameters: one for the local bifurcation and two for the global reinjection (this setup is also inspired in the numerical observations in (Zimmermann et al., 997)). Our main result is to assess the conditions under which heteroclinic cycles with arbitrary number of global excursions occur. We discuss

3 3 the bifurcations of periodic orbits, the occurrence of multi-loop homoclinic and heteroclinic orbits, as well as the onset of chaotic dynamics related to the occurrence of Smale's horsehoe. These features contrast with those of the codim-/codim- cycle (Chow et al., 99). In the next Section we discuss the detailed setup of the problem and summarize the main results of this work. In Sec. 3 we derive the corresponding Poincare map. The coming three Sections contain the proof of the main results. In Sec. 4 we study -loop homoclinic connections, in Sec. 5 we study the periodic orbits and their bifurcations, while in Sec. 6 the multi-loop global connections are considered. In Sec. 7 we discuss geometrically the onset of chaotic dynamics. We present also some numerical examples. Finally, Sec. 8 is devoted to discussion. Setup of the Problem and Main Results The overall scenario that is suggested by the numerical studies constitutes the motivating setup for this article. We investigate the existence of multi-loop homo- and heteroclinic connections between two xed points that (i) are involved in a local transcritical bifurcation (or saddle-node, see below) and (ii) are connected by a global, twisted, zero-loop heteroclinic orbit (degenerated at the local bifurcation point). Let us consider a suciently smooth 3-dim vector eld with real parameters ; ; b. Assume that _u = f(u; ; ; b) () A The vector eld f(u; ; ; b) has locally a transcritical (or saddle-node) bifurcation for b =. We will assume for simplicity that we have two xed points for b >. The case where the xed points occur for b < can be handled similarly. The normal form of such a local ow taking the center manifold to coincide locally with the y axis, assuming that we are away from resonances, can be written as (Ilyashenko & Yakovenko, 99): _x = (b; y) x _y =?by + y + a(b)y 3 () _z = 3 (b; y) z where i are smooth functions and i (; ) = i 6=. Under general conditions, any ow undergoing a transcritical bifurcation is C k locally smoothly equivalent (k ) to Eq. (). For the sake of the physical motivation we will introduce an additional simplifying assumption: A b > (but suciently small) and a() is suciently small, so that the (eventual) additional xed points of the system lie far away from the region of interest in phase space. Since we are explicitly interested in dealing with b > but small, the saddle-node and transcritical cases can be treated as one, since their normal forms dier in a linear change of

4 4 coordinates. Dierences arise when one studies how the features of the dynamics change as b \crosses" zero. In the few cases we deal with such questions we will be more specic. We choose to consider explicitly the transcritical setup because of its relevance in applications (e.g., (Zimmermann et al., 997)). The previous assumption allows us to simplify the normal form when dealing with the region of phase space dened by y [; b]. Within such region the ow can be linearized around each xed point (Ilyashenko & Yakovenko, 99) and a featureless ane map can describe the dynamic evolution in the region y [; b? ] with < b. Such dynamics can via a smooth change of coordinates be rewritten as: ( i R; i = ; 3). _x = x _y =?by + y (3) _z = 3 z This vector vector eld has a saddle-xed point X A at u = (; ; ) with real eigenvalues ( ;?b; 3 ) and another saddle-xed point X B at u = (; b; ) with real eigenvalues ( ; b; 3 ). We assume also, A 3 3 > > Therefore, the unstable manifold of X A, W u (X A ), is -dim and locally coincides with the z axis, while the stable manifold of X B is also -dim and locally coincides with the line y = b; z =. The two dimensional stable manifold of X A for b >, W s (X A ), lies locally on the plane z = ; y < b, and the two dimensional unstable manifold of X B lies on the orthogonal plane x = ; y >. Thus the local bifurcation structure gives, for small enough b, a structurally stable codimension heteroclinic connection? i BA from X B to X A. See Fig. b. In general, global connections between xed points enter (leave) a neighbourhood of the equilibrium tangent to the so-called stable (unstable) leading subspace, namely the subspace corresponding to the eigenvalues with negative (positive) real part closest to zero. The other directions are referred to as the strong (non-leading) stable or unstable subspaces, denoted by W ss and W uu, respectively. If, for example, for certain parameters the global connection enters a neighbourhood of the xed point tangent to the non-leading subspace, it corresponds to an orbit-ip global bifurcation, where the orbit switches the branch of the leading subspace to which it becomes asymptotic. This bifurcation changes the twist around the homoclinic orbit, see (Champneys & Kuznetsov, 994) for a review on degeneracies of homoclinic orbits. We have, A 4 For = = and b = the non-hyperbolic xed point X AB at u = (; ; ) has a degenerate codimension- global connection? in W u (X AB ) T W s (X AB )) (see Fig. a). Moreover this global connection is persistent in a non-degenerate way for b >. Although we are interested in the case b >, this assumption describes a reasonably general behaviour for all b. For = = and b = we have a transcritical (saddlenode) local bifurcation and a two-fold degenerate homoclinic orbit: rst, the xed point is Eq. (3) is also the topological normal form of Eq. () under assumption (A) ( Sositasvili, 973). Hence, topological properties of our results can be extended to b =.

5 5 a u W (X AB) b u W (X A) z cu W (X ) AB X AB y Π W ss z X A u W (X ) B y W uu XB Π s W (X AB) x δ η cs W (X ) AB Π x δ + b s W (X A) η u W (X ) B s W (X B) Π u W (X A) u W (X A) Figure : Schematic diagram of manifold organization of T-point bifurcation with saddle-node (transcritical) bifurcation. (a) b =. The center stable and unstable manifolds are denoted by W cs (X AB ) and W cu (X AB ) respectively. Note that a homoclinic orbit to the non-hyperbolic equilibrium can only exist for. (b) b >. W u (X A ) and W s (X B ) are one dimensional, while W s (X A ) and W u (X B ) are two dimensional manifolds. non-hyperbolic and second, the orbit undergoes an orbit-ip global bifurcation. Persistent, in this context means that taking a section transverse to the homoclinic orbit on the incoming side, by changing ; the connection covers completely an open ball around q = T?. The non-degeneracy assumption means that for an open ball around = = the global connection enter (leaves) the xed point along the leading manifold and, A 5 For an open neigbourhood around = = we have (a) For b > (b = ), consider a curve D in W s (X A ) (W cs (X AB )) intersecting? transversely. The time-evolved curve D t approaches W ss (X A ) (W s (X AB )) for t?. (b) (Twisted reinjection): Consider a section transverse to? on the outgoing side. For b >, the intersection of with W u (X B ) is a curve D ~. The time-evolved curve ~D t builds a Mobius band along the ow. Property (a) is called the strong inclination property (Deng, 989; Deng, 99), and for our purposes it guarantees that the inclination (Champneys & Kuznetsov, 994; Homburg et al., 994) of the global reinjection does not change in the parameter region of interest. We restrict our studies to twisted reinjections (property (b)) motivated by numerical observations in application problems (Zimmermann et al., 997). With this basic set-up we will be interested in the qualitative features of the dynamics of such a system as we change b, and. The results to be stated below do not provide the complete unfolding of a system fullling (A){(A5), but rather concentrate in presenting sucient conditions for the existence of dierent kinds of homo- and heteroclinic loops, motivated by the numerical observations in this direction discussed in (Zimmermann et al., 997). We now proceed to present our main results:

6 6 Result For Eq. () with xed b > : (a) There exists a heteroclinic cycle consisting of a (local) codim-? i BA codim-? AB occurring in parameter space at the point h. AB and a (global) (b) The unfolding of this cycle gives rise to two branches h and A h B of homoclinic connections to X A and X B (labeled? A and? B ) originating from the point h AB (the T-point). (c) There exists also a structurally stable (codim-) heteroclinic connection? BA, where the -dim manifolds W s (X A ) and W u (X B ) intersect, in a parameter region bounded by h A and h B. (d) Both homoclinic branches undergo an orbit-ip bifurcation at points denoted F A and F B respectively. In (Bykov, 993) it is shown that (under general nondegeneracy assumptions) (a))(b) and that if there exists more than one heteroclinic cycle of type (a) then chaotic dynamics of horseshoe-type is present. Bykov discusses in that article the unfolding of a general codim- /codim- heteroclinic cycle between two saddle xed-points with real eigenvalues. The result mentioned here corresponds to the choice of eigenvalues that ts our setup. For other choices of eigenvalues (that are out of the scope of this article), Bykov nds that -loop homoclinic orbits and curves of saddle-node and period-doubling of periodic orbits branch out of the T-point as well. Result For b 6= suciently small and? > 3 b > there is a point F A on h A and a point F B on the branch h B where curves of saddle-node and period-doubling bifurcations of -loop periodic orbits originate. These orbit-ip points are also important in the organization of multi-loop heteroclinic orbits. We dene a n-loop heteroclinic orbit as that performing n global excursions (see below). In this sense, the orbits in Result are -loop heteroclinics. Consequently, we will denote a n-loop heteroclinic orbit as? n AB, and in parameter space it will be denoted as h n AB; analogously for n-loop homoclinics, e.g.,? n and A hn A. Note that because we are close to a local bifurcation with real eigenvalues we cannot have multiple codim- local heteroclinic connections of type? i BA. The main result of this article is: Result 3 For > 3 b > there exists an open set in the space of global connections where n-loop heteroclinic connections of codim- occur, for arbitrary n. The validity limits for this result will be discussed in the next section and will be more precisely stated in Theorem 4 below. It will be seen along the discussion that (A5) is crucial for this result. See also the remark at the end of Sec. 3. We will argue that as b all heteroclinic and homoclinic orbits collapse to (; ; b) = (; ; ) in parameter space. In the saddle-node case, neither xed points nor associated homo/heteroclinic orbits exist for b <. Hence, the appearance of a multitude of homoclinics and heteroclinics can be regarded as an explosion (Sparrow, 98), meaning that for a very small change of b, arbitrarily many of such bifurcations take place (each bifurcation at its own (; ) value, though).

7 7 3 The Model We will describe Eq. () constructing a geometric model via the composition of a local map around the local vector eld Eq. () and a global map which can be suitably approximated by an ane map (see (Wiggins, 988; Wiggins, 99)). The model is taken in such a way that any ow having properties (A){(A5) will dier from it in an exponentially small quantity. We will henceforth prove our results on the model rather than in general ows. This method of discussion goes back to ( Sil'nikov, 965) and has been formalized and improved in (Deng, 989) (naming it the exponential expansion). We begin by dening two Poincare sections around the local part of the ow. By a suitable rescaling of x and z we can dene these planes at, = f(x; y; z)j x = g ; = f(x; y; z)j z = g (4) The local map L : can be obtained integrating Eq. (3) starting from an initial condition on : z B C B C L y H(y; z) (5) z A A where and because of (A3), byz H(y; z) = b? y + yz ; b 6= ; (6) =? = 3 > ; = b= 3 : (7) To compute L we use that the time of passage between the sections and is T = 3 ln(=z). The global map G : is constructed assuming that we can (to lowest order) model the ow between the two control planes in a small region around the -dim manifold W u (X A ) with an ane map: G : x y C A 7 A B C D C B x y C B A b + C A = y z C A (8) where A; B; C; D are constants and AD? BC 6=. (A5)a requires D 6= so that the mapping of W u (X B ) T with G is not tangent to W s (X A ) T. Also, (A5)b demands D <, what will be henceforth assumed. The parameter unfolds a homoclinic to X AB (occurring for b =, < ) while unfolds the orientation of the reinjection ((A4), see Fig. b). To complete the construction we compose G and L to obtain the return map R : (dropping the trivial component): y R(y; z; ; ; b) : z = A B z + b + (9) C D H(y; z) Note that a single-loop periodic orbit (with a single global excursion) of the ow corresponds to a xed point of the Poincare map Eq. (9), while n-loop orbits will correspond to period-n orbits of the map.

8 8 Because of our choice of maps and coordinates, we will henceforth assume that detm < so that R can be regarded as the return map of a ow (see below). The domain of this return map consists of all points z >. In particular, we will be most interested in the region y b between both xed points although this is not a restriction. Note however that the map is continuous in this region but the limit lim (y;z)(b;) R(y; z; ; ; b) does not exist. Points (y; z) such that R(y; z; ; ; b) = (w; ) correspond in the ow to orbits entering X A (on W s (X A )) for w < b and on W s (X B ) for w = b. In this sense, periodic points of the map (periodic orbits) that approach z = as parameters are changed may have one of three \destinies": become homoclinic to X A, homoclinic to X B or heteroclinic. Remark : The map dened above can be regarded as having 7 coecients. These coecients, however do not have all the same status. We are interested in and as bifurcation parameters, responsible for the codimension of our bifurcation sets for b > suciently small. The coecients A, B, C, D, on the contrary, model the global excursions and have a dierent status. We are interested in results that are somehow independent of the specic values for A, B, C, D. See the next remark for a more specic statement. Remark : Theorem 4 below is stated and proved for an open subset of global parameter values A, B, C, D. Apart for the twisting requirement D < reecting (A5)b, the method of proof requires technical assumptions on the relative sizes of these coecients. Hence, as stated above, our results do not constitute the complete description of a family of systems fullling (A){(A5). Rather, we state sucient conditions for the existence of properties suggested by numerical observations. Remark 3 (Validity): The construction of the return map as the composition of a global ane map and the map arising from the local system Eq. (3) is valid for b suciently small. More explicitly, b min(b ; b ), where b is such that the ane approximation of the global map holds for a disc of size b around the unstable manifold of X A and b is such that Eq. (3) is a proper description of a general smooth ow of type (A), (A) in a ball of size b around X A (i.e., having only a local transcritical (or saddle-node) bifurcation where the singularities originating in the higher-order terms lie well outside this ball). Since we are interested in nondegenerate homo- and heteroclinic loops (arising as transverse intersections of invariant manifolds) the results we nd on the model problem will be persistent under small perturbations. Remark 4: All our results are discussed assuming > = O(b). The case < can be treated similarly, noting that by reversing the time direction the set (X A ; X B ; ; ) transforms into (X B ; X A ; = ; = ). We obtain hence the same results, with the role of A and B interchanged and a reversed ow direction in phase space. 4 -loop Global Connections The setup of the model, based on (A4) allows for an immediate proof of Result. Let us take any xed (and small) b > for the rest of the discussion. Since the global connection

9 9 of (A4) is of codim- and non-degenerate, there exists a (; ) such that the global map G maps W u (X A ) T on W s (X B ) T. We have hence chosen coordinates such that this heteroclinic connection occurs at = =, so h AB= (; ). The -loop homoclinic orbits to X A can be studied by checking where the image of W u (X A ) T intersects W s (X A ) T (i.e., z = ). Hence, the locus of h A in parameter space parameterized by s < b is given by: h A : = s; = () Similarly the homoclinic orbits to X B may be studied by checking when W u (X B ) T intersects W s (X B ) T. Hence, b = G t Bt + + b () Dt + = gives the locus of h B parameterized by t >. In parameter space, h B : =?Bt; =?Dt () In parameter space, both codim- curves h A and h B terminate on h AB. This point was called T-point in (Glendinning & Sparrow, 986). These results have been shown previously in (Bykov, 993) to hold true under similar assumptions. Because of (A4), the homoclinic orbits are non-degenerate (Deng, 989) along an open segment on h and A h away from B h AB, i.e., they enter (leave) the xed points along the leading stable (unstable) directions. The orbit-ip condition for h A corresponds to a reinjection entering the local region at y = (along the strong stable manifold). Hence, F A = (; ) = (?b; ). In the same manner, the orbit-ip condition for h B is F B = (; ) = (?B b;?d b). Taking t > and w < b in the l.h.s. of (), there exists also a codim- heteroclinic connection? (i.e., existing in an open region of parameters,, b), bounded by BA? A and? B. This last result should be compared with the codim- heteroclinic connections discussed in (Doelman & Holmes, 996). This completes the discussion of Result. We have shown that: Theorem For Eq. (), for any xed b >, under (A){(A5): (a) There exists a heteroclinic cycle consisting of a (local) codim-? i BA codim-? AB occurring in parameter space at the point h AB. and a (global) (b) The unfolding of this cycle gives rise to two branches h and A h B of homoclinic connections to X A and X B (labeled? A and? B ) originating from the point h AB. (c) There exists also a structurally stable (codim-) heteroclinic connection? BA, where the -dim manifolds W s (X A ) and W u (X B ) intersect, in a parameter region bounded by h A and h B. (d) Both homoclinic branches undergo an orbit-ip bifurcation at points denoted F A and F B respectively.

10 5 Periodic Orbit Organization We discuss here Result. Recall that the relationship of the eigenvalues correspond to >. For notational convenience we dene M = A B so that the xed point C D equation, Eq. (9), reads: y z = M + b + (3) z H(y; z) where The Jacobian around a xed point (y ; z ) and we used the short hand H y b (y; z) DR(y ; z ) = J J J J reads: J = B H y ; J = A z? + B H z J = D H y ; J = C z? + D H z j y=y ;z=z (4) z (b? y + yz ; H ) (y; z) = b(b? y) yz? (b? y + yz (5) ) The Floquet multipliers of the periodic orbit now correspond to the eigenvalues of DR(y ; z ), s = Tr(DR) Tr(DR)? det(dr) (6) 4 We notice that as det(dr) > for a return map of a ow, this implies detm < : det(dr) =? b detm (b? y + yz ) z +? (7) For the next two Subsections we will need the following two asymptotic limits for z. Given that y(z) = b + az for a constant a and > then, (? H(y(z); z) = b a z? + : : : ; > b + az? (8) + : : :; < and for y(z) = a z, H(y(z); z) = az + + : : : (9) 5. Saddle-node bifurcations We will consider the saddle-node bifurcation of xed points for b > ( > ). The strategy will be to parameterize the bifurcation curve and the location of the xed points by the single variable z.

11 First solve Eq. (3) for (y; z) and (y; z), (y; z) = y? b? A z? B b yz b? y + yz (y; z) = z? C z? D b yz b? y + yz () A necessary condition for a saddle-node bifurcation is that one of the Floquet multipliers = +, i.e.,?(tr)(dr) + det(dr) + = () is satised. The bifurcation condition Eq. () can be expressed as a polynomial in y, where c (z) y + c (z) y + c 3 (z) = () c (z) = (? z ) (? + C z? )? b D z? c (z) =?b (? z )(? + C z? ) + b D z? c 3 (z) = b (? + C z? + B z + detm z +? ) It is also required that b? y(z) + y(z) z 6=, or equivalently that y(z) is away from the singularity at b y sing =? z : (3) We notice that the singularity lies outside the rectangle < z <, y b. Solving y(z) from Eq. (), s y(z) =? c c? c 3 (4) c c c and plugging it in Eq. () gives the saddle-node bifurcation curve, in terms of z. ( SN (z); SN (z)) (5) The asymptotic solution of Eq. (4) will depend on the relation of the eigenvalues ( ; ). We will consider the case > >, c c =? b? z? D + C D z? + : : : ; so the asymptotic limit z of y(z) is, c 3 c = b D z?? b C D z?? b B D z + : : : (6) y(z) = ( b + B D z + : : : + D z?? C D z?? B D z + : : : (7) Taking the rst root in Eq. (7), the saddle-node bifurcation asymptotic expansion is, SN (z) =?B b? B D z? + : : :; (8) SN (z) =?D b? B z? + : : : (9)

12 so ( SN (z); SN (z)) F B when z. Taking the second root y(z) = O(z? ), the saddle-node bifurcation in (; ) parameter space, using Eq. (9), becomes, SN (z) =?b + z? therefore ( SN (z); SN (z)) F A when z. D + : : : ; (3) SN (z) = + (? ) z + : : : (3) Finally, note that for B > the saddle-node bifurcation curve produces a cusp, since there exists a z > such that d=dz = = d=dz. From Eq. () this condition corresponds to dy=dz = (B +det M z? )=D. The l.h.s. is large and negative for small z and becomes positive for large z while for B > and suciently small det M, the r.h.s. is negative for a large range of z. 5. Period-doubling bifurcations In the same lines as above we can study the asymptotic limit of the period-doubling bifurcation of xed points of the return map. In this case the bifurcation condition corresponds to =? and Tr(DR) + det(dr) + = (3) which can be expressed as a polynomial in y: c (z) y + c (z) y + c 3(z) =, where c (z) = (? z ) ( + C z? )? b D z? c (z) =?b (? z )( + C z? ) + b D z? c 3 (z) = b ( + C z? + B z? detm z +? ) and b? y(z) + y(z) z 6=. Solving y(z) as before and evaluating its asymptotic limit as z for > >, we arrive to, ( b + B y(z) = D z + : : :? D z?? C D z?? B D z + : : : (33) Plugging in Eq. () gives the period-doubling bifurcation curves: PD (z) =?b? z? so ( PD (z); PD (z)) F A when z, and, D + : : : ; (34) PD (z) = + ( + ) z + : : : (35) PD (z) =?b B? B D z? + : : :; (36) PD (z) =?b D? z? + : : : (37) so ( PD (z); PD (z)) F B when z. This nishes the proof of Result.

13 3 Note that for the period-doubling curve dy=dz is positive and very large for z small and hence for B > the curve will not develop a cusp in the way the saddle-node curve does. On the other hand, if there exists a z such that the discriminant root in y(z) is zero at z and real in z [; z ], both period-doubling curves will join continuously in one curve..8 F B sn η.6.4 h B. sn F A δ pd h A h AB Figure : Bifurcation set showing the saddle-node (sn) and perioddoubling (pd) bifurcations of -loop periodic orbits. Both curves approach the orbit-ip point F B for z with the same slope (see text). The saddlenode curve has a cusp (see text) and lies to the left of the period-doubling curve for z. Also the homoclinics h A, h B and the T-point h AB are shown. = :3, = :, b = :, A = :3, B = :6, C = :, D =?: Discussion and examples The results given above are derived assuming >. Also, our work focuses in the case (near the local bifurcation) which corresponds to the rst case studied by Bykov ( < and < in (Bykov, 993)). For > Bykov nds that the saddle-node and period-doubling bifurcation curves may originate at h AB but this case is out of the scope of this article. As a numerical example of the results in this Section we present a computation of the locus of saddle-node and period-doubling bifurcations in parameter space in Fig. for B >. The tangency between the saddle-node and period-doubling curves at F B needs not subsist when higher-order corrections are taken into account. Notice that the saddle-node bifurcation curve presents a cusp at (; ) ' (?:47; :65) while the period-doubling is a continuous curve from one orbit-ip point to the other. In the region bounded by the cusp and h B there exist three -loop periodic orbits.

14 4 6 Multi-loop Global Connections The (n+)-loop periodic orbits are solutions of the system of equations R n (y ; z ; ; ; b) = (y ; z ). For? n+ we have to solve R n (y A ; z ; ; ; b) = (y n ; ) with y = b +, z =, z k >, k < n and y n < b, while for? n+ we have y AB n = b instead. An analogous but more involved expression holds for? n+ B. We will study the occurrence of multi-loop heteroclinic and homoclinic orbits for > in the rest of this Section. 6. -loop global bifurcations. For the case n = the h AB, h BA, h A and h B bifurcation sets can be worked out explicitly. Consider the system of equations: s z = M (t) + + b (38) H(y (t); z (t)) with (y ; z )(t) = ( + b + Bt; + Dt), where s and t are parameters in [; b]. We have that s < b parameterizes W s (X A ) in and t > parameterizes W u (X B ) in. For? we AB take s = b and t =, for? BA we take s < b and t >. Equation (38) is well dened for + Dt ; + b + Bt. Solving for H(:) in the second component of Eq. (38) and replacing in the rst, we arrive to an expression for (; b; s; t), (; b; s; t) = s? b + B D? detm D ( + Dt) (39) Replacing this in the second component of Eq. (38) we arrive at an implicit equation for : Db =?( + C( + Dt) (; b; s; t) + Bt )? b + (; b; s; t) + Bt ( + Dt)? L(; ; b; s; t) (4) The solution of the above equation denes all the -loop connections present in our problem. Let us start with? AB (s = b; t = ). We rst observe that L as. Now we compute the derivative of L in the limit and obtain, dl d =? + (? ) B b D?? C? + O(? ) (4) Hence, Eq. (4) will have a solution for b suciently small whenever D < : Theorem For the return map Eq. (9) with D < (and hence for a ow satisfying (A){ (A5)) there exists always at least one? AB. For b, the heteroclinic becomes degenerate into (; ) = (; ) h AB. We note also that there may exist more than one h AB if the r.h.s. of Eq. (4) has a minimum. A minimum can occur for B=D >. For D > the occurrence of h AB cannot be guaranteed in general, what shows the relevance of (A5)b. Since? AB is a T-point in the same sense as in Theorem it is clear that there will be two associated homoclinic branches. These branches occur at the solutions of Eq. (38)

15 5 with s < b and t = (h A) or s = b and t > (h B). Anyway, to show the existence of these curves we can use the implicit function theorem, knowing from the previous result that for D <, Eq. (4) has a solution for s = b; t =. It is straightforward to notice that the derivative of that equation with respect to on the solution is non-zero (the proof mimics that of Lemma in the Appendix) at least for B > > D. Hence, there exists a function (s; t), such that (b; ) = and (s; t) satises Eq. (4). The limit of h A as s + can be studied directly from the second component of Eq. (38). It is observed that the solution ( ; ) (?b; + ) F A. Similarly when t b?, then ( ; ) (?b B;?b D) F B. We summarize these observations in: Theorem 3 With the assumptions of Theorem and B >, there exist, for suciently small, codim- curves of homoclinic orbits h A and h B terminating at h AB and a codim- surface of heteroclinic orbits h BA. Also h A and h B can be continued up to F A and F B respectively. Figure 3 displays an example of bifurcation set of the periodic orbits. There are two homoclinics? A and? B which originate at? AB and collide at F A and F B respectively..8 F B η.6.4 h B h B. F A h A δ h A h AB h AB Figure 3: Bifurcation set including the -loop and -loop global bifurcations. Note how close the h B lies to h B. = :, = :, b = :, A = :3, B = :6, C = :, D =?:9. 6. (n+)-loop global bifurcations Let us discuss now the main result of this work, namely Theorem 4 For the return map Eq. (9) (and hence for a ow satisfying (A){(A5)), and for, > + and an open set in A, B, C, D-space (such that AD? BC <, (B + D)(? jb=dj) > (A + C), C > and <?D < B < ) there exist codim- (n + )-loop heteroclinic and codim- homoclinic orbits for arbitrary n.

16 6 Remarks: (i) The requirements on A, B, C, D other than negative determinant and twisted reinjection (which is already needed for Theorem ) are technicalities due to the method of proof. Numerical experiments suggest that the result holds for a larger region in A, B, C, D space, but the mechanism of the proof fails outside the area indicated above. (ii) The hypotheses (B + D)(? jb=dj) > (A + C), and > + may be rephrased as >, (B + D) > (A + C) and b suciently small. (iii) The case < can be reduced to the case >, see Remark 4 in Section 3. Proof: The statement is already shown for n = ;. Let us for simplicity change variables to s = y? b, so that the iterates of the unstable manifold of X A read: s =, z =, s z k+ = + A B C D z b(b+s)z?s+(b+s)z Our goal is to show that for any b > there exists (; ) such that for any n we have z n = = s n while all iterates with k < n satisfy z k > > s k. The strategy of the proof is as follows.. Let =?Bb and =?Db. Consider. For = the return map Eq. (9) has one xed point while for < it has two xed points.. For = we are in the orbit-ip condition for X B. In such a case the following results hold: (a) Lemma : s n < < z n < js n j ) s n+ < < z n+ < js n+ j. We defer the proof of this lemma to the Appendix. Since?Bb = s < < z =?Db < Bb, we have by induction that s n < < z n < js n j, 8n. (b) There is a trapping region bounded by the curves s = Bz=D, s =?z and a suitable curve joining these two which passes through s =?Bb. (c) Inside the trapping region there exists a stable xed point of the return map. 3. For we have that s m < 8m k ) z m > 8m k. The proof is as follows: < z. Also for k > it holds that Ds k? Bz k = det(m)z, so if s k? k < < z k? we have that Bz k = Ds k? det(m)z > because of the signs of the coecients. We k? notice in the same way that if s k < < z k then js=zj < jb=dj. 4. Decreasing the stable xed point eventually disappears in a saddle-node bifurcation. Let be the value where this bifurcation occurs. 5. Let n <. Adjusting n adequately, there exists n > such that s n = < z n while s k < < z k for k < n. This fact is immediate by continuity since for suciently small, s n becomes positive. 6. Lemma : dsn d ( n) < < dzn d ( n). We discuss this Lemma in the Appendix. Hence, by moving slightly above n, s n can be made negative and arbitrarily close to zero, i.e., js n j z n, while s n+ is positive. By increasing even more, s n+ becomes eventually zero, since for all s-iterates are negative. This shows that adjusting adequately, one can obtain for any n > that s n = < z n and s k < < z k for k < n. k (4)

17 7 7. Consider now =?Db?, with >. By the implicit function theorem, n =@ 6= (in fact it is? + O(b)), there exists a function () such that () = n and s n (()) =. The value z n varies continuously with and for suciently large, z n becomes negative. Hence, there exists n ; ( n ) such that s n = = z n and thus a n + -loop codim- heteroclinic orbit occurs. 8. Since each n + -loop heteroclinic orbit satises the conditions of (Bykov, 993) or Theorem, there exist branches of n + -loop homoclinic orbits to X A and X B. In the coming Section we give a geometrical interpretation of this result, while in Fig. 4 we show a numerical computation of a few multi-loop heteroclinics. η F B h B sn h 5 AB h 4 AB δ h 8 AB h 7 AB h 6 AB Figure 4: Bifurcation set showing the locus of heteroclinic orbits h n AB for n = 4; ::; 8 and the saddle-node bifurcation orginating from F B. = :63, b = :, = :, = :7, A =?:, B = :8, C = :, D =?:3 7 Geometrical Approach In this Section we will describe the organization of the ow on from a geometrical viewpoint and further show how a horseshoe is formed in a region of parameter space. Recall that we are assuming D < and B > throughout. As a preparation we compute the pre-image of W s (X A ) T with the Poincare map. Setting z = in the second component of Eq. (9) and solving for y, we obtain: y = f(z; ) The following properties hold for f: b ( + Cz ) (? z )( + Cz )? b Dz (43) For z then f(z; ), and for > then f(; ) = b. discontinuous at (; ). Hence, f(z; ) is

18 8 For Cz, f(z; ) ' b ( + +bd z + : : :). An iteration of the return map (y ; z ) = R(y; z; ; ; b) with y < f(z; ) implies z >. Likewise y > f(z; ) implies z <. The y value such that z = is also required for the geometric characterization. Hence, together with f we dene a function g(z; ; ; b) = y n+ (f(z; ); z): where y = g() = + b? B D and g(z ) = b for z = g(z; ; ; b) + b? B D + detm D z (44) D?B = detm. We are now ready to study the iterations of W u W u (X A ) S W u (X B ) on the Poincare section. For easier notation we will label the n-th iteration of W u T with the map R as w n = (y n ; z n ), where y n and z n are coordinates on. These curves will be parameterized by a parameter t. Hence, w corresponds (y ; z )(t) ( + b + B t; + D t); in particular, w () corresponds to the -dim W u (X A ) T. In what follows we will give a account of how w n (t) behaves when parameters (; ) are changed. From Sec. 4 we know that the locus of? A is < b; =. Let us consider a parameter cut with small and constant = >, and choose < such that there is no intersection between the graph of f and w (see Fig. 5a). This can always be achieved since f(z; ) does not depend on, while y does. This manifold conguration corresponds to? BA. Let us denote t =?=D the value of t such that z (t ) =, so w (t) has positive z-component for t < t. Let us consider as well w = R(w ), the image of w () by the Poincare map. w lies to the right of w and close to it. Their separation is of the order of z which is small for jzj < and >. Then by the properties of f, all the curve w will have positive z-component. Notice that lim R(w (t)) = w (); (45) tt and w () is mapped with R to the second iteration of the -dim W u (X A ) on, w (). Increasing, the set w approaches the graph of f (which remains xed). At a certain, w () = (f(z ); z ), i.e., the \tip" of w touches the graph of f at some point z (see Fig. 5b). It is clear then that z () = and by Eq. (44), y () = g(z ()). If y () < b then we obtain the -loop homoclinic connection to X A,?. Increasing further, A w () crosses the z = axis and? is obtained. For even higher, BA w may intersect W s (X A ) creating?. B Let us make a continuation of? A by increasing and adjusting such that the point w () belongs to the graph of f. As increases, f becomes less steep, so it is required to increase to maintain the intersection. At a certain tan a tangency between the graph of f and w occurs at t = (see Fig. 5c). To compute the curve tan () where any point of w is tangent to the graph of f, we must solve: f(z) = + b + B (z? ); D f (z) = B D (46) where the r.h.s. of the rst equation comes from eliminating t and solving for y(z) in (y; z) = w (t). If we further increase > tan, then w will cross the graph of f twice,

19 9 w (t) (δ+b,η) z w (t) y g(z) f(z) a s W (X B) w (t) w (t) b (δ+b,η tan) w (t) w (t) c s W (X A) y=b d e f w (t) w (t) w (t) w (t) w (t) w (t) g w (t) h w (t) w (t) w (t) w (t) w (t) w (t) Figure 5: Manifold organization on. w (t) is the projection of W u (X A ) S W u (X B ) in. Note that w () = ( + b; ) and w (t ) = ( + b? B D ; ). (a) corresponds to? BA ; (b),(c) and (d) to? A and? BA ; (e) corresponds to? AB and? BA ; (f) and (h) corresponds to?3 A,? BA and? BA ; (g) corresponds to? 3 BA,? AB and? BA. Only a portion of w is shown.

20 which means that a part of w (t) will have a negative z-component and will have the shape of a hook (Fig. 5d). Increasing further and adjusting so that we continue? A, w () will intersect W s (X A ) creating? AB, Fig. 5e. In this case (y ; z )() = (g(z ); f(z )). In this situation, we can decrease slightly such that w still intersects twice the graph of f, and adjust such that w () intersects f once again, Fig. 5f (notice that this can be done by perturbations in ; as small as necessary). In this case, w () w () = (g(z ()); ) and a 3-loop homoclinic to X A, is created from h AB (only a portion of w is shown in the gure). If w () crosses once again the graph of f (see Fig. 5h) a 3-loop homoclinic to X A is again obtained and in this case w will also be a hook. The transition between these two versions of? 3 A occurs when the tip of w is tangent to f. Note that the tangency between the tip of w n and f corresponds to an inclination-ip global bifurcation (Champneys & Kuznetsov, 994; Homburg et al., 994). For n =, this bifurcation is illustrated in Fig. 5b-d. In this case, the? is not twisted while A w is not tangent to f (Fig. 5b). After the tangency, w crosses f twice and? A becomes twisted. In connection with the inclination-ip bifurcation horseshoe-like dynamics and innitely many homoclinic orbits have been found (Deng, 993; Homburg et al., 994). We will not go into this topic in detail, but only sketch the geometry of the horseshoe construction in our setup. 7. Horseshoe construction close to -loop global bifurcations Consider a parameter region where w crosses twice the graph of f close to the creation of h A such that w () lies above the curve f, as Fig. 6 shows. From the denition of (y ; z ) = w (t), we can eliminate t and solve for y (z ) = + b + B D (z? ). Let z L be such that f(z L ) = y (z L ), i.e., the intersection of w and the graph of f closest to z =. Let z R be such that < z R < z L, and label y R = f(z R ). Consider a region U in (see Fig. 6a) bounded by g() y y R (side a), (y(z); z) with z z L (side b), (f(z); z) with z R z z L (side c), and the segment (y R ; z) with z z R (side d). z R is chosen suciently small, such that the \hook" that w forms lies inside the region U. Let us iterate U twice with the Poincare map. The rst iteration with the Poincare map of U, produces a wedge U = R(U), Side a is contracted to w () = ( + b; ). Side c is mapped to (y; ), with g(z R ) y g(z L ). Side b is stretched to side b' where (f(z R ); z R ) (g(z R ); ) with g() g(z R ) y R. Side d is stretched to side d' where (f(z L ); z L ) (g(z L ); ) with g() g(z R ) < g(z L ) y R. The next iteration, U = R (U), produces a horseshoe (Fig. 6b): Side a' w () w () and side c' is mapped to w (). Side d' and b' cross the graph of f twice so the image with R will have a portion with negative z-component

21 a' a c'' b b' d' R(U) a'' b U c d b U c d c' a b'' a d'' R (U) Figure 6: Horseshoe construction. (a) First iteration of R(U) crosses twice the graph of f. (b) R (U) intersects again R and forms a horseshoe. Notice that the side identication reveals an overall rotation of U with respect to U. Also if (y R ; z R ) (b; ) then side d' w and side d" w. From this we can identify the necessary conditions to have a topological horseshoe: (; ) are such that w intersects the graph of f twice, i.e., w has a hook shape with the \tip" of the hook above the graph of f (Fig. 5g). This occurs for > tan (), and 3 (), where 3 () is the locus of h 3 A in parameter space (Fig. 5f). The intersection of side b" with z = has to have y y R < b. To compute this, nd z such that y (z ) = f(z ) corresponding to the \top" intersection of side b' with the graph of f. The requirement is that g(z ) < b. Note that as g(z ) b we approach h AB. For xed b > and an open set of choices of A, B >, C, D < (compatible with AD? BC < ) there is an open region in (; ) where these conditions are met. 7. (n+)-loop heteroclinics revisited. We give here an account of how the n +?loop heteroclinics are formed. Let = and consider > such that the stable xed point lying in the trapping region is about to undergo a saddle-node bifurcation. Since one branch of the stable manifold of the xed point crosses the line w and w () lies inside the trapping region, all iterates w n will allso intersect the stable manifold of the xed point. Since all w n are z -close to each other and w crosses the curve f, all w n will cross this curve. Moreover, the unstable manifold of the saddle-companion of the stable xed point will have the same property. For < and arbitrarily close to there are only a nite number of iterates of w () with s <. However, this number can be made arbitrarily large by increasing. Varying one of this iterates can be made to cross s = < z or the curve f, giving hence the conditions for the proof in the previous Section. Increasing, all z i became smaller until eventually the \last" one hits z =.

22 F B η h 4 AB h 4 B h 3 B h 3 AB h 4 A h 3 A h B h B h A h AB F A h A δ Figure 7: Schematic view of the homoclinic and heteroclinic bifurcation sets. Shadowed regions correspond to the occurrence of Smale's horseshoe. The codim- h n BA heteroclinic orbits are not shown. 7.3 Overall picture The results of Secs. 6 and 7 can be put together in order to establish the overall picture of the homoclinic and heteroclinic bifurcations. The results of (Sandstede, 993) (see (Champneys & Kuznetsov, 994) for a discussion) about homoclinic bifurcations near the orbit-ip points apply to our system. Suppose we have a real saddle equilibrium u with eigenvalues satisfying < < <. When the homoclinic to u undergoes an orbit-ip degeneracy, then the two interesting cases are: (a) < j j, (b) j j < < j j. The codim- unfolding for case (a) has a saddle-node and period-doubling of periodic orbits and a secondary homoclinic-doubling bifurcation. There are no other periodic orbits or homoclinic bifurcations in the vicinity of the degeneracy. For case (b) is was determined that apart from saddle-node and period-doubling, there are countably many secondary homoclinic bifurcations with arbitrary number of loops, accumulating to the primary bifurcation. Also there is a region in which the system likely (Champneys & Kuznetsov, 994) contains Smale's horseshoe. Translating these results to the case studied here ( > > ), we nd that F A corresponds to case (a), while F B satises case (b) (after time-reversal). Hence, innitely many multi-loop homoclinics h n B branch out from F B. In Fig. 7 we show a schematic view of the bifurcation sets. h A and h A and their counterparts to X B are easily drawn from the results derived in the previous Sections. From Fig. 5 we infer that h A goes over from non-twisted to twisted and eventually terminates in h. Continuating the \tip" of AB w along f, we see that h 3 is born from A h AB as a non-twisted orbit. Eventually, when the tip of w has undergone a tangency similar to w, h 3 A becomes twisted and ends in h 3. The region above the non-twisted AB h3 A in Fig. 7 corresponds to the formation of a hook in w and hence to the occurrence of Smale's horseshoe. This process repeats itself for all higher n provided the w k develop \hooks". The reason for the dierent behaviour of h A and h n A, n > can be traced to the fact that f for = has dierent asymptotics than for >. Concerning the homoclinic orbits to X B, only h B can be explicitly computed, and it

23 3 is shown to exist all the way up to F B. Of the other h n B, n > orbits, we only know their existence but not any eventual ulterior bifurcation. However from the above results of (Sandstede, 993) we may safely conjecture that all the h n B orbits will eventually terminate at F B. 8 Discussion Recently in the complex Ginzburg-Landau (CGL) amplitude equations, a similar breakup of a heteroclinic cycle between two real saddle xed points in the traveling-wave ODE system has been encountered (Doelman, 993; Doelman & Holmes, 996). In their study, Doelman and Holmes derive a Poincare map perturbing from a Hamiltonian system specic to their problem, nding global bifurcations similar to those discussed in this article. However, their system contains 4 xed points, which can annihilate by pairs in saddle-node bifurcations and hence their setup has dierent features than ours. They discuss in detail the multi-loop codim- heteroclinic connections, analogous to our? n BA, nding an explosion around an orbit-ip global bifurcation for a situation which corresponds to D >. Also, heteroclinic connections to a periodic orbit were found and analyzed. As mentioned in Section, our initial motivation was to study the T-points close to a non-hyperbolic xed point, inspired in (Bykov, 993) and motivated by the numerical observations on the traveling-wave ODE associated to a FitzHugh-Nagumo reaction-diusion problem (RD) (Zimmermann et al., 997). The multitude of homoclinic and heteroclinic orbits as well as the locus of the orbit-ip points F A and F B discussed in this article occur in a region of size b both in parameter space (b; ; ) and phase space ((y; z) ), for any small enough b >. All these features \collapse" to the (doubly) degenerate homoclinic orbit? for b =. We do not enter closely in the problem of how this collapse takes place. Anyway, for the case of a saddle-node bifurcation, no homoclinic or heteroclinic orbits (not even xed points) exist for b <. Theorem 4 states that for a vector eld (e.g., the traveling-wave ODE of RD) where the non-degeneracy conditions (A){(A5) hold and close enough to the local bifurcation, there exist n-loop T-Point bifurcations, for all n >, accumulating close to the homoclinic branch h B whenever the reinjection is adequate. In terms of the solutions of the RD problem, the boundary of existence, in parameter space, of traveling pulses is given by the locus of the n-loop T-Points bifurcations. In (Zimmermann et al., 997) -loop homoclinic orbits where studied and several candidates of multi-loop heteroclinics were found close to one of the -loop homoclinic branches. Also single-, double- and triple-loop T-Points orbits were continued with a numerical continuation package (Doedel & Kernevez, 986) and were seen to approach each other, both in phase space and in parameter space, as they approached the local bifurcation. However the catalysis traveling-wave has a third (saddle-focus) xed point X C. In particular, a codim- heteroclinic orbit? AC between X C and one of the xed points of the local bifurcation was found. Both the global and local maps may eventually be inuenced by the manifolds of this xed point if b is not small enough, possibly taking the system outside the validity range specied in the theorems. Finally, we want to point out that the horseshoe-like dynamics addressed in this paper