The bifurcation diagram of cubic polynomial vector fields on $\mathbb{C}\mathbb{P}^1$

In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$ for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of $\epsilon_1,\epsilon_0\in\mathbb{C}$. The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure allows describing it for parameter values in $\mathbb{S}^3$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.

1 Introduction e study of polynomial vector elds on CP was initiated by Douady and Sentenac in [ ]. e point at in nity is a pole of order k − with k separatrices. In their long paper, Douady and Sentenac describe how the phase portrait can be obtained from the position of the separatrices of ∞. is allows them to describe at length the geometry of the generic vector elds in this family, which are those for which there are no homoclinic connections between any two separatrices of ∞. ey also show that there are exactly C(k) = k k (k + ) connected components of generic vector elds in the parameter space є = (є k− , . . . , є ), and that these are simply connected. (C(k) is the k-th Catalan number.) eir proof is extremely ingenious and uses that any generic polynomial vector eld ( . ) is completely determined (possibly up to some symmetry by a rotation of order k) by the following: • A topological invariant describing how the separatrices of in nity end generically at nite singular points, thus dividing CP in k connected components. e union of the separatrices is called the connecting graph. • An analytic invariant given by k numbers τ , . . . , τ k ∈ H, where H ⊂ C is the upper half-plane, and essentially equivalent to the collection of eigenvalues at each singular point. ese numbers correspond to "complex travel times" along curves joining sectors to sectors at ∞, without cutting the connecting graph (see details below).
C. Rousseau e results of Douady and Sentenac were further generalized to nongeneric vector elds by Branner and Dias [ ], who introduced invariants for nongeneric vector elds, but the paper [ ] does not describe the bifurcation diagram.
One of the goals of Douady and Sentenac was to develop tools for analyzing the unfoldings of parabolic points of germs of di eomorphisms on (C, ) (i.e., xed points with multiplier equal to ). One of the rst attempts to understand the unfolding of a codimension parabolic point goes back to Martinet [ ], where he considered unfoldings for parameter values for which the unfolded xed points are hyperbolic. He could show that the comparison of the two linearizing charts at each xed point converges to the Ecalle-Voronin modulus of the parabolic point: indeed the domains of linearization tend to the sectors used in the de nition of the Ecalle-Voronin modulus. But, until the thesis of Lavaurs [ ], no method would allow study of the sector of parameter values containing the values for which the unfolded xed points have multipliers on the unit circle. ere, a new idea was introduced, namely to unfold the sectors allowing us to de ne the Ecalle-Voronin modulus into sectors with vertices at the two singular points on which almost unique changes of coordinates to the normal form exist. Lavaurs' thesis covered a sector of parameter values complementary to the one studied by Martinet. By generalizing the method of Lavaurs, the article [ ] nally gave a complete modulus for the unfolding of a germ of codimension di eomorphism. e further generalization to codimension k > requires precisely the study of the polynomial vector elds on CP initiated by Douady and Sentenac, and this was the motivation for their study. A work in progress by Colin Christopher and the author on the realization for the parabolic point of codimension is using exactly the bifurcation diagram for cubic vector elds presented in this paper.
It is striking that the geometry of the family of polynomial vector elds ( . ) is relevant for a large class of bifurcation problems, namely the bifurcations in the unfoldings of several resonant singularities of codimension k, which all have the property that the change of coordinates to normal form is divergent but k-summable. Such singularities include the codimension k parabolic xed point of a -dimensional complex di eomorphism (studied in [ ]), the resonant saddle point in C (and hence the Hopf bifurcation), the saddle-node (studied in [ ]), and the nonresonant irregular singular point of Poincaré rank k (studied in [ ]). e works [ ] and [ ] give analytic moduli of classi cation for the analytic unfoldings of the codimension k singularities, but none of them present the bifurcation diagram. e bifurcation diagram of cubic vector elds presented here is also interesting in itself. e phase portraits are organized by the pole at in nity. In its neighborhood, the trajectories are organized as for a saddle with two attracting and two repelling separatrices. e only (real) codimension bifurcations are bifurcations of homoclinic connections between the separatrices of ∞. e higher codimension bifurcations are bifurcations of multiple singular points and simultaneous bifurcations of lower codimension. e parameter space is -dimensional and the bifurcation diagram has a conic structure, allowing us to describe it for parameter values in S . e main di culty of the study is understanding the nontrivial spatial organization of the Note that these phase portraits naturally de ne four quadrants in CP .
bifurcation surfaces in parameter space. e highest codimension bifurcations are the organizing centers of the bifurcation diagram. It is also natural to consider ( . ) as an ODE with complex time. If z , . . . , z k+ are the singular points, then Ω = CP ∖ {z , . . . , z k+ } is the orbit of a single point. But one has to pay attention that the time is rami ed when k > . For generic vector elds, Douady and Sentenac exploited this idea by covering Ω with sectors corresponding to strips in the time variable t ∈ C. In the case k = , we show how to generalize their description when the vector eld has a homoclinic connection.

The Study of Cubic Polynomial Vector Fields
In this section, we study the di erent bifurcations of the real trajectories of the vector eld is vector eld has three singular points counted with multiplicity in the nite plane. (ii) A simple singular point in the nite plane is a strong focus, a center, or a radial node. A su cient condition for a singular point z j to be a center is that its eigenvalue belongs to iR * . In that case, the center is isochronous and its period (given by the residue theorem) is π i P ′ є (z j ) . (iii) A polynomial vector eld ( . ) has no limit cycle.

Preliminaries on Polynomial Vector Fields in
(iv) A homoclinic loop between two separatrices of ∞ splits the singular points into two groups {z j ∶ j ∈ I } and {z j ∶ j ∈ I }, such that ∑ j∈I P ′ є (z j ) ∈ iR * (and hence . In the particular case I ℓ = , then, the corresponding singular point is a center. (v) A separatrix of ∞ either ends in a nite singular point or makes a homoclinic loop by merging with a second separatrix of ∞.

The Vector Field at ∞
When є = , the vector eld ( . ) has a triple singular point at the origin and a pole of order at in nity. e pole has two attracting and two repelling separatrices converging to the origin (see Figure ).
For any є, the pole at in nity and its separatrices organize the phase space as noted by Douady and Sentenac in [ ].
e full phase portrait on CP is obtained by gluing the phase portrait near ∞ given in Figure (a) with the phase portrait on a disk in the nite plane.

The Conic Structure
Hence, it is su cient to describe the intersection of the bifurcation diagram with a -dimensional real sphere S = {є ∶ є = Cst}, or use charts of the form є j = Cst. All strata will be cones on this bifurcation diagram and will be adherent to є = . In particular, we can always suppose that є = . Of course any Cst can be taken. Depending on the context, we can choose di erent values so as to simplify the computations.
Note that S is toplogically equivalent to the completion of R with a point at innity, which we will denote R .

Bifurcations of Real Codimension
Because of the special form of the system, all bifurcations of multiple singular points are of complex codimension or , hence of real codimension or . erefore, the only bifurcations of real codimension are the bifurcations of homoclinic loop when two separatrices of the pole at in nity coalesce. ere are four types of connections depending on the quadrant where they occur: we denote them (H j ), j = , . . . , ; see Figures (a) and .
Each time a homoclinic loop occurs, it surrounds a singular point with a pure imaginary eigenvalue. Each connection corresponds to a bifurcation of real codimension . Because there are four separatrices, there can be up to two simultaneous homoclinic loops forming a gure-eight loop; this will be a bifurcation of real codimension . Considering that the system is invariant under Figure : e four homoclinic loops in the four quadrants in CP inside a disk around ∞.
the study of one bifurcation surface (H j ) allows us to determine the others. ese bifurcations are represented by surfaces in S . e equation of the bifurcation locus is implicit, and hence not easy to visualize. It is of the form P ′ є (z j ) ∈ iR * , with boundary points when P ′ є (z j ) = .

Bifurcation of Parabolic Points
is occurs when two singular points coalesce in a double singular point (parabolic point), namely when the discriminant ∆ vanishes, where Note that this local bifurcation is of complex codimension , hence real codimension , but it is represented by a curve on S because of the conic structure. A generic bifurcation of double singular point occurs when the third singular point is not a center, i.e., its eigenvalue is not on the imaginary axis.

Proposition .
(i) In S , ∆ = can be represented as a ∶ torus knot on the torus є = є (i.e., a closed curve turning twice (resp. thrice) around in є (resp. є )). Four points of this knot are not generic, i.e., correspond to higher order bifurcations. ese four points divide ∆ = in four arcs of regular (generic) points.
(ii) e bifurcation diagram at a regular point of ∆ = (when the third singular point is not a center) is given in Figure . In particular, two surfaces of homoclinic bifurcations in adjacent quadrants end along the regular arcs of ∆ = (see Figure (a)).
Proof When ∆ = , then є = √ ( є ) . Hence, instead of cutting the bifurcation diagram by a sphere є = Cst, we prefer to use a chart є = (i.e., є ∈ S ). At a point of ∆ = corresponding to a particular є ′ , P є ′ (z) can be written as and a general unfolding for є close to є ′ has the form It is easily checked that the change (є , є ) ↦ (η , η ) is an analytic di eomorphism in the neighborhood of є ′ when a = . Note that P ′ є ′ (− a) = a . Hence, P ′ є ′ (− a) ∈ iR if and only if a ∈ ( ± i)R. Since є = − a , when we suppose є = , this yields e Bifurcation Diagram of Cubic Polynomial Vector Fields four points on the torus knot: a = ±( ± i)

√
, symmetric to each other under ( . ). ese four points in parameter space have codimension , since the third singular point is a center surrounded by a homoclinic loop. ey cut ∆ = into four regions, which are sent one to the other under ( . ). At points of ∆ = , the homoclinic connections become heteroclinic connections through the parabolic point in phase space ( Figure ). At a regular point of ∆, there are two homoclinic connections in adjacent sectors (Figure (a)). Near such a point, two half-surfaces (H j ) and (H j+ ) merge on ∆ = (indices are (mod )). ese half-surfaces are tangent on ∆ = . Indeed, they correspond to Let us put √ η = x + i y and let (a + η ) = α + iβ. Since, by hypothesis, P ′ є (− a) ∉ iR for η = η = , then α±β = for small η , η . We have that Re(P ′ є (a+η ± √ η )) = if and only if ±(x − y )+αx −β y = . Because α = ±β, these two hyperbolas are distinct and have a quadratic tangency at the origin in the (x, y)-space, i.e., in the √ η -space.
en the corresponding curves in η -space are also tangent. A ner analysis would show that the tangency is of order .

The Other Bifurcations
Apart from the codimension bifurcation at є = , there remains two bifurcations, one of codimension , and one of codimension . e codimension bifurcation is the gure-eight loop, i.e., the intersection of (H j ) and (H j+ ). It occurs when two (and hence three) eigenvalues are pure imaginary, i.e., all singular points are centers.

Proposition .
e intersection of the codimension bifurcation (H j ) ∩ (H j+ ) (denoted (H j, j+ )) with є = , occurs along the two line segments with endpoints on ∆ = . e bifurcation diagram is given in Figure . Proof We can suppose that for some є ′ , Hence, . Since the sum of the inverses of these derivatives vanishes, as soon as two of them belong to iR, the third belongs to iR. Moreover, Using the symmetry ( . ), we can suppose a = α( + i), b = β( + i), α, β > . It is easily checked that when the singular points are distinct, the unfolding for є near є ′ has the form ere are surfaces of homoclinic loop bifurcations corresponding to the three conditions We want to show that these surfaces are transversal when the singular points are distinct. For this, we let Two of these surfaces are transversal if the corresponding × Jacobian with respect to (ν , ν ) does not vanish. e three surfaces are two by two transversal when e last bifurcation occurs for nonregular points of ∆ = . At these points, the system has a parabolic point and a center; see Figure (b).

Proposition .
e bifurcation diagram for a codimension point as in Figure (b) is given in Figure . Proof is situation corresponds to a nonregular point of ∆ = , which is an endpoint of a segment of double homoclinic curve ( gure-eight loop), since it can be unfolded into an (H j, j+ ).
e only bifurcation surfaces are the four homoclinic loop surfaces which all merge along the two bifurcations (H j, j+ ), j = , . In the limit, some become tangent because the transversality condition ( . ) is violated at the limit.  e gure is on CP minus a point.

The Two Open Sets W and W of Generic Values
Before putting together the bifurcation diagram, we must describe the generic situation, which was studied in great detail by Douady and Sentenac [ ]. When є is not a bifurcation value, the separatrices of ∞ land at singular points. ere are two generic ways in which this can occur: • the two repelling separatrices land at the same attracting singular point, and the two other separatrices each land at a di erent repelling point (see Figure ); let us call W the open set of these parameter values; • the two attracting separatrices land at the same repelling singular point, and the two other separatrices each land at a di erent attracting point; let us call W this open set.
W and W are simply connected sets in C . is is a highly nontrivial result, which would be very di cult to prove by "classical" methods, including visualizing the two domains in S . Indeed, these domains are limited by the four surfaces (H i ) of homoclinic connections, and the boundary of these surfaces is the union of the curve ∆ = with the two segments (H i ,i+ ) where double homoclinic connections occur (see Figure (a)). is means that these surfaces are folded in S . Fortunately, Douady and Sentenac produced a very clever argument allowing to show that W and W are simply connected. Indeed, for each є ∈ W j , we can compute the "complex time" to go from ∞ to ∞ along a loop not cutting the connecting graph (i.e., the union of the separatrices). Up to homotopy, there are exactly two such loops, providing two nonreal times, and we can orient the loops so that the complex times both belong to the upper halfplane H (see Figure ). ese complex times are given (up to sign depending on the e Bifurcation Diagram of Cubic Polynomial Vector Fields orientation of the loop) by where z j is the singular point inside the loop. In particular, the complex times depend only on the homotopy class of the loop. Let us call , , , , the four quadrants when turning in the positive direction. Each loop joins two adjacent quadrants i, j. Let τ i , j be the travel time from quadrant i to quadrant j along a curve from ∞ to ∞ not intersecting the connecting graph. We have two times, τ i , j and τ i ′ , j ′ , corresponding to the two nonhomotopic loops. e very subtle theorem of Douady and Sentenac follows.

eorem . ([ ])
e map F∶ W j → H , given by F(є) = (τ i , j , τ i ′ , j ′ ), is a holomorphic di eomorphism. e tuple (τ i , j , τ i ′ , j ′ ) is the Douady-Sentenac analytic invariant announced in the introduction. We will come back to the idea of the proof in Section .

Corollary .
W and W are simply connected open sets in C . Because of the conic structure of the bifurcation diagram, their intersection with S is also simply connected.

The Bifurcation Diagram for є ∈ S
We now have all the ingredients to give the global bifurcation diagram for parameter values inside the sphere S . Let us rst introduce the following notation.

Notation .
• (H i j ), with j = i + , denotes the transversal intersection of (H i ) with (H j ), whose bifurcation diagram appears in Figure (   Proof Each W j is conformally equivalent to H . e codimension part of the boundary is included in the two pieces R × H and H × R. But { } × H and H × { } are excluded, since these would correspond to a singular point merging with in nity, which is forbidden in our normalized family. Hence the boundary has four open pieces of the form R ± × H and H × R ± , corresponding to (H j ) for j = , . . . , ; note that these pieces are simply connected, and so will be their intersection with a sphere. Looking at the intersection of the parameter space with S , which is homeomorphic to the completion of R with one point, then each W j is homeomorphic to an open ball of dimension . e two are separated by the bifurcation diagram that is topologically like a -dimensional sphere S (but with gluing some curves on it), which we call the "bifurcation sphere". e bifurcation sphere is a patchwork of four open pieces corresponding to the (H j ) for j = , . . . , . ese four pieces, as well as the recipe to glue them for constructing S appear in Figures and ; the gluing is similar to the type of gluing used when constructing abstract Riemann surfaces. In practice however, these pieces are quite twisted in the -dimensional sphere. Figure (a) represents the codimension bifurcations, namely the torus knot ∆ = and the two segments of double homoclinic bifurcations; the endpoints of these segments, called (H j ) divide the torus knot into the four pieces (H j, j+ ).
e key to understanding Figure is  e "polyhedron" of Figure has eight vertices, ten edges, and four faces. Four of the vertices are attached to three edges and four of them are arti cial ones, since attached to only two edges. Hence, the topological sphere S is homeomorphic to a tetrahedron, with four of its edges cut in two parts as in Figure . Figure : In (a), the torus knot representing ∆ = and the two segments of double homoclinic bifurcations. e four endpoints of the two segments divide the torus knot into four pieces. (Note that the torus is not part of the bifurcation diagram.) In (b) the boundary of (H ) containing two pieces of the torus knot separated by one segment on the right. At the le end, the second segment is traveled twice, once in each direction: this back and forth movement is natural if we remember that we are following the boundary of (H ) along (H ) in Figure . e boundaries of the other (H j ) are obtained through rotations of one fourth of a turn.

The Construction of Douady and Sentenac
Let us consider a generic vector eld for є ∈ W or є ∈ W . e idea of Douady and Sentenac is that, when the time is extended to the complex domain, then all points of the phase space, except the singularities, belong to the "complex trajectory" of a single point. Hence, it is natural to reparameterize the phase space by the time t, where t(z) = z ∞ dζ p є (ζ) , but we have to pay attention, since the function t(z) is multivalued. e phase plane minus the separatrices is the union of two open simply connected regions (see Figure ). Each of these regions is the image of a strip in the t-coordinate. e boundary of these strips are simply the inverse images of the separatrices (see Figure ), and the singular points have disappeared at in nity. Note that when a singular point is