## Abstract

Let us consider incompressible and inviscid flows in two-dimensional domains with multiple obstacles. The instantaneous velocity field becomes a Hamiltonian vector field defined from the stream function, and it is topologically characterized by the streamline pattern that corresponds to the contour plot of the stream function. The present paper provides us with a procedure to construct structurally stable streamline patterns generated by finitely many point vortices in the presence of the uniform flow. Starting from some basic structurally stable streamline patterns in a disc of low genus, i.e. a disc with a small number of holes, we repeat some fundamental operations that append a streamline pattern by increasing one genus to them. Owing to the inductive procedure, one can assign a sequence of operations as a representing word to each structurally stable streamline pattern. We also give the canonical expression for the word representation, which allows us to make a catalogue of all possible structurally stable streamline patterns in a combinatorial manner. As an example, we show all streamline patterns in the discs of genus 1 and 2.

## 1. Introduction

Vortex structures in the presence of uniform flow in exterior domains with multiple obstacles can be observed in flow phenomena arising in environmental flows and biofluids. For instance, it is important to predict the diffusion of contaminants advected by the flow in rivers and ocean flows, in which the flow domain contains many obstacles, such as waterbreaks, sandbanks and islands. Johnson & McDonald [1] considered the motion of vortices near multiple slit-shaped gaps to recognize the importance of the gaps in mid-ocean barriers to ocean circulations. They also studied the motion of a vortex near two cylindrical islands to understand how the topography of the flow domain affected the evolution of the vortex, and thus they suggested that it played a significant role in mass transportation in ocean flows [2]. In biofluids, it has been recognized numerically that interaction between the wings and the vortices shed from them gives an efficient flight of butterflies [3] and a slow vertical descend of plant seeds [4].

As a mathematical model to deal with these problems, we consider incompressible and inviscid flows in multiply connected domains. We impose the slip boundary condition to the flow along the boundaries of the domain. The inviscid model remains valid for the slightly viscous flows unless the flow is highly turbulent, since point vortices can approximate the vortex structures shed from the boundaries owing to the formation of thin boundary layers. Moreover, the model has a theoretical advantage owing to its mathematical simplicity. That is to say, when we identify the two-dimensional space with the complex plane , the inviscid flow in the two-dimensional space is represented by an analytic function called *the complex potential*, say *F*(*z*), from which the two-dimensional velocity field (*u*,*v*) is recovered owing to the formula *u*−*iv*=*F*^{′}(*z*). The imaginary part of the complex potential *ψ*(*z*)=Im[*F*(*z*)], which is known as *the stream function*, also has a significant physical meaning, since a contour line of this function is identical to a streamline of the flow. In addition, the motion of a particle located at (*x*_{0}(*t*),*y*_{0}(*t*)) at time *t* is described by d*x*_{0}/d*t*=∂*ψ*/∂*y* and d*y*_{0}/d*t*=−∂*ψ*/∂*x*, which indicates that *ψ*(*z*) gives rise to a Hamiltonian vector field for the particle evolution.

Complex potentials for a given multiply connected domain are obtained from those for a canonical multiply connected domain with the same multiplicity by constructing conformal mapping between these two domains, since the complex potentials are conformally invariant. Crowdy & Marshall [5] considered a domain in the unit circle that contains many circular boundaries, called *a circular domain*, for which the analytic formula of the complex potential for a point vortex has been provided. The complex potential for the uniform flow in exterior circular domains has also been constructed by Crowdy [6]. These complex potentials are described in terms of a transcendental function, called *the Schottky–Klein prime function*, that is defined associated with the radii and centres of the circular boundaries.

In view of the applications to the fluid problems stated above, we are interested in the potential flows generated by many point vortices in multiply connected exterior domains in the presence of uniform flow, which we refer to as *vortex flows*. In particular, in the present paper, we are concerned with the global topological structure of the streamline patterns generated by vortex flows and then we classify them in a unified manner. The importance of topological studies of irrotational flows has already been realized by Klein [7]. Furthermore, among all possible streamlines, we focus on *structurally stable* vortex flows. The structural stability here means that the qualitative behaviour of the streamlines is unchanged under small continuous perturbations, whose exact definition will be provided in the following section. The structurally stable flows are also physically significant since they are more likely to be observed in many real fluid flows. Topological classification of the streamline patterns has been investigated for vortex flows in an unbounded plane without boundaries [8] and on a sphere [9], although uniform flow was not contained in these studies. The present paper is not only an extension of the preceding works to vortex flows in multiply connected domains, but it also has a theoretical significance because of the following three reasons. First, the uniform flow adds a new streamline structure to the flow, which has not been considered so far. Secondly, owing to the existence of boundaries in the domain, we need to consider a streamline attached to these boundaries that gives rise to another structurally stable streamline pattern. Finally, even though we restrict our attention to the structurally stable velocity field, we obtain many non-trivial streamline patterns, since it has been shown that homoclinic connections generated by the divergence-free vector fields on two-dimensional manifolds are structurally stable [10]. Consequently, the structurally stable streamline patterns generated by the vortex flows with uniform flow in multiply connected domains can have various topological patterns, whose classification provides us with a significant catalogue of flow patterns arising in biofluids and environmental flows.

Construction of the streamline patterns proceeds as follows. Starting from some basic structurally stable flow patterns in a disc of low genus, we obtain the flow patterns in the multiply connected domain with higher genus by repeating fundamental operations that append a structurally stable streamline pattern by increasing one genus in the domain. Accordingly, one can assign a sequence of operations for a streamline pattern, which we call a *word representation of the streamline pattern*. Thanks to the word representation, we can make a catalogue of all possible structurally stable streamline patterns generated by the vortex flows in a combinatorial way.

This paper consists of five sections. In the next section, after reviewing some known results on complex potentials for a point vortex and uniform flow in multiply connected circular domains, we give two streamline patterns in the disc of genus 0 and 1 pattern in the annulus of genus 1. We show that they are only structurally stable streamline patterns to which the construction procedure is applied. In §3, we introduce five fundamental operations that add a structurally stable streamline pattern with one genus to the flow, with which we show how to construct streamline patterns in multiply connected domains with higher genus and how to assign their word representations. In §4, applying the construction procedure, we give all possible structurally stable streamline topologies and their word representation generated by the vortex flows in multiply connected domains of genus 1 and 2. The last section contains a summary and discussion.

## 2. Structurally stable vortex flows in multiply connected domains

Let us review some known results on complex potentials in multiply connected domains. Suppose that a circular domain, say , in the unit circle of the complex *ζ*-plane contains *M* circular boundaries. Then, one can introduce a transcendental function *ω*(*ζ*,*α*) for *ζ*, as
in which *Θ*^{′′} denotes an infinite set of Möbius maps defined from the radii and the centres of the circular boundaries. See Crowdy & Marshall [5,11] and Baker [12] for detailed definitions and properties.

Crowdy & Marshall [5] have provided analytic formulae of the complex potential *W*_{V}(*ζ*;*α*_{V}) for a point vortex located at with strength *κ*, which is represented by
Since as , the complex potential generates infinite circular closed streamlines in the neighbourhood of the point vortex, as in figure 1*a*. Note that as long as we are concerned with the topological structures of the streamlines, an elliptic centre and a circular boundary cannot be distinguished from a point vortex, since the circular closed streamlines around them are topologically equivalent to those around a point vortex.

Let denote an exterior multiply connected domain with *M*+1 circular boundaries in the complex *z*-plane. Then, the complex potential for uniform flow in the domain is obtained from a complex potential function in the circular domain by constructing a conformal mapping *z*=*g*(*ζ*) from to . Since is unbounded, the conformal mapping *g*(*z*) maps some point in , say *ζ*=*β*_{U}, to infinity of . Thus, the conformal mapping has the following form:
with a real constant *a*. Crowdy [6] showed that the complex potential *W*_{U}(*g*^{−1}(*z*);*β*_{U}) gave uniform flow with speed *U* and an inclined angle to the real axis *ϕ* of the complex *z*-plane, in which
Since the uniform flow *W*_{U}(*g*^{−1}(*z*);*β*_{U}) is analytic everywhere in except at infinity, it is represented by as . Since we are just interested in the topological structure of the streamlines, we may assume that *ϕ*=0 and *β*_{U}=0 without loss of generality. Thus, the complex potential *W*_{U}(*ζ*;0) behaves asymptotically as as . Since the leading singular term 1/*ζ* represents the complex potential for a source–sink pair at the origin, the streamlines in the neighbourhood of the source–sink pair correspond to the contour lines of the stream function in polar coordinates (*r*,*θ*) around the origin, which consist of infinite orbits departing from and returning to the source–sink pair as in figure 1*b*. Generally, the complex potential 1/*ζ*^{n} (*n*≥1) gives rise to the flow of the *n*-tuple source–sink pair located at the origin, whose stream function is given by in polar coordinates. Thus, we can generalize the definition of the complex potential for the *n*-tuple source–sink pair in the multiply connected circular domains as follows.

### Definition 2.1

A point is said to be an *n*-source–sink point, if is a vector field on generated by a stream function is denoted by *ψ*, for which there is a pair of a neighbourhood *U* of *p* and a homeomorphism *h* from *U* to the unit disc *D* with *h*(*p*)=0, such that in the polar coordinates associated with the disc *D*.

For the sake of later reference, we call a standard disc. It is easy to see that the complex potential *W*_{U}(*ζ*,0) gives the 1-source–sink point located at the origin of . Strictly speaking, since the set of streamlines converging to the *n*-source–sink point has a non-empty interior point, it is unable to define a Hamiltonian vector field at this point, which requires us to loosen the definition of Hamiltonian vector fields. Thus, we say a vector field *V* is a Hamiltonian vector field with the *n*-source–sink point *p*, if is a Hamiltonian vector field on . Hence, in order to discuss the stability of the streamline topologies of the Hamiltonian vector fields with the *n*-source–sink point on , we need to give the exact definition of their structural stability. Denote by the set of *C*^{r}-Hamiltonian vector fields with an *n*-source–sink point on with *C*^{r}-topology (*r*≥1). Then, we have the following definitions on the stability of the vector field in .

### Definition 2.2

For *s*≤*r*, is locally (*C*^{s})-structurally stable at in , if for any neighbourhood *U* of *p* and any Hamiltonian vector field , which is *C*^{s}-near of *V* in , is topologically equivalent to *V* . In other words, there is a homeomorphism such that *h* maps each orbit of *V* |_{U} to that of homeomorphically and it preserves the orientation of the orbits.

### Definition 2.3

is (*C*^{1})-structurally stable, if any vector field , which is *C*^{1}-near of *V* in , is topologically equivalent to *V* .

Let us note that we use the standard definition of the structurally stability for the *C*^{r}-Hamiltonian vector field, denoted by , without the *n*-source–sink point as will be stated in theorem 3.1. With these definitions, we prove the following proposition, which assures structural stability in the neighbourhood of the *n*-source–sink point locally in .

### Proposition 2.1

For *r*≥1, is locally structurally stable at the *n*-source–sink point *p*.

### Proof.

Let *H* be a Hamiltonian of *V* on and *D*_{h} a standard disc for *p*. Each streamline of *V* , except *p*, corresponds to a level set of *H* and it connects between either *p* and the boundary of *D*_{h}, or *p* and itself in *D*_{h}\{*p*}. Since *p* is the *n*-source–sink point and its stream function is represented by Im[1/*r*^{n}] in its neighbourhood, the absolute value of the gradient of the Hamiltonian is strictly positive, i.e. |∇*H*|≥*n*>0 in *D*_{h}\{*p*}. Hence, even if the vector field *V* is perturbed, holds in for its perturbed Hamiltonian and the perturbed *n*-source–sink point . Therefore, there is a small neighbourhood of and a topological equivalence , such that *h*|_{Dh\{p}} maps *V* to the Hamiltonian vector field of .

It is easy to see that the flows around a point vortex, an elliptic centre and a circular boundary are locally structurally stable in and . There is another different structurally stable streamline pattern in the neighbourhood of the boundary, which consists of the streamlines connected to 2*n* saddle points at the boundary for *n*≥1, as in figure 1*c*.

We consider the complex potential *W*(*ζ*) in the circular domain that consists of the 1-source–sink point at the origin and *N* point vortices as
2.1in which *α*_{m} denotes the location of the *m*th point vortex with strength *κ*_{m}. One can add the complex potential generating the circulation around the boundary, whose analytic formula has also been provided by Crowdy [6]. However, it is unnecessary to consider the complex potential, since it just generates closed orbits around the boundary that are topologically equivalent to those generated by a point vortex, and thus it adds no new topological streamline structure to the flow.

We are interested in the global topology of structurally stable streamline patterns of the flow generated by the complex potential (2.1). The genus *M* of the flow domain is equivalent to the sum of the numbers of elliptic centres, point vortices and circular boundaries contained in the domain. However, for the sake of simplicity, we regard *M* as the number of circular boundaries when considering streamline patterns, since closed circular orbits around an elliptic centre or a point vortex are topologically indistinguishable from those around a circular boundary. Let us also note that it is sufficient to classify the streamline topologies in the bounded circular domain , since it can be mapped to an exterior domain by conformal mapping, and the streamline patterns generated by *W*(*ζ*) are kept unchanged topologically under the action of the conformal mapping.

For the sake of reference, we name all components of the structurally stable streamlines generated by the complex potential *W*(*ζ*) in the multiply connected domain (figure 2). Streamlines departing from and returning to the 1-source–sink point are called *ss-orbits*. When an orbit connects between the 1-source–sink point and a saddle point at the boundary, we refer to the orbit and the saddle point as an *ss-∂-saddle connection* and an *ss-∂-saddle*. A ∂-saddle is a saddle at the boundary that is not connected to the 1-source–sink point. If two ∂-saddles are connected by a streamline, we say the connecting orbit is a *∂-saddle connection*. A streamline connecting a saddle point with the 1-source–sink point is called an *ss-saddle connection*. The other orbits consist of circular closed orbits, saddles and their homoclinic saddle connections.

Since we construct the structurally stable streamline patterns in from those in inductively, as will be explained in §3, we need initial streamline patterns. There are two fundamental streamline patterns in , both of which contain the 1-source–sink point at the origin. The first pattern in figure 3*a* contains two ss-∂-saddles that are connected to the 1-source–sink point via ss-∂-saddle connections, whose streamline topology is denoted by *I*. In the second pattern, there contains no ss-∂-saddle connection, but one saddle point with a homoclinic saddle connection and two ss-saddle connections, as shown in figure 3*b*, which is symbolized by *II*. Recall that a closed disc has an Euler number of 1, the indices of an *n*-source–sink point, a saddle and a ∂-saddle are 2*n*, −1 and , respectively, and that the Euler number equals the sum of indices of (∂-)saddles by the Poincaré–Hopf theorem. Then, it is easy to show that *I* and *II* are the only structurally stable streamline patterns in , since we have either one saddle or two ∂-saddles to satisfy the Poincaré–Hopf theorem. In terms of the equality of the indices, the streamline patterns *I* and *II* correspond to 1=2+(−1) and , respectively. The streamline patterns *I* and *II* are substantially different in a mathematical sense in that they cannot be transformed to each other continuously. In the circular domain , we have one initial structurally stable streamline pattern that cannot be constructed from the patterns *I* and *II*. The streamline pattern is denoted by *O*, which consists of regular closed orbits around the circular boundary, as in figure 3*c*. The streamline patterns *I* and *II* (respectively, *O*) are the initial structurally stable patterns in (respectively, ) to which we apply the procedure to construct streamline patterns in multiply connected domains with higher genus. In what follows, for the sake of simplicity to draw streamline patterns, we show no ss-orbits and closed orbits, but we pay attention to the global topological structure of ss-∂-saddle connections, ∂-saddle connections, homoclinic saddle connections and ss-saddle connections, since these orbits are substantial to distinguish the streamline topologies. In addition, the 1-source–sink point is schematically symbolized by as in figure 3*d*,*e*.

Let us finally note a topological equivalence between two streamline patterns. Figure 4 shows two equivalent patterns in . They consist of the 1-source–sink point with two ss-∂-saddle connections and one saddle point with a homoclinic saddle connection and two ss-saddle connections. They look different at a glance, but they have the same topological structure, if we identify the outer boundary of figure 4*a* with the inner boundary of figure 4*b* and vice versa. In other words, there is a homeomorphism of a closed annulus exchanging the boundaries, which maps (*a*) to (*b*). In the topological classification of streamline patterns, these two patterns are identified as one pattern. This fact plays an important role in assigning words for streamline patterns.

## 3. Word representation of streamline topologies

We introduce the following five operations to build up structurally stable streamline patterns in from those in by adding one genus to the domain. Schematic diagrams for these operations are shown in figure 5.

(

*A*_{0}) An ss-orbit is replaced by a saddle with a homoclinic saddle connection enclosing a circular boundary, and two ss-saddle connections.(

*A*_{2}) An ss-orbit is replaced by two ∂-saddles at a circular boundary that connect to the 1-source–sink pair with two ss-∂-saddle connections.(

*B*_{0}) A closed orbit is replaced by a saddle with two homoclinic saddle connections, i.e. a figure of eight saddle connection.(

*B*_{2}) A closed orbit is replaced by two ∂-saddles at a circular boundary connected by a ∂-saddle connection.(

*C*) Two ∂-saddles with a ∂-saddle connection are added to a circular boundary equipped with 2*k*∂-saddles (*k*>0). The ∂-saddle connection and the boundary enclose a circular boundary.

The operations *A*_{0} and *A*_{2} can be applied to ss-orbits, whereas *B*_{0} and *B*_{2} are applied to closed orbits. The operation *C* cannot be applied to boundaries without ∂-saddles. We shall prove later that these are the only operations by which we obtain the structurally stable streamline patterns from the initial patterns *O*, *I* and *II*.

First, we construct structurally stable streamline patterns without the 1-source–sink point starting from the initial pattern *O* in figure 3*f* by applying the operations defined above repeatedly. Since the flow contains no 1-source–sink point, it is sufficient to recall the characterization of structural stability for Hamiltonian vector fields on a two-dimensional compact submanifold of a closed orientable connected surface [10].

### Theorem 3.1 (theorem 2.3.8, p. 74 [10])

*Suppose that V is a C*^{r}*-Hamiltonian vector field on a compact orientable surface. V is structurally stable in* *if and only if V is regular and all (∂-)saddle connections are self-connected.*

In addition, it is well known that any divergence-free vector field is determined by the union of saddle connections and the ∂-saddle connections, called *the saddle connection diagram*, up to topological equivalence [10]. The operations *A*_{0} and *A*_{2} are unable to be applied to the pattern *O*, since it never contains the 1-source–sink point and thus no ss-orbits. Hence, we assign a sequence of the operations, say *OO*_{1}*O*_{2}⋯*O*_{k} with *O*_{i}∈{*B*_{0},*B*_{2},*C*} for , as a representing word of the streamline pattern. We now prove that *B*_{0}, *B*_{2} and *C* are the only operations in order to construct structurally stable streamline patterns from *O*.

### Corollary 3.1

*Suppose that the Hamiltonian vector field V is structurally stable in* . *Then, V can be represented by a sequence of operations starting from the initial pattern O*.

### Proof.

By theorem 3.1, the saddle connection diagram consists of homoclinic saddle connections and ∂-saddle connections that connect two ∂-saddles at the same boundary. Recall that the Euler number of is 1−*M*. Since *V* is regular, we can suppose that there are *k*_{1} saddles and *k*_{2} ∂-saddles. Using the Poincaré–Hopf theorem, these numbers satisfy *M*−1=*k*_{1}+*k*_{2}/2.

We will show the assertion by induction on *M*. Suppose that *M*=1. Then, *k*_{1}=*k*_{2}=0, and so there are no saddles and ∂-saddles. Hence, *V* is regular and we have *O*. Suppose that *M*>1. We say that a connected component in the saddle connection diagram is innermost, if it bounds no other saddles and ∂-saddles. If an innermost component in the saddle connection diagram of *V* in is a homoclinic saddle connection (respectively, a ∂-saddle connection with two ∂-saddles), then *V* is obtained by the operation *B*_{0} (respectively, *B*_{2}) from a structurally stable Hamiltonian vector field on . By the inductive hypothesis, suppose that is represented by *OO*_{1}⋯*O*_{M−2}, then *V* has a word representation *OO*_{1}⋯*O*_{M−2}*B*_{0} (respectively, *OO*_{1}⋯*O*_{M−2}*B*_{2}). Otherwise, all innermost components are ∂-saddle connections with more than two ∂-saddles. Then, *V* is obtained by the operation *C* from a structurally stable Hamiltonian vector field on . Hence, *V* has a word representation *OO*_{1}⋯*O*_{M−2}*C*.

We notice that there must be *B*_{2} before *C* appears in the sequence of operations, since the pattern *O* has no ∂-saddles, and thus *C* can only be applied to a boundary with ∂-saddles, which is created as a result of *B*_{2}. Hence, we have the following lemma.

### Lemma 3.1

*Let OO*_{1}⋯*O*_{M−1} *be a sequence of operations, where O*_{i}∈{*B*_{0},*B*_{2},*C*}. *Then, the following are equivalent*:

(1)

*the sequence is a word representation for a structurally stable Hamiltonian vector field in**and*(2)

*for any i with O*_{i}=*C*,*there is some j*<*i**such that O*_{j}=*B*_{2}.

We call the sequence of operations an *O*-word for a given structurally stable Hamiltonian vector field. According to lemma 3.1, either *B*_{0} or *B*_{2} can follow the initial word *O*. Hence, the structurally stable streamline patterns in are *OB*_{0} and *OB*_{2}, whose saddle connection diagrams are shown in figure 6*a*,*c*, respectively.

Now, we consider the structurally stable streamline patterns with the 1-source–sink point constructed from the initial patterns *I* and *II*. We need to modify the structural stability of the Hamiltonian vector field due to the existence of the 1-source–sink point *V* in . In order to do this, let us first note how to characterize streamline patterns *I* and *II* generated by the 1-source–sink point. The saddle connection diagram for *OB*_{0} shown in figure 6*a* is topologically equivalent to that shown in figure 6*b* for the same reason as we discussed in figure 4. Then, replacing the two circular obstacles by two centre points and collapsing these centres to one point on the homoclinic orbit between them, we obtain the streamline pattern *II*. In a similar manner, there is a homeomorphism of *OB*_{2} exchanging the boundaries, which maps (*c*) to (*d*). Thus, the streamline pattern *I* is obtained by collapsing two centres to one point on the ∂-saddle connection in the middle. Accordingly, the streamline patterns *I* and *II* are constructed from the singular pattern *OB*_{2} and *OB*_{0} by identifying two centres to one point, which is the location of the 1-source–sink point.

The structural stability for the Hamiltonian vector fields with the 1-source–sink point is characterized as follows.

### Theorem 3.2

*The Hamiltonian vector field* *is structurally stable, if and only if*

(1)

*the restriction of V on the complement of the 1-source–sink point is regular;*(2)

*all saddle connections are homoclinic connections; and*(3)

*all ∂-saddle connections connect two ∂-saddles located at the same boundary.*

### Proof.

Obviously, the regularity is necessary. Therefore, we may assume that the restriction of *V* on the complement of the 1-source–sink point *p* is regular. Now suppose that there is a heteroclinic saddle connection between two distinct saddles *p* and *q*. Then, *H*(*p*)=*H*(*q*) is satisfied where *H* represents the Hamiltonian for . We will show that the energy equality does not hold when we perturb the vector field. In order to accomplish it, we introduce a Hamiltonian vector field as follows. Let *b*:[0,1]→[0,1] be a smooth non-increasing function such that *b*(*r*)=1 for , *b*(*r*)=0 for , *b*′(*r*)<0 for , with which we define an axisymmetric function *f*:*U*→[0,1] by *f*(*r*,*θ*):=*b*(*r*) in the polar coordinates (*r*,*θ*) of the unit disc *U*. Let *V* _{f} be a Hamiltonian vector field on *U* defined from *f*. Then, a contour line of *f*(*r*,*θ*) for any is a closed orbit. For arbitrary *ε*>0, we define a function on by on the open unit disc *U* around *p* and otherwise. Then, is a smooth function, and so define the Hamiltonian vector field . Since , there are no orbits connecting *p* and *q*. Hence, *V* is not structurally stable.

If there is a ∂-saddle connection between two ∂-saddles *p* and *q* at different boundaries with the same energy level *H*(*p*)=*H*(*q*), then we can show the vector field *V* is not structurally stable as follows. Considering an annulus [0,1)×*S*^{1} in around the circular boundary with *p*, we can define the perturbed Hamiltonian vector field obtained from *H* by perturbing it in the annulus by using the function *b*(*r*), for which the energy equality no longer holds.

Conversely, suppose that (1), (2) and (3) hold. By proposition 2.1, *V* is locally structurally stable at the 1-source–sink point. Let be a small perturbed vector field of *V* . Then, we may assume that there is a small neighbourhood *U* of the 1-source–sink point on which *V* and its small perturbation are identical. Hence, it suffices to show that any small perturbation of which fixes ∂*U* is topological equivalence to . Since the streamlines in the neighbourhood of the 1-source–sink point can be obtained from the streamlines around two centres by identifying the centres as we discussed in figure 6, we can replace *V* |_{U} with some vector field on *U* with two centres. Then, all ss-(∂-)saddle connections are replaced by (∂-)saddle connections. By theorem 3.1, the resulting vector fields of *V* are structurally stable. Hence, is topological equivalent to .

It follows from theorem 3.2 that any structurally stable Hamiltonian vector field with the 1-source–sink point is determined by the union of saddle connections, ∂-saddle connections, ss-saddle connections and ss-∂-saddle connections up to topological equivalence, which is called the *ss-saddle connection diagram*. While all of the operations can be applied to the initial pattern *I*, operation *A*_{2} is not allowed for the initial pattern *II*, which is proved as follows.

### Lemma 3.2

*The structurally stable Hamiltonian vector field V with the 1-source–sink point can be represented by either a sequence IO*_{1}⋯*O*_{k} *or a sequence* , *where O*_{i}∈{*A*_{0},*A*_{2},*B*_{0},*B*_{2},*C*} *and* *for i*=1,…,*k*.

### Proof.

Let us first replace the 1-source–sink point with two centres as in the proof of theorem 3.2. Then, the resulting vector field belongs to , and thus it is represented by an *O*-word, say, with for *i*≥2. Now, we recover the 1-source–sink point by collapsing the two centres again. Then, the header of the *O*-word becomes *OB*_{2}=*I* or *OB*_{0}=*II* as we discussed in figure 6, and the other words change to either *A*_{0}, *A*_{2}, *B*_{0}, *B*_{2}, or *C* for *i*≥2. Moreover, if we operate *A*_{2} to *II* at some step, then the 1-source–sink point has ∂-saddles and so *V* can be constructed from *IA*_{0} as we already confirmed with the identity *IA*_{0}=*IIA*_{2} in figure 4.

The proof of this lemma indicates that the difference in the patterns *I* and *II* originates from the fact that there are two topologically different *O*-words, which are *OB*_{0} and *OB*_{2}, in . Hence, we can assign a sequence of the operations, *IO*_{1}*O*_{2}⋯*O*_{k} with *O*_{i}∈{*A*_{0},*A*_{2},*B*_{0},*B*_{2},*C*} and with for *i*≤*k* to represent the structurally stable streamline patterns starting from *I* and *II*, respectively. In the sequence of operations from *I*, *A*_{0} or *C* should appear before *B*_{0} and *B*_{2}, since there is no closed orbit in the initial pattern *I* to which *B*_{0} and *B*_{2} are applied. This fact is stated as follows.

### Lemma 3.3

*Let IO*_{1}⋯*O*_{k} *be a sequence of operations, where O*_{i}∈{*A*_{0},*A*_{2},*B*_{0},*B*_{2},*C*} for *i*=1,…,*k*. *Then the following are equivalent*:

(1)

*the sequence is a word representation for a structurally stable Hamiltonian vector with the 1-source–sink point in**and*(2)

*for any i*>1*with O*_{i}=*B*_{0}or*B*_{2},*there is some j*<*i such that O*_{j}=*A*_{0}*or C*.

Regrading a sequence of operations from *II*, *B*_{2} should be followed by *C*, which is given as follows.

### Lemma 3.4

*Let* *be a sequence of operations, where* for *i*=1,…,*k*. *Then the following are equivalent:*

(1)

*the sequence is a word representation for a structurally stable vector field with the 1-source–sink point in**and*(2)

*for any i*>1*with**there is some j*<*i such that*.

We refer to the sequence of words for a given structurally stable vector field with the 1-source–sink point from *I* and *II* as *a I-word* and

*a*, respectively. A word can be assigned for a structurally stable streamline pattern as a sequence of operations. However, we must note that there are many streamline patterns represented by one word. For example, in the initial pattern

*II*-word*II*, ss-orbits exist above and below the 1-source–sink point to both of which

*A*

_{0}is applied, which gives rise to different streamline patterns represented by

*IIA*

_{0}. On the other hand, we also note that there are several word representations for one structurally stable streamline pattern. Indeed, the words

*IA*

_{0}

*A*

_{2}and

*IA*

_{2}

*A*

_{0}represent the same streamline patterns. Based on these observations, it is useful to discuss the commutativity of the operations in the sequence. Obviously, the same operations commute. The commutativity of

*A*

_{0}and

*C*for

*I*-words and

*II*-words are shown as follows.

### Lemma 3.5

*The streamline patterns with the word representations O*_{0}*O*_{1}⋯*O*_{i}*A*_{0}*CO*_{i+3}⋯*O*_{k} *and* *O*_{0}*O*_{1}⋯*O*_{i}*CA*_{0}*O*_{i+3}⋯*O*_{k} *with O*_{0}∈{*I*,*II*} *are equivalent. Namely, A*_{0} *and C commute in the word representation*.

### Proof.

Since *A*_{0} does not increase boundaries with ∂-saddles and *C* just changes a boundary with ∂-saddles, these operations are applied independently. □

The operation *A*_{2} commutes with *A*_{0}, *B*_{0} and *B*_{2} for *I*-words.

### Lemma 3.6

*The streamline patterns with the word representations IO*_{1}⋯*O*_{i}*A*_{2}*O*_{i+2}⋯*O*_{k} *and IO*_{1}⋯*A*_{2}*O*_{i}*O*_{i+2}⋯*O*_{k} *with O*_{i}∈{*A*_{0},*B*_{0},*B*_{2}} *are equivalent*.

### Proof.

Since *A*_{0} and *A*_{2} are applied to different ss-orbits, *A*_{0} and *A*_{2} commute. The operation *A*_{2} increases no closed orbits to which *B*_{0} and *B*_{2} are applied. Conversely, *B*_{0} and *B*_{2} add no ss-obits. Hence, *B*_{0} and *B*_{2} commute with *A*_{2}.

The equivalence between the streamline patterns with respect to the exchange of operations in the sequence proved in the above lemmas are symbolically denoted by *A*_{0}*C*=*CA*_{0}, *A*_{2}*A*_{0}=*A*_{0}*A*_{2}, *A*_{2}*B*_{0}=*B*_{0}*A*_{2} and *A*_{2}*B*_{2}=*B*_{2}*A*_{2}. In order to discuss the commutativity for the other operations, we need to define an inclusion relation between two word representations. If the streamline patterns represented by a word *O*_{0}*O*_{1}⋯*O*_{i−1}*O*_{i}*O*_{i+1}*O*_{i+2}⋯*O*_{k} are included by those by a word *O*_{0}*O*_{1}⋯*O*_{i−1}*O*_{i+1}*O*_{i}*O*_{i+2}⋯*O*_{k}, then the inclusion relation is symbolized by *O*_{i}*O*_{i+1}≤*O*_{i+1}*O*_{i}. Then, we have the following inclusion relations.

### Lemma 3.7

*The inclusion relations B*_{0}*A*_{0}≤*A*_{0}*B*_{0}, *B*_{2}*A*_{0}≤*A*_{0}*B*_{2}, *CA*_{2}≤*A*_{2}*C*, *B*_{2}*B*_{0}≤*B*_{0}*B*_{2}, *B*_{0}*C*≤*CB*_{0} *hold for the exchange of two operations in the sequence*.

### Proof.

First, *A*_{0} is independently applied to an ss-orbit, even if *B*_{0} and *B*_{2} exist in the sequence before *A*_{0}. On the other hand, *A*_{0} adds new closed orbits to which *B*_{0} and *B*_{2} are applied. Hence, we have *B*_{0}*A*_{0}≤*A*_{0}*B*_{0} and *B*_{2}*A*_{0}≤*A*_{0}*B*_{2}. Second, *C* does not affect *A*_{2}, since *C* creates no new ss-orbits. Conversely, *A*_{2} adds a new boundary with two ∂-saddles to which *C* can be applied. Thus, *CA*_{2}≤*A*_{2}*C* holds. Third, *B*_{0} increases new closed orbits to which *B*_{2} is applied, but *B*_{2} does not. Hence, we have *B*_{2}*B*_{0}≤*B*_{0}*B*_{2}. Finally, *CB*_{0}≥*B*_{0}*C* holds, since *C* increases new closed orbits and *B*_{0} adds no boundary with ∂-saddles.

Note that we cannot determine the inclusion relation for exchange of *B*_{2} and *C* in the sequence of word representations, which is symbolized by *B*_{2}*C*||*CB*_{2}. The commutativity of the five operations in word representations and their inclusion relations are summarized in table 1. Owing to the order relation ≤, we can show the existence of a maximal word for any structurally stable streamline patterns.

### Lemma 3.8

*Each structurally stable streamline pattern on* *has a maximal word representation*.

### Proof.

Note that the relation ≤ implies the reflexive and transitive relation on the set of *O*-words (respectively, *I*-words, *II*-words). Since the number of *O*-words (respectively, *I*-words, *II*-words) is finite, each word is less than or equal to some maximal word.

Let us note that this lemma concludes nothing about the uniqueness of the maximal word representation. As a matter of fact, the expression of the maximal word depends on how we exchange the words in the sequence following the rules in table 1. Hence, we need to specify how to exchange the words in order to obtain the unique expression of the maximal word representation. Thus, in the rest of this section, we derive canonical expressions for maximal *O*-words, *I*-words and *II*-words. Let us define a block component *W*(*s*,*t*,*u*) by *W*(*s*,*t*,*u*)=(*B*_{0})^{s}(*B*_{2})^{t}(*C*)^{u} for non-negative integers *s*,*t* and *u*. Then, we have the following theorem.

### Theorem 3.3

*For any maximal O-word for a structurally stable streamline pattern in* *there exist integers k≥1, s*_{m}*,t*_{m}*≥0 for m=1,…,k and u*_{m}*>0 for m=1,…,k−1 such that
*3.1*where t*_{m}*>0 for any m<k with* *.*

### Proof.

Let us first note that it is not possible to exchange *CB*_{0} and *B*_{2}*C* in the sequence of any *O*-word to obtain the maximal *O*-word owing to *B*_{2}*C*||*CB*_{2} and *B*_{0}*C*≤*CB*_{0}. We show that every *O*-word assigned to the structurally stable streamline pattern in can be reduced to the maximal *O*-word inductively as follows.

Starting the initial word *O*, we look for the location where (*C*)^{u1} first appears in the sequence of the *O*-word. If there exists no operation *C* in the sequence, i.e. *u*_{1}=0, then the *O*-word consists of *B*_{0} and *B*_{2}, and thus it can be reduced to *O*(*B*_{0})^{s1}(*B*_{2})^{s2}=*OW*(*s*_{1},*t*_{1},0) for some *s*_{1},*t*_{1}≥0 by exchanging *B*_{0} and *B*_{2} owing to *B*_{2}*B*_{0}≤*B*_{0}*B*_{2}, which ends the proof. On the other hand, if *u*_{1}≠0, we can rearrange the sequence with *B*_{0} and *B*_{2} between *O* and (*C*)^{u1} by the block component *W*(*s*_{1},*t*_{1},*u*_{1}) for some *t*_{1}>0. This is because if *t*_{1}=0, we have *O*(*B*_{0})^{s1}(*C*)^{u1}⋯≤ *O*(*C*)^{u1}(*B*_{0})^{s1}⋯ owing to (*B*_{0})^{s1}(*C*)^{u1}≤(*C*)^{u1}(*B*_{0})^{s1}, which is not an *O*-word. Hence, the *O*-word is expressed as *OW*(*s*_{1},*t*_{1},*u*_{1})⋯ with *t*_{1},*u*_{1}>0.

Now suppose that *u*_{m}≠0 and the *O*-word is reduced to *OW*(*s*_{1},*t*_{1},*u*_{1})⋯*W*(*s*_{m},*t*_{m},*u*_{m})⋯ with *t*_{i}>0 for any *i*≤*m*. Then, we look for the location of (*C*)^{um+1} in the sequence beyond *W*(*s*_{m},*t*_{m},*u*_{m}). If *u*_{m+1}=0, then the *O*-word can be reduced to
for some *s*_{m+1},*t*_{m+1}≥0 by exchanging *B*_{0} and *B*_{2}, which concludes the proof. Otherwise the sequence of *B*_{0} and *B*_{2} between *W*(*s*_{m},*t*_{m},*u*_{m}) and (*C*)^{um+1} is reduced to the block component *W*(*s*_{m+1},*t*_{m+1},*u*_{m+1}) for some *s*_{m+1}≥0 and *t*_{m+1}>0. The positivity of *t*_{m+1} is assured as follows. If *t*_{m+1}=0, we have
owing to *B*_{0}*C*≤*CB*_{0}. Then, with , the sequence is reduced to
for which we can repeat the process again.

### Theorem 3.4

*Let p,q,r be non-negative integers. Then, for any maximal I-word for a structurally stable streamline pattern in* *there exist integers k≥1, s*_{m}*,t*_{m}*≥0 for m=1,…,k and u*_{m}*>0 for m=1,…,k−1 such that
*3.2*where t*_{m}*>0 for any m<k with* *if p+r>0. Otherwise, it is represented by I(A*_{2})^{q} *with M=q.*

### Proof.

For a given *I*-word representing a structurally stable streamline pattern in , we can move all *A*_{0} and *A*_{2} in the sequence of operations before *B*_{0}, *B*_{2} and *C* by exchanging the order of the operations owing to *A*_{0}*A*_{2}=*A*_{2}*A*_{0}, *CA*_{2}≤*A*_{2}*C*, *B*_{2}*A*_{2}=*A*_{2}*B*_{2}, *B*_{0}*A*_{2}=*A*_{2}*B*_{0}, *CA*_{0}=*A*_{0}*C*, *B*_{0}*A*_{0}≤*A*_{0}*B*_{0} and *B*_{2}*A*_{0}≤*A*_{0}*B*_{2}. Hence, the *I*-word is reduced to *I*(*A*_{0})^{p}(*A*_{2})^{q}*O*_{p+q+1}⋯*O*_{M} for some *p*,*q*≥0, in which *O*_{i}∈{*B*_{0},*B*_{2},*C*} for *p*+*q*<*i*≤*M*.

Suppose first that *p*=0. If *O*_{q+1}≠*C*, namely *r*=0, then there are no *B*_{0} and *B*_{2} in the following sequence, since they cannot be applied without *A*_{0} or *C* in the sequence of *I*-words due to lemma 3.3. Hence, if *p*+*r*=0, the *I*-word is represented by *I*(*A*_{2})^{q} and *M*=*q*. On the other hand, if *O*_{q+1}=*C*, then there exists *r*>0 such that the *I*-word is represented by *I*(*A*_{2})^{q}(*C*)^{r}*O*_{q+r+1}⋯*O*_{M}, in which *O*_{i}∈{*B*_{0},*B*_{2},*C*} for *q*+*r*<*i*≤*M*. The remaining sequence *O*_{q+r+1}⋯*O*_{M} can be reduced to some block components of *B*_{0}, *B*_{2} and *C* by using the same procedure as for *O*-words in theorem 3.3. That is, there exist integers *k*≥1, *s*_{m},*t*_{m}≥0 for *m*=1,…,*k* and *u*_{m}>0 for *m*=1,…,*k*−1 such that the *I*-word is expressed by
where *t*_{m}>0 for any *m*<*k*. This is the maximal expression for the *I*-word for *p*=0 and *r*>0.

Next, we assume *p*≠0. Then, *B*_{0} and *B*_{2} can exist in the remaining part of the sequence *I*(*A*_{0})^{p}(*A*_{2})^{q}⋯ to which we can apply the same procedure for the remaining sequence of *B*_{0}, *B*_{2} and *C* as used for *O*-words. Hence, we have the maximal expression (3.2).

### Theorem 3.5

*Let p be a non-negative integer. Then, for any maximal II-word for a structurally stable streamline pattern in* *there exist integers k≥1, s*_{m}*,t*_{m}*≥0 for m=1,…,k and u*_{m}*>0 for m=1,…,k−1 such that
*3.3*where t*_{m}*>0 for any m<k with* *.*

### Proof.

The proof is carried out in a similar manner to theorem 3.4, since lemma 3.4 indicates that *II*-words have the same rule as *O*-words in terms of the order of operations *B*_{0}, *B*_{2} and *C* in the sequence.

## 4. Structurally stable streamline patterns in and

We give all possible streamline patterns for the vortex flows in the presence of the uniform flow in and and their corresponding maximal word representations using (3.1)–(3.3). In order to obtain the canonical expressions, we need to determine the indices *p*, *q*, *r*, the number of block components *k* and its corresponding integers *s*_{m}, *t*_{m} for *m*=1,…,*k* and *u*_{m} for *m*=1,…,*k*−1 in the combinatorial way. Let us note that the number of block components *k* satisfies 2(*k*−1)≤*M* for *I*-words and *II*-words, and 2(*k*−1)≤*M*−1 for *O*-words, since we have *u*_{m}=*t*_{m}=1 for 1≤*m*≤*k*−1 at least.

First, we consider all maximal *O*-words. Since according to theorem 3.3, the maximal *O*-word for *M*=1 is *O*. When *M*=2, we have *k*=1, and thus we determine the combinations of the indices (*s*_{1},*t*_{1}) in the block component. Since we have (*s*_{1},*t*_{1})=(1,0) and (0,1), the maximal *O*-words for *M*=2 are *OB*_{0} and *OB*_{2}, whose corresponding streamline patterns have already been shown in figure 6*a*,*c*.

For maximal *I*-words, since by theorem 3.4, *k*=1 is allowed for *M*=1. When *p*=*r*=0, we have *q*=1, whose maximal *I*-word becomes *IA*_{2}. If *p*+*r*>0, we have either *p*=1 or *r*=1. Hence, the maximal *I*-words for *M*=1 are given by *IA*_{0}, *IA*_{2} and *IC* whose corresponding streamline patterns are shown in figure 7. For *M*=2, if *k*=2, then we have *t*_{1}=*u*_{1}=1 and thus *p*=*r*=0 due to *M*=2. However, if *p*=*r*=0, then we have *s*_{1}=*t*_{1}=0, which is a contradiction. Hence, we have *k*=1 and thus we distribute *M*=2 to the set of indices (*p*,*q*,*r*,*s*_{1},*t*_{1}). First, when *p*=*r*=0, we have the maximal *I*-word *IA*_{2}*A*_{2} for *q*=2. There are two streamline patterns represented by *IA*_{2}*A*_{2}, since ss-orbits to which *A*_{2} is applied exist in the two disjoint regions separated by the ss-∂-saddle connections. If we apply *A*_{2} to two ss-orbits on the different regions, the streamline pattern for *IA*_{2}*A*_{2} becomes figure 8*a*. Applying *A*_{2} to two ss-orbits on the same side, we obtain the other streamline pattern represented by *IA*_{2}*A*_{2}, as shown in figure 8*b*.

When *p*+*r*>0, the combinations of the indices (*p*,*q*,*r*,*s*_{1},*t*_{1}) and their corresponding maximal *I*-words are listed in table 2 whose representing streamline patterns are shown in figures 8 and 9. There are several two-streamline patterns represented by *IA*_{0}*A*_{0} (figure 8*c*,*d*) depending on whether or not *A*_{0} are applied to an ss-orbit on the same side where the saddle created by *A*_{0} is located. Regarding *IA*_{0}*A*_{2}, there are three kinds of different streamline patterns denoted by *IA*_{0}*A*_{2} shown in figure 8*e*–*g*, since *A*_{2} can be applied to an ss-orbit outside or inside of the saddle created by *A*_{0}. Figure 8*h*,*i* shows the streamline patterns represented by *IA*_{0}*C*, which correspond to the cases when we apply operation *C* to a boundary with ∂-saddles on the same side where the saddle created by *A*_{0} is located on the different sides. Since there is only one region in *IA*_{0} that contains closed orbits to which *B*_{0} and *B*_{2} are applied, each of *IA*_{0}*B*_{0} and *IA*_{0}*B*_{2} represents one streamline pattern whose corresponding ss-saddle connection diagrams are shown in figure 9*a*,*b*, respectively. For *IA*_{2}*C*, we obtain two streamline patterns when we apply *C* to a boundary with two ∂-saddles on different sides (figure 9*c*) or on the same side (figure 9*d*) of the boundary created by *A*_{2}. The maximal *I*-words *ICB*_{0} and *ICB*_{2} represent the streamline patterns shown in figure 9*e*,*f*, respectively, since *B*_{0} and *B*_{2} are applied to a closed orbit around the boundary created by *C*. We have three streamline patterns for *ICC* in which two ∂-saddle connections are on different sides (figure 9*g*), on the same side (figure 9*h*) and one ∂-saddle connection encloses the other ∂-saddle connection (figure 9*i*).

Finally, we consider the maximal *II*-words in and . Owing to *k*=1 for *M*=1, we have three combinations of the indices (*p*,*s*_{1},*t*_{1})=(1,0,0), (0,1,0) and (0,0,1), whose corresponding maximal *II*-words are *IIA*_{0}, *IIB*_{0} and *IIB*_{2}, whose corresponding ss-saddle connection diagrams are shown in figure 10*a*–*d*, respectively. There are two patterns for *IIA*_{0}, since the initial pattern *II* has two disjoint regions containing ss-orbits to which *A*_{0} is applied.

Now suppose that *M*=2. All streamline patterns represented by the maximal *II*-words are drawn in figures 11 and 12. Since ss-orbits exist in the two disjoint regions in pattern *II*, we obtain three streamline patterns when we apply the operations *A*_{0}*A*_{0} in the different regions (figure 11*a*) and in the same region (figure 11*b*,*c*). On the other hand, as shown in figure 10*a*,*b*, *IIA*_{0} represents the two streamline patterns that have two disjoint regions containing closed orbits to which we can apply *B*_{0} and *B*_{2}. Hence, we have four streamline patterns for *IIA*_{0}*B*_{0} shown in figure 11*d*–*g* and *IIA*_{0}*B*_{2} shown in figure 11*h*–*k*. Since *IIB*_{0} has three disjoint regions with closed orbits to which we can apply *B*_{0} and *B*_{2}, we have three streamline patterns for *IIB*_{0}*B*_{0} and *IIB*_{0}*B*_{2}. However, for *IIB*_{0}*B*_{0}, two have the same topology, *IIB*_{0}*B*_{0} represents two streamline patterns. Thus all ss-saddle connection diagrams for *IIB*_{0}*B*_{0} and *IIB*_{0}*B*_{2} are shown in figure 12*a*,*b* and figure 12*c*–*e*, respectively. The maximal *II*-word *IIB*_{2}*B*_{2} represents one streamline pattern in figure 12*f*. When the number of block components is *k*=2, we have only one maximal *II*-word *IIB*_{2}*C* for *t*_{1}=*u*_{1}=1, whose representing streamline pattern is given in figure 12*g*. For *k*=1, table 3 is a list of combinations of indices (*p*,*s*_{1},*t*_{1}) and their word representations.

## 5. Summary and discussion

This paper has shown how to assign a sequence of words, called the maximal word, to every structurally stable streamline pattern generated by potential flows consisting of uniform flow and point vortices in two-dimensional multiply connected domains. Owing to the maximal word representation, we not only classify structurally stable streamline topologies, but we also make a complete catalogue of all possible streamline patterns in a combinatorial manner. Let us note that the canonical expressions of the maximal words (3.1)–(3.3) represent a family of streamline patterns. On the other hand, for a sequence of words assigned to every streamline pattern, we can convert it uniquely into a canonical maximal word representation following the procedures given in the proofs of theorems 3.3–3.5.

The classification of streamline patterns for vortex flows with uniform flow is an extension of work by Aref & Brøns [8]. As a matter of fact, the maximal *O*-words can represent structurally stable streamline patterns without uniform flow in the unbounded plane when we regard circular boundaries as point vortices. Since there is no circular boundary, the operations *B*_{2} and *C* are not allowed. Hence, the maximal *O*-word for the streamline pattern becomes for some *q*≥0. Let us note that Aref & Brøns have considered structurally unstable patterns that contain heteroclinic connections between saddles, which cannot be represented by maximal *O*-words. Furthermore, maximal word representations of structurally stable streamline patterns add the following new aspects, which that are of significance from the application point of view. First, one can obtain all possible streamline patterns with their maximal word representations in multiply connected exterior domains in the presence of uniform flow, which are applicable to the classification of flow patterns arising in biofluids and environmental flows. Second, we pay attention to structurally stable flows, which are more likely to be observed in real fluid problems.

In the (ss-)saddle connection diagrams shown in this paper, all obstacles contained in the multiply connected domain are represented by *M* circular holes. However, these circular obstacles can be replaced by point vortices and elliptic stagnation points, since they generate closed circular orbits around them that are topologically equivalent to each other. Thus, the genus *M* of the domain is equivalent to the sum of the numbers of circular holes, point vortices and elliptic centres. Let us also note that the topological classification of the streamline patterns is independent of the shapes of the boundary of the obstacles, since the complex potentials are conformally invariant under the action of conformal mapping between any multiply connected domains and canonical circular domains with the same multiplicity.

Let us finally note that it is important to consider structurally unstable streamline patterns that contain heteroclinic orbits and multiple homoclinic saddle connections and so on. Since the structurally unstable streamline patterns are reduced to some structurally stable ones under small perturbations, the unstable streamline pattern is regarded as a marginal state between two stable streamline patterns with the maximal word representations. Consequently, through the unstable streamline patterns, we can describe the transition between two maximal words in a combinatorial way, which will be reported in the near future.

- Received September 20, 2012.
- Accepted November 8, 2012.

- © 2012 The Author(s) Published by the Royal Society. All rights reserved.