The lifted field

An elegant, geometric approach to the study of IDEs (1.4) is given in [69] (see also [1]).

Consider P T^∗R² = R² × RP , the manifold of contact elements to the plane (the projectivised cotangent bundle to R²). We take an affine chart p = dy/dx for RP , so the manifold of contact elements is R³ endowed with the canonical contact structure determined by the 1-form dy − pdx. The projection associated to the contact structure is

π : R³ → R² (x, y, p) 7→ (x, y).

If the coefficients of a BDE (1.5) do not vanish simultaneously we may assume a 6= 0 and write (1.5) as F = 0 where

F (x, y, p) = a(x, y)p²+ 2b(x, y)p + c(x, y),

and p = dy/dx. The set of directions defined by (1.5) form a surface M = F⁻¹(0) ⊂ R³.

In general M is smooth, and the image of π|_M is a two-fold covering of regions where b² − ac > 0. The critical set of π|_M given by F = F_p = 0 is called the criminant.

There is an involution σ on M that interchanges points with the same image under π.

The bivalued direction field given by (1.5) lifts to a single valued field ξ on M obtained by intersecting the contact planes with the tangent planes to M .

Lemma 1.3.1 ([8]) A suitable lifted field is given by ξ = F_p ∂

∂x + pF_p ∂

∂y − (F_x+ pF_y) ∂

∂p. Proof : Suppose that

ξ = A ∂

∂x + B ∂

∂y + C ∂

∂p,

for some smooth functions A,B,C. We require that (dy − pdx)(ξ) = 0. It follows that B = pA.

The normal to the surface M is given by (F_x, F_y, F_p). Since ξ is tangent to M we have that

(A, pA, C) · (F_x, F_y, F_p) = 0,

that is, CFp = −A(Fx+ pFy). The result follows. 2 The configuration of the solution curves of (1.5) at a point on the discriminant are determined by the pair (ξ, σ).

If the contact plane at a point is tangent to M then the lifted field vanishes, that is, has a singularity. This happens generically at isolated points, including at the lift of folded singularities. The projection of such points are called well-folded singularities if ξ has an elementary zero with separatrices transverse to the criminant and tangents not projecting to zero.

If the vector field ξ is regular then a smooth model (in the neighbourhood of a regular point on the discriminant) is given by

dy²− xdx² = 0

(see [26]). The integral curves are a family of cusps. A smooth model of in a neighbourhood of a well-folded singularities is given by

dy²− (y − λx²)dx² = 0

(see [27, 30]). We refer to the smooth modulus λ as the index modulus of the singularity.

There are three stable topological models: well-folded saddles (λ < 0), nodes (0 < λ < 1/16) and foci (λ > 1/16), occurring when the lifted field ξ has a saddle, node or focus respectively. These are illustrated in Figure 1.1. The index of the lifted field at the singular point is given by sign(λ).1.

Figure 1.1: A well-folded saddle (left), well-folded node (centre) and well-folded focus (right).

The lifted field field method is illustrated in Figure 1.2.

The lifted field method may be extended to BDEs with vanishing coefficients (see for example [20]). In this case we consider the surface

M = {(x, y, [α : β]) ∈ R˜ ², 0 × RP¹|aα² + 2bαβ + cβ² = 0}.

Observe that all directions are solutions where the coefficients vanish. We will consider BDEs for which this occurs at an isolated point which we we will always take to be the origin. The set π⁻¹(0) = {0} × RP¹ is called the exceptional fibre.

We use the term criminant for the closure of the set π⁻¹(∆) \ ({0} × RP¹).

Consider the affine chart for RP¹ p = β/α (we also consider the chart q = α/β) and set

F (x, y, p) = a(x, y)p²+ 2b(x, y)p + c(x, y).

The bi-valued direction field defined by the BDE in the plane lifts to the single field ξ on ˜M given in Lemma 1.3.1. The vector field ξ extends smoothly to the exceptional fibre, which is an integral curve of ξ.

M π

M π

Figure 1.2: The lifted field method: the criminant (top left), a well-folded saddle (top right), a well-folded node (bottom left) and a well-folded focus (bottom right).

From Lemma 1.3.1 we have that zeros of ξ are given by

F = F_p = F_x+ pF_y = 0. (1.6)

When the BDE has an isolated singularity we may restrict attention to zeros of ξ on the exceptional fibre. Observe that F (0, 0, p) = F_p(0, 0, p) = 0. It follows that the zeros of ξ on the exceptional fibre are given by the roots of the cubic

φ(p) = (F_x+ pF_y)(0, 0, p).

It shown in [20] that the surface ˜M is smooth if and only if δ has a Morse singularity. Furthermore, it is shown in [11] that if δ has an A_k-singularity then the

surface ˜M has an isolated A_k−1-singularity on the exceptional fibre and is smooth elsewhere. Thus, in the case of BDEs with discriminant having a cusp singularity, the surface ˜M has a Morse singularity. As the entire exceptional fibre {0} × RP¹ lies on the surface ˜M it follows that ˜M has an A⁻₁ (cone) singularity.

The term separatrix is used ambiguously in the study of IDEs; we make the following definition to avoid confusion.

Definition 1.3.2 The images under the projection π of the stable, unstable and centre curves of a zero of the lifted field ξ are called the separatrices of the singularity.

In the case of a folded saddle or node, the stable or unstable curves are the only integral curves passing through the singularity. They are smooth and transverse to each other and to the criminant. It follows (see Lemma 2.6 in [23]) that the separatrices are the only smooth curves passing through a folded singularity and that they are tangent to the discriminant. In the case of a saddle-node there is one stable or unstable manifold and a centre manifold tangent to the eigenvector associated to the zero eigenvalue.

Remark 1.3.3 The ambiguity of the term separatrix arises from the fact that sep-aratrices (in the sense of Definition 1.3.2) do not always separate distinct sectors.

For example, in the case of a folded node, the projections of the weak separatrices of the node do not separate distinct sectors. Conversely, in certain cases (for example, cusp type 2 BDEs as defined in Chapter 7) there are curves that do separate distinct sectors that are not separatrices.