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

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 [3]: 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 [25].

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 [25] 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.