UNFOLDING THE CUSP-CUSP BIFURCATION OF PLANAR ENDOMORPHISMS

UNFOLDING THE CUSP-CUSP BIFURCATION OF PLANAR ENDOMORPHISMS BERND KRAUSKOPF, HINKE M OSINGA, AND BRUCE B PECKHAM Abstract In man applications of practical interest, for eample, in control theor, economics, electronics and neural networks, the dnamics of the sstem under consideration can be modelled b an endomorphism, which is a discrete smooth map that does not have a uniquel defined inverse; one also speaks simpl of a noninvertible map In contrast to the better known case of a dnamical sstem given b a planar diffeomorphism, man questions concerning the possible dnamics and bifurcations of planar endomorphisms remain open In this paper we make a contribution to the bifurcation theor of planar endomorphisms Namel we present the unfoldings of a codimension-two bifurcation, which we call the cusp-cusp bifurcation, that occurs genericall in families of endomorphisms of the plane The cusp-cusp bifurcation acts as an organising center that involves the relevant codimension-one bifurcations The central singularit involves an interaction of two different tpes of cusps Firstl, an endomorphism tpicall folds the phase space along curves J where the Jacobian of the map is ero The image J 1 of J ma contain a cusp point, which persists under perturbation; the literature also speaks of a map of tpe Z 3 < Z 1 The second tpe of cusp occurs when a forward invariant curve W, such as a segment of an unstable manifold, crosses J in a direction tangent to the ero eigenvector Then the image of W will tpicall contain a cusp This situation is of codimension one and genericall leads to a loop in the unfolding The central singularit that defines the cusp-cusp bifurcation is, hence, defined b a tangenc of an invariant curve W with J at the pre-image of the cusp point on J 1 We stud the bifurcations in the images of J and the curve W in a neighborhood of the parameter space of the organiing center where both images have a cusp at the same point in the phase space To this end, we define a suitable notion of equivalence that distinguishes between the different possible local phase portraits of the invariant curve relative to the cusp on J 1 Our approach makes use of local singularit theor to derive and anale completel a normal form of the cusp-cusp bifurcation In total we find eight different two-parameter unfoldings of the central singularit We illustrate how our results can be applied b showing the eistence of a cusp-cusp bifurcation point in an adaptive control sstem We are able to identif the associated two-parameter unfolding for this eample and provide all different phase portraits Ke words Discrete-time sstem, noninvertible planar map, invariant curve, unstable manifold, codimension-two bifurcation AMS subject classifications 37D1, 37M2, 65P3 1 Introduction Man situations of practical interest are modelled mathematicall b a map on a suitable phase space In this case time is thought to be discrete and the time evolution of an initial point is given b the iterates of the map We consider here the case that the dnamics is generated b an endomorphism, that is b a smooth map that does not have a (uniquel defined) inverse As is common in the literature, we simpl speak of a noninvertible map Note that the theor of endomorphisms of the real line is well developed, with the logistic map being the most famous eample However, much less is known about the possible dnamics and bifurcations of endomorphisms on R n for n 2 We are concerned here with the case of a noninvertible planar map that maps R 2 to itself Such maps arise naturall as models in several areas of application, including control theor [1, 8, 9], economics [2, 3], radiophsics [22] and neural networks [33] Noninvertibilit easil occurs in applications that feed back sampled data using too large a sampling time This is Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, Universit of Bristol, Bristol BS8 1TR, UK Department of Mathematics and Statistics, Universit of Minnesota Duluth, Duluth, MN 55812, USA 1

2 B Krauskopf, H M Osinga and B B Peckham effectivel equivalent to the case that a vector field model is integrated with a too large integration step [21] In a region where a planar endomorphism has a well-defined branch of an inverse it ma displa all the dnamical compleit of a planar diffeomorphism, that is, of a smooth planar map with a smooth inverse As is now well known, planar diffeomorphisms tpicall show complicated dnamics, including chaos; see, for eample, [13, 29] as entr points to the etensive literature Famous eamples of planar diffeomorphisms are the Hénon map [16] and the Ikeda map [17], as well as Poincaré maps of periodicall driven sstems, such as the forced pendulum, Van der Pol and Duffing oscillators [13, 37] The main question in the stud of planar endomorphisms is what etra dnamical features ma occur beond what is known for planar diffeomorphisms While there have been quite a number of studies of specific planar endomorphisms, much less is known about the generic dnamics and bifurcations of planar endomorphisms that must be epected in a tpical eample In short, there is, as et, no sstematic bifurcation theor for planar endomorphisms To be specific, consider a famil of endomorphisms of the plane (11) f : R 2 R m R 2 (, λ) f(, λ) where λ R m is an m-dimensional parameter, and f is a smooth map We consider the case that f is not a diffeomorphism, which means that the Jacobian Df of f has a nonero kernel We define the (non-empt) singular locus (12) J := ker(df ) = { R 2 Df () is singular } The image J 1 := f(j ) is called the critical locus In the literature the critical locus J 1 is also referred to as critical curve [28] or Ligne Critique (LC) [25], while J is also referred to as the curve of merging pre-images [28] or LC 1 [25] Technicall, LC J 1 and LC 1 J, but for this paper it is sufficient to consider them equal This is justified b their following properties, which follow immediatel from the implicit function theorem Proposition 11 (Generic properties of J and J 1 ) Genericall, that is, for a generic famil f and parameter λ in general position, the singular locus J is a smooth curve where dim(ker(df )) = 1 The eigenvector of the eigenvalue is transverse to J ecept at isolated points, called pre-cusp points, where it is tangent to J Therefore, the critical locus J 1 consists of smooth curve segments that meet at isolated cusp points, which are the images of the pre-cusp points Generall speaking, noninvertibilit gives rise to regions with different numbers of inverses The critical locus J 1 divides the phase plane R 2 into regions with a constant number of pre-images These regions are usuall labeled b Z k, where k is the number of pre-images in that region [1, 25] Genericall, that is, ecept at the cusp points, the map f folds the phase plane along a smooth curve in J, which is mapped to J 1 Therefore, the number of pre-images differs locall b two on either side of J 1 [4] As one moves from one region into the net b crossing a fold curve the number of pre-images changes to k ± 2 In the interior of a region Z k for k 1 one can select a single branch of the inverse, so that the map f is locall a diffeomorphism New phenomena, which do

The cusp-cusp bifurcation of planar endomorphisms 3 not occur for diffeomorphisms, ma arise when a (forward) invariant object, such as a fied point, periodic orbit or invariant manifold, interacts with the boundar of the region of definition of the inverse branch Therefore, the bifurcation theor of endomorphisms can be thought of as the stud of the interaction of dnamical objects and their images with J and J 1, respectivel The literature on noninvertible planar maps consists largel of case studies in specific eamples that reveal specific phenomena and a number of codimension-one bifurcations A lot of attention has been devoted to the structure of basins of attraction Because a basin ma consist of disconnected regions or be multipl connected, it is often referred to as a sea with possibl islands in it, or as land with lakes [18, 25]; connectivit properties are characterised with an island number or lake number [25] However, as was shown in [6, 7], changes to the basin boundar are due to two tpes of tangencies, called an inner and an outer tangenc, of a stable set (the generaliation of the stable manifold of a saddle point) with J 1 Another topic that received a lot of interest as tpical for noninvertible maps are self-intersections and associated loops of (forward) invariant sets; see [1, 11, 23, 26] and [21], where loops are referred to as antennae The main interest is in bifurcations leading to the destruction of an invariant curve (also called IC or torus ), which is the closure of the unstable manifolds of a suitable periodic orbit if the dnamics on the curve is phase locked To define an unstable manifold of an endomorphism consider a generic saddle point p of f (or a suitable iterate of f) Genericit of p means in particular that p J Therefore, there eists the local unstable manifold Wloc u (p) associated with the unique inverse branch that fies p The global unstable manifold W u ( ) can then be defined as (13) W u (p) = n=1 f n (W u loc (p)) Note that forward images under f are unique, so that W u (p) is indeed forward invariant Furthermore, W u (p) is genericall an immersed manifold; see, for eample, [36] Even though W u (p) ma have structurall stable transverse self-intersections, it is justified to speak of W u (p) as the global unstable manifold In particular, W u (p) can be computed numericall b an algorithm that works for diffeomorphisms; see also [6, 7] Indeed, depending on how an unstable manifold or other invariant curve crosses J, a loop in its image ma be the result The codimension-one bifurcation that creates a small loop was analsed in [1, 11, 23]; we refer to it as the loop-creation bifurcation and it is discussed in more detail in Sec 2 Due to forward invariance of W u (p), tpicall an infinite number of loops is created in the loop-creation bifurcation, which ma give rise to a rather spectacular loss of smoothness of an invariant curve In combination with other mechanisms (as known for diffeomorphisms) of the break-up of tori, one finds phenomena such as the appearance of loops on an unstable manifold, and the reappearance of an attractor, this time chaotic with loops [11, p 17]; this tpe of attractor has been called a weakl chaotic ring [1, 26] In this paper we make a contribution to the bifurcation theor of planar endomorphisms b providing the unfolding of a codimension-two bifurcation, which we call the cusp-cusp bifurcation Our stud was motivated b the properties of a well-referenced eample of a noninvertible sstem, namel the discrete-time adaptive control sstem discussed in [1, 8, 9] More specificall, in [1] a loop of the unstable manifold of a saddle point is found that surrounds a cusp point C 1 on J 1 Several other codimensionone bifurcations are found near this situation, including a loop-creation bifurcation

4 B Krauskopf, H M Osinga and B B Peckham and the passing of the invariant curve through the cusp point These bifurcations and the generic phase portraits near them are illustrated in [1] near the cusp point C 1 on J 1 as well as near the pre-cusp point C on J, where f(c ) = C 1 We identif the cusp-cusp bifurcation as the organising center of the observed dnamics in the sense that the known codimension-one bifurcations occur in its unfolding Specificall, at a cusp-cusp bifurcation there is a quadratic tangenc of an invariant curve W with the singular locus J eactl at the pre-cusp point C This situation corresponds to a cusp of f(w ) eactl at the cusp C 1 on J 1, hence, the name of this bifurcation; see alread Fig 31(a) and Fig 63(a) and (b) The situation is of codimension two because one parameter is needed to ensure a tangenc of W with J, while another parameter must be adjusted to ensure that this tangenc occurs at C Recall that the eistence of an isolated cusp point on J 1 is a generic propert, that is, a cusp point is stable under small parameter variations Therefore, we assume that the unfolding parameters of the cusp-cusp bifurcation do not lead to bifurcations of the cusp point (such as the disapearance of the cusp point in a swallowtail bifurcation) We derive a normal form of the cusp-cusp bifurcation b considering the interaction of a parabola (representing W locall) under the action of a normal-form map given b projections via the graph of the function f in R 4 near a generic cusp point This surface can best be viewed in projection onto R 3, where it takes the form of a generic two-dimensional surface with a cusp as known from singularit theor [4]; see also Fig 23 In this setup both the cusp point C 1 and the pre-cusp point C are at the origin and the curve J is a fied parabola with C as its maimum The unfolding or primar parameters in the normal form are the vertical and horiontal positions of the maimum or minimum of the parabola representing W, while its steepness (relative to the curve J ) plas the role of a higher-order or secondar parameter in the normal form Depending on its value we distinguish eight different kinds of cuspcusp bifurcations with associated two-parameter unfoldings All unfoldings (cases i to viii) are presented as two-parameter diagrams with associated representative phase portraits, that is, configurations of W near J and f(w ) near J 1 We also show that the phase portraits found in [1] are organied b a cusp-cusp bifurcation of tpe vii Singularit theor is the main tool in our analsis of the cusp-cusp bifurcation It has proven its use in the bifurcation analsis of invertible maps and vector fields, where classic singularities occur in the product of phase space and parameter space; see [12] and also [5] for a recent eample The question in the present setting is how a noninvertible planar map is folding the phase plane over itself In other words, one needs to stud projections via a surface, namel the smooth image of the domain, in a higher-dimensional space This point of view has been adopted before, for eample, in [38] The action of a noninvertible planar map near an of the fold curves that make up J 1 can be understood b considering projections via a surface in R 3 with a generic fold line [14, 15] Planar quadratic maps with bounded critical sets are considered in [27] However, the use of singularit theor for a sstematic bifurcation analsis of noninvertible maps is quite new The papers [3, 32] stud connections between the comple quadratic famil and more general two-parameter families maps of the plane R 2 In [7] projections via a folded surface were used to identif two tpes of codimension-one tangenc bifurcations of a stable set with J 1 that are responsible for rearrangements of the basins of attractions of a noninvertible map The stud presented here is ver much in the same spirit Note that near folds and cusps, the situations considered here, the surface via which one projects can be embedded in R 3, rather than in R 4, which greatl helps with visualising the local map

The cusp-cusp bifurcation of planar endomorphisms 5 W W W E E E J (a) J (b) J (c) Z Z Z f( ) f(w ) f( ) f( ) J 1 Z 2 J 1 f(w ) Z 2 J 1 f(w ) Z 2 Fig 21 Sketches of the situation before (a), at (b), and after (c) a codimension-one loop creation, where the invariant curve W becomes tangent at to the line field E (green curves) (top row), which results in the creation of a little loop of f(w ) (bottom row) Another important ingredient of our stud is a suitable notion of equivalence that allows one to define codimension-one bifurcations in this contet We consider here equivalence as given b the position of the oriented invariant curve relative to J 1 ; see Definition 34 The paper is organied as follows In Sec 2 we discuss some mathematical background, including the normal form of a smooth map near a cusp point In Sec 3 we introduce the normal-form setting of the cusp-cusp bifurcation, as well as our notion of equivalence of invariant curves relative to J 1 In Sec 4 we discuss the codimensionone bifurcations that occur near a cusp-cusp bifurcation, and in Sec 5 we present all cases of two-parameter unfoldings The adaptive control sstem from [1] is studied in Sec 6 to show how our results manifest themselves in practice We conclude in Sec 7, where we also point out directions for future research The numerical methods that we use in Sec 6 are discussed in the Appendi A 2 Mathematical setting and cusp normal-form map In order to understand the folding properties of f we consider the line field E accociated with J, defined b (21) E = {l(e (), ) J }, where l(v, p) is the line through the point p given b the vector v, and e () is the eigenvector of the eigenvalue of Df () As stated in Proposition 11, ever line E := l(e (), ) is transverse to J at generic J Now consider a curve W crossing J transversel at a generic point in the sense of Proposition 11 If W is not tangent to E then the image f(w ) under f is quadraticall tangent to J 1 at the point f( ) This tangenc at f( ) is structurall stable, because it corresponds to the projection of a curve that crosses a fold [4] The

6 B Krauskopf, H M Osinga and B B Peckham (a) (b) C Ĵ C 1 Z 1 J E U J 1 Z 3 V (c) (d) Z 1 C Ĵ C 1 E J J 1 Z 3 Fig 22 The line field E at J and its image f(e) at J 1 (green curves) in neighborhoods U of the pre-cusp point C (a), and V of the cusp point C 1 (b) Panels (c) and (d) show the situation for the normal-form map (22), where J and bj are parabolas and the leaves of E and f(e) are given b { = const} eceptional case is given b the line field E An curve W that crosses J tangent to E has its tangent vector mapped to the ero vector b Df ( ) Then f(w ) will have a cusp at f( ), unless the two branches of W on either side of J happen to map eactl on top of each other (which is not generic) The direction of the cusp, that is, the limiting direction of the induced tangent vector to f(w ) as the cusp point f( ) is apprached, is determined b the curvature of W To illustrate the concept we present in Fig 21 the one-parameter unfolding of the loop creation bifurcation; the top row shows a neighborhood of a transverse intersection point of a forward invariant curve W with J, and the bottom row a neighborhood around f( ) on J 1 Note that both W and f(w ) have a direction, indicated b arrows, which need to be specified to understand the action of the map Genericall, the tangent dw ( ) of W at is transverse to the line E, which means that f(w ) has a quadratic tangenc with J 1 ; see Fig 21(a) and (c) However, when l(dw ( ), ) = E then f(w ) has a cusp at f( ) J 1 ; see Fig 21(b) In the unfolding of this codimension-one situation a small loop of f(w ) is created (or destroed) near f( ), which eplains the name of this bifurcation 21 Characteriation of a cusp of J 1 We now consider the basic setting of this paper, namel, an isolated generic cusp point C 1 on J 1 and, thus, an isolated precusp point C on J The situation is sketched in Fig 22(a) and (b) in neighborhoods U and V around C and C 1, respectivel Note that J 1 has a second pre-image, denoted Ĵ f 1 (J 1 ), in the neighborhood U, which is tangent to J at the pre-cusp point C The three regions below J and in between Ĵ and J get mapped to the single region with three pre-images, which is the region under the cusp point C 1 and

The cusp-cusp bifurcation of planar endomorphisms 7 bounded b J 1 This situation is referred to in the literature as a map of tpe Z 3 < Z 1 We remark that an map of tpe Z k+2 < Z k for k 1 has the same properties locall near the cusp point, but with (k 1) additional sheets of inverses that pla no role in the local unfolding Figure 22 also shows the line field E near J Notice that the line E C is not transverse to J Therefore, the defining propert of the cusp-cusp bifurcation, namel a quadratic tangenc of an invariant curve W with J at C, can be interpreted as defining a degenerate loop bifurcation 22 Cusp normal-form map As is known from singularit theor [4], a generic smooth planar map with a cusp point can locall near the pre-cusp point be brought to the normal form (, ) (u, v) = (a b 3, ) b a smooth change of variables, for an positive nonero constants a and b We make a convenient choice, namel we consider the normal-form map defined as ( ) ( ) ( ) F : = 3 3 (22) The (, )-plane corresponds to a local neighborhood of C and the (, )-plane to a local neighborhood of C 1 Both C and C 1 are located at the origin The eact form of (22) was chosen such that the Jacobian ( ) [ ] 3 DF = 2 3 3 1 is singular along the particularl simple critical curve (23) J := { = 2 } The image of the parabola J is the standard cusp (24) J 1 := { = ±2( ) 3 } A straightforward calculation shows that it has the second pre-image (25) Ĵ := { = 1 4 2 } The line field E does not depend on and consists simpl of the lines l((1, ) T, ) The situation for the normal-form map F is shown in Fig 22(c) and (d) in neighborhoods around C and C 1, respectivel Note that the coordinate change that transforms a generic map f near a cusp point into its normal form F deforms the curves J and Ĵ to parabolas and the curve J 1 to a standard cusp Furthermore, it straightens out the line field E to horiontal lines Figure 23 shows how the action of the map F can be interpreted geometricall as a projection via the cusp surface S in (,, )-space given b 3 3 = Note that S is a graph over the (, )-plane, but not over the (, )-plane An point (, ) in the (, )-plane corresponds to a unique point on S under projection in the -direction It is then mapped to the unique point F (, ) under projection in the -direction Conversel, a point (, ) in the (, )-plane lifts to a single point on S in the region of unique pre-images, to two points on S if (, ) J 1, and to

8 B Krauskopf, H M Osinga and B B Peckham C W J S Ĵ f(w ) C 1 J 1 Fig 23 Visualiation in (,, )-space of how the normal-form map (22) can be interpreted as a projection via the cusp surface S The curve W in the (, )-plane is mapped to the curve F (W ) in the (, )-plane; the eample is phase portrait 31 of Fig 52 three points on S in the region of three pre-images Notice also the (projection of) the curves J and Ĵ on S Figure 23 also shows how a parabola W in the (, )-plane around the pre-cusp point C maps under F The result is a curve F (W ) in the (, )-plane with a selfintersection and two tangencies with J 1, one at either side of the cusp point C 1 This can be understood b considering the projection of the parabola onto the cusp surface S, and then down to the (, )-plane 3 Normal-form setting of the cusp-cusp bifurcation At the cusp-cusp bifurcation there is a tangenc of an invariant curve W with the curve J at a precusp point C As a consequence, f(w ) has a cusp eactl at the cusp point C 1 on J 1 This codimension-two bifurcation can be classified as a global bifurcation, because it involves an invariant curve Nevertheless, we can consider a normal-form setting in the small neighborhoods U and V of C and C 1, respectivel, b replacing f near C with the normal-form map F as given b (22) Then the curves J and Ĵ are the parabolas given b (23) and (25) and the points C and C 1 are at the origin of the (, )- and (, )-planes, respectivel Furthermore, the tangenc of W with J at C is genericall quadratic As a consequence, in an unfolding the intersections of W with J and Ĵ in a sufficientl small neighborhood U are determined entirel b the quadratic nature of the curve W Therefore, from now on we consider the quadratic normal form for W in the (, )-plane given b (31) W := { = γ( a) 2 + b}

The cusp-cusp bifurcation of planar endomorphisms 9 Here the parameters a, b R are the primar unfolding parameters and γ R is a higher-order or secondar parameter that determines the steepness of W relative to the fied parabolas J and Ĵ The invariant curve in the neighborhood V of C 1 is simpl given b F (W ) Note that the maimum (for γ < ) or minimum (for γ > ) of W lies at (, ) = (a, b); the cusp-cusp bifurcation occurs for a = b = The normal-form setting of the cusp-cusp bifurcation is now given as the stud of all possible different configurations of F (W ) relative to J 1 and C 1 as a function of the unfolding parameters a and b, and for fied generic values of the higher-order term γ These different configurations correspond one-to-one to different tpes of intersections of W with J and Ĵ; see alread Definition 34 Throughout this paper we use the convention that the orientation of W is from left to right, that is, in the direction of increasing The orientation of F (W ) is induced b this convention Notice that the map F reverses the orientation of the invariant curve in Fig 23, which is illustrated b the arrows of W and F (W ) We have the following general result Proposition 31 (Orientation of F (W )) For an a and b the image F (W ) of an (oriented) parabola W of the form (31) is oriented from left to right if γ < 1 3 and from right to left for γ > 1 3 ; see also Fig 31(c) (e) Proof The curve (32) R := { = 1 3 2 } in the (, )-plane is mapped under the normal-form map F in a two-to-one fashion to the line segment { = and } in the (, )-plane The region above R is mapped to the (, )-plane in an orientation reverserving wa, that is, positive are mapped to negative B contrast, the region below R is mapped to the (, )-plane in an orientation preserving wa The result follows, since the segments for sufficientl large of a parabola of the form (31) both lie in either one of the two regions 31 The cusp-cusp singularit The nature of the cusp-cusp singularit for (a, b) = (, ) depends on the higher-order term γ of (31) For (a, b) = (, ) and for generic γ the cusp point C 1 is the onl intersection point of F (W ) and J 1 The eception is that F (W ) J 1, which happens when either W = J or W = Ĵ We further distinguish different cases of the cusp-cusp singularit depending on the orientation of F (W ), whether the curve F (W ) is below or above J 1, and whether or not the cusp of F (W ) at C 1 points upwards or downwards Proposition 32 (Cusp-cusp singularit) There are five intervals on the γ-line of generic cusp-cusp singularities for (a, b) = (, ) in (31) The boundar points between these intervals are given b γ = 1, γ = 1 3, γ = 1 4, and γ = Proof According to Proposition 31 the orientation of F (W ) changes when W crosses R, which gives rise to the boundar point γ = 1 3 Furthermore, for W = J or W = Ĵ we have that F (W ) = J 1, which gives rise to the boundar points γ = 1 and γ = 1 4, respectivel For W above Ĵ, that is, for γ > 1 4 the curve F (W ) lies above J 1 It follows immediatel from (22) that the cusp of F (W ) is approached from the direction of negatitive for an γ < and from the direction of positive for γ > The situation for γ = is degenerate in that F (W ) does not have a cusp at all

1 B Krauskopf, H M Osinga and B B Peckham (a) i (b) (f) (c) ii iii iv (g) vi (d) (h) (e) v (i) vii viii Fig 31 Sketches of all generic and non-generic cases of the cusp-cusp singularit; see Proposition 32 The left-hand panels show W (red curve) near the pre-cusp point C, and the right-hand panels shown f(w ) (red curve) in relation to J 1 The curves J and J 1 are black, J b is gre, and R is dashed and double-covers the dashed straight line; the arrows show the direction of parametriation b Shown are γ < 1 (a), γ = 1 (b), 1 < γ < 1 3 (c), γ = 1 3 (d), 1 3 < γ < 1 4 (e), γ = 1 4 (f), 1 < γ < (g), γ = (h), and < γ (i) For the generic cases in panels (a), (c), (e), g) and (i) 4 the associated two-parameter unfoldings of Fig 51 are indicated Proposition 32 is illustrated in Fig 31, where panels (a), (c), (e), (g) and (i) show the generic cases for (a, b) = (, ) Note that panels (a) and (c) do not differ either in position or orientation in the (, )-plane The difference is that the corresponding curve lies on different sheets of the cusp surface S (see also Fig 23) and, as we will see in Sec 5, this leads to different two-parameter unfoldings Figure 31(b), (d), (f) and (h) show the degenerate situations at the boundar points, namel, the situations when W = J, W = R, W = Ĵ and W, respectivel 32 Notion of equivalence In order to speak of tpical or generic situations and their bifurcations we now define what we mean b equivalence Our definition is in the spirit of [32] where a notion of equivalence was defined with respect to certain bifurcation phenomena Specificall, we formalie the approach from the literature of distinguishing between different configurations of an invariant curve relative to J 1 and

The cusp-cusp bifurcation of planar endomorphisms 11 C 1 Namel, we consider a curve f(w ) in a neighborhood of C 1, which is the image of a curve W in a neighborhood of the associated pre-cusp point C For notational convenience we give the definition in the contet of the normal-form setting, which has the advantage of leaving J and J 1 fied However, it can be etended in a straightforward manner to define equivalence of two general invariant curves relative to J 1 in two respective neighborhoods of a cusp point and the corresponding pre-cusp point Definition 33 (Generic event) Consider the image F (W ) of the (oriented) parabola W given b (31) under the normal-form map F A generic event is either a quadratic tangenc of F (W ) with J 1 at a generic point on J 1 (in a neighborhood of which J 1 is locall a smooth curve), or a transverse intersection of F (W ) with J 1 at a generic point on J 1 an isolated self-intersection of F (W ), In the first two cases we sa that two such generic events are of the same tpe if the respective tangenc or intersection points 1 both lie on the same side of the cusp point C 1 on J 1, that is, both occur for < or both for >, and 2 the are passed b F (W ) (locall) in the same direction, that is, both towards increasing or both towards decreasing We are now able to define equivalence based on the notion of generic events Note that a cusp on W is not a generic event Definition 34 (Equivalence of an invariant curve relative to J 1 ) Consider two images F (W ) and F ( W ) of two (oriented) parabolas W and W given b (31) under the normal-form map F We sa that the curves F (W ) and F ( W ) are equivalent with respect to J 1 if the 1 have the same induced orientation, and 2 both encounter the same tpes of generic events in the same order The task for the remainder of this paper is to find all equivalence classes of phase portraits according to this notion of equivalence Specificall, we need to find all generic two-parameter unfoldings in the (a, b)-plane for different choices of the higher-order term γ in (31) We start with a straightforward result on the possible number of events Proposition 35 (Number of events) In the normal-form setting of the cuspcusp bifurcation, that is, for W given b (31) and an a, b and generic γ, that is, for γ { 1, 2 3, 1 2, 1 3, 1 4,, 1}, there are at most two (quadratic) tangencies of F (W ) with J 1, two transverse intersection of F (W ) with J 1, and one self-intersection of F (W ) Proof The curves W, J and Ĵ are parabolas and graphs over the -ais Therefore, for an a, b and γ, W can have at most two intersections with J or Ĵ, which limits the respective number of tangencies and intersections of F (W ) with J 1 to two as well An self-intersection of F (W ) must lie in the region under J 1, that is, in the -interval between the two (if the eist) unique folds of F (W ) with respect to Furthermore, a self-intersection is due to a pair of points on W with the same -value According to (31) and (22) the curve F (W ) covers its -range in a two-to-one fashion

12 B Krauskopf, H M Osinga and B B Peckham with a single maimum or a single minimum (depending on the sign of γ) Since F (W ) has either ero or two folds with respect to, there can be at most one such pair of points with the same -value, namel, eactl when the single maimum/minimum of F (W ) lies in between the two fold points 4 Codimension-one bifurcations Definition 34 gives rise to codimensionone bifurcations that correspond to a transition between equivalence classes of equivalence of the curve F (W ) relative to J 1 There are five basic codimension-one bifurcations where the change is local, that is, it occurs in a small neighborhood of the bifurcation point in phase cross parameter space Furthermore, we find bifurcations at infinit and a transition due to the degenerac of the parametriation (31) 41 The basic codimension-one bifurcations A great advantage of the normal-form setting is that it is possible to compute the loci of all these bifurcations eplicitl Proposition 41 (Basic codimension-one bifurcations) In the normal-form setting of the cusp-cusp bifurcation there are eactl five codimension-one bifurcations for (a, b) (, ) in (31) that correspond to a local change of events as defined in Definition 34 The are illustrated in Figs 41 and 42 1 The cusp transition, denoted b C, where the curve W passes through J at the pre-cusp point C, which means that F (W ) passes eactl through the cusp point C 1 on J 1 ; see Fig 41(a) The locus of this bifurcation in the (a, b)-plane is the parabola (41) b = c C (γ) a 2 = γ a 2 2 The loop-creation bifurcation, denoted b L, where W crosses J tangent to the (horiontal) line field E; see Fig 41(b) The locus of this bifurcation in the (a, b)-plane is the parabola (42) b = c L (γ) a 2 = a 2 3 The intersection-at-tangenc bifurcation, denoted b I, where F (W ) selfintersects at a tangenc point with J 1 ; see Fig 41(c) The locus of this bifurcation in the (a, b)-plane is the parabola (43) b = c I (γ) a 2 = (9γ + 4) a 2 4 The tangenc-creation bifurcation, denoted b T, where W is tangent to J ; see Fig 42(a) The locus of this bifurcation in the (a, b)-plane is the parabola (44) b = c T (γ) a 2 = γ 1 + γ a2 5 The enter-eit bifurcation, denoted b E, where W is tangent to Ĵ; see Fig 42(b) The locus of this bifurcation in the (a, b)-plane is the parabola (45) b = c E (γ) a 2 = γ 1 + 4γ a2 Proof Each of these bifurcations is defined b a codimension-one condition on the parabola W as given b (31), so that their loci can be computed eplicitl

The cusp-cusp bifurcation of planar endomorphisms 13 C C (a) Ĵ J J 1 L L (b) Ĵ J J 1 I I (c) Ĵ J J 1 Fig 41 Arrangement of W and F (W ) at the cusp-transition bifurcation C (a), the loopcreation L (b), and the intersection-at-tangenc bifurcation I (c) 1 B definition of the cusp-transition bifurcation, W passes through the pre-cusp point C at (, ) = (, ) Substitution into (31) gives the locus 2 Since the line field E of the normal-form map F consists of horiontal lines, a loop-creation bifurcation occurs when the etremum at (, ) = (a, b) of W lies on J Substitution into (31) gives the locus 3 A transversal crossing of W and J, sa, at (, ), corresponds to a tangenc of F (W ) with J 1 Hence, an intersection-at-tangenc bifurcation occurs when W intersects Ĵ at the same -value, that is at some point ( 1, ) From (22), (23) and (25) we conclude that then 1 = 2, and we determine from γ( a) 2 + b = γ( 2 a) 2 + b as = 2a Hence, in order for (, ) to lie on J, a and b should be such that = γ( a) 2 + b = 2 = 9γa 2 + b = 4a 2, which gives the locus 4 At the tangenc-creation bifurcation W is tangent to J at, sa, ( t, t ) J W

14 B Krauskopf, H M Osinga and B B Peckham T T (a) Ĵ J J 1 E E (b) Ĵ J J 1 Fig 42 Arrangement of W and F (W ) at the tangenc creation T (a), and the enter-eit bifurcation E (b) Since the slopes of W and J at ( t, t ) must be equal we conclude from (23) that { t = γ( t a) 2 + b = 2 t, 2γ( t a) = 2 t, { t = γ a2 (1+γ) + b = a2 γ 2 2 t = aγ 1+γ, (1+γ) 2, and this gives the locus 5 The locus of tangenc of W with Ĵ can be derived eactl as for the tangenc with J, b using (25) instead of (23) Finall, there are onl these five codimension-one bifurcations because, genericall in the normal form setting, W can onl either intersect J and Ĵ transversel, or have a quadratic tangenc with J or Ĵ In particular, higher-codimension singularities, for eample, cubic tangencies, are not possible in the normal-form setting Indeed the codimension-one bifurcations of Proposition 41 give rise to different configurations of F (W ) and J 1 The first three bifurcations involve transverse intersection of W and J At the cusp transition bifurcation, a tangenc point of F (W ) with J 1 moves from the left to the right of the cusp point C 1 At the loop-creation bifurcation a small loop, that is, a self-intersection, is created; see also Fig 21 At the intersection-at-tangenc bifurcation there is an echange of the order along F (W ) between a self-intersection of F (W ) and a (generic) tangenc between F (W ) and J 1 The other two bifurcations correspond to codimension-one tangencies of W with J and Ĵ At a tangenc-creation two transverse intersections of W and J and, hence, two (generic) tangencies between F (W ) and J 1, are created At the enter-eit bifurcation two transverse intersections of F (W ) and J 1 are created

The cusp-cusp bifurcation of planar endomorphisms 15 (a) (b) C W C F (W ) J Ĵ U J 1 V (c) W (d) F (W ) U J Ĵ V J 1 Fig 43 The local interaction of W with J 1 and b J in a neighborhood U (a) corresponds to the the local interaction of F (W ) with J 1 in a neighborhood V (b) Panels (c) and (d) show the same situation sketched on the Poincaré disk The eample is phase portrait 31; compare with Fig 23 42 Representation on the Poincaré disk and bifurcations at infinit Events as defined in Definition 34 can move outside a fied neighborhood of interest, which changes the equivalence tpe of F (W ) inside this neighborhood b changing the number of events that one encounters In order to represent the entire phase portrait in a compact region, we now introduce the representation of the (, )- and (, )-planes on the Poincaré disk The representation on the Poincaré disk is quite popular in bifurcation theor, because an dnamicall relevant object can be kept track of even if the bifurcate at (the circle representing) infinit; see, for eample, [2] The Poincaré disk is obtained b closing off the plane R 2 with a circle that represents the asmptotic directions of curves at infinit The phase plane is then represented on the unit disk, where points e iφ on the unit circle correspond to the directional limits φ In the present situation we are concerned with the curve W relative to J and Ĵ, all of which are parabolas and graphs over the horiontal - ais Hence, the directional limits of J and Ĵ for ± are π/2, so that these curves start and end on the Poincaré disk at the point i = e πi/2 Similarl, the curve W starts and ends at i for γ < and at i for γ > The curve J 1, on the other hand, has the directional limits and π, and the same is true for the image of an parabola as given b (31) for an values of a, b and γ Therefore, J 1 and F (W ) have the limits 1 and +1 on the Poincaré disk The situation is illustrated in Fig 43 for the phase portrait from Fig 23, that is, γ > Panel (a) of Fig 43 shows the local picture near J and panel (c) the corresponding picture on the Poincaré disk The respective images are shown in panels (c) and (d) This is illustrated in Fig 43(d) for the local picture in panel (b) We have the following result concerning bifurcations of phase portraits on the Poincaré disk

16 B Krauskopf, H M Osinga and B B Peckham Proposition 42 (Bifurcations at infinit) In the normal-form setting of the cusp-cusp bifurcation with W given b (31) there are three bifurcations at infinit on the Poincaré disk that change the equivalence class according to Definition 34 1 The tangenc at infinit, where a generic tangenc of F (W ) with J 1 moves via infinit from one side of the cusp point C 1 to the other The locus of this bifurcation is given b γ = 1, which is the vertical asmptote of the coefficient of the tangenc creation c T (γ) = γ 1+γ ; see (44) and, for eample, the transition between phase portraits 2 and 9 in Fig 52 2 The intersection at infinit, where a transverse intersection of F (W ) and J 1 moves via infinit from one side of the cusp point C 1 to the other The locus of this bifurcation is given b γ = 1 4, which is the vertical asmptote of the γ 1+4γ coefficient of the enter-eit bifurcation c E (γ) = ; see (45) and, for eample, the transition between phase portraits 15 and 22 in Fig 52 3 The self-intersection at infinit, where a self-intersection of F (W ) bifurcates at the boundar of the Poincaré disk This happens b a swap between the begin and end points of F (W ) at ±1, and this also results in a change of direction of F (W ) The locus of this bifurcation is given b γ = 1 3, which corresponds to the crossing of W and R ; see (32) and, for eample, the transition between phase portraits 7 and 15 in Fig 52 Proof The respective changes (of phase portraits) and their loci follow immediatel from the fact that the loci of the tangenc bifurcation and the enter-eit bifurcation have asmptotes For an fied values of a and b the limits for ± of the parabola W lie below the curve R if γ < 1 3, and above the curve R if γ > 1 3 This means that the directional limits of F (W ) change when γ crosses 1 3 Effectivel, the two ends of F (W ) reconnect differentl to the points ±1 on the boundar of the Poincaré disk Note that for the degenerate case of γ = 1 3 the both approach the curve { = }, that is, the point i on the boundar of the Poincaré disk One can view this as an ambivalent connection to the points ±1 via the lower quarters of the circle at infinit 43 Inversion near the cusp As we discuss now, there is one more possibilit for transitions between equivalence classes of F (W ) Recall that according to Proposition 32 the passage through the degenerate case γ = changes the tpe of the central singularit for (a, b) = (, ) This also has consequences for F (W ) if (a, b) (, ) Proposition 43 (Inversion near cusp) The parabola W given b (31) has a maimum for γ < and a minimum for γ > As a result, near the cusp point C 1 on J 1 the curve F (W ) has a local maimum for γ < and a local minimum for γ > We call the transition through the degenerate case γ = the inversion near the cusp, and it results in a change of the phase portrait in the case that F (W ) interacts with J 1 ; see, for eample, the transition between phase portraits 24 and 29 in Fig 52 Note that the inversion near the cusp is not a bifurcation in a classical sense Rather it is a degenerac of the parametriation, which changes the direction in which F (W ) approaches a neighborhood of the cusp point C 1 This manifests itself quite dramaticall as a tpe of inversion in this neighborhood There are actuall onl three different transitions featuring an inversion near the cusp; see alread Fig 51 and Fig 52, and compare panels 24 to 29, 25 with 3, and 26 with 31 We refer to the inversion near the cusp loosel as a bifurcation (albeit an unusual one) of

6 c (γ) 4 2 2 c C (γ) c E (γ) c L (γ) c T (γ) The cusp-cusp bifurcation of planar endomorphisms 17 i ii iii iv v vi vii viii 2 3 4 5 1 8 9 1 11 7 15 14 24 25 12 12 25 3 4 19 6 26 c I (γ) 2 2 31 1 3 4 1 6 2 1 3 2 1 2 1 3 4 1 1 γ 2 21 29 1 9 11 1 1 2 7 13 14 1 11 17 18 15 22 16 23 17 1 3 1 4 24 28 21 24 29 Fig 51 Division of the (γ, c (γ))-plane into regions of different phase portraits for a >, which are shown in Fig 52; see Proposition 51 The γ-ais is divided into regions of different twoparameter unfoldings, cases i viii, which are shown individuall in Figs 53 to 51 The colors of the dividing codimension-one bifurcations from Proposition 41 are as introduced in Figs 41 and 42; the dash-colored vertical curves are the corresponding bifurcations at infinit from Proposition 42; the vertical dark blue line is the inversion near the cusp from Proposition 43 27 32 codimension-one, because it forms a codimension-one boundar between equivalence classes of F (W ) in (a, b, γ)-space 5 Two-parameter unfoldings in the (a, b)-plane The five codimension-one bifurcations of Sec 41 are all given b parabolas b = c (γ) a 2 through the origin Hence, the bifurcation diagram in the (a, b)-plane is determined entirel b the ordering of the respective bifurcations, which is given b the sies of the coefficients c (γ) In Fig 51 we show all information on the cusp-cusp unfolding (in the normal-form setting) in a convenient and ver condensed representation b plotting the graphs of the functions c (γ) This corresponds geometricall to showing how the slices {a = ±1} in the (a, b)-plane change with γ The different bifurcations are shown in different colors as introduced in Figs 41 and 42 In this representation the other bifurcations in Secs 42 and 43 are vertical lines since the do not depend on a and b The set of all bifurcation curves divides the (γ, c (γ))-plane into 32 regions of equivalent phase portraits; the insets show enlargements of two rather small regions Representatives of all 32 phase portraits that one finds for a > are shown as sketches

18 B Krauskopf, H M Osinga and B B Peckham 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 21 22 23 24 25 26 27 28 29 3 31 32 Fig 52 Sketches on the Poincaré disk of all generic phase portraits for a > of F (W ) (red curve) in relation to J 1 (black curve); the arrows show the direction of parametriation b The respective phase portraits for a < are obtained b the operation of conjugation

The cusp-cusp bifurcation of planar endomorphisms 19 on the Poincaré disk in Fig 52 Since all bifurcation curves in the (a, b)-plane are parabolas through the origin, the bifurcation diagram is smmetric under the operation a a It follows directl from (22) and (31) that the corresponding phase portraits for a < are obtained b reflecting the curve F (W ) on the -ais on the (, )-plane without changing its orientation, so that all events (see Definition 33) now occur on the opposite side of C 1 and in the reverse order We call the thus obtained phase portrait the conjugate phase portrait and denote it b a bar across the number Note that a phase portraits and its conjugate are generall not equivalent The unfoldings represented in Figs 51 and 52 were obtained b a combination of analsis and topological arguments guided b eploration on the computer We stress again that in the normal-form setting all bifurcation curves can be found analticall Furthermore, phase portraits can be eplored simpl b plotting F (W ) relative to J 1 for appropriate values of a, b and γ The overall result can be summaried as follows Proposition 51 (Two-parameter unfoldings) There are eight two-parameter unfoldings in the (a, b)-plane, denoted i viii, as shown in Figs 51 The boundar points between the respective intervals of the γ-line are 1, 2 3, 1 2, 1 3, 1 4,, and 1 Proof According to Propositions 42 and 43, the points γ = 1, γ = 1 3, γ = 1 4, and γ = give rise to changes of the equivalence class of phase portraits and, hence, of neighboring two-parameter unfoldings in the (a, b)-plane It follows from (41) and (43) that c C (γ) and c I (γ) intersect at γ = 1 2, where the cusp transition and the intersection-at-tangenc bifurcation change their order Similarl, from (41) and (42), c C (γ) and c L (γ) intersect at γ = 1, where the cusp transition and the loop creation change their order Finall, from (43) and (44) we conclude that c I (γ) is tangent to c T (γ) at γ = 2 3 This means that the intersection-at-tangenc happens at a double-tangenc point on J 1 Therefore, the intersection at γ = 2 3 is a genuine codimension-two point that leads to a change in the phase portraits Indeed, for (a, b) in between the tangenc-creation and intersection-at-tangenc bifurcation curves, the self-intersection of F (W ) lies closer to C 1 than the two tangencies of F (W ) with J 1 if γ < 2 3, while it lies furthest awa from C 1 if γ > 2 3 ; compare phase portraits 8 and 13 in Figs 54 and 55, respectivel The individual two-parameter unfoldings i viii in the (a, b)-plane are shown individuall in Figs 53 to 51 In each figure, the middle panel shows how the colored codimension-two bifurcation curves of Proposition 41 divide the (a, b)-plane into regions of different phase portraits The surrounding panels on the left show the respective configurations of W near C and those on the right that of F (W ) near C 1 Both the bifurcation curves and the configurations of W and F (W ) were drawn in Matlab directl from their respective formulas In this wa, one gets an impression of actual shapes and sies In particular, we chose a fied neighborhood of C and its image, which is a fied neighborhood of C 1 Hence, all panels in Figs 53 to 51 can be directl compared in terms of scale; where necessar, insets provide enlargements of how F (W ) lies relative to J 1 A slight disadvantage of choosing a fied neighborhood for all cases is that an intersection of W with J or Ĵ, and thus the corresponding generic event, ma have moved outside of a panel All phase portraits shown in Figs 53 to 51 are for a >, as was the case for the sketches in Fig 52 (which contain all occurring generic events) Note that crossing the line a = above or below all bifurcation curves does not constitute a bifurcation, so that the respective phase

2 B Krauskopf, H M Osinga and B B Peckham 2 1 1 2 3 2 b 2 1 1 2 i 3 3 3 4 4 4 4 5 5 6 6 2 2 a 2 5 6 6 5 Fig 53 Case i: < γ < 1 The two-parameter unfolding in the (a, b)-plane with the bifurcation curves C (blue), L (black), I (purple), T (orange), and E (green); compare with Figs 41 and 42 Representative phase portraits near C are presented anti-clockwise from the second image at the top, and near C 1 clockwise from the third image at the top; shown are W and F (W ) (red curves), J and J 1 (black curves), and bj (gre curve) The data is for γ = 15, with phase portraits for (a, b) = (2, 17), (a, b) = (7, 17), (a, b) = (2, 17), (a, b) = (2, 17), (a, b) = (7, 11), and (a, b) = (2, 17), respectivel portraits must be invariant under conjugation The fact that this is indeed the case is evidence of the consistenc of the unfoldings in Figs 53 to 51 As mentioned, the phase portraits for a < can be obtained b conjugation, but a conjugate phase portrait ma be equivalent to a different phase portrait for a > The following phase portraits do not result in new phase portraits under conjugation with respect to the original list of 32 for a > in Fig 52 invariant under conjugation: 1, 6, 7, 12, 15, 2, 21, 26, and 31 identities under conjugation: 22 = 27, 23 = 28, and 24 = 29 As a result, there are 17 new classes of phase portraits that one obtains under conju-