Visibility of heteroclinic networks

TL;DR

This work identifies a gap where classical stability notions fail to predict what is actually observed near heteroclinic networks and introduces visibility as a rigorous, complementary framework. By analyzing canonical networks—the GH cycle, KS network, and RPSSL system—the authors show that even stable networks can conceal observable dynamics, such as selective subcycles or aperiodic switching, depending on parameters and resonance. They formalize visibility with Lyapunov, quasi, and asymptotic variants, including almost/essential/fragmentary prefixes, and apply these to concrete examples to distinguish what is observable after transients. The findings have implications for understanding outcomes in applications like population dynamics and game theory, and point toward future work on proving visibility and extending to higher‑dimensional unstable manifolds using modern Poincaré‐map and piecewise‑dynamics techniques.

Abstract

The concept of stability has a long history in the field of dynamical systems: stable invariant objects are the ones that would be expected to be observed in experiments and numerical simulations. Heteroclinic networks are invariant objects in dynamical systems associated with intermittent cycling and switching behaviour, found in a range of applications. In this article, we note that the usual notions of stability, even those developed specifically for heteroclinic networks, do not provide all the information needed to determine the long-term behaviour of trajectories near heteroclinic networks. To complement the notion of stability, we introduce the concept of visibility, which pinpoints precisely the invariant objects that will be observed once transients have decayed. We illustrate our definitions with examples of heteroclinic networks from the literature.

Figures (7)

• Figure 1: (a) Sketch of the Guckenheimer--Holmes heteroclinic 'network', with the cycle in the positive orthant highlighted. (b) Sketch of the dynamics in the positive orthant when $\rho_{123}>1$. (c) Sketch of the dynamics in the positive orthant when $\rho_{123}=1$; note that there exists an infinite family of periodic orbits.
• Figure 2: Trajectories near the Guckenheimer--Holmes cycle \ref{['eq:RPS']}. In (a) the initial conditions have positive values for all coordinates, and the trajectory approaches the cycle between $+\xi_1$, $+\xi_2$ and $+\xi_3$. In (b), the initial condition has a negative value of $x_2$ and the trajectory asymptotes onto a cycle between $+\xi_1$, $-\xi_2$ and $+\xi_3$. Parameters are the same in both panels and are given in table \ref{['tab:parametervalues']} in Appendix A.
• Figure 3: (a) The Kirk--Silber network, and (b) the Rock--Paper--Scissors--Spock--Lizard network, both represented as directed graphs. Dots indicate equilibria, which lie on co-ordinate axes, and arrows represent heteroclinic connections. In (a) all connections are one-dimensional and contained in the same plane as the equilibria they connect. The colours in (b) indicate two different types of connection: type A (amber) is a two-dimensional connection with one connecting trajectory in the plane containing the equilibria it connects and the remaining connecting trajectories in three-dimensional space; type B (blue) are one-dimensional connecting trajectories in the plane containing the equilibria. See Section \ref{['sec:LS']} and Figure \ref{['fig:2Dwu']} for more detail on the type A (amber) connections.
• Figure 4: Trajectories near the Kirk--Silber network. In each case, the coordinates are plotted on a logarithmic axis, each colour corresponding to one coordinate as indicated. Parameters are given in table \ref{['tab:parametervalues']} in appendix A. In (a) (section \ref{['sec:KS1']}), the trajectory is approaching the $\xi_1$-$\xi_2$-$\xi_3$ cycle: the time spent near, e.g., $\xi_1$ increases on each loop around the cycle, the minimum value of the $x_1$, $x_2$ and $x_3$ coordinates decreases on each loop, and additionally the $x_4$ coordinate is decaying. In (b) (section \ref{['sec:KS2']}), the trajectory initially appears to approach $\xi_1$-$\xi_2$-$\xi_3$: again the time spent near $\xi_1$ increases on each loop around the cycle. However, now the $x_4$ coordinate grows and the trajectory switches to the $\xi_1$-$\xi_2$-$\xi_4$ cycle. In (c) and (d) the $\xi_1$-$\xi_2$-$\xi_4$ cycle is at resonance. In (c) (section \ref{['sec:KS3']}), the trajectory starts near the $\xi_1$-$\xi_2$-$\xi_4$ cycle and the $x_3$ coordinate decays. However, trajectories do not get closer to the $\xi_1$-$\xi_2$-$\xi_4$ within that subspace as they remain near a periodic orbit. In (d) (section \ref{['sec:KS4']}), the trajectory starts near the $\xi_1$-$\xi_2$-$\xi_3$ cycle and then switches to the $\xi_1$-$\xi_2$-$\xi_4$ cycle; the $\xi_1$-$\xi_2$-$\xi_4$ cycle is at resonance so the trajectory remains a finite distance away.
• Figure 5: Trajectories near the Kirk--Silber network: the same data as in Figure \ref{['fig:KS_example']} but on a linear scale.

Theorems & Definitions (14)

• Definition 1
• Definition 2: adapted from Kuznetsov Kuznetsov1998
• Definition 3: adapted from Glendinning Glendinning1994
• Definition 4: Milnor Milnor1985
• Definition 5: Definition 1.1 in Brannath Brannath1994
• Definition 6: Definition 1.2 in Brannath Brannath1994
• Definition 7: Podvigina Podvigina2012
• Definition 1
• Definition 2
• Definition 3