A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields

TL;DR

This paper characterizes how complex polynomial vector fields undergo bifurcations that preserve equilibrium multiplicities, showing they can be decomposed into simpler, well-understood bifurcations.

Contribution

It provides a detailed characterization and decomposition of multiplicity-preserving bifurcations in complex polynomial vector fields of arbitrary degree.

Findings

Any such bifurcation can be realized as a composition of simpler bifurcations.

The simpler bifurcations involved are explicitly characterized.

The results apply to polynomial vector fields of any degree d.

Abstract

For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points can be realized as a composition of a finite number of simpler bifurcations, and these bifurcations are characterized.

Equations

$${\xi }_P=P(z)\frac{d}{dz},z\in \mathbb{C},$$

$$\dot{z}=P(z),\gamma (0,z)=z,z\in \mathbb{C},t\in \mathbb{R},$$

$$\dot{x}$$

$$\dot{y}$$

$$\dot{z}=\frac{\lambda }{z^m},\lambda \in {\mathbb{C}}^{\ast },m\in \mathbb{N},$$

$$(\underline{\alpha },\underline{\tau })=\left({\int }_{T_{k,j}}\frac{dz}{P(z)},\dots ,{\int }_{s_{k,j}}\frac{dz}{P(z)}\right)\in {\mathbb{H}}^s\times {\mathbb{R}}_{+}^h,$$

$$\left(\underline{\tilde{\alpha }},\underline{\tilde{\tau }}\right)=\left({\int }_{{\gamma }_1}\frac{dz}{P_{\xi }(z)},\dots ,{\int }_{{\gamma }_{s+h}}\frac{dz}{P_{\xi }(z)}\right)\in {\mathbb{H}}^s\times V_{+}^h,$$

$$s_{k,j_0}=s_{k_1,j_1},s_{k_2,j_2},\dots ,s_{k_n,j_n}=s_{k_0,j},n\ge 2.$$

$$({\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_s,{\tilde{\tau }}_1,\dots ,{\tilde{\tau }}_h)=\left({\int }_{{\gamma }_1}\frac{dz}{P_{\xi }(z)},\dots ,{\int }_{{\gamma }_{s+h}}\frac{dz}{P_{\xi }(z)}\right).$$

$${\dim }_{\mathbb{R}}(\mathcal{C})=2s+h,\text{and}{\mathrm{codim}}_{\mathbb{R}}(\mathcal{C})=2m^{\ast }+h.$$

$${\mathrm{codim}}_{\mathbb{R}}(\mathcal{C})$$

$$=2d-2-2(d-m^{\ast }-1-h)-h$$

$$=2m^{\ast }+h.$$

$$a_{m+1}$$

$$a_{m+1}+a_{m+2}$$

$$\dots$$

$$a_{m+1}+a_{m+2}+\cdots +a_{m+n+1}$$

$$\dots$$

$$a_{m+1}+a_{m+2}+\cdots +a_{m+n+1}+\cdots +a_{m+n+\ell +1}$$

$$\dots$$

$$a_{m+1}+a_{m+2}+\cdots +a_{m+n+1}+\cdots +a_{m+n+\ell +1}+\cdots +a_N$$

$$a_{m+1}+a_{m+2}+\cdots +a_{m+n+1}+\cdots +a_{m+n+\ell +1}+\cdots +a_N+a_1$$

$$a_{m+n+\ell +1}$$

$$\dots$$

$$a_{m+n+\ell +1}+\cdots +a_N+a_1$$

$$\dots$$

$$a_{m+n+\ell +1}+\cdots +a_N+a_1+\cdots +a_{m+n}$$

$$a_{m+n+\ell +1}+\cdots +a_N+a_1+\cdots +a_{m+n}+a_{m+n+1}$$

Full text

1 Introduction

Bifurcations, the qualitative change in dynamics produced by varying parameters, are fundamental to the analysis of any family of dynamical systems but are notoriously difficult to describe in any generality. This paper makes a significant step towards a complete description of the bifurcations of the global topological structure of the integral curves of the complex polynomial vector fields

[TABLE]

or equivalently, the maximal solutions $\gamma(t,z)$ to the associated autonomous ordinary differential equation (ODE)

[TABLE]

where $P(z)=z^{d}+a_{d-2}z^{d-2}+\dots+a_{0}$ is a monic and centered polynomial of degree $d\geq 2$. Namely, we characterize the multiplicity-preserving bifurcations, i.e. bifurcations where the multiplicities of the equilibrium points (the zeros of $P$) are preserved under small perturbation. For significant work on parabolic bifurcations (which do not preserve multiplicities) of complex vector fields, see MRR2004 , CR2014 , and Rousseau2015 .

Complex polynomial ODEs of the form (2) are a subset of the $\mathbb{R}^{2}$ systems

[TABLE]

whose global qualitative structure in general remains a fundamental open problem in dynamics. Famously, part of Hilbert’s 16th problem inquires to the number and configurations of limit cycles in the plane for each degree $d$ polynomial system in two real variables. Even though holomorphic vector fields, which lack limit cycles (e.g. Sve1978 ; Nee1994 ), may seem distant from Hilbert’s 16th problem, experts in this area have shown that a significant class of perturbations to study are non-holomorphic perturbations of holomorphic polynomial vector fields with centers AGP2010 , LS04 , ALS2005 . Creating a complete description of bifurcations for complex polynomial vector fields would not only answer a fundamental question about holomorphic systems in their own right, but understanding the holomorphic perturbations may reduce the complexity of the analysis of non-holomorphic perturbations.

Complex vector fields are also linked to other areas of mathematics. In particular, the properties of single-variable complex vector fields have been utilized in proving prominent results in iterated complex dynamics, including the study of parabolic bifurcations (e.g. Shi , Ben1993 , Outhesis , BT2007 ) and in the proof that there exist quadratic polynomial Julia sets of positive Lebesgue measure BC2005 . Single-variable complex vector fields are being used to study higher dimensional complex systems RT2008 , as well as quadratic and Abelian differentials on Riemann surfaces. The integral curves of holomorphic vector fields on Riemann surfaces foliate the Riemann surface, and they are similar in structure to the trajectories of these differentials. Understanding the bifurcations of complex ODEs may contribute to one of these fields in a way that has not yet been explored.

Even when restricting to the study of holomorphic or meromorphic planar systems in their own right, a comprehensive study of the bifurcations remains elusive. Researchers have made headway by considering lower-degree polynomial systems so that the bifurcation diagram can be visualized GH ; LS04 ; RT2008 , or describing bifurcations induced by rotations of vector fields JMRCVV95 . The aim of this paper is to make to make a significant step towards a comprehensive analysis of all global bifurcations for polynomial vector fields in $\mathbb{C}$ of arbitrary degree by characterizing the bifurcations where the multiplicity of the equilibrium points is preserved.

This paper is organized as follows. Sections 2 and 3 review basic properties of complex ODEs and introduce parameter space and bifurcations for the systems under consideration. Section 4 describes deformations in rectifying coordinates and proves that these are enough to study the multiplicity-preserving bifurcations. Rank $k$ bifurcations are defined in Section 5, and the first main theorem of this paper is proved:

**Theorem **5.1

Every multiplicity-preserving bifurcation can be realized as a composition of rank 1 bifurcations.

Section 6 characterizes the rank 1 bifurcations to show that they are all essentially the same type. The informal statement of the second main theorem in this paper is:

**Theorem 6.1 **(informally)

Every rank 1, multiplicity-preserving bifurcation is of the type where a sequence of $n\geq 1$ homoclinic separatrices (satisfying some technical conditions) break, and $n-1$ (specified) homoclinic separatrices form under perturbation.

2 Preliminaries

The complex ODEs (2) inherit the general properties of the real planar ODEs (1), including existence and uniqueness of solutions, dependence on initial conditions, Hartman-Grobman theorem (linearization), and long-term behavior described by the Poincare-Bendixon theorem.

Equilibrium points

Classification of the local dynamics of holomorphic vector fields near an equilibrium point, a zero $\zeta$ of $P$, is known to depend only on the order of the zero and the dynamical residue $\textrm{Res}(1/P,\zeta)$ JAJ ; Ben1991 ; BT76 ; GGJ2000 ; OH1966I ; OH1966II ; OH1966III ; Sve1978 . The simple equilibrium points come in three types: sink (attracting), center (rotational), and source (repelling). There can not be saddle points in the usual sense, since the Cauchy-Riemann equations force the eigenvalues of the Jacobian to be of the form $\lambda=\alpha\pm i\beta$, which can not have real parts with opposite sign. Near zeros of multiplicity $m>1$, there are $m-1$ attracting and $m-1$ repelling directions and $2(m-1)$ elliptic sectors. See Figure 1 for an example.

Poles

Another type of singularity for the systems (2) is the pole at infinity. In more general families, e.g. rational vector fields, there may be more than one pole. The local dynamics near a pole only depends on the order of the pole. Indeed, the phase portrait of a vector field in the neighborhood of a pole of order $m\in\mathbb{N}$ is shown in Sverdlove Sve1978 to be topologically equivalent to the phase portrait of the differential equation

[TABLE]

in a neighborhood of $z=0$, and Garijo, Gasull, and Jarque GGJ2000 extend the result to conformal (biholomorphic) conjugacy. Even though holomorphic vector fields can not have saddle points in the usual sense, the local behavior of Equation (4) resembles that of a saddle point in the following way. It has $m+1$ attracting and $m+1$ repelling straight trajectories meeting at $z=0$, alternating in orientation, and the other trajectories of (4) in neighborhood of $z=0$ are hyperbolic in form Nee1994 (see Figure 1).

No limit cycles

A marked difference between holomorphic and general real planar vector fields is that there can not be isolated periodic trajectories, hence no limit cycles, for the former, due to the identity theorem of complex analysis: consider a period $T$ solution $\gamma(t,z_{0})$. Since $\gamma(T,z)-z$ is holomorphic in $z$ and $=0$ on an arc of $\gamma$, it is identically [math] in a neighborhood of $z_{0}$.

Separatrices

The global structure of the integral curves up to topological equivalence is determined by the separatrices, the maximal trajectories that are incoming to or outgoing from the poles DN1975 ; ALGM1973 . Therefore, unsurprisingly, the separatrices will play a vital role in understanding the bifurcations. The reader may refer to Figure 1 to guide his or her understanding of the details below which are needed for this paper. Polynomials of degree $d>2$ have a pole of order $d-2$ at the point at infinity, hence there are $2(d-1)$ separatrices. Separatrices for monic polynomials have asymptotic directions $\ell\pi/(d-1),\ \ell=0,\dots,2d-3$ at infinity. The separatrices come in two types for polynomial vector fields: landing separatrices, which have $\alpha$- or $\omega$-limit at an equilibrium point for $t\rightarrow-\infty$ or $t\rightarrow+\infty$ respectively; and homoclinic separatrices which join infinity to itself and are defined for $t$ in a finite interval. Separatrices $s_{\ell}$ are labelled according to their asymptotic directions. Outgoing separatrices are denoted $s_{k}$ with odd $k$, incoming separatrices are denoted $s_{j}$ with even $j$. Homoclinic separatrices are labelled $s_{k,j}$, with odd $k$ and even $j$ corresponding to its outgoing and incoming asymptotic directions at infinity. Near infinity, the complement of the separatrices has $2(d-1)$ connected components, which define $2(d-1)$ accesses to infinity. The ends can informally be thought of as $2(d-1)$ points at infinity, counted as distinct from within each access. Formally, the ends $e_{\ell}$, $\ell=0,\dots,2d-3$ are the principal points of the prime ends (in the sense of Carathéodory) of these regions. The label is chosen such that the access corresponding to $e_{\ell}$ is between the separatrices with labels $\ell-1$ and $\ell$ mod $2(d-1)$ (again, see Figure 1 for an example).

The separatrix structure can be represented in a separatrix disk model, by blowing up the point at infinity to $\mathbb{S}^{1}$, labelling the points $\exp\left(\frac{2\pi{\rm i}\ell}{2d-2}\right)$, $\ell=0,\dots,2d-3$ on $\mathbb{S}^{1}$ by $s_{\ell}$, and embedding the separatrix graph in the disk (see again Figure 1).

2.1 Rectifying Coordinates and Transversals

The separatrices as explained above provide a decomposition of the dynamical plane into open sets, called zones, whose boundaries contain a union of separatrices. The results in this paper rely on decomposing the dynamical plane into these zones, which are building blocks that glue together to give a polynomial vector field. We will look at deformations of these glued building blocks to understand the nearby polynomial vector fields, i.e. the bifurcations. The key players here are the rectifying coordinates, elaborated on below. Technical details are given in this subsection for self-contained reading, but the main points which are needed for this paper are:

there are three zone types for polynomials: strip, cylinder, and half-plane; and 2. 2.

to understand the statement of Theorem 6.1, one needs to understand the labelling of separatrices on the boundaries of these strips, cylinders, and half-planes as in Remark 1.

Rectifying Coordinates

It is well-known that for polynomials, the connected components of $\mathbb{C}$ minus the separatrix graph (called zones) come in three types: center zones, sepal zones, and $\alpha\omega$-zones. In the last case, there are exactly two singular points on the boundary of the zone, and all trajectories inside the zone have their $\alpha$-limit at one singular point and $\omega$-limit at the other. These three zone types are isomorphic to half-planes, strips, or cylinders via the rectifying coordinates $\Phi(z)=\int_{z_{0}}^{z}\frac{dw}{P(w)}$ (called the distinguished or natural parameter in the literature on quadratic differentials JAJ ). Under $\Phi$, trajectories are pushed forward to horizontal lines (see Figures 2, 3, and 4).

The basin of a center, called a center zone, is isomorphic to an upper or lower half-infinite cylinder by the rectifying coordinates $\Phi$ (see Figures 2 and 5). Trajectories are pushed forward under $\Phi$ to horizontal circles on the cylinder. The boundary of a center zone consists of a sequence $s_{k_{1},j_{1}},s_{k_{2},j_{2}},\dots,s_{k_{n},j_{n}}$ of one or several homoclinic separatrices, together with the ends between them. A counterclockwise center zone is isomorphic to an upper half-infinite cylinder (as in Figure 2) with the rectified homoclinic separatrices on the lower boundary. The indices of the $s_{k_{i},j_{i}}$ on the boundary satisfy $k_{i+1}=j_{i}+1$, where subindexes $i$ are $i=1,\dots,n$ mod $n$, and indices $k,\ j=0,\dots,2d-3$ are mod $2(d-1)$. A clockwise center zone is isomorphic to a lower half-infinite cylinder with the sequence of homoclinics on the upper boundary. The indices of the $s_{k_{i},j_{i}}$ on this upper boundary satisfy $k_{i+1}=j_{i}-1$. For a center zone, the first separatrix $s_{k_{1},j_{1}}$ in the sequence is not well-defined, since the sequence of homoclinics closes to a circle. The order is otherwise well-defined by the orientation of the separatrices.

A sepal zone, i.e. an elliptic sector, is isomorphic to an upper or lower half-plane by $\Phi$ (see Figure 3). Trajectories are pushed forward under $\Phi$ to horizontal lines. The boundary of a sepal zone consists of a sequence of separatrices $s_{j_{0}}$, $s_{k_{1},j_{1}},s_{k_{2},j_{2}},\dots,s_{k_{n},j_{n}}$, $s_{k_{0}}$ and the ends between them. The separatrices $s_{j_{0}}$ and $s_{k_{0}}$ are landing, and between them are $n\geq 0$ homoclinic separatrices. An upper half-plane has the rectified separatrices on the lower boundary, whose indices satisfy $k_{i+1}=j_{i}+1$, $i=0,\dots,n-1$ and $k_{0}=j_{n}+1$, $k_{i},\ j_{i}=0,\dots,2d-3$ mod $2(d-1)$. A lower half-plane (as in Figure 3) is similar except the index relationship of separatrices on the upper boundary follows the rule $k_{i+1}=j_{i}-1$, $i=0,\dots,n-1$ and $k_{0}=j_{n}-1$. The order of the separatrices on the boundary is well-defined by the orientation of the separatrices in rectifying coordinates, left to right.

A region with distinct $\alpha$- and $\omega$-limit points on the boundary is isomorphic to a horizontal strip under $\Phi$ (see Figure 4). Trajectories are pushed forward under $\Phi$ to horizontal lines. The boundary of a strip zone has two components. The upper component consists of a sequence of separatrices $s^{+}_{j_{0}}$, $s^{+}_{k_{1},j_{1}},s^{+}_{k_{2},j_{2}},\dots,s^{+}_{k_{n},j_{n}}$, $s^{+}_{k_{0}}$ of two landing separatrices, $n\geq 0$ homoclinic separatrices, and the ends between them. Similarly, the lower boundary component consists of a sequence of separatrices $s^{-}_{j_{0}}$, $s^{-}_{k_{1},j_{1}},s^{-}_{k_{2},j_{2}},\dots,s^{-}_{k_{m},j_{m}}$, $s^{-}_{k_{0}}$, $m\geq 0$. The relationship between the indices on the upper boundary component is $k^{+}_{i+1}=j^{+}_{i}-1$, $i=0,\dots,n-1$ and $k^{+}_{0}=j^{+}_{n}-1$, and on the lower boundary component is $k^{-}_{i+1}=j^{-}_{i}+1$, $i=0,\dots,m-1$ and $k^{-}_{0}=j^{-}_{m}+1$. The order of the separatrices on each boundary component is well-defined by the orientation of the separatrices in rectifying coordinates, left to right.

Remark 1.

The index relationship of the sequence of separatrices on the boundary of a zone does not depend on the type of zone (center, elliptic, or strip). It only depends on whether the sequence is on an upper boundary or lower boundary. Omitting details that can be found in the paragraphs above, the relationship can be summarized as: $k_{i+1}=j_{i}-1$ on an upper boundary, and $k_{i+1}=j_{i}+1$ on a lower boundary.

Transversals

There are a number of geodesics in $\allowbreak\mathbb{C}\setminus\allowbreak\{\text{equilibrium points}\}$ that join the point at infinity to itself, with respect to the metric with length element $\frac{|dz|}{|P(z)|}$. This is seen in rectifying coordinates as straight line segments which connect ends to ends. There are $s+h$ among these that are singled out: the $h$ homoclinic separatrices and the $s$ distinguished transversals $T_{k,j}$. The distinguished transversals are a unique and well-defined choice of one transversal for each of the $s$ strips. This choice is the transversal $T_{k,j}$ which joins $e_{k}$ to $e_{j}$, where $e_{k}$ is the right-most end on the lower boundary of the strip and $e_{j}$ is the left-most end on the upper boundary of the strip (see Figures 4 and 5). The reason for this choice is not needed in this paper, but it is explained in detail in KDcomb .

2.2 Classification and Pseudo-Invariants

Classification

One way to classify the polynomial vector fields, i.e. define a complete set of realizable invariants, is to express the topological structure via an admissible gluing of strips, cylinders, and half-planes, and the geometry by the analytic invariants

[TABLE]

where the set of $s_{k,j}$ and $T_{k,j}$ are the homoclinic separatrices and the distinguished transversals for $\xi$. That is, assign $h$ real numbers $\tau=\int_{s_{k,j}}\frac{dz}{P(z)}>0$ to the homoclinics and $s$ complex numbers $\alpha=\int_{T_{k,j}}\frac{dz}{P(z)}\in\mathbb{H}$ to the distinguished transversals (see Sent and BD09 for more details). The analytic invariants $\tau\in\mathbb{R}_{+}$ that correspond to homoclinic separatrices are the Euclidean lengths of the rectified homoclinic separatrices on the boundaries of the strips, cylinders, and half-planes; and the analytic invariants $\alpha\in\mathbb{H}_{+}$ are complex numbers that record the height and shear of each strip (the complex ”length” of $T_{k,j}$). There are other ways to present the classification of polynomial vector fields which will not be explained in this paper.

The implication of the classification that is required for this paper is that it provides a decomposition into strips, cylinders, and half-planes, identified along their boundaries in the appropriate way, and conversely, any (admissible) gluing of strips, cylinders, and half-planes gives rise to a polynomial vector field BD09 . This construction of a polynomial vector field from gluings of the building blocks is (briefly) as follows. The strips, cylinders, and half-planes are glued together to construct a Riemann surface which is conformally equivalent to the Riemann sphere minus punctures, and endowing the strips, cylinders, and half-planes with the vector field $\frac{d}{dz}$ leads to a polynomial vector field on Riemann sphere, with equilibrium points at the punctures. We will understand perturbations (bifurcations) of polynomial vector fields by analyzing perturbations in this construction. While the invariants (5) change discontinuously under perturbation (to be explained), one can show that certain perturbations of the building blocks, preserving the original gluing, give rise to polynomial vector fields that are nearby the initial one. This will be explained in further detail in the paragraph below and in Theorem 3.2.

Pseudo-invariants

The analysis in this paper relies heavily on deformations of the building blocks (strips, cylinders, and half-planes) in rectifying coordinates. More specifically, let $\xi_{0}\in\Xi_{d}$ be the vector field to be perturbed. The classification gives a configuration of half-planes, strips, and cylinders, whose boundaries consist of rectified separatrices and where the vector field is $\frac{d}{dz}$. Next, the rectified zones are deformed (as sets) by piecewise linear mappings given by sending horizontal segments corresponding to the rectified $s_{k,j}$ for $\xi_{0}$ to non-horizontal segments (see Figure 6).

As sets, the configuration of the gluing of the rectified zones has not changed, but endowing the distorted zones with $\frac{d}{dz}$ will, to be explained by Theorem 3.2, results in another $\xi\in\Xi_{d}$ close to $\xi_{0}$, but where $\xi$ has qualitatively different dynamics Even though the configuration of rectified zones has not changed as sets, when the dynamics are considered, the zones have changed type (e.g. Figure 8 shows a deformed cylinder that becomes part of a strip zone dynamically). In terms of $(\underline{\alpha}_{0},\underline{\tau}_{0})$, this corresponds to allowing one or more of the $\tau_{0}$ to move off the positive real axis. However, we can not use $(\underline{\alpha},\underline{\tau})$ to denote the perturbation of $(\underline{\alpha}_{0},\underline{\tau}_{0})$, since the qualitative change means the distinguished transversals $T_{k,j}$ and homoclinic orbits $s_{k,j}$ change and hence by definition, the analytic invariants (5) will change discontinuously. This leads us to define the correct quantities to consider to correspond to deformations in rectifying coordinates:

Definition 1.

The pseudo-invariants of $\xi$ with respect to $\xi_{0}$ are

[TABLE]

where $V_{+}$ is an $\epsilon$ neighborhood of $\mathbb{R}_{+}$, $\gamma_{1},\dots,\gamma_{s+h}$ are the curves (as sets) in $\mathbb{C}$ which coincide with the $s$ distinguished transversals and $h$ homoclinic orbits for $\xi_{0}$ and are fixed under perturbation, and $P_{\xi}$ is a perturbation of $P_{0}$.

We emphasize that $\left(\underline{\widetilde{\alpha}},\underline{\widetilde{\tau}}\right)$ are in general not the analytic invariants for the perturbed vector field (credit is due to the referee for pointing out this important distinction). They do correspond to deformations in rectifying coordinates since the initial configuration of half-planes, strips, and cylinders are from the topology of $\xi_{0}$, and this configuration as sets has not changed under deformation, corresponding to fixing the curves $\gamma_{1},\dots,\gamma_{s+h}$ under perturbation. Note that the pseudo-invariants are only locally defined.

3 Parameter Space and Bifurcations

This section reviews known results on the space $\Xi_{d}$ of degree $d$ monic, centered complex polynomial vector fields and its decomposition into loci of qualitatively same dynamics. The ideal question to be answered is stated, as well as the restricted question that will be answered in this paper, i.e. describing the multiplicity-preserving bifurcations. Then we review a result proving the manifold structure of each locus, where the construction of the proof shows that deformations in rectifying coordinates give ”nearby” vector fields.

Combinatorial Classes

To understand the possible bifurcations, we examine the space $\Xi_{d}$ of degree $d$ monic, centered complex polynomial vector fields and partition $\Xi_{d}$ into combinatorial classes $\mathcal{C}$ such that all $\xi\in\mathcal{C}$ have the same (labelled) separatrix graph with labelling of the asymptotic directions. Using the (labelled) separatrix graph as the equivalence relation is dynamically meaningful since it implies topological equivalence of all $\xi\in\mathcal{C}$, i.e. there is a homeomorphism which sends trajectories of $\xi\in\mathcal{C}$ to trajectories of $\xi^{\prime}\in\mathcal{C}$ which respects the orientation of but not necessarily the parameterization by time. Adding the further restriction that the labelling must be the same, there are generally distinct combinatorial classes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ where all $\xi_{1}\in\mathcal{C}_{1}$ are topologically equivalent with all $\xi_{2}\in\mathcal{C}_{2}$.

The space $\Xi_{d}\cong\mathbb{C}^{d-1}$ since the monic, centered polynomials $P(z)=z^{d}+a_{d-2}z^{d-2}+\dots+a_{0}$ can be parameterized by the coefficients $(a_{0},\dots,a_{d-2})\in\mathbb{C}^{d-1}$.

Since the separatrix structure gives topological equivalence of trajectories, a change in separatrix structure under small perturbation therefore gives a bifurcation. Understanding bifurcations is therefore about understanding how the classes $\mathcal{C}$ fit together in parameter space. This leads us to ask the following question:

Question 1.

For every $\xi\in\Xi_{d}$, what are the $\mathcal{C}$ that intersect every neighborhood of $\xi$?

In this paper, the author has chosen to express the answer to this question by showing that each bifurcation is a composition of a finite number of ”moves,” that is, simpler bifurcations whose types are characterizable.

Stability

The structurally stable vector fields in $\Xi_{d}$, i.e. those that do not change qualitatively under perturbation, are those with neither homoclinic separatrices nor multiple equilibrium points Sent ; TLD09 and are of full dimension in parameter space. The bifurcation locus in $\Xi_{d}$ consists of all vector fields with at least one homoclinic separatrix or multiple equilibrium point Sent ; TLD09 . One can see this intuitively by imagining a multiple equilibrium point splitting into several equilibrium points or a homoclinic separatrix breaking under perturbation.

Bifurcations of (2) can only involve breakings of homoclinics and/or splittings of multiple points due to the following result from TLD09 , which proves that landing separatrices are stable. In other words, an equilibrium point can not ”lose” a landing separatrix under small perturbation, as long as its multiplicity is preserved.

Theorem 3.1 (Dias, Tan).

Let $\zeta^{0}$ be an equilibrium point for $\xi_{0}\in\Xi_{d}$, and let $\zeta$ be an equilibrium point for $\xi\in\Xi_{d}$ in a sufficiently small neighborhood of $\xi_{0}$ such that $\text{mult}(\zeta^{0})=\text{mult}(\zeta)$ and $\lim\limits_{{P}\rightarrow{P_{0}}}\zeta=\zeta^{0}$. If the separatrix $s_{\ell}$ for $\xi_{0}$ lands at $\zeta^{0}$, then the separatrix $s_{\ell}$ (same $\ell$) for $\xi$ lands at $\zeta$.

Multiplicity-preserving Bifurcations

The bifurcations which allow splitting of multiple equilibrium points are more complicated than the multiplicity-preserving bifurcations, and not only because they involve the variation of more parameters. For splitting bifurcations, the possible changes in topological structure may depend on the initial analytic data, in addition to the initial topological data TLD09 . The bifurcations with this added complexity are to be studied in a future paper; the aim in this paper is to analyze the restricted question:

Question 2.

Given any point $\xi\in\Xi_{d}$ in the bifurcation locus, what are the possible bifurcations such that the multiplicities of the equilibrium points are preserved?

Such bifurcations are called multiplicity-preserving bifurcations.

Definition 2.

Let $\zeta^{0}_{i}$, $i=1,\dots,N$ be the equilibrium points for $\xi_{0}\in\mathcal{C}_{0}$, not counting multiplicity. The local multiplicity-preserving set of $\xi_{0}$, denoted $\mathbf{LMP}(\xi_{0})$, is the set of all vector fields $\xi\in\Xi_{d}\setminus\mathcal{C}_{0}$, in a sufficiently small neighborhood of $\xi_{0}\in\Xi_{d}$ such that $\text{mult}(\zeta_{i})=\text{mult}(\zeta^{0}_{i})$, $i=1,\dots,N$.

Definition 3.

The multiplicity-preserving set of $\xi_{0}$, denoted $\mathbf{MP}(\xi_{0})$, is the set of all combinatorial classes $\mathcal{C}$ whose intersection with $\mathbf{LMP}(\xi_{0})$ is non-empty.

Note that the notation above differs from that in TLD09 .

Remark 2.

If $\xi$ has at least one homoclinic separatrix, then $\mathbf{MP}(\xi)$ is non-empty.

Remark 3.

Since the multiplicities are preserved, $\mathbf{LMP}(\xi_{0})$ can be locally parameterized by the $N$ roots (not counting multiplicity), and in fact can be locally parameterized by $N-1$ roots due to centering of the polynomial.

3.1 Deformations Give Bifurcations

In this subsection, we recall a result from TLD09 , which proves that each $\mathcal{C}$ is a manifold. While the result as stated is of minor importance for this paper, the construction in the proof implies that piecewise-linear deformations in rectifying coordinates do give ”nearby” vector fields in our space $\Xi_{d}$, and these deformations are (locally) parameterized by the pseudo-invariants (6).

Parameterizing a Class

Within a class $\mathcal{C}$, vector fields have the same zone types (strip, half-plane, cylinder) with the same labelling, but the analytic invariants $\alpha_{i}\in\mathbb{H}_{+}$, $i=1,\dots,s$, and $\tau_{i}\in\mathbb{R}_{+}$, $i=1,\dots,h$, will be different (see Figure 7).

This idea is formalized in the following theorem from TLD09 .

Theorem 3.2 (Dias).

There exists a real analytic isomorphism $F:\mathbb{H}^{s}\times\mathbb{R}_{+}^{h}\rightarrow\mathcal{C}$, which is $\mathbb{C}$-analytic in the first $s$ coordinates and $\mathbb{R}$-analytic in the last $h$ coordinates. It is the restriction of a holomorphic mapping in $(s+h)$ complex variables: $\widetilde{F}:\mathbb{H}^{s}\times V_{+}^{h}\rightarrow\Xi_{d}$, where $V_{+}$ is an $\epsilon$ neighborhood of $\mathbb{R}_{+}$.

In particular, $\widetilde{F}:(\underline{\widetilde{\alpha}},\underline{\widetilde{\tau}})\mapsto\xi_{(\underline{\widetilde{\alpha}},\underline{\widetilde{\tau}})}\in\Xi_{d}$ is holomorphic (in $s+h$ variables) in a neighborhood of $(\underline{\alpha}_{0},\underline{\tau}_{0})\in\mathbb{H}^{s}\times\mathbb{R}_{+}^{h}\subset\mathbb{C}^{s+h}$.

The proof of Theorem 3.2 is briefly summarized for its use in this manuscript. For details, see TLD09 . A vector field $\xi\in\Xi_{d}$ is in a combinatorial class $\mathcal{C}$ and has a corresponding set of analytic invariants $(\underline{\alpha},\underline{\tau})\in\mathbb{H}^{s}\times\mathbb{R}_{+}^{h}$. This determines a glued configuration of half-planes, strips, and cylinders, in which the vector field is $\frac{d}{dz}$. The rectified zones (as sets) are deformed by piecewise linear mappings given by allowing the $\tau$ to leave the real axis (see Figure 6), corresponding to a set of pseudo-invariants $\left(\underline{\widetilde{\alpha}},\underline{\widetilde{\tau}}\right)$. The gluing gives a Riemann surface conformally isomorphic to the Riemann sphere with punctures. Endowing the deformed zones with $\frac{d}{dz}$ leads to a polynomial vector field $\xi_{(\underline{\widetilde{\alpha}},\underline{\widetilde{\tau}})}\in\Xi_{d}$ on the Riemann sphere with equilibrium points at the punctures. The holomorphic dependence of parameters in the Measurable Riemann Mapping Theorem gives that $\xi_{(\underline{\widetilde{\alpha}},\underline{\widetilde{\tau}})}$ depends holomorphically on the $(\underline{\widetilde{\alpha}},\underline{\widetilde{\tau}})$.

4 Deformations in Rectifying Coordinates

The results in the above section show that deformations in rectifying coordinates correspond to bifurcations in the family $\Xi_{d}$, in particular, those where homoclinic separatrices break in various ways. We will see in examples below that the resulting change in separatrix structure can be read from the deformed rectifying coordinates. We will prove in this section that these deformations ”cover” the local multiplicity-preserving set, meaning that every multiplicity-preserving bifurcation can be seen as a deformation in rectifying coordinates.

Example 1 - The Breaking of one Homoclinic Separatrix

Consider the example depicted in Figure 8. The top row corresponds to a combinatorial configuration (in the disk and in rectifying coordinates) in $\Xi_{3}$ which has one sink, one source, and one center. The separatrices $s_{2}$ and $s_{3}$ are landing, and there is one homoclinic separatrix $s_{1,0}$. Perturbing the single real analytic invariant $\tau$ to have non-zero imaginary part results in breaking the single homoclinic separatrix. The center becomes a sink or source, and one can see where the separatrices now land by looking in the rectifying coordinates.

We explain the general situation when exactly one homoclinic separatrix $s_{k,j}$ breaks. A homoclinic separatrix $s_{k,j}$ is on the boundary of exactly two zones: one ”upper” and one ”lower” (in rectifying coordinates). Allowing the single $\widetilde{\tau}$ corresponding to $s_{k,j}$ to vary holomorphically causes the separatrices $s_{k}$ and $s_{j}$ to land after perturbation. If $\widetilde{\tau}\in\mathbb{H}_{+}$, the separatrix $s_{k}$ lands at the equilibrium point on the boundary of the lower zone, and $s_{j}$ lands at the equilibrium point on the boundary of the upper zone. If $\widetilde{\tau}\in\mathbb{H}_{-}$, the separatrix $s_{k}$ lands at the equilibrium point on the boundary of the upper zone, and $s_{j}$ lands at the equilibrium point on the boundary of the lower zone. The equilibrium point at which $s_{k}$ (resp. $s_{j}$) lands after perturbation is either a sink (resp. source) or multiple equilibrium point, depending on whether the zones having $s_{k,j}$ on the boundary are vertical half-strips or strips in the first case, or in the latter case, a half-plane.

Example 2 - Forming a New Homoclinic Separatrix

If more than one pseudo-invariant $\widetilde{\tau}$ vary at the same time, more complicated things can happen. In particular, new homoclinic separatrices can form. Figure 9 shows an example of a degree $d=4$ combinatorial configuration before perturbation (left) and after perturbation (right), in the disk model (top) and in rectifying coordinates (bottom). Before perturbation, there is one double equilibrium point, receiving two landing separatrices $s_{2}$ and $s_{3}$, and two centers, whose boundaries consist of one homoclinic separatrix each: $s_{1,0}$ and $s_{5,4}$ with lengths $\tau_{1}$ and $\tau_{2}$ respectively. In rectifying coordinates, this corresponds to two half-planes (only one is shown) and two cylinders (the dashed vertical half lines should be seen as identified under horizontal translation). Perturbing such that $\Im(\widetilde{\tau_{1}})+\Im(\widetilde{\tau_{2}})=0$ (right), $s_{1,0}$ and $s_{5,4}$ both break, and a new homoclinic separatrix $s_{1,4}$ forms. The separatrices $s_{0}$ and $s_{5}$ land after perturbation at a source and a sink respectively, which were centers before perturbation.

We now describe the general requirement for a homoclinic separatrix to form under perturbation. In order to understand this situation, we need to define H-chains.

Definition 4.

An H-chain of length $n$ is a sequence of $n$ homoclinic separatrices $\{s_{k_{i},j_{i}}\}$, $i=1,\dots,n$, such that for each $i=1\leq n-1$, either $k_{i+1}=j_{i}+1$ or $k_{i+1}=j_{i}-1$ mod $2(d-1)$ (see Figure 5, where $\{s_{7,6},s_{5,4}\}$ is an H-chain of length $2$).

That is, for each $i=1,\dots,n-1$, the pair $s_{k_{i},j_{i}}$ and $s_{k_{i+1},j_{i+1}}$ are on the boundary of the same zone and are consecutive on the boundary reading left to right in rectifying coordinates (see Figure 10).

The H-chains tell us exactly which homoclinic separatrices can form under perturbation, as explained in the proposition below.

Proposition 1.

Let $\xi_{0}\in\Xi_{d}$ have at least two homoclinic separatrices, and let $\xi\in\Xi_{d}$ be a perturbation of $\xi_{0}$. Suppose $s_{k,j_{0}}$ and $s_{k_{0},j}$ are two homoclinic separatrices for $\xi_{0}$. The separatrix $s_{k,j}$ for $\xi$, where $k$ and $j$ are the same as in $s_{k,j_{0}}$ and $s_{k_{0},j}$, can form under perturbation if and only if $s_{k,j_{0}}$ and $s_{k_{0},j}$ belong to a common H-chain where $s_{k,j_{0}}$ comes before $s_{k_{0},j}$ in the H-chain.

For an example, see Figure 12 where $s_{5,4}$ and $s_{13,12}$ have an H-chain in common before perturbation, and $s_{5,12}$ forms after perturbation.

Proof.

Without loss of generality, we may assume the H-chain is of the form

[TABLE]

For $i=2,\dots,n$, there is a sequence $I_{i}$ of length $n-1$ with elements in $\{+,-\}$ corresponding to whether $k_{i+1}=j_{i}+1$ or $k_{i+1}=j_{i}-1$, $i=1,\dots,n-1$ ($I_{1}$ not defined). If there are $q$ sign changes in this itinerary, then there are $q+1$ zones that $s_{k}$ needs to pass through to reach $s_{j}$ (see Figure 11). The separatrix $s_{k,j}$ will form under perturbation if the following conditions on perturbations of the associated $\tau_{i}$, $i=1,\dots,n$ are satisfied: For $i=1,\dots,n-1$,

• •

if $I_{i+1}=+$, then $\sum\limits_{j=1}^{i}\Im(\widetilde{\tau}_{j})<0$;

• •

if $I_{i+1}=-$, then $\sum\limits_{j=1}^{i}\Im(\widetilde{\tau}_{j})>0$;

• •

and $\sum\limits_{j=1}^{n}\Im(\widetilde{\tau}_{j})=0$,

(see again Figure 11). These conditions on the partial sums ensure that the new separatrix will not leave a zone before it is desired.

If there is no H-chain where $s_{k,j_{0}}$ comes before $s_{k_{0},j}$, then there is no overlapping sequence of zones in rectifying coordinates through which $s_{k}$ can have access to $s_{j}$. The converse then follows from Proposition 2 below, which proves that any multiplicity-preserving bifurcation can be realized as a deformation in rectifying coordinates. ∎

In general, several homoclinic separatrices can form simultaneously under perturbation, if the conditions on the partial sums as in Proposition 1 are all satisfied (see Figure 12).

4.1 Deformations in Rectifying Coordinates Cover the Local Multiplicity-preserving Set

We prove that all multiplicity-preserving bifurcations can be seen by deformations in the rectifying coordinates. In other words, we want to prove that varying the $\widetilde{\tau}$ covers all multiplicity-preserving bifurcations.

Proposition 2.

Let $N$ be the number of equilibrium points of $\xi_{0}$, not counting multiplicity. The map $(\widetilde{\alpha}_{1},\dots,\widetilde{\alpha}_{s},\widetilde{\tau}_{1},\dots,\widetilde{\tau}_{h})\mapsto(\zeta_{1},\dots,\zeta_{N-1})\in\mathbf{LMP}(\xi_{0})$, the local multiplicity-preserving set at $\xi_{0}$, is locally surjective.

Proof.

Due to centering, only $N-1$ roots are independent, hence one can locally parameterize the local multiplicity-preserving set by $N-1=s+h$ of the roots. It is enough to show that the map $(\zeta_{1},\dots,\zeta_{N-1})\mapsto(\widetilde{\alpha}_{1},\dots,\widetilde{\alpha}_{s},\widetilde{\tau}_{1},\dots,\widetilde{\tau}_{h})$ is (locally) well-defined and continuous for all $(\zeta_{1},\dots,\zeta_{N-1})\in\mathbf{LMP}(\xi_{0})$, the local multiplicity-preserving set of $\xi_{0}$. This is trivially true by noting that by Definition 6,

[TABLE]

These integrals are locally well-defined and depend continuously on the $(\zeta_{1},\dots,\zeta_{N-1})$.

∎

Therefore, to understand the multiplicity-preserving bifurcations, it is enough to analyze deformations in rectifying coordinates.

5 Multiplicity-preserving Bifurcations Can Be Decomposed into Compositions of Rank 1 Bifurcations

In this section, it is proved that every multiplicity-preserving bifurcation can be realized as a composition of simpler bifurcations, the rank 1 bifurcations (to be defined). The rank 1 bifurcations will be characterized in Section 6, showing they can not be arbitrarily complicated.

5.1 Dimension and Codimension of a Class

Note that Theorem 3.2 gives the (real) dimension of a combinatorial class.

Proposition 3.

For a combinatorial class $\mathcal{C}\in\Xi_{d}$, with $s$ strips, $h$ homoclinic separatrices, and $m^{\ast}=\sum_{i}(\text{mult}(\zeta_{i})-1)$,

[TABLE]

Proof.

Since $\Xi_{d}\simeq\mathbb{C}^{d-1}$, it follows $\dim_{\mathbb{R}}(\Xi_{d})=2d-2$. A combinatorial class $\mathcal{C}$ is analytically isomorphic to $\mathbb{H}^{s}\times\mathbb{R}_{+}^{h}$, giving $\dim_{\mathbb{R}}(\mathcal{C})=2s+h$. Let $N$ be the number of equilibrium points not counting multiplicity. Then $m^{\ast}=d-N$. Putting this together with $s+h=N-1$ gives $s=d-m^{\ast}-1-h$. Therefore, the (real) codimension of each class as a subset of parameter space is

[TABLE]

∎

Definition 5.

The boundary $\partial\mathcal{C}$ of a combinatorial class $\mathcal{C}$ is defined in the usual sense: the closure $\overline{\mathcal{C}}$ in $\Xi_{d}$, minus $\mathcal{C}$.

The following lemma proves a statement crucial to the proof of Theorem 5.1. Namely, for multiplicity-preserving sets, if one class intersects the boundary of another, then it must be entirely contained in the boundary. This does not generally hold for bifurcations which allow splitting of multiple points and would not be directly applicable to that case.

Lemma 1.

Let $\xi_{0}\in\mathcal{C}_{0}\subset\Xi_{d}$ have at least one homoclinic separatrix so that $\mathbf{MP}(\xi_{0})$ is non-empty. For every $\mathcal{C}\subset\mathbf{MP}(\xi_{0})$, it holds that $\mathcal{C}_{0}\subset\partial\mathcal{C}$.

Proof.

Necessarily $\mathcal{C}_{0}\cap\partial\mathcal{C}\neq\emptyset$. Then $\mathcal{C}_{0}\subset\partial\mathcal{C}$, since by Theorem 3.2 the combinatorics of the multiplicity-preserving bifurcations do not depend on initial homoclinic length, but only the relative imaginary parts of the perturbed $(\widetilde{\alpha}_{1},\dots,\widetilde{\alpha}_{s},\widetilde{\tau}_{1},\dots,\widetilde{\tau}_{h})$. ∎

Corollary 1.

It follows that $\mathcal{C}_{0}$ has strictly greater codimension than all $\mathcal{C}\subset\mathbf{MP}(\xi_{0})$.

5.2 Rank $k$ Bifurcations

The following notation will be used: a subscript of [math] denotes before perturbation, and a subscript of $1$ denotes after perturbation.

Definition 6.

A rank $k$ bifurcation is a bifurcation from $\xi_{0}\in\mathcal{C}_{0}$ to $\xi_{1}\in\mathcal{C}_{1}$ such that $\dim_{\mathbb{R}}(\mathcal{C}_{1})-\dim_{\mathbb{R}}(\mathcal{C}_{0})=k$.

Remark 4.

The author prefers the terminology ”rank” over ”codimension” of a bifurcation, since bifurcations of complex polynomial vector fields that allow splitting of multiple points (to be analyzed in a future paper) can have ”rank 0” bifurcations, i.e. two loci with the same dimension can be adjacent, and using the terminology ”codimension 0” would suggest that no bifurcation occurs in this case.

The rank $k$ bifurcations can be loosely characterized as follows by using the codimension. Recall that for all vector fields in a combinatorial class $\mathcal{C}$, $s$ is the number of strips, $h$ is the number of homoclinic separatrices, and $m^{\ast}=\sum_{i}(\text{mult}(\zeta_{i})-1)$, where $s,\ h,\ m^{\ast}\in\{0,\dots,d-1\}$. Proposition 3 states that $\dim_{\mathbb{R}}(\mathcal{C})=2s+h$ and ${\rm codim}_{\mathbb{R}}(\mathcal{C})=2m^{\ast}+h$. For analyzing multiplicity-preserving bifurcations, the codimension proves more useful. Using Proposition 3, a rank $k$ bifurcation is when $2(m^{\ast}_{0}-m^{\ast}_{1})+(h_{0}-h_{1})=k$. Since multiplicities are constant, i.e. $m^{\ast}_{0}=m^{\ast}_{1}$, this simplifies to $h_{0}-h_{1}=k$. In words, the number of homoclinic separatrices before bifurcation is $k$ greater than the number of homoclinic separatrices after bifurcation. Therefore, a rank $k$ multiplicity-preserving bifurcation is such that $h\leq h_{0}$ homoclinic separatrices break and $h-k$ new homoclinic separatrices form.

The behavior can be further narrowed down by the following proposition.

Proposition 4.

There are no rank $k\leq 0$ multiplicity-preserving bifurcations.

Proof.

There can not be more homoclinic separatrices after perturbation than existed initially, i.e. $h_{1}\leq h_{0}$. This is because there is a fixed number $2(d-1)$ of asymptotic directions and landing separatrices are stable when multiplicities are fixed by Theorem 3.1. Therefore, there can be no rank $k<0$ multiplicity-preserving bifurcations. There can be no rank $k=0$ multiplicity-preserving bifurcations, since Corollary 1 implies that the dimension of $\mathcal{C}_{0}$ is strictly less than the dimension of $\mathcal{C}_{1}$. ∎

5.3 Compositions of Rank 1 Bifurcations

By Definition 6, a rank 1 bifurcation is a bifurcation from $\xi_{0}\in\mathcal{C}_{0}$ to $\xi_{1}\in\mathcal{C}_{1}$ such that $\dim_{\mathbb{R}}(\mathcal{C}_{1})-\dim_{\mathbb{R}}(\mathcal{C}_{0})=1$. The multiplicity-preserving, rank 1 bifurcations are the bifurcations such that $h_{0}-h_{1}=1$, meaning that $h\leq h_{0}$ homoclinic separatrices break and $h-1$ new ones form. See Figures 8, 9, 13, and 14 for examples.

Theorem 5.1.

Every multiplicity-preserving bifurcation can be realized as a composition of rank 1 bifurcations.

The idea of the proof is to show that every multiplicity-preserving bifurcation is accessible through a sequence of rank 1, multiplicity-preserving bifurcations. To do this, we will show the existence of a sequence of classes such that

all are contained in the multiplicity-preserving set, 2. 2.

each class is contained in the boundary of the next, and 3. 3.

each differs in dimension by 1 (see Figure 15).

We first prove that if $\mathcal{C}\subset\mathbf{MP}(\xi_{0})$, then all of $\mathcal{C}$’s relevant boundary components are also in the multiplicity-preserving set $\mathbf{MP}(\xi_{0})$.

Lemma 2.

For all $\mathcal{C}\subset\mathbf{MP}(\xi_{0})$, it holds that every component $\mathcal{C}_{1}$ of $\partial\mathcal{C}$ with $\xi_{0}\in\partial\mathcal{C}_{1}$, $\mathcal{C}_{1}\subset\mathbf{MP}(\xi_{0})$.

Proof.

It will prove to be almost trivially true by the observation that equilibrium points can not coalesce under small perturbation. The contrapositive is proved: for every $\mathcal{C}\subset\Xi_{d}$ with $\xi_{0}\in\partial\mathcal{C}$, and for every $\mathcal{C}_{1}$ such that $\mathcal{C}_{1}\cap\partial\mathcal{C}\neq\emptyset$ and $\xi_{0}\in\partial\mathcal{C}_{1}$, if $\mathcal{C}_{1}\not\subset\mathbf{MP}(\xi_{0})$, then $\mathcal{C}\not\subset\mathbf{MP}(\xi_{0})$. Assume $\mathcal{C}_{1}$ intersects the boundary of $\mathcal{C}$, and they both have $\xi_{0}$ in the boundary. If $\mathcal{C}_{1}$ is in the splitting set of $\xi_{0}$, then it means that any vector field in $\mathcal{C}_{1}$ has more zeros than $\xi_{0}$. Since $\mathcal{C}_{1}$ intersects (or is contained in) the boundary of $\mathcal{C}$, then there are points of $\mathcal{C}$ arbitrarily close to points in $\mathcal{C}_{1}$. Hence the number of zeros for a vector field in $\mathcal{C}$ must be greater than or equal to the number of zeros of a vector field in $\mathcal{C}_{1}$ since equilibrium points can not merge under perturbation. Therefore, the number of zeros in $\mathcal{C}$ are greater than the number in $\xi_{0}$, proving that $\mathcal{C}$ belongs to the splitting set of $\xi_{0}$. ∎

Now we can finish the main proof.

Proof of Theorem 5.1.

We will show that for every $\mathcal{C}\subset\mathbf{MP}(\xi_{0})$, there exists a sequence of classes $\mathcal{C}_{i}$, $i=1,\dots,k-1$ such that $\mathcal{C}_{i}\subset\mathbf{MP}(\xi_{0})$ and $\partial\mathcal{C}\supset\mathcal{C}_{k-1},\dots,\partial\mathcal{C}_{2}\supset\mathcal{C}_{1},\partial\mathcal{C}_{1}\supset\mathcal{C}_{0}$ and $\mathcal{C}_{i-1}$ to $\mathcal{C}_{i}$ and $\mathcal{C}_{k-1}$ to $\mathcal{C}$ are rank 1 bifurcations. Specifically, it will be proved that for every $\mathcal{C}$ with $\xi_{0}\in\partial\mathcal{C}$ and $\dim_{\mathbb{R}}(\mathcal{C})-\dim_{\mathbb{R}}(\mathcal{C}_{0})=k$, $k>1$, there exists $\mathcal{C}_{1}\subset\mathbf{MP}(\xi_{0})$ such that:

$\dim_{\mathbb{R}}(\mathcal{C})>\dim_{\mathbb{R}}(\mathcal{C}_{1})>\dim_{\mathbb{R}}(\mathcal{C}_{0})$, and 2. 2.

$\mathcal{C}_{1}\subset\partial\mathcal{C}$.

Let $V_{0}$ be a neighborhood of $\xi_{0}$ in $\Xi_{d}$. The set $\mathcal{C}_{0}$ can not be a dense subset of $\partial\mathcal{C}\cap V_{0}$ by the assumption that $\dim_{\mathbb{R}}(\mathcal{C})-\dim_{\mathbb{R}}(\mathcal{C}_{0})>1$. There must therefore be some other $\mathcal{C}_{1}$ such that $\partial\mathcal{C}\cap\mathcal{C}_{1}\neq\emptyset$ and $\xi_{0}\in\partial\mathcal{C}_{1}$. Next, $\mathcal{C}_{1}\subset\mathbf{MP}(\xi_{0})$ by Lemma 2, and item 2. follows from Lemma 1. Finally, $\mathcal{C}_{1}\subset\partial\mathcal{C}$ implies $\dim_{\mathbb{R}}(\mathcal{C}_{1})<\dim_{\mathbb{R}}(\mathcal{C})$ by Corollary 1. ∎

6 Characterization of the Rank 1 Multiplicity-preserving Bifurcations

The result of Theorem 5.1 is not helpful if the rank 1 bifurcations can be arbitrarily complicated. We characterize the rank 1 bifurcations in this section to show that this is not the case. It will be proved that all rank 1 bifurcations are of essentially the same type as the examples in Figures 13 and 14. We first characterize that type.

Definition 7.

A chained homoclinic breaking is a bifurcation where a sequence $s_{k_{1},j_{1}},...,s_{k_{n},j_{n}}$ of $n\geq 1$ homoclinic separatrices break such that

for each $i=1,\dots,n-1$, $s_{k_{i},j_{i}}$ and $s_{k_{i+1},j_{i+1}}$ are on the boundary of the same zone before perturbation, 2. 2.

$n-1$ homoclinic separatrices form: $s_{k_{1},j_{2}},s_{k_{2},j_{3}},...,s_{k_{n-1},j_{n}}$, 3. 3.

$s_{j_{1}}$ lands at the $\alpha$-limit (equilibrium) point on the boundary of the zone to the right of $s_{k_{1},j_{1}}$, and $s_{k_{n}}$ lands at the $\omega$-limit (equilibrium) point on the boundary of the zone to the left of $s_{k_{n},j_{n}}$ if $\widetilde{\tau}_{k_{1},j_{1}}\in\mathbb{H}_{-}$ (similar for $\widetilde{\tau}_{k_{1},j_{1}}\in\mathbb{H}_{+}$), and 4. 4.

the separatrix graph is otherwise unchanged.

Interestingly, one can also see that this type of bifurcation inserts a strip zone with the new homoclinics on its boundary (see Figures 13 and 14).

We can now present the second main theorem of this paper.

Theorem 6.1.

Every rank 1, multiplicity-preserving bifurcation is a chained homoclinic breaking.

In order to prove this theorem, we will build new objects: homoclinic graphs (H-graphs, for short). These graphs are to be defined such that an admissible path in the H-graph corresponds to a homoclinic separatrix that can appear under perturbation, and the vertices of this path correspond to the homoclinic separatrices that must break to form that new homoclinic.

6.1 H-graphs

H-graphs are directed graphs, based on the initial homoclinic separatrix configuration, to help organize the possible multiplicity-preserving bifurcations. They are to be defined such that:

• •

an admissible path (to be defined) corresponds to a homoclinic separatrix that can appear under perturbation, and

• •

the vertices of this path correspond to the homoclinic separatrices that must break in order for the new homoclinic to form.

E.g. a single admissible path using $k+1$ vertices corresponds to a rank $k$ bifurcation, since $k+1$ homoclinics break and only one new one forms.

Definition 8.

An H-graph (see Figure 16) corresponding to a combinatorial class $\mathcal{C}$ is a directed graph which is embedded in the combinatorial disk model such that:

The vertices of the H-graph correspond to the homoclinic separatrices for $\mathcal{C}$. 2. 2.

Each edge is contained in a connected component of the disk minus the homoclinic separatrices 3. 3.

There is an edge directed from $v_{1}$ to $v_{2}$ if there is an H-chain containing both corresponding homoclinic separatrices, where the homoclinic for $v_{1}$ comes before the homoclinic for $v_{2}$.

Remark 5.

Within a connected component that does not correspond to a cylinder, all of the directed edges of the H-graph in that component must run in the same direction as the H-chain on the boundary (see the component in Figure 16 with $s_{5,6}$, $s_{7,0}$, and $s_{1,2}$ on the boundary). In a component that does correspond to a cylinder, some edges in the H-graph may run contrary to the orientation of the H-chain, since one can ’wrap around’ the cylinder (see Figure 16 where the directed edge $s_{7,0}\rightarrow s_{17,12}$ runs in the same direction as the H-chain along the boundary, and the directed edge $s_{17,12}\rightarrow s_{7,0}$ runs contrary to the orientation of the H-chain on the boundary).

Admissible Paths

Consider the case where a homoclinic separatrix can form from homoclinics on the boundary of a single zone. Though more homoclinics may break under perturbation, only two are required to break for this new homoclinic to form (see Figure 14). If a homoclinic forms from an H-chain that involves more than one zone, two homoclinic separatrices from each zone are required to break since in total: the start homoclinic separatrix, the end homoclinic separatrix, and all the homoclinic separatrices in between that join the consecutive zones (one to enter the zone, one to leave the zone) must break (see Figures 11 and 12). These observations lead us to define admissible paths in the H-graph (see Figure 17).

Definition 9.

An admissible path is a path in the H-graph such that it respects the orientation of the edges of the H-graph and uses only one edge per connected component of the disk minus the homoclinic separatrices.

E.g. the path in Figure 16 from $s_{5,6}\rightarrow s_{7,0}\rightarrow s_{1,2}$ would not be an admissible path since it uses two edges in a single connected component.

Simultaneous Paths

The H-graph was designed so that paths correspond to single homoclinic separatrices that can appear under perturbation. To consider the possibility when more than one homoclinic may form at once, simultaneous paths in the H-graph need to be considered (see Figure 17). Union path graphs are defined as allowed unions of admissible paths.

Definition 10.

A union graph is a union of admissible paths such that:

• •

No pair of paths can share a start or end vertex, i.e. there can not be a vertex which is a start vertex for two paths or an end vertex for two paths (see Figure 18).

• •

Each vertex in the union of these admissible paths has a well-defined direction through it, explained further in the following. Consider the vertex in question of the union of admissible paths. This vertex corresponds to a homoclinic separatrix in the initial configuration, and there are two connected components (zones) in the disk that have this homoclinic on the boundary. By ”well-defined direction” through the vertex, we mean that in one connected component, all edges of the union of paths must be incoming to the vertex, and in the other connected component, all edges of the union of paths must be outgoing from that vertex (see Figure 19 for a violation of this criterion).

We explain the need for the two conditions in the definition. If two paths shared a start (resp. end) vertex, it would be the same as saying $s_{k,j_{1}}$ and $s_{k,j_{2}}$ (resp. $s_{k_{1},j}$ and $s_{k_{2},j}$) could form simultaneously, but it is not possible for two homoclinic separatrices to share an asymptotic direction. If the vertices do not have a well-defined direction through them, it corresponds to a single $\widetilde{\tau}$ taking on a value in $\mathbb{H}_{+}$ and $\mathbb{H}_{-}$ simultaneously, which is not possible (see Figures 19 and 18).

Remark 6.

We do not claim that every union graph defined as above corresponds to a possible simultaneous formation of homoclinic separatrices, only that a simultaneous formation of homoclinics must be of that form.

Proposition 5.

The union graph contains neither directed nor undirected cycles.

Proof.

Suppose the union graph does contain a cycle. Pick a vertex $v$ in the cycle and let $e_{1}$ and $e_{2}$ be the edges in the cycle that meet at $v$.

Case I:

The cycle contains a vertex $v$ such that $e_{1}$ is incoming to $v$ and $e_{2}$ is outgoing from $v$ (see left of Figure 20). Now $e_{1}$ and $e_{2}$ must be on opposite sides of the homoclinic that goes through $v$, otherwise this would violate direction flow (the second condition in Definition 10). Since each homoclinic must traverse the entire disk, it must cut the cycle somewhere else besides at $v$. We arrive at a contradiction since a homoclinic can not cross two vertices, nor can it cross an edge in the H-graph (see left of Figure 20). Therefore, there can be no cycles in the union graph with such a vertex. This eliminates the possibility of a cycle having an odd number of edges or exactly two edges, which would necessarily have such a vertex.

Case II:

For all vertices $v$ in the cycle, either both $e_{1}$ and $e_{2}$ are incoming to $v$ or both are outgoing from $v$ (since not in Case I). This implies there are an even number of edges in the cycle, and the orientation of the edges is alternating (see right of Figure 20). Furthermore, any such cycle would have to be entirely contained in a single connected component of the disk minus the initial homoclinic configuration. Otherwise, a homoclinic would cross its interior, and this is impossible by the same argument as in Case I. The orientation of all homoclinics on the boundary of the connected component is the same (connected component to the left of all homoclinics, or the connected component is to the right of all homoclinics on the boundary). In fact, this connected component must be a cylinder zone, since the cycle has alternating orientation of edges and therefore must contain edges contrary to the orientation of the H-chain on the boundary (see Remark 5). We will use two edges in the cycle which have ’opposite’ orientation to the orientation of the H-chain on the boundary to arrive at a contradiction. There must exist at least two such edges in every cycle with an even number of edges greater than 2 or we would be in Case I (see Figure 21).

In particular, any such cycle contains a segment of three edges in the H-graph, where the first and third are contrary to the orientation of the H-chain on the boundary and the second has the same orientation as the H-chain on the boundary (see Figure 21). We will focus on the two contrary edges in such a segment. Each of those two edges corresponds to two different new homoclinics that form, since no single path in the H-graph can contain two edges from the same connected component (see Definition 9). Pick one edge that has opposite orientation to the H-chain (e.g. $s_{m+1}\rightarrow s_{1}$ in Figures 21 and 22), and cut the corresponding cylinder in rectifying coordinates such that the first homoclinic in the H-chain is the starting homoclinic (vertex) for that edge in the H-graph (see top of Figure 22 where the cut is made at $s_{m+1}$).

If this edge ($s_{m+1}\rightarrow s_{1}$) is in the union graph, it corresponds to a new homoclinic (path in H-graph) which enters the cylinder at $s_{m+1}$ and leaves the cylinder at $s_{1}$ (possibly beginning at $s_{m+1}$ and/or stopping at $s_{1}$). In order for the new homoclinic (path in H-graph which contains the edge $s_{m+1}\rightarrow s_{1}$) to pass through $s_{m+1}$ and $s_{1}$ without leaving the cylinder in between, the ordered partial sums from left to right of the analytic invariants for the H-chain need to all be greater than $b\geq 0$, the height of the new homoclinic relative to the first vertex in the cut cylinder ($b$ would be determined by the sums of the other analytic invariants in earlier parts of the chain), and the final partial sum needs to be less than or equal to $b$ (see the top of Figure 22). The same condition must apply to the other opposite oriented edge in the cycle segment. This will lead to a contradiction. Let $s_{m+1}$ and $s_{1}$ be the corresponding start and end homoclinics for an opposite oriented edge in the H-graph (see Figure 21). Let $a_{m+1},\dots,a_{1}$ be the imaginary parts of the corresponding analytic invariants after perturbation. For the directed edge $s_{m+1}\rightarrow s_{1}$ to be realized, we need

[TABLE]

Now consider the other opposite oriented edge in the cycle segment with starting and ending homoclinics $s_{m+n+\ell+1}$ and $s_{m+n+1}$ respectively (see Figure 21). Let $c\geq 0$ be the height of the new homoclinic relative to the first vertex in this cut of the cylinder. Than for directed edge $s_{m+n+\ell+1}\rightarrow s_{m+n+1}$ to be realized, we need

[TABLE]

By the first set of inequalities that $a_{m+1}+\cdots+a_{m+n+\ell}>b$, and by the second set of inequalities, $a_{m+n+\ell+1}+\cdots+a_{N}+a_{1}>c$. Then the last inequality in the first set becomes $(>b)+(>c)\leq b$, a contradiction.∎

of Theorem 6.1.

Assume throughout the rest of the proof that only rank 1, multiplicity-preserving bifurcations are considered.

In order for a rank 1 bifurcation to occur, there must be $n$ admissible (simultaneous) paths that altogether use $n+1$ vertices. The union graph for rank 1 bifurcations must be connected, since any two (or more) disjoint paths would require at least 4 vertices, corresponding to rank 2 or more. The union graph is therefore a tree since it has no cycles by Proposition 5. The union graph for rank 1 bifurcations must actually be a path itself (only two leaves) by the following (see Figure 23). Each homoclinic separatrix has two asymptotic directions: one outgoing from infinity (odd), and one incoming to infinity (even). It follows that if $n+1$ homoclinics break, and $n$ are to form, only one odd and one even asymptotic direction can be lost in total. There can only be two leaves since each leaf of the union tree corresponds to a lost asymptotic direction. Furthermore, all the vertices besides the two leaves in the union of paths must be both the end of a path and the beginning of a (different) path.

Since each vertex has a well-defined direction through it, the union path has an orientation. Choose the leaf that is the tail of a directed edge – it is the beginning of a path. This path can only consist of one edge, since the second vertex must have a path end there and it can not come from the other direction because of the orientation of the union path. Unless the second vertex is also the last vertex of the union path, a second path must start at the second vertex, and end at the third vertex for the same reason. The same argument must hold for subsequent vertices, so all paths must be of length 1. This corresponds exactly to the type of bifurcation as described in the statement of the theorem. ∎

Acknowledgements.

There are many people to thank for patiently listening to various stages of this work and providing helpful suggestions. In particular, the author is indebted to Bodil Branner, Frederick Gardiner and the other participants in the Extremal Length seminar at CUNY Graduate Center, Christian Henriksen, Poul G. Hjorth, Louis Pedersen, and Tan Lei. The author would also like to express gratitude to the anonymous referee.

Conflict of interest

The authors declare that they have no conflict of interest.