Stability of heteroclinic cycles in rings of coupled oscillators

From the coupled map lattices proposed in the 1980s to the modern discipline of complex networks the study of simple systems connected in some way forms a fundamental paradigm in dynamical systems. Applications are plentiful and diverse, and include spatially extended systems, chemical reactions, biological and ecological networks, and span many length scales[1]. In many cases, a goal of study is spatiotemporal chaos[2], and this might typically mean computing a long-term average (for example, a Lyapunov exponent). However, it is well-known that a class of networks — those with invariant subspaces forced by symmetries in the system — permit heteroclinic cycles[3], that is, trajectories along which time averages do not converge, instead slowing down as they repeatedly and systematically get closer and closer to fixed points. We investigate a family of coupled map lattices defined on ring networks and establish stability properties of the many possible families of heteroclinic cycles.

Specifically, we consider how the structure of the architecture, or topology, of the network of physical nodes determines the architecture of a heteroclinic network in phase space between fixed points. We note that this question was asked in the reverse by Ashwin and Postlethwaite [4], who showed how to construct a system of ordinary differential equations into which was embedded a heteroclinic network of any specified topology.

In particular, we consider systems of the form

where $\gamma\leq 0$ is a parameter. Each of the $x_{i}^{(k)}\in[0,1]$ measures the activity at time $i$ of the $k$th ‘node’ in a network, and $A_{\hat{k}k}$ is an adjacency matrix which determines the connectivity between the different nodes. We take the function $f$ to be the logistic map

where $r\in(0,4]$ is a parameter. The dynamics of the uncoupled system with $\gamma=0$ is well known [5], but briefly, for $r\in(0,1]$, the origin is an asymptotically stable fixed point. At $r=1$ there is a transcritical bifurcation creating a second fixed point at $x=\frac{r-1}{r}$. This fixed point is asymptotically stable for $r\in(1,3]$, and at $r=3$ undergoes a period-doubling bifurcation which leads to a period-doubling cascade followed by the onset of chaos at $r\approx 3.56995$.

When $x^{(\hat{k})}$ is non-zero for some $\hat{k}$, then the term $e^{-\gamma\sum_{\hat{k}}A_{\hat{k}k}x^{(\hat{k})}}$ in (1), with $\gamma>0$, has an inhibitory effect upon any node connected to $\hat{k}$, i.e. those for which $A_{\hat{k}k}=1$. Specifically, if $\gamma x^{(\hat{k})}$ is large enough, then this term can have the same effect as reducing the value of $r$ in the $x^{(k)}$ equation to less than 1, and hence causing the values of those $x^{(k)}$ to decrease towards zero. Heuristically, it is then clear that oscillatory behaviour is possible, as nodes can alternately be active (have a non-zero value), and hence inhibit those nodes they are connected to; decay, when other nodes in turn inhibit them; and finally grow again to an active state as the nodes inhibiting them decay in turn.

In figure 1 we show a time series from a cycle of three such coupled nodes, specifically, the set of equations

In panel (a), we use $r=2$, so the dynamics of the uncoupled system contains a stable fixed point. The time series clearly shows the trajectory cycling between three fixed points, in a manner essentially identical to that seen in the well-known Guckenheimer–Holmes heteroclinic cycle[3]. In panel (b), we use $r=3.5$, so the uncoupled system is in the chaotic regime, and we see cycling between three chaotic attractors. This phenomena was previously described by Ashwin, Rucklidge and Sturman [6, 7].

In this paper, we extend the work of Ashwin et al. [6, 7] and consider larger networks of coupled systems in the form of (1). We refer these equations as describing the network of connections between nodes in physical space, and for the remainder of the paper, refer to this network as instead a directed graph with directed edges between nodes. We begin in section II, by considering another example: a five-node ring graph with one-way nearest-neighbour coupling. We determine the fixed points and the heteroclinic connections which exist between them. We refer to this network of connection as the phase space network, or heteroclinic network, which has heteroclinic connections (or sometimess simply connections) between fixed points. In section III, we consider general systems of the form of (1) and describe how to find the fixed points and heteroclinic connections for such a system. In general, this procedure results in a very complex heteroclinic network that is difficult to analyse, so in section IV we look in detail at $N$-node directed graphs with one-way nearest neighbour coupling in the physical space. Here, as well as determining the structure of the heteroclinic network in phase space, we are able to analyse the dynamic stability of subcycles within the network. We use results from Podvigina [8], and some classical results on solutions to polynomials [9, 10] to prove theorem 1, which shows that only one of the subcycles can ever be stable, and then, only if $\gamma$ is large enough. In section V we make some conjectures about larger networks. Section VI concludes.

We give, in this section, an example of a five node directed graph with one-way nearest-neighbour coupling. We determine the possible fixed points in phase space, and the heteroclinic connections between them. We show time series of typical trajectories close to the resulting heteroclinic network but defer the computation of the stability of each of the sub-cycles to section IV.

We begin with a few formal definitions. Consider the map

with fixed points $\zeta_{1},\dots,\zeta_{M}$. A heteroclinic connection between $\zeta_{j}$ and $\zeta_{j^{\prime}}$ is a solution to (3) for which $x_{i}\rightarrow\zeta_{j}$ as $i\rightarrow-\infty$ and $x_{i}\rightarrow\zeta_{j^{\prime}}$ as $i\rightarrow\infty$. Suppose that there exist heteroclinic connections between $\zeta_{j_{k}}$ and $\zeta_{j_{k+1}}$ for $k=1,\dots,M-1$, and also one between $\zeta_{j_{M}}$ and $\zeta_{j_{1}}$. Then the set $H$ consisting of the fixed points $\zeta_{j}$ and the connecting orbits is a heteroclinic cycle. A heteroclinic network [11] is usually defined to be a union of heteroclinic cycles. In this paper, we relax the definition somewhat: we allow a heteroclinic network to consist of a set of fixed point solutions, and heteroclinic connections between them, which contains at least one heteroclinic cycle. Note that this means that not every heteroclinic connection in the network need be part of a cycle.

For the example we consider in this section, the network of nodes in physical space is shown in figure 2(a). The equations governing this system are

which are equivariant with respect to a rotation symmetry of the coordinates.

Equation (4) has two different types of fixed points solution in phase space which are of interest to us, namely, those with one node active (that is, with a single component that is $O(1)$), or those with two nodes active. More precisely, assume for now that $r\in[1,3]$, and let $\hat{x}=\frac{r-1}{r}$. Then, using coordinates $(x^{(1)},x^{(2)},x^{(3)},x^{(4)},x^{(5)})$, we label the fixed points with only $x^{(1)}$ active as:

and similarly we have $\xi_{2},\dots,\xi_{5}$, where for each $\xi_{j}$, the $j$th component is equal to $\hat{x}$, and the remainder are equal to zero. Next, we label

and similarly define $\xi_{j,m}$, which has the $j$th and $m$th components equal to $\hat{x}$, where $|j-m|\neq 1$ (i.e. $j$ and $m$ are not adjacent nodes in the ring graph). Note that $\xi_{j,m}\equiv\xi_{m,j}$ but we typically list $j$ and $m$ in increasing numerical order.

Note that in this example, there cannot be any fixed points with more than two components equal to $\hat{x}$, because of the connectivity of the graph: two nodes which are connected by an edge cannot both be active at a fixed point. In larger, more general graphs, we would expect to see fixed points with more active components (see section III for the general setup).

We next consider the dynamics of (4) in two two-dimensional subspaces, and show that there exist heteroclinic connections from $\xi_{j}$ to $\xi_{j-1}$ and from $\xi_{j}$ to $\xi_{j,j\pm 2}$ (where indices are taken $\mod 5$).

For the first, consider the dynamics in the subspace where $x^{(2)}=x^{(3)}=x^{(4)}=0$, namely the system

System (5) has fixed points at $(x^{(1)},x^{(5)})=(\hat{x},0)\equiv\xi_{1}$ and $(x^{(1)},x^{(5)})=(0,\hat{x})\equiv\xi_{5}$. Consider an initial condition close to $\xi_{1}$, but with $x^{(5)}_{0}\neq 0$. Since the $x^{(5)}$ equation is decoupled from $x^{(1)}$, then it behaves as it would in the uncoupled logistic map, specifically, the $x^{(5)}$ component initially grows and approaches the value $\hat{x}$. As $x^{(5)}$ grows, the coupling term in the $x^{(1)}$ equation has the effect of essentially reducing the $r$ value of the logistic map in the $x^{(1)}$ equation to, eventually, $re^{-\gamma\hat{x}}$. Thus, if $re^{-\gamma\hat{x}}<1$, then $x^{(1)}$ will eventually decay to zero, and the trajectory approaches $\xi_{5}$. There is thus a heteroclinic connection from $\xi_{1}$ to $\xi_{5}$, and by symmetry, heteroclinic cycles from $\xi_{j}$ to $\xi_{j-1}$.

For the second type of connection, consider the dynamics in the subspace where $x^{(2)}=x^{(4)}=x^{(5)}=0$, namely the system

In this subspace, both $x^{(1)}$ and $x^{(3)}$ are decoupled from each other. There are three fixed points, $(x^{(1)},x^{(3)})=(\hat{x},0)\equiv\xi_{1}$, $(x^{(1)},x^{(3)})=(0,\hat{x})\equiv\xi_{3}$, and $(x^{(1)},x^{(3)})=(\hat{x},\hat{x})\equiv\xi_{1,3}$. Both $\xi_{1}$ and $\xi_{3}$ are saddle points, and perturbations close to these fixed points will result in trajectories which approach $\xi_{1,3}$. There are thus heteroclinic connections between $\xi_{1}$ and $\xi_{1,3}$, $\xi_{3}$ and $\xi_{1,3}$, and by analogy, heteroclinic connections between any $\xi_{j}$ and $\xi_{j,m}$ or $\xi_{m,j}$.

Finally, we consider the dynamics of (4) in a three-dimensional subspace, and show that there is a heteroclinic connection from $\xi_{j,j+2}$ to $\xi_{j-1,j+2}$. Consider the dynamics in the subspace where $x^{(2)}=x^{(4)}=0$, namely the system

There are, as before, fixed points in this system with one component non-zero, but of interest right now are the two fixed points with $(x^{(1)},x^{(3)},x^{(5)})=(\hat{x},\hat{x},0)\equiv\xi_{1,3}$ and $(x^{(1)},x^{(3)},x^{(5)})=(0,\hat{x},\hat{x})\equiv\xi_{3,5}$. Note first that the $x^{(3)}$ and $x^{(5)}$ components are decoupled. Both have stable fixed points at $x^{(i)}=\hat{x}$. Thus perturbations close to $\xi_{1,3}$ will have an $x^{(3)}$ component which remains close to $\hat{x}$, but, as in the case of the connection between $\xi_{1}$ and $\xi_{5}$, the $x^{(5)}$ component will grow, and again, so long as $re^{-\gamma\hat{x}}<1$, the $x^{(1)}$ component will decay to zero.

Due to the rotational symmetry of the system (4), we thus have three families of heteroclinic connections, namely:

Note that here, indices are taken $\mod 5$. In later sections, indicies are taken $\mod n$, where $n$ is the number of nodes in the graph. We refer to the families of heteroclinic connections as $1\rightarrow 1$ connections, $1\rightarrow 2$ connections and $2\rightarrow 2$ connections respectively, where a $p\rightarrow q$ connection is a transition from a fixed point with $p$ nodes active to a fixed point with $q$ nodes active.

The complete set of connections between fixed points is shown in panel (b) of figure 2. Notice that there are two heteroclinic cycles, one between fixed points of type $\xi_{j}$, with $1\rightarrow 1$ connections, and the other between fixed points of type $\xi_{j,k}$, with $2\rightarrow 2$ connections. The $1\rightarrow 2$ connections are not part of any cycles. In figure 3 we show numerical simulations showing trajectories close to each of these cycles. We will compute the stability of cycles of these types in general in section IV that follows. We shall see that the cycle between fixed points of type $\xi_{j,k}$ can have some form of stablility, if parameters are chosen correctly, specifically, if $\gamma>\frac{3\log r}{2\hat{x}}$, but the cycle between fixed points of type $\xi_{j}$ can never be stable. However, if initial conditions are chosen carefully (in a manner described in that section), we can, as seen in panel (b) of figure 3, observe this cycle for a reasonably long period of time.

In this section, we describe how to find the fixed points and heteroclinic connections in phase space for any directed graph with inhibitory coupling. In section IV that follows, we apply this to an $n$-node graph with nearest-neighbour coupling.

In this section we consider the case of a general $n$-node ring graph, with one-way nearest-neighbour coupling. That is, the system (1) with $A$ a cyclic permutation matrix, given by equations

We refer to these graphs as $(N,1)$-graphs.

Note that the system (9) is equivariant with respect to the group $\mathbb{Z}_{N}$, generated by the element $\sigma$, which has action

As for the five-node system, equations (9) have $n$ fixed point solutions $\xi_{j}$ each with the $j$th component equal to $\hat{x}$, and all other components zero. Other fixed points are found using the method in section III.1¹¹1For the $(N,1)$-graph case the fixed points can be enumerated directly with the Lucas numbers $L_{n}$, given by $L_{1}=1,L_{2}=3,L_{n}=L_{n-1}+L_{n-2},n\geq 3$. The $(N,1)$-graph has $L_{n}$ fixed points, including the configuration with $N$ zero components.. Note that for any set of nodes $Z_{+}$, $Z_{s}(Z_{+})=|Z_{+}|$, that is, the number of suppressed nodes is the same as the number of active nodes, and each suppressed node has a suppression number of one.