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# From coupled networks of systems to networks of states in phase space

• Dynamical systems on graphs can show a wide range of behaviours beyond simple synchronization - even simple globally coupled structures can exhibit attractors with intermittent and slow switching between patterns of synchrony. Such attractors, called heteroclinic networks, can be well described as networks in phase space and in this paper we review some results and examples of how these robust attractors can be characterised from the synchrony properties and how coupled systems can be designed to exhibit given but arbitrary network attractors in phase space.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation: • • Figure 1.  (a) Coupled cell structure of the Guckenheimer-Holmes system, see Example 2. (b) The Guckenheimer-Holmes heteroclinic cycle in phase space as the limiting set of a trajectory between the saddle equilibria $(x,y,z) = (\xi,0,0)$, $(0,\xi,0)$ and $(0,0,\xi)$ for some $\xi>0$, with active cells along the cycle indicated. Note that this is part of a larger network of twelve connections between the six equilibria $(\pm \xi,0,0)$, $(0,\pm \xi,0)$ and $(0,0,\pm \xi)$; typical initial conditions limit to one of eight possible cycles. (c) Corresponding time-series showing asymptotic slowing down as the trajectory approaches the heteroclinic cycle.

Figure 2.  A coupled cell network with two cell types and three edge types, graphically indicated by different box and arrow styles. As in $1.$ of Def. 3.1, edges of the same type have equivalent sources and targets. Note also that the input sets $I(c_1)$ and $I(c_2)$, and $I(c_3)$ and $I(c_4)$ respectively, consist of the same number of edges of each type, which is the content of $2.$ in Def. 3.1.

Figure 3.  The three cell network architecture of $\mathcal{N}_3$ from : note the presence of two edge types.

Figure 4.  (a) The heteroclinic cycle of Example 4 seen in phase space, with two nodes and four connections. (b) Time-series for a trajectory along the cycle in (a) with added independent noise of amplitude $10^{-7}$ in all components and initial condition $(x,y,z) = (1,1.001,0.999)$. Observe heteroclinic switching between two synchronized states $x = y = z = \pm 1$ but two types of switching owing to the two branches of the unstable manifold shown in Fig. 5: calculations using xppaut .

Figure 5.  The system (2, 5) in the synchrony subspace ${\bf{P}}_2$. Green lines show the nullcline for the $y$-component and red lines show the nullcline for the $x$-component, with parameters as in text. Blue curves are trajectories that approximate of the unstable manifold of $(x,y,z) = (-1,-1,-1)$: note that both branches of the unstable manifold are asymptotic to the sink at $(1,1,1)$. The numerical integration of (2, 5) is performed using xppaut  and a Runge-Kutta integrator with time step $0.05$.

Figure 6.  Time-series showing the $p_i$ components of trajectories exploring a realization of the Kirk-Silber network using (6, 7), with low amplitude noise. (a) shows a typical time-series for a heteroclinic network realization - note that the system moves around between four equilibria $P_i$, where $p_i = 1$ and $p_j = 0$ for $j\neq i$. The network is the union of the two cycles $P_1\rightarrow P_2\rightarrow P_4\rightarrow P_1$ and $P_1\rightarrow P_3\rightarrow P_4\rightarrow P_1$. (b) shows the case where one connection from $P_1$ has been made more unstable and in addition the connection from $P_3$ has been made excitable rather than heteroclinic. In particular, the noise-induced escape times for the heteroclinic transitions are fairly uniform while those for excitable transition are more widely distributed, consistent with exponential distribution.

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