Suppose we have a dynamical system given by a system of first order differential equations
where $x\in {\mathbb{R}}^{n}$ and $f\in \mathcal{X}$, the set of ${C}^{1}$ vector fields on ${\mathbb{R}}^{n}$ with bounded global attractors.^{1} We say an invariant set Σ is a heteroclinic cycle (HC) if it consists of a union of hyperbolic equilibria $\{{x}_{i}:i=1,\dots ,p\}$ and connecting orbits ${s}_{i}\subset {W}^{u}({x}_{i})\cap {W}^{s}({x}_{i+1})$.^{2} We say that a heteroclinic cycle Σ is robust to perturbations in$\mathcal{Y}\subset \mathcal{X}$ if $f\in \mathcal{Y}$ and there is a ${C}^{1}$neighbourhood of f such that all $g\in \mathcal{Y}$ within this neighbourhood have a heteroclinic cycle that is close to Σ.
Let us suppose that
$f\in \mathcal{X}$ has a HC Σ between equilibria
${x}_{i}$. As the connection
${s}_{i}$ is contained within
${W}^{u}({x}_{i})\cap {W}^{s}({x}_{i+1})$, this implies that
$dim({W}^{u}({x}_{i})\cap {W}^{s}({x}_{i+1}))\ge 1$. In order for the connection from
${x}_{i}$ to
${x}_{i+1}$ to be robust with respect to arbitrary
${C}^{1}$ perturbations it is necessary that the intersection is transverse [
12], meaning that
$dim({W}^{u}({x}_{i}))+dim({W}^{s}({x}_{i+1}))\ge n+1.$
(2)
Using the fact that
$dim({W}^{u}({x}_{i}))+dim({W}^{s}({x}_{i}))=n$ for any hyperbolic equilibrium and adding these for all equilibria along the cycle, we find that
$\sum _{i=1}^{p}[dim({W}^{u}({x}_{i}))+dim({W}^{s}({x}_{i+1}))]=pn.$
(3)
This implies that it is not possible for Equation 2 to be satisfied for all connections. Hence our first statement is the following (which can be thought of a special case of the KupkaSmale Theorem [12], see also [13]).
Proposition 1 A heteroclinic cycle between$p>0$hyperbolic equilibria is never robust to general${C}^{1}$perturbations in$\mathcal{X}$.
The heteroclinic cycle may however be robust to a constrained set of perturbations. We explore this in the following sections.
2.1 Conditions for robustness of heteroclinic cycles with constraints
A subset
$I\subset {\mathbb{R}}^{n}$ is an
affine subspace if it can be written as
$I:=\{x\in {\mathbb{R}}^{n}:Ax=b\}$ for some realvalued
$n\times n$ matrix
A and vector
$b\in {\mathbb{R}}^{n}$ (this is a linear subspace if
b can be chosen to be zero). For a given phase space
${\mathbb{R}}^{n}$, suppose that we have a (finite) set of nonempty affine subspaces
$\mathcal{I}=\{{I}_{1},\dots ,{I}_{d}\}$
(4)
that are closed under intersection; i.e. the intersection
${I}_{j}\cap {I}_{k}$ of any two subspaces
${I}_{j},{I}_{k}\in \mathcal{I}$ is an element of
$\mathcal{I}$ unless it is empty. We include
${I}_{1}={\mathbb{R}}^{n}$, which is trivially invariant, so
$\mathcal{I}$ is always nonempty. For a given
$\mathcal{I}$, we define the set of
vector fields (in$\mathcal{X}$) respecting$\mathcal{I}$ to be
${\mathcal{X}}_{\mathcal{I}}:=\{f\in \mathcal{X}:f(I)\subset I\text{for all}I\in \mathcal{I}\}$
(5)
and call the subspaces in $\mathcal{I}$invariant subspaces in the phase space of the dynamical systems described by $f\in {\mathcal{X}}_{\mathcal{I}}$.
A set of invariant affine subspaces $\mathcal{I}$ may arise from a variety of modelling assumptions; for example,

If f is a LotkaVolterra type population model that leaves some subspaces corresponding to the absence of one or more “species” invariant then $f\in {\mathcal{X}}_{\mathcal{I}}$ where $\mathcal{I}$ is the set of the invariant subspaces forced by the absence of these species.

If f is symmetric (equivariant) for some group action G and $\mathcal{I}$ is the set of fixed point subspaces of G then $f\in {\mathcal{X}}_{\mathcal{I}}$ because fixed point subspaces are invariant under the dynamics of equivariant systems [14], Theorem 1.17]. Note that for an orthogonal group action, the fixed point subspaces are linear subspaces. It is known that symmetries impose further constraints on the dynamics such as repeated eigenvalues or missing terms in Taylor expansions [14] but we focus here only on the invariant subspaces.

If f is a realization of a particular coupled cell system with a given coupling structure then $f\in {\mathcal{X}}_{\mathcal{I}}$ where $\mathcal{I}$ corresponds to the set of possible cluster states (also called synchrony subspaces or polydiagonals in the literature [15–17]).
Note that ${\mathcal{X}}_{\mathcal{I}}$ inherits a subset topology from $\mathcal{X}$; for a discussion of homoclinic and heteroclinic phenomena in general and their associated bifurcations in particular, we refer to the review [13].
Suppose that for a vector field
$f\in {\mathcal{X}}_{\mathcal{I}}$ we have a heteroclinic cycle Σ between hyperbolic equilibria
$\{{x}_{i}\}$ (
$i=1,\dots ,p$) with connections
${s}_{i}$ from
${x}_{i}$ to
${x}_{i+1}$. We define
${I}_{c(i)}:=\bigcap _{\{c:{s}_{i}\subset {I}_{c}\in \mathcal{I}\}}{I}_{c}$
(6)
i.e. the smallest subspace in
$\mathcal{I}$ that contains
${s}_{i}$. The invariant set
${I}_{c(i)}$ is clearly well defined because
$\mathcal{I}$ is closed under intersections. We define the
connection scheme of the heteroclinic cycle to be the sequence
${x}_{1}\stackrel{{I}_{c(1)}}{\to}{x}_{2}\stackrel{{I}_{c(2)}}{\to}\cdots \stackrel{{I}_{c(p)}}{\to}{x}_{1}.$
(7)
The following theorem gives necessary and sufficient conditions for such a heteroclinic cycle to be robust to perturbations in
${\mathcal{X}}_{\mathcal{I}}$, depending on its connection scheme (we will require robustness to preserve the connection scheme). More precisely it depends on the following equation being satisfied:
$dim({W}^{u}({x}_{i})\cap {I}_{c(i)})+dim({W}^{s}({x}_{i+1})\cap {I}_{c(i)})\ge dim({I}_{c(i)})+1$
(8)
for each i. Note that there is a slight complication for the sufficient condition  it may be necessary to perturb the system slightly within ${\mathcal{X}}_{\mathcal{I}}$ to unfold the intersection to general position and remove a tangency between ${W}^{u}({x}_{i})$ and ${W}^{s}({x}_{i+1})$. This complication has the benefit that it allows us to make statements about particular connections without needing to verify that the intersection of manifolds is transverse.
Theorem 1 Let Σ
be a heteroclinic cycle for$f\in {\mathcal{X}}_{\mathcal{I}}$between hyperbolic equilibria$\{{x}_{i}:i=1,\dots ,p\}$with connection scheme Equation 7.
 1.
If the cycle Σ is robust to perturbations in${\mathcal{X}}_{\mathcal{I}}$then Equation 8is satisfied for$i=1,\dots ,p$.
 2.
Conversely, if Equation 8is satisfied for$i=1,\dots ,p$then there is a nearby$\tilde{f}\in {\mathcal{X}}_{\mathcal{I}}$ (with$\tilde{f}$arbitrarily close to f) such that Σ is a heteroclinic cycle for$\tilde{f}$that is robust to perturbations in${\mathcal{X}}_{\mathcal{I}}$.
Proof We will abbreviate ${I}_{c}:={I}_{c(i)}$. Because ${s}_{i}$ is a connection from ${x}_{i}$ to ${x}_{i+1}$, there is a nontrivial intersection of ${W}^{u}({x}_{i})\cap {W}^{s}({x}_{i+1})$ within ${I}_{c}$. As ${I}_{c}$ is the smallest invariant subspace containing ${s}_{i}$, typical points $y\in {s}_{i}$ will have a neighbourhood in ${I}_{c}$ that contain no points in any other ${I}_{j}$. In a neighbourhood of this y, perturbations of f in ${\mathcal{X}}_{\mathcal{I}}$ have no restriction other than they should leave ${I}_{c}$ invariant.
The stability of the intersection of the unstable and stable manifolds depends on the dimension of the unstable manifolds (also called the
Morse index[
13]) for these equilibria for the vector field restricted to
${I}_{c}$. Pick any codimension one section
$P\subset {I}_{c}$ transverse to the connection at
y. We have
$dim(P)=dim({I}_{c})1$
(9)
and within
P, the invariant manifolds have dimensions
$\begin{array}{rcl}dim({W}^{u}({x}_{i})\cap P)& =& dim({W}^{u}({x}_{i})\cap {I}_{c})1,\\ dim({W}^{s}({x}_{i+1})\cap P)& =& dim({W}^{s}({x}_{i+1})\cap {I}_{c})1.\end{array}$
(10)
The intersection of these invariant manifolds may not be transverse within
P, but it will be for a dense set of nearby vector fields. In particular, if
$dim({W}^{u}({x}_{i})\cap P)+dim({W}^{s}({x}_{i+1})\cap P)<dim(P)$
(11)
then there will be an open dense set of perturbations of f that remove the intersection, giving lack of robustness of ${s}_{i}$ and hence we obtain a proof for case 1. On the other hand, if Equation 11 is not satisfied, we can choose a vector field $\tilde{f}$ that is identical to f except on a small neighbourhood of y  there it is chosen to preserve the connection but to perturb the manifolds so that the intersection is transverse. Transversality of the intersection then implies robustness of the connection and hence we obtain a proof for case 2. □
Note that caution is necessary in interpreting this result for a number of reasons:
 1.
Just because a given heteroclinic connection is not robust due to this result does not necessarily imply that there is no robust connection from ${x}_{i}$ to ${x}_{i+1}$ at all. Indeed, it may be [18] that there are several connections from ${x}_{i}$ to ${x}_{i+1}$ and that perturbations will break some but not all of them. In this sense, it may be that at the same time, one heteroclinic cycle is not robust, but another heteroclinic cycle between the same equilibria is robust.
 2.
We consider robustness to perturbations that preserve the connection scheme  there are situations where a typical perturbation may break a connection but preserve a nearby connection in a larger invariant subspace. This situation will typically only occur in exceptional cases.
 3.
The structure of general robust heteroclinic cycles may be very complex even if we only examine cases forced by symmetries  they easily form networks with multiple cycles. There may be multiple or even a continuum of connections between two equilibria, and they may be embedded in more general “heteroclinic networks” where there may be connections to “heteroclinic subcycles” [16, 19, 20].
 4.
Theorem 1 does not consider any dynamical stability (attraction) properties of the heteroclinic cycles.
 5.
In what follows, we slightly abuse notation by saying that a heteroclinic cycle is robust if the cycle for an arbitrarily small perturbation of the vector field is robust.
If ${W}^{u}({x}_{i})$ is not contained in ${W}^{s}({x}_{i+1})$ then the heteroclinic cycle Σ cannot be asymptotically stable. We say that an invariant set Σ is a regular heteroclinic cycle if it consists of a union of equilibria and a set of connecting orbits ${s}_{i}\subset {W}^{u}({x}_{i})$ with ${W}^{u}({x}_{i})\subset {W}^{s}({x}_{i+1})$. The following result is stated in [13] for the case of symmetric systems.
Theorem 2 Suppose that Σ
is a regular heteroclinic cycle for$f\in {\mathcal{X}}_{\mathcal{I}}$between hyperbolic equilibria$\{{x}_{i}:i=1,\dots ,p\}$.
Suppose that${x}_{i+1}$is a sink for the dynamics reduced to${I}_{c(i)}$,
i.
e.
$dim({W}^{s}({x}_{i+1})\cap {I}_{c(i)})=dim({I}_{c(i)})$
(12)
for all i. Then the heteroclinic cycle is robust to perturbations within${\mathcal{X}}_{\mathcal{I}}$.
Proof Suppose that ${W}^{s}({x}_{i+1})\supset {I}_{c(i)}$. Since ${W}^{u}({x}_{i})$ is contained in ${W}^{s}({x}_{i+1})$ by regularity of the HC, and because ${I}_{c(i)}\supseteq {s}_{i}={W}^{u}({x}_{i})$, we find $dim({W}^{u}({x}_{i})\cap {I}_{c(i)})+dim({W}^{s}({x}_{i+1})\cap {I}_{c(i)})=dim({W}^{u}({x}_{i}))+dim({I}_{c(i)})\ge dim({I}_{c(i)})+1$. Hence, Equation 8 follows and we could apply Theorem 1 case 2. In fact this is a simpler case in that because $dim({W}^{s}({x}_{i+1})\cap {I}_{c(i)})=dim({I}_{c(i)})$ the intersection must already be transverse  one does not need to consider any perturbations to force transversality of the intersection. □
2.2 Cluster states for coupled systems
RHCs may appear in coupled systems due to a variety of constraints from the coupling structure  these are associated with cluster states (also called synchrony subspaces [
15] or polydiagonals for the network [
21]). Consider a network of
N systems each with phase space
${\mathbb{R}}^{d}$ and coupled to each other to give a set of differential equations on
${\mathbb{R}}^{n}$, with
$n=Nd$, of the form
$\frac{d{x}_{i}}{dt}={f}_{i}({x}_{1},\dots ,{x}_{N})$
(13)
for
${x}_{i}\in {\mathbb{R}}^{d}$$i=1,\dots ,N$. We write
$f:{\mathbb{R}}^{Nd}\to {\mathbb{R}}^{Nd}$ with
$f(x)=({f}_{1}(x),\dots ,{f}_{N}(x))$. We define a
cluster state for a class of ODEs to be a partition
${\mathcal{P}}_{i}$ of
$\{1,\dots ,N\}$ such that the linear subspace
${I}_{i}:=\{({x}_{1},\dots ,{x}_{N}):{x}_{j}={x}_{k}\iff \{j,k\}\text{are in the same part of}{\mathcal{P}}_{i}\}$
is dynamically invariant for all ODEs in that class. For a given symmetry or coupling structure, we identify a list of possible cluster states and use these to test for robustness of any given heteroclinic cycle using Theorem 1.
We remark that the simplest (and indeed only, up to relabelling) coupling structure for a network of three identical cells found by [
15] to admit heteroclinic cycles can be represented as a system of the form
$\begin{array}{rcl}\dot{x}& =& f(x;y,z),\\ \dot{y}& =& f(y;x,z),\\ \dot{z}& =& f(z;y,x).\end{array}$
(14)
For an open set of choices of $f(x,y,z)$, the heteroclinic cycle involves two saddles within the subspace ${I}_{1}:=\{x=y=z\}$ and connections that are contained within ${I}_{2}:=\{x=y\}$ in one direction and ${I}_{3}:=\{x=z\}$ in the other. This represents a system of three identical units coupled in a specific way, where each unit has two different input types; we refer to [15] for details. It can be quite difficult to find a suitable function f that gives a robust heteroclinic cycle in this case. Nevertheless, once one has found a heteroclinic cycle, it can be shown to be robust using Theorem 1 (case 2).
Other examples of robust heteroclinic cycles between equilibria for systems of coupled phase oscillators are given in [22, 23]. For such systems the final state equations are obtained by reducing the dynamics to phase difference variables. In this case, each equilibrium represents the oscillatory motion of oscillators with some fixed phase difference.
2.3 Robust heteroclinic cycles between periodic orbits
In cases where a phase difference reduction is not possible, one may need to study heteroclinic cycles between periodic orbits in order to explain heteroclinic behaviour. Unlike heteroclinic cycles between equilibria, heteroclinic cycles between periodic orbits can be robust under general perturbations since for a hyperbolic periodic orbit p, $dim({W}^{u}(p))+dim({W}^{s}(p))=n+1$. Hence, the condition Equation 2 can be satisfied. For instance, consider a system on ${\mathbb{R}}^{3}$ with two hyperbolic periodic orbits p and q for which the stable and unstable manifolds ${W}^{s}(p)$, ${W}^{u}(p)$, ${W}^{s}(q)$, and ${W}^{u}(q)$ are twodimensional. In this case, ${W}^{u}(p)$ and ${W}^{s}(q)$ (and similarly, ${W}^{u}(q)$ and ${W}^{s}(p)$) intersect transversely, and therefore, a heteroclinic cycle between p and q can exist robustly. However, for this heteroclinic cycle only one orbit connects p to q, whereas infinitely many orbits which are backward asymptotic to p move away from the heteroclinic cycle. As a result, such a robust heteroclinic cycle cannot be asymptotically stable.
To overcome this difficulty we assume that the connections of a heteroclinic cycle between periodic orbits consist of unstable manifolds of periodic orbits and these are contained in the stable manifold of the next periodic orbit. Namely, we say an invariant set Σ is a heteroclinic cycle that contains all unstable manifolds if it consists of a union of periodic orbits and/or equilibria $\{{x}_{i}:i=1,\dots ,p\}$ and a set of connecting manifolds ${S}_{i}={W}^{u}({x}_{i})$ with ${W}^{u}({x}_{i})\subset {W}^{s}({x}_{i+1})$.
Theorem 3 Suppose that Σ
is a heteroclinic cycle that contains all unstable manifolds for$f\in {\mathcal{X}}_{\mathcal{I}}$between hyperbolic equilibria or periodic orbits$\{{x}_{i}:i=1,\dots ,p\}$.
If there exists a finite sequence$\{{I}_{c(1)},\dots ,{I}_{c(p)}\}$of elements in$\mathcal{I}$such that${I}_{c(i)}\supset {S}_{i}$and$dim({W}^{s}({x}_{i+1})\cap {I}_{c(i)})=dim({I}_{c(i)})$
(15)
(in other words, ${x}_{i+1}$is a sink for the dynamics reduced to${I}_{c(i)}$) for all$i=1,\dots ,p$then Σ is robust to perturbations within${\mathcal{X}}_{\mathcal{I}}$.
Proof Consider a unique orbit ${s}_{i}\subset {S}_{i}$. Since ${W}^{s}({x}_{i+1})$contains a neighbourhood of${x}_{i+1}$in${I}_{c(i)}$, ${s}_{i}$ is robust by the same reasoning as in the proof of Theorem 2. This implies that the manifold of connections ${S}_{i}$ is robust for all i. □
Note that a heteroclinic cycle may contain all unstable manifolds but not be attracting even in a very weak sense (essentially asymptotically stable [24]). Conversely, a heteroclinic cycle may not contain all unstable manifolds but may be essentially asymptotically stable.