Krauskopf, B., Lee, CM., & Osinga, HM. (2008). Codimension-one tangency bifurcations of global Poincaré maps of four-dimensional vector fields.

Transcription

1 Kraskopf, B, Lee,, & Osinga, H (28) odimension-one tangency bifrcations of global Poincaré maps of for-dimensional vector fields Early version, also known as pre-print Link to pblication record in Explore Bristol Research PF-docment University of Bristol - Explore Bristol Research General rights This docment is made available in accordance with pblisher policies Please cite only the pblished version sing the reference above Fll terms of se are available:

2 odimension-one tangency bifrcations of global Poincaré maps of for-dimensional vector fields Bernd Kraskopf, lare Lee and Hinke Osinga Bristol entre for Applied Nonlinear athematics, epartment of Engineering athematics, University of Bristol, Bristol BS8 TR, United Kingdom Abstract When one considers a Poincaré retrn map on a general nbonded (n)-dimensional section in a vector field in R n there are typically points where the flow is tangent to the section The only notable exception is when the system is (eqivalent to) a periodically forced system The tangencies case bifrcations of the Poincaré retrn map when the section is moved when there are no bifrcations in the nderlying vector field The interaction of invariant manifolds and the tangency loci on the srface gives rise to discontinities of the Poincaré map and there can be open regions where the map is not defined We stdy the case of the for-dimensional phase space R 4 Specifically, we make se of tools from singlarity theory and flowbox theory to present normal forms of the codimension-one tangency bifrcations in the neighborhood of a tangency point AS classification scheme nmbers: 37, 37G5 37G25, 58K5 ntrodction n many applications one is faced with the task of nderstanding the dynamics of a mathematical model given by a system of ordinary differential eqations Written as an atonomos vector field, sch a dynamical system takes the general form ẋ = f(x,λ), x X, λ R m, () where X is the phase space, λ is a (vector-valed) parameter, and f : X X is sfficiently smooth n this paper we are interested in the important case that the phase space X is Eclidean and of dimension at most for, that is, X = R n for n 4 The dynamics is given by the flow Φ associated with () An important tool for analysing the dynamics of a vector field is to consider the dynamics of a Poincaré retrn map P on a sitable codimension-one section Σ, (ie Σ is of dimension n ) The classic application of the Poincaré map is the stdy of periodic orbits One chooses a small, local section transverse to a given periodic orbit, then the image of a point x Σ nder P is given as the next intersection of the orbit corresponding athor; larelee@bristolack

3 Tangency bifrcations of global Poincaré maps of 4 vector fields 2 of x with Σ, formally, P : Σ Σ x P(x) := Φ tx (x), (2) where t x is the time to the next intersection with Σ The retrn map back to the local section is a local diffeomorphism (a smooth map with a smooth inverse) in a neighborhood of the intersection point of the periodic orbit with the section, which is a fixed point nder this Poincaré map Hence, the stability analysis of the periodic orbit in the fll phase space is redced to the stdy of the fixed point of the Poincaré map; see standard textbooks sch as [7, 9, 9] Poincaré maps are also sed to stdy other invariant sets, both theoretically and in experiments, for example see [4, 5] One typically records all intersection points, possibly after transients died down, of some orbit with a sitable section (typically some hyperplane) The reqirement here is that the orbit actally intersects the section repeatedly Note that one often records maxima and/or minima of a time series, which is eqivalent to considering the intersection points of the derivative of the time series with the section where the vale is zero n other words, one is typically interested in the dynamics on an nbonded or global section, and not only on some small, local neighborhood Except in the case that one considers the stroboscopic map of a periodically forced system, the Poincaré map is not a diffeomorphism on the entire global section [22] Note that P in (2) is well defined at x only if inf{t > Φ t (x) Σ} < n general there may be points in the section Σ whose trajectories do not retrn to Σ in either forward or backward time, meaning that the Poincaré map or its inverse is ndefined Bt even if the Poincaré map is well defined, another isse arises, namely, for a general (non-periodically forced) vector field and any chosen global section Σ, there is a nonempty set of points where the flow is tangent to the section This set is called the critical tangency locs, and it is formally given as := {x Σ f(x) n Σ (x) = }, (3) where n Σ (x) is the nit normal to Σ at the point x mportantly, the Poincaré map, defined as the kth retrn to Σ (for fixed k ), is discontinos at any point on the critical tangency locs This is becase the nmber of intersections increases or decreases by one at the tangency; [, 2, 3, 4] The kth iterate of the global Poincaré map can be extended continosly across by considering the (k ± )st iterate of P in the adjoining region The choice of the sign follows from the condition that t x, the time to the next intersection, depends continosly on x across We remark that this extension is not niqe as it depends on which open region of Σ \ one starts in Frthermore, the continos extension of P is not differentiable at, meaning that P cannot be extended across as a diffeomorphism What is more, the existence of the set gives rise to bifrcations of invariant sets in the section Σ that are not bifrcations of the nderlying vector field itself Since the Poincaré map is sed as a tool to stdy

4 Tangency bifrcations of global Poincaré maps of 4 vector fields 3 the dynamics of the vector field, it is important to classify this type of bifrcation of only the Poincaré map itself We are concerned with an important class of sch bifrcations, which we refer to as tangency bifrcations A tangency bifrcation is characterised by the tangency of a smooth invariant manifold of a given dimension with the section Σ t can be broght abot by changing either a system parameter or by moving Σ We stdied this type of bifrcation in [] for vector fields p to dimension n = 3, that is, for Poincaré maps on sections of dimension one and two n this paper we present normal forms for all codimension-one tangency bifrcations of vector fields in R 4 n light of the difficlty of visalising dynamics in for space dimensions, considering a Poincaré map on a three-dimensional section is particlarly sefl in this case Hence, the list of tangency bifrcations presented here is of direct se for the interpretation of the dynamics of Poincaré maps on global three-dimensional sections Any tangency bifrcation necessarily takes place at some point in the critical tangency locs, and it can be classified by its order of tangency; see section 2 for details We consider the sitation locally near the tangency point and assme that the respective local piece of the invariant manifold be part of an invariant periodic orbit, invariant tors, or of a stable or nstable manifold Specifically, the normal forms of the codimension-one tangency bifrcations are presented in a standard flowbox with parallel flow along one of the axes A flowbox is simply a domain in phase space that does not contain any compact invariant sets, sch as eqilibria, is bonded by orbit segments and transverse in- and ot-sets of codimension one Note that it is possible to find a flowbox near any tangency bifrcation which, according to the Flowbox Theorem [3], can be mapped diffeomorphically to the standard flowbox by straightening ot the flow At the moment of tangency bifrcation there is one orbit of (), which is part of some invariant manifold, that is tangent to the three-dimensional section precisely at the tangency point associated with the bifrcation t is the order of this tangency that determines the geometry of the section within the standard flowbox Note that the critical tangency locs on the three-dimensional section is of dimension two The order of contact in the directions perpendiclar to this orbit determine the geometry of the manifold inside the flowbox n fact it sffices to classify the cases when the manifold is also a codimension-one hypersrface, ie of dimension three in a for-dimensional flowbox The one parameter nfolding of the tangency bifrcation is given by moving the section p and down t is very difficlt to show the interaction of two three-dimensional hypersrfaces in a for-dimensional flowbox However, a tangency bifrcation can be characterised and nderstood by the interaction of the two-dimensional projections of the section and the manifold in the three-dimensional in-set of the flowbox Namely, the sitation in the in-set determines the geometry of the two-dimensional intersection of the manifold with the three-dimensional section Hence, each codimension-one tangency bifrcation is presented by images that show the flowbox in the in-set and in the section before, at and after the bifrcation The figres in this paper have been rendered directly from

5 Tangency bifrcations of global Poincaré maps of 4 vector fields 4 the respective normal forms This paper is organised as follows n section 2 we introdce some notation as needed to provide backgrond information on tangency bifrcations There we already provide the list of all codimension-one tangency bifrcations for vector fields of dimension p to for Section 3 smmarises the reslts from [] on tangency bifrcations for vector fields of dimension p to three; in particlar, we emphasise how the respective manifolds in the in-set determine the sitation in the one- or twodimensional section The normal forms of the tangency bifrcations for vector fields on R 4 are presented one by one in section 4 Finally, we discss or reslts and conclde in section 5 2 Backgrond and notation We are interested in how the flow of a vector field on R n interacts globally with a Poincaré section Σ By this we mean that Σ is a smooth manifold of dimension n that is nbonded in all directions n other words, Σ divides the phase space R n into two disjoint parts The notion of a global section is qite natral, and the easiest and often sed example is that Σ is an (n )-dimensional hyperplane We consider here the general case of a system that is not periodically forced Hence, the critical tangency locs defined in (5) is nonempty [22] At a reglar point in the flow makes a qadratic tangency with the section Σ t follows from the mplicit Fnction Theorem that consists generically of smooth codimension-one sbmanifolds of Σ The codimension-one sbmanifolds of that correspond to reglar points meet in sbmanifolds of higher codimension, which correspond to higher-order tangencies To be specific, for n = 2 the set consists generically of isolated points in the onedimensional section Σ For n = 3 the critical tangency locs consists generically of smooth one-dimensional crves of reglar points in the two-dimensional section Σ that meet at isolated csp points, where the flow has a cbic tangency with the section Σ For the case n = 4, which is the main sbject here, consists generically of smooth two-dimensional srfaces in the three-dimensional section Σ Two srfaces meet along a one-dimensional crve of csp points On the crve of csp points one may find isolated points where the flow has a qartic tangency (ie, of order for) with Σ; sch a point is also known as a swallowtail point Overall, the critical tangency locs divides the section Σ into open regions Neighboring regions (divided by reglar points of ) correspond to different directions of the flow, with respect to the normal to Σ Note that Σ is orientable, so that its nit normal n Σ (x) can be transported nambigosly along Σ by moving the point x 2 lassification of tangency bifrcations Or interest is in how a smooth invariant manifold, of dimension l composed of a family of orbits of vector field (), interacts with a Poincaré section Σ n order

6 Tangency bifrcations of global Poincaré maps of 4 vector fields 5 to introdce or notion of bifrcation, we first recall from [] the relevant notion of topological eqivalence efinition onsider two open neighborhoods U and U 2 of R n Sppose that Σ U and Σ 2 U 2 are (n)-dimensional smooth sections that divide U and U 2 into two parts and have critical tangency loci and 2, respectively Sppose frther, that there is an l-dimensional invariant manifold U, and an l-dimensional invariant manifold 2 U 2 We say that the flow Φ on U is Σ--topologically eqivalent to the flow Φ 2 on U 2 if there exists a homeomorphism h : U U 2 sch that (E) h maps orbits of Φ in U to orbits of Φ 2 in U 2 and respects the direction of time; (E2) h maps Σ to Σ 2 and to 2 ; (E3) h maps to 2 Note that (E2) and (E3) ensre that h Σ maps Σ to 2 Σ 2 The notion of topological eqivalence in efinition implies notions of strctral stability Namely, a phase portrait (a particlar arrangement of Σ, and ) is strctrally stable if all sfficiently small (and smooth) pertrbations are topologically eqivalent At a bifrcation point the phase portrait is not strctrally stable n this paper we stdy bifrcations that are not bifrcations of the nderlying flow, yet change the topological strctre of the invariant objects of the Poincaré map on the section Σ That is, we consider the case that there is a homeomorphism h that satisfies (E) and (E2), bt not (E3) To be even more specific, we exclde the case that an eqilibrim crosses Σ nstead consider the case of a tangency bifrcation of an l-dimensional smooth invariant manifold of vector field () A tangency bifrcation arises when an orbit O(x ) = {Φ t (x ) t R} (4) is tangent to the given (n )-dimensional Poincaré section Σ at a point x Σ Note that a manifold of dimension one consists of a single orbit However, for l 2 the manifold consists of a smooth (l )-dimensional family of orbits Therefore, to classify the tangency bifrcation we need to take into accont both the order of the tangency of the orbit O(x ) as well as the natre of the tangency between and Σ in the family direction of, which is transversal to the direction of the flow Following [], we se the classification of tangency bifrcations, which is based on the intersection of the tangent spaces T (x ) and T Σ (x ) efinition 2 Let be an l-dimensional invariant manifold of a vector field f on R n with a given (n )-dimensional global section Σ Sppose that the following conditions are satisfied: (B) there is a point x ch that the orbit O(x ) has a tangency of degree d N with Σ at x, where we assme that the tangency is at least qadratic, that is, d 2;

7 Tangency bifrcations of global Poincaré maps of 4 vector fields 6 (a) (b) T (x ) O(x ) O(x ) Σ Σ x Σ T (x ) x Figre Two cases where an orbit O(x ) on a two-dimensional manifold (prple) has a qadratic tangency with a planar section Σ (green) The sitation in panel (a) is strctrally stable, while that in panel (b) is of codimension one; which case occrs is determined by the dimension of the intersection of the tangent space T (x ) of (ble) with the tangent space T Σ (x ) = Σ (B2) The dimension of the orthogonal complement N of f(x ) in T (x ) T Σ (x ) is p; and (B3) The point x is a critical point of codimension q of the restriction ϑ N to N of the local chart ϑ : T (x ) R n of the manifold at x Then we say that and Σ have a d-tangency with singlarity dimension p and singlarity codimension q at x (or d-p-q-tangency or short) To clarify efinition 2, figre shows two examples of tangency bifrcations of vector fields in R 3 n both panels (a) and (b) a two-dimensional invariant manifold (prple) has an orbit O(x ) that has a qadratic tangency with a two-dimensional planar Poincaré section Σ (green) This qadratic tangency takes place at a reglar point x of the critical tangency locs The difference between the two cases is the following n figre (a) the manifold is not tangent to Σ in the family direction, which means that the tangent space T (x ) (ble) of at x intersects T Σ (x ) = Σ in a one-dimensional line, which is the tangent f(x ) to the orbit O(x ) at x Hence, this is a strctrally stable codimension-zero sitation As figre (a) shows, when the section (or the manifold) is moved p or down we still find a niqe point in Σ with the same properties as x f were to be a one-dimensional manifold that only contains the orbit O(x ) then this wold be a 2-- tangency, which is of codimensionone By contrast, in figre (b) the tangent spaces T (x ) and T Σ (x ) coincide, so that their intersection is two-dimensional This means that the manifold has a second direction of tangency, apart from the direction f(x ) of the flow We are dealing with a 2-- tangency according to efinition 2 n particlar, p = becase the manifold

8 Tangency bifrcations of global Poincaré maps of 4 vector fields 7 is tangent to Σ in N, which in trn means that x is indeed a critical point of ϑ N in (B3) Since the tangency with respect to the one-dimensional orthogonal complement N is qadratic, the codimension of this critical point is q = Notice that the sitation in figre (b) is no longer strctrally stable nstead, the 2-- tangency in R 3 is of codimension one t can be nfolded by moving either the section or the manifold p and down The notion of codimension can be formlated for any d-p-q-tangency of an l- dimensional manifold with a Poincaré section of a vector field f on R n We have the following reslt Proposition A d-p-q-tangency of an l-dimensional invariant manifold R n with a global section Σ is of codimension one if l = d + q, where p < l < n This proposition is a special case of the general formla for codimension given in [] Notice that the codimension does not depend on the dimension n of the phase space Therefore, it is most convenient to stdy a given d-p-q-tangency in the smallest possible phase space, which is that of dimension n = l+ n other words, it is possible to list the codimension-one tangency bifrcations by the dimension n of the phase space, where one only needs to consider a manifold of dimension l = n Hence, the qestion is how the two (n )-dimensional smooth manifolds and Σ interact near a tangency point x This gives rise to the following classification of codimension-one tangency bifrcations for n 4 n R 2 for a manifold of dimension l = : (T) the 2---tangency: a qadratic tangency of the orbit that defines ; the nfolding is presented in section 3 n R 3 for a manifold of dimension l = 2: (T2) the 2---tangency: a qadratic tangency of an orbit on and a qadratic tangency with respect to the one-dimensional space N There are two cases, depending on the relative directions of the two qadratic tangencies; the nfoldings are presented in section 32 (T3) 3---tangency for l = 2: a cbic tangency of an orbit on ; the nfolding is presented in section 33 n R 4 for a manifold of dimension l = 3: (T4) the tangency: a qadratic tangency of an orbit on with qadratic tangencies in the two additional tangent directions that span the two-dimensional space N There are two cases, depending on the relative directions of the qadratic tangencies; the nfoldings are presented in section 4 (T5) the 2--2-tangency: a qadratic tangency of an orbit on that makes a cbic tangency with respect to the one-dimensional space N; the nfoldings are presented in section 42

9 Tangency bifrcations of global Poincaré maps of 4 vector fields 8 (T6) the 3---tangency: a cbic tangency of an orbit on that makes a qadratic tangency with respect to the one-dimensional space N There are two cases, depending on the relative directions of the csp and the qadratic tangency; the nfoldings are presented in section 43 (T7) 4---tangency for l = 3: A qartic tangency of an orbit on ; the nfolding is presented in section Normal form setting in standard flowbox A codimension-one tangency bifrcation can be nfolded by changing a single system parameter to alter the position of the manifold Alternatively, one can keep the vector field fixed and change the position of the section While the first option appears to be the more natral one in many applications, changing only the section gives topologically eqivalent nfoldings This second option is sed here as the more convenient choice to define normal forms of the codimension-one tangency bifrcations We consider from now on a one-parameter family of global sections, where the dependence on the parameter s is smooth Each of these sections has a critical tangency locs = (s) that depends on s To help s nderstand the geometry of the flow we introdce another geometrical object: the extended critical locs, defined as = (s) (5) s Since the dependence of on s is smooth, is a smooth codimension-one sbmanifold of the phase space R n Hence, its dimension is n, the same as that of the section and the manifold Generically, the extended critical locs is transverse to Therefore, knowing properties of the flow throgh gives new geometric insight We introdce the critical tangency locs on, which is given by := {x f(x) n (x) = }, (6) where n (x) is the nit normal to at the point x As with on Σ the tangency locs in generically consists of codimension-one sbmanifolds that divide into regions with opposite directions of the flow relative to the normal bndle of Becase a tangency bifrcation does not involve eqilibria in Σ, it is possible to consider the phase portrait in a flowbox near the point x at which the tangency takes place This flowbox is given by an in-set and an ot-set transverse to the flow, which are connected by orbit segments According to the Flowbox Theorem [3] any flowbox can be mapped diffeomorphically to a standard flowbox in R n, which we define here as { =, (7) v =, where [, ] and v [, ] n t follows that the in-set and the ot-set of the standard flowbox are = { = } and O = { = +} (8)

10 Tangency bifrcations of global Poincaré maps of 4 vector fields 9 To realise a d-p-q-tangency bifrcation in the standard flowbox given by (7), one needs to provide formlae for the family of sections and the manifold ore specifically, for any s the section is a hypersrface whose geometry is determined by the order d of the tangency of the orbit O(x ) The manifold in the flowbox is also a hypersrface, bt its geometry determines the natre of the tangency in the other directions (with respect to the space N of efinition 2), as encoded by p and q Or approach of constrcting normal forms is in the spirit of singlarity theory [5], which sggests how the respective srfaces shold be chosen However, one also has to deal with the flow in the flowbox The sitation is very similar to the one encontered when one considers flows on phase spaces with bondaries [8, 2, 6] or when stdying piecewise-smooth vector fields [2] As explained in [], transformations can be constrcted (while observing the restrictions imposed by the presence of the flow) that allow one to bring a given tangency bifrcation into its normal form in the standard flowbox; see also [8, ] n or approach the family is a standard singlarity srface given by the tangency-order d that moves in the v n -direction with the parameter s For the list in section 2 we need to consider the three cases for d = 2, 3 and 4, which give rise to fold, csp and swallowtail srfaces, respectively; compare, for example, [7, section 94] From the definition of one immediately obtains formlae for the critical tangent locs, the extended critical locs and its critical tangency locs Or family of (qadratic) fold srfaces in R n for n 2 is given by = {(,,,v n ) R n v n 2 s = }, (9) (s) = {(,,,v n ) = and v n = s}, () = {(,,,v n ) R n = }, () = Or family of (cbic) csp srfaces in R n for n 3 is given by (2) = {(,,,v n ) R n 3 + v n s = }, (3) (s) = {(,,,v n ) = 3 2, v n = s 2 3 }, (4) = {(,,,v n ) R 4 = 3 2 }, (5) = {(,,,v n ) =, = } (6) Or family of (qartic) swallowtail srfaces in R n for n 4 is given by = {(,,,v n ) R n v n s = }, (7) (s) = {(,,,v n ) = 2 4 3, v n = s}, (8) = {(,,,v n ) R n = }, (9) = {(,,,v n ) = 6 2, = 8 3 } (2) Now that families have been determined for the different cases of tangency, the task is to define manifolds that represent the different tangencies in the additional

11 Tangency bifrcations of global Poincaré maps of 4 vector fields directions Note that we have chosen to define the section sing v n and the lowest v i coordinates This means that an increase in the dimension only changes to the extent that a new coordinate is added As a reslt, the manifolds that are defined in are also defined in terms of v n and the lowest v i -coordinates with all other v i being set to zero mportantly the geometry of the manifolds on the in-set of the standard flowbox determines the invariant set Σ in the Poincaré section, for any given tangency bifrcation n the next section we present the tangency bifrcations (T) (T3) for n = 2 and n = 3 from [] from this point of view Section 4 then presents the tangency bifrcations (T4) (T7) for n = 4 Or images of nfoldings in the next sections are necessarily increasing in complication To help their interpretation we se a single color code throghot for the different objects (and their projections): the manifold is prple, the section is green, the critical tangency locs and the extended critical tangency locs are grey, and the critical tangency locs is white The projection of onto the in-set divides into regions Orbits originating in one sch region have the same nmber of intersections with before exiting the flowbox n or figres each region of whose orbits intersect at least once is shown in green; frthermore, the nmber of intersections is indicated by circled nmbers ➀ ➃ A region of whose orbits do not intersect at all is white Frthermore, we indicate the direction of the flow on Σ \ and on \ as follows The normal n Σ () to Σ at the tangency point x = always lies in the v n -direction, and we choose n Σ () sch that it points in the direction of positive v n Frthermore, we choose the normal n () to sch that n Σ (), n () and f() form a right-handed coordinate system The direction of the normals n Σ (x) and n (x) at some general point x of or, respectively, is then obtained by the condition that it agrees with that of n Σ () and n () when it is moved to along the srface (which is orientable) n all dimensions and for all degrees of tangency, we denote a region of \ with the symbol when the flow is sch that f(x) n Σ (x) >, and with the symbol otherwise Similarly, regions of \ are denoted with the symbols and, depending on whether f(x) n (x) > or not The direction of the flow is readily determined for the three different cases of sections from formlae (9) (7) 3 Tangency bifrcations in dimensions one and two The tangency bifrcations (T) (T3) for n = 2 and n = 3 from section 2 were the topic of [] They are presented here for completeness, bt also to help with the interpretation of the tangency bifrcations for n = 4 in the next section oreover, the presentation differs slightly from that in [] in that the emphasis is now on how the projections of the section and the manifold in the in-set of the standard flowbox determine the invariant set Σ =

12 Tangency bifrcations of global Poincaré maps of 4 vector fields (a) O (b) = m, O (c) m m O Figre 2 The nfolding of the 2-- tangency bifrcation of a one-dimensional invariant manifold (prple) with a global section (green) in the standard flowbox for n = 2 Also shown are the projections onto both the in-set (left bondary of the panels) and the section (bottom of the panels); from (a) (c), s = 5, s =, and s = 5 3 The 2-- tangency bifrcation in R 2 The first codimension-one tangency bifrcation is a qadratic tangency of a single orbit segment on a one-dimensional manifold As is discssed in [], the classic case is that the piece of manifold in the flowbox is part of a periodic orbit t has been observed that this qadratic tangency may give rise to additional branches in a bifrcation diagram when a periodic orbit changes shape with parameter variation [6, 8] Similarly, when maxima of a time series are recorded, a new maximm arises at an inflection point of the time series, that is, at a tangency bifrcation of the derivative of the time series [2] The 2-- tangency bifrcation of a one-dimensional manifold is of codimension one for any n Proposition 2 A 2-- tangency of a one-dimensional manifold in R n is given in the standard flowbox (7) by the family (9) of qadratic sections with the manifold = {(,,,v n ) R n v i = for i =,,n } (2) The normal form is given by n = 2 The nfolding of the 2-- tangency is illstrated in figre 2 with three panels of the standard flowbox for n = 2 before, at and after the bifrcation The critical tangency locs on the qadratic section is at = As s is changed, moves down and throgh the fixed manifold For s = the two crves and are tangent

13 Tangency bifrcations of global Poincaré maps of 4 vector fields 2 As we explain now, this nfolding can be nderstood by observing only the sitations in the in-set and in the section The in-set can be identified with the -space, and it is shown on the left bondaries of the flowbox panels in figre 2 The projection of onto is shown as the green segment where each orbit intersects twice t is bonded by the projection (grey dot) of the critical tangency locs onto, which is the point = s The manifold is niqely determined by, which is the point = (prple dot) When s is changed, the projection of onto changes becase moves relative to As a reslt, the invariant set Σ = changes within Since is a fnction over the variable, it can be identified with the -axis as illstrated at the bottom bondaries of the flowbox panels in figre 2 To the left of the flow points pward throgh the section and to the right of it points downward This is denoted with the symbols and respectively This choice of symbol is more intitive in the one-dimensional context than the symbols and Figre 2 illstrates that the sitation in indeed determines the sitation in Specifically, in panel (a) s > so that is below and not in the green region of, meaning that does not intersect and the set Σ is empty Panel (b) shows the moment of bifrcation when s = so that = ; hence, is tangent to and the set Σ consists of a single point (prple dot) Finally, in panel (c) s < so that is above and, hence, in the green region of two intersections; as a reslt, Σ consists of two points (prple dots) 32 The 2-- tangency bifrcations in R 3 The 2-- tangency bifrcation is the first example of a qadratic tangency bifrcation that occrs in vector fields with phase space of dimension at least three Proposition 3 A 2-- tangency bifrcation of a two-dimensional manifold in R n is given in the standard flowbox (7) by the family (9) of qadratic sections with the manifold = {(,,,v n ) R n v n = ± ; v i = for all other i } (22) The mins-sign in (22) is referred to as the minimax case and the pls-sign as the saddle case of the 2-- tangency bifrcation The normal form is given by n = 3 As with the normal form of the 2-- tangency bifrcation for n = 2, we can describe the nfolding of the 2-- tangency bifrcation for n = 3 by considering only the in-set and the section, which can again be identified (by projection) with the (, )-plane Figres 3 and 4 show the minimax case and the saddle case, respectively, on the level of the in-set (left colmn) and the section (right colmn), rather than the entire flowbox The three rows show the phase portraits before, at and after the bifrcation Note that we consider in Proposition 3 a crved manifold on the in-set to represent the tangency in the N-direction perpendiclar to the flow This representation differs from, bt is eqivalent to that of figres 9 and in [], where is represented as a plane Specifically, the in-set is now given by the (, )-plane, and both cases are

14 Tangency bifrcations of global Poincaré maps of 4 vector fields 3 5 (a) 5 (a2) (b) (b2) 5 5 Σ (c) 5 (c2) Σ Figre 3 Unfolding of the minimax case of the 2-- tangency bifrcation of a two-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn) and the section (right colmn); from (a) (c), s = 5, s =, and s = 5 characterised by a qadratic tangency between and in As the panels in the left colmns of figres 3 and 4 show, the difference between the two minimax and saddle cases lies in the direction of the crvatre of in relation to and the projection of Both figres again illstrate that the sitation in determines the sitation in For the minimax case in figres 3, crves away from and only enters the projection of for s This means that the set Σ is created (or destroyed) in the form of a topological circle n row (a), s > so does not intersect or the projection of in, which is why Σ is empty ecreasing s to the bifrcation point for s = in row (b), becomes tangent to at a single point, meaning that Σ consists of a single point (prple dot) Finally, in row (c), s < so enters the projection of in and, hence, the set Σ is a circle For the saddle case in figres 4, crves towards and, therefore, enters the

15 Tangency bifrcations of global Poincaré maps of 4 vector fields 4 5 (a) 5 (a2) Σ (b) (b2) Σ (c) 5 5 (c2) 5 Σ Figre 4 Unfolding of the saddle case of the 2-- tangency bifrcation of a two-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn) and the section (right colmn); from (a) (c), s = 5, s =, and s = 5 projection of for all s Hence, Σ for all s since intersects the projection of on for all s The intersection of with in row (a) consists of two arcs that both cross, each corresponding to one of two parts of inside the projection of onto At the moment of bifrcation in row (b), is totally contained in the projection of bt is tangent to This means that the two arcs of Σ meet on to form a cross For s < in row (c), is still totally contained in the projection of bt no longer intersects Therefore, Σ again consists of two arcs bt they do not intersect 33 The 3-- tangency bifrcation in R 3 The 3-- tangency bifrcation is the first tangency bifrcation that involves a cbic tangency of an orbit on the manifold

16 Tangency bifrcations of global Poincaré maps of 4 vector fields (a) (b) (c) ➀ ➂ ➀ ➀ ➂ 5 5 ➂ Σ Σ (a2) (b2) (c2) 5 Σ (a3) (b3) (c3) 5 5 Figre 5 Unfolding of the 3-- tangency bifrcation of a two-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn), the section (middle colmn) and the extended critical locs (right colmn); from (a) (c), s = 5, s =, and s = 5 Proposition 4 A 3-- tangency bifrcation of a two-dimensional manifold in R n is given in the standard flowbox (7) by the family (3) of cbic sections with the manifold = {(,,,v n ) R n v n = ; v i =, for all other i } (23) The normal form is given by n = 3 Note that the manifold in Proposition 4 is chosen to be in general position with respect to, which reflects the fact that p = n particlar, cannot be parallel to either the v n - or the v n 2 -direction; see [] As before, we consider in figre 5 the nfolding of this bifrcation by showing the in-set and the section, which can again be identified (by projection) with the (, )-plane For the first time the extended tangency locs is nontrivial: it is the parabolic cylinder given by (5) that is divided by the crve of (6) into two parts with different flow directions To help with the

17 Tangency bifrcations of global Poincaré maps of 4 vector fields 6 nderstanding of the 3-- tangency bifrcation we also show in figre 5 the sitation in, which can be identified (by projection) with the (, )-plane The 3-- tangency bifrcation for n = 3 is shown in figre 5 before, at and after the bifrcation The critical tangency locs divides into two regions of different directions of the flow e to the cbic shape of, the in-set is divided by the projection into two regions ➀ and ➂ where orbits intersect once or three times, respectively Notice the csp in the crve As s changes, and the region of three intersections with move and interact with the manifold Figre 5 illstrates that the sitation in determines not only the sitation in Σ, bt also that in n row (a) for s >, does not intersect in panel (a), meaning that all orbits starting on only intersect Σ once within the flowbox ndeed Σ does not intersect in panel (a2) and the intersection of = Σ in is above in panel (a3) At the moment of bifrcation for s = in row (b), intersects exactly once at the csp point; see panel (b) As a reslt, becomes tangent to in panel (b2), which can also be seen in in panel (b3) Notice that the cbic tangency of and corresponds to a qadratic tangency of Σ and in Σ, and of and in ndeed, the cbic tangency can only take place at the intersection with and For s <, as in row (c), enters the region of where orbits intersect the section three times within the flowbox Specifically, the segment of Σ with > to the left of is mapped to inside the parabola, which in trn is mapped to the right of before exiting the flowbox 4 Tangency bifrcation in dimension for We now present all codimension-one tangency bifrcations that appear for the first time in vector fields of dimension at least for; these are cases (T4) (T7) in the list in section 2 According to Proposition, they involve the interaction of a three-dimensional manifold with a three-dimensional section Σ inside the standard flowbox of dimension for Since it is practically impossible to visalise this in for dimensions we present the respective nfoldings on the level of the in-set, the section and the extended critical tangency locs, as was done in section 3 Now all images show spaces of dimension three 4 The tangency bifrcation in R 4 As or first tangency bifrcation in for dimensions we consider a qadratic tangency of the flow where the tangent spaces of and coincide Hence, there are two frther directions along which the manifold and the section have a qadratic tangency, so that is given by a qadratic srface in the in-set As was the case with the 2-- tangency bifrcation of a two-dimensional manifold, there are two different cases depending on the relative directions of the corresponding qadratic crves: either they are all on the same side of, or one of them is on one side and the other two are on the side

18 Tangency bifrcations of global Poincaré maps of 4 vector fields 7 (a) (a2) (b) (b2) Σ (c) (c2) Σ Figre 6 Unfolding of the minimax case of the tangency bifrcation of a three-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn) and the section (right colmn); from (a) (c), s = 5, s =, and s = 5 Proposition 5 A tangency bifrcation of a three-dimensional manifold in R n is given in the standard flowbox (7) by the family (9) of qadratic sections with the manifold = {(,,,v n ) R n v n = ± 2 ; v i = for all other i } (24) The mins-sign in (24) is referred to as the minimax case and the pls-sign as the saddle case of the tangency bifrcation The normal form is given by n = 4

19 Tangency bifrcations of global Poincaré maps of 4 vector fields 8 (a) (a2) Σ (b) (b2) Σ (c) (c2) Σ Figre 7 Unfolding of the saddle case of the tangency bifrcation of a three-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn) and the section (right colmn); from (a) (c), s = 5, s =, and s = 5 The two cases of tangency bifrcation are shown in figres 6 and 7 before, at and after the bifrcation in terms of the three-dimensional in-set and the threedimensional section For any vale of the parameter s, the section is divided by the critical tangency locs (the grey plane given by = ) into two parts with different directions of the flow as indicated The set (grey) divides into two regions: in the green region onto which projects, each orbit has two intersections with, and in the white region none The difference between the two cases arises from the fact that

20 Tangency bifrcations of global Poincaré maps of 4 vector fields 9 the two-dimensional intersection of the manifold (prple) in the in-set is either a maximm or a saddle srface For the minimax case in figre 6 the manifold in the in-set is a paraboloid with a maximm (or a minimm) There are no intersections between and for positive vales of s, so that Σ = ; see row (a) ecreasing s to the moment of bifrcation at s = in row (b), and become tangent at a single point of ; hence, Σ now consists of a single point (prple dot) in (b2) As s is decreased frther, Σ expands into a topological sphere; see row (c) This sphere is bisected by, and points to the left of are mapped to points to the right of Figre 6 shows that the minimax case of the tangency bifrcation describes how a compact piece of invariant manifold emerges in or disappears from a three-dimensional Poincaré section For the saddle case in figre 7 the manifold in the in-set is a saddle srface, which intersects for any vale of s For positive s the saddle point of in lies below, which means that Σ is composed of two domed srfaces, known as a two-sheeted circlar hyperboloid Each dome is bisected by and the points on Σ to the left of retrn on the same dome to the right of ; see (a2) As s decreases the two domes become more pointed ntil they form a cone at the bifrcation point for s = ; see (b2) The bifrcation arises becase the saddle point of is now contained in ; see (b) When s is decreased frther, as in row (c), the saddle point of moves into the (green) region of where points with two intersections with originate Hence, the cone opens p into a one-sheeted hyperboloid; see (c3) Figre 7 shows that the saddle case of the tangency bifrcation describes how two pieces of invariant manifold in a three-dimensional Poincaré section come together and merge into a single piece 42 The 2--2 tangency bifrcation in R 4 We now discss the sitation where there is a qadratic tangency in the direction of the flow, bt the intersection of the tangent spaces of and Σ is of dimension two, rather than three as in section 4 As there is a single additional direction of tangency between and Σ, this tangency is necessarily cbic if the bifrcation is to be of codimension one, meaning that is given by a csp srface in the in-set Proposition 6 A 2--2 tangency bifrcation of a three-dimensional manifold in R n is given in the standard flowbox (7) by the family (9) of qadratic sections with the manifold = {(,,,v n ) R n v n = ; v i = for all other i } (25) The normal form is given by n = 4 The 2--2 tangency bifrcation is shown in figre 8 before, at and after the bifrcation in terms of the three-dimensional in-set and the three-dimensional section As before, is divided by into two parts with different directions of the flow as indicated, while (grey) divides into two regions with either two or zero intersections with The manifold is a csp srface (prple) that intersects for any vale of

21 Tangency bifrcations of global Poincaré maps of 4 vector fields 2 (a) (a2) Σ (b) (b2) Σ (c) (c2) Σ Figre 8 Unfolding of the 2--2 tangency bifrcation of a three-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn) and the section (right colmn); from (a) (c), s =, s =, and s = the parameter s For positive s, as in row (a) of figre 8, the csp point lies in the region of below with no intersections, and Σ consists of two disjoint parts At the bifrcation for s = in row (b) the csp point of lies in, so that the two parts of Σ connect at a single point When s is decreased the csp point of lies in the (green) region of with two intersections with, and Σ is now a single connected srface

22 Tangency bifrcations of global Poincaré maps of 4 vector fields 2 Figre 8 shows that the 2--2 tangency bifrcation provides a different mechanism of how two pieces of invariant manifold in a three-dimensional Poincaré section may merge into a single piece ore specifically, the merging of two pieces of invariant manifold in the Poincaré section does not reqire that the two tangent spaces of and coincide as is the case for the 2--2 tangency bifrcation from section 4 The 2--2 tangency bifrcation manifests itself in the critical tangency locs as the nfolding of a pitchfork bifrcation when the normal form symmetry is broken; see, for example, [9] This is again different from the saddle case of the 2--2 tangency bifrcation, which is characterised by a saddle transition in The fact that we obtain a pitchfork bifrcation here is de to the invariance of the standard csp srface in (25) nder the transformation (,, ) (,, ), which is a rotation abot the -axis over π When the manifold is deformed away from the normal form the symmetry in the pitchfork may be lost 43 The 3-- tangency bifrcations in R 4 We now consider a tangency bifrcation of a three-dimensional manifold with an orbit that has a cbic tangency with the section Σ To obtain a bifrcation of codimension one, there is a single additional direction along which has a qadratic tangency with Σ There are two cases that depend on whether the qadratic tangency is away from or towards the critical tangency locs Proposition 7 A 3-- tangency bifrcation of a three-dimensional manifold in R n is given in the standard flowbox (7) by the family (3) of cbic sections with the manifold = {(,,,v n ) R n v n = ± 2 ; v i = for all other i } (26) The mins-sign in (24) is referred to as the minimax case and the pls-sign as the saddle case of the 3-- tangency bifrcation The normal form is given by n = 4 The two cases of 3-- tangency bifrcation are shown in figres 9 and before, at and after the bifrcation in terms of the three-dimensional in-set, the three-dimensional section, and the three-dimensional extended tangency locs For any vale of the parameter s the section is divided by the critical tangency locs (the grey parabolic cylinder) into two part with different directions of the flow as indicated The extended tangency locs is divided by its critical tangency locs (the grey plane given by = ) into two part with different directions of the flow as indicated The intersection = Σ is the cbic green srface given by v n = 3 s The set (grey) consists of two sheets that meet along a crve of csps (white line) t divides into two regions, where each orbit has one or three intersections with, marked ➀ and ➂ respectively The difference between the two cases arises from the fact that the manifold in either intersects region ➂ for all s or there are vales of s for which does not intersect this region

23 Tangency bifrcations of global Poincaré maps of 4 vector fields 22 (a) ➂ (a2) Σ ➀ (b) (b2) Σ ➀ ➂ (c) (c2) Σ ➀ ➂ Figre 9 Unfolding of the minimax case of the 3-- tangency bifrcation of a three-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn) and the section (right colmn); from (a) (c), s = 5, s =, and s = 5 For the minimax case in figre 9 the manifold lies entirely in the region of one intersection with when s is positive; see row (a) At the bifrcation point for s = in row (b), toches the csp line on in, meaning that Σ becomes tangent to at a point of its tangency locs ; see (b2) When s is decreased, as in row (c), intersects in a closed crve with two csp points, so that a part of lies now in region ➂; see (c) As a conseqence, Σ intersects in a closed crve; see (c2) and (c3) Figre 9 shows that the minimax case of the 3-- tangency bifrcation describes

24 Tangency bifrcations of global Poincaré maps of 4 vector fields 23 (a3) (b3) (c3) Figre 9 (ontined) Unfolding of the minimax case of the 3-- tangency bifrcation; shown is the extended critical locs ; from (a) (c), s = 5, s =, and s = 5 how an invariant manifold of a Poincaré map in a three-dimensional section may develop (or lose) a patch that has two images inside the flowbox nder the Poincaré map Note that this transition manifests itself as a minimax transition between the two srfaces Σ and in For the saddle case in figre the manifold always has a part in region ➂ For positive s there are two intersection crves of with, which both cross the csp line of ; see (a) As a reslt, there is a part of Σ, in between the two crves Σ, that does not get mapped to another part of Σ nder the Poincaré map; see (a2) and (a3)

25 Tangency bifrcations of global Poincaré maps of 4 vector fields 24 (a) ➂ (a2) Σ (b) ➀ (b2) ➂ Σ (c) ➀ (c2) Σ ➀ ➂ Figre Unfolding of the saddle case of the 3-- tangency bifrcation of a three-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn) and the section (right colmn); from (a) (c), s = 5, s =, and s = 5 At the bifrcation for s =, as in row (b), the manifold has a tangency with the csp line of, which means that the manifold Σ becomes tangent to at a point of ; see (b2) and (b3) After the bifrcation for negative s, as in row (c), there are again two intersection crves of with, bt they now do not cross the csp line instead they lie entirely in one of the sheets of ; see (c) Therefore, in the critical tangency locs divides the manifold into three pieces: the left-most piece (for negative ) is mapped to the middle piece (arond = ), which is mapped to the right-most piece

26 Tangency bifrcations of global Poincaré maps of 4 vector fields 25 (a3) (b3) (c3) Figre (ontined) Unfolding of the saddle case of the 3-- tangency bifrcation; shown is the extended critical locs ; from (a) (c), s = 5, s =, and s = 5 (for positive ) Figre shows that the saddle case of the 3-- tangency bifrcation describes how an invariant manifold of a Poincaré map in a three-dimensional section may develop (or lose) a patch that has no images inside the flowbox nder the Poincaré map Note that this transition manifests itself as a saddle transition between the two srfaces Σ and in

27 Tangency bifrcations of global Poincaré maps of 4 vector fields The 4-- tangency bifrcation in R 4 The 4-- tangency bifrcation is the first tangency bifrcation with a qartic tangency of an orbit on the manifold Sch a tangency point may occr generically in R 4 at an isolated point The section Σ in the flowbox is a swallowtail srface; compare with [] For this bifrcation to be of codimension one, we mst consider the interaction with a three-dimensional manifold in general position, that is, the tangent spaces of and Σ only intersect in the direction given by the flow at the tangency point Proposition 8 A 4-- tangency bifrcation of a three-dimensional manifold in R n is given in the standard flowbox (7) by the family (7) of qartic sections with the manifold = {(,,,v n ) R n v n = ; v i = for all other i } (27) The normal form is given by n = 4 The 4-- tangency bifrcation is shown in figres before, at and after the bifrcation in terms of the three-dimensional in-set, the three-dimensional section, the three-dimensional extended tangency locs, and the (, ) plane The latter represents the two-dimensional critical tangency locs, which is a parabolic cylinder The set (grey) in consist of three sheets that meet along two csp crves (white), which in trn meet at a swallowtail point This srface divides into three regions, where each orbit has zero, two or for intersections with, respectively For any vale of the parameter s the section is divided into two by the critical tangency locs (the grey csp srface) with different directions of flow in each part as indicated The flow is tangent to along a fold crve which has a csp point at = Similarly, the extended tangency locs is divided by its critical tangency locs (the grey parabolic cylinder) into two parts with different directions of the flow as indicated The intersection = Σ in is the qartic green srface given by v n = 4 s n figre the manifold always intersects the srface For positive s, as in row (a), intersects the srface well away from the swallowtail point, so that Σ intersects well way from the csp point on Hence, divides Σ into two pieces; the piece for smaller is mapped nder the Poincaré map to the piece for larger Looking at the extended tangency locs and its tangency locs this means that and do not intersect At the bifrcation point the plane goes exactly throgh the swallowtail point; see (b) This means that the manifold Σ is tangent to the fold crve at the csp point; see (b2) This corresponds to a qadratic tangency of with in ; see (b3) and (b4) After the bifrcation for negative s, as in row (c), the plane intersects all three regions of Therefore, in the section the set Σ crosses the fold crve on twice; see (c2) This also means that the crve in now intersects the crve = Σ at two points; see (c3) and (c4) Figre shows that the 4-- tangency bifrcation describes how an invariant manifold that is mapped to itself once nder the Poincaré map in a three-dimensional section may develop (or lose) a patch that has two images inside the flowbox nder the Poincaré map Note that

28 Tangency bifrcations of global Poincaré maps of 4 vector fields 27 (a) (a2) ➃ Σ Σ (b) (b2) ➃ Σ Σ (c) (c2) ➃ Σ Σ Figre Unfolding of the 4-- tangency bifrcation of a three-dimensional invariant manifold (prple) with a global section (green); shown are the in-set (left colmn) and the section (right colmn); from (a) (c), s = 5, s =, and s = 5 this transition manifests itself as a csp bifrcation between the two srfaces Σ and in, which in trn is a qadratic tangency between and in the two-dimensional critical tangency locs of

29 Tangency bifrcations of global Poincaré maps of 4 vector fields 28 (a3) (a4) (b3) (b4) (c3) (c4) Figre (ontined) Unfolding of the 4-- tangency bifrcation of a three-dimensional invariant manifold (prple) with a global section (green); shown are the extended critical locs (left colmn) and the intersection of Σ with = Σ in the two-dimensional space as represented by the (, )-plane (right colmn); from (a) (c), s = 5, s =, and s = 5 5 iscssion and otlook We considered tangency bifrcations between an invariant manifold and a Poincaré section in vector fields of dimensions p to for This type of bifrcation of the associated Poincaré map occrs becase the flow becomes tangent to the section The normal forms for all codimension-one tangency bifrcations were presented within a standard flowbox by a one-parameter family of sections and a fixed invariant manifold Each