## Mathematics and the World: London Mathematical Laboratory

### Stability of Heteroclinic Cycles in Rings of Coupled Oscillators

[from the London Mathematical Laboratory]

Complex networks of interconnected physical systems arise in many areas of mathematics, science and engineering. Many such systems exhibit heteroclinic cyclesdynamical trajectories that show a roughly periodic behavior, with non-convergent time averages. In these systems, average quantities fluctuate continuously, although the fluctuations slow down as the dynamics repeatedly and systematically approach a set of fixed points. Despite this general understanding, key open questions remain concerning the existence and stability of such cycles in general dynamical networks.

In a new paper [archived PDF], LML Fellow Claire Postlethwaite and Rob Sturman of the University of Leeds investigate a family of coupled map lattices defined on ring networks and establish stability properties of the possible families of heteroclinic cycles. To begin, they first consider a simple system of N coupled systems, each system based on the logistic map, and coupling between systems determined by a parameter γ. If γ = 0, each node independently follows logistic map dynamics, showing stable periodic cycles or chaotic behavior. The authors design the coupling between systems to have a general inhibitory effect, driving the dynamics toward zero. Intuitively, this should encourage oscillatory behavior, as nodes can alternately be active (take a non-zero value), and hence inhibit those nodes to which it is 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 the simple case of N = 3, for example, this dynamics leads to a trajectory which cycles between three fixed points.

The authors then extend earlier work to consider larger networks of coupled systems as described by a directed graph, describing how to find the fixed points and heteroclinic connections for such a system. In general, they show, this procedure results in highly complex and difficult to analyze heteroclinic network. Simplifying to the special case of N-node directed graphs with one-way nearest neighbor coupling, they successfully derive results for the dynamic stability of subcycles within this network, establishing that only one of the subcycles can ever be stable.

Overall, this work demonstrates that heteroclinic networks can typically arise in the phase space dynamics of certain types of symmetric graphs with inhibitory coupling. Moreover, it establishes that at most one of the subcycles can be stable (and hence observable in simulations) for an open set of parameters. Interestingly, Postlethwaite and Sturman find that the dynamics associated with such cycles are not ergodic, so that long-term averages do not converge. In particular, averaged observed quantities such as Lyapunov exponents are ill-defined, and will oscillate at a progressively slower rate.

In addition, the authors also address the more general question of whether or not a stable heteroclinic cycle is likely to be found in the corresponding phase space dynamics of a randomly generated physical network of nodes. In preliminary investigations using randomly generated Erdős–Rényi graphs, they find that the probability of existence of heteroclinic cycles increases both as the number of nodes in the physical network increases, and also as the density of edges in the physical network decreases. However, even in cases where the probability of existence of heteroclinic cycles is high, there is also a high chance of the existence of a stable fixed point in the phase space. From this they conclude that the question of the stability of the heteroclinic cycle is important in determining whether or not the heteroclinic cycle, and associated slowing down of trajectories, will be observed in the phase space associated with a randomly generated graph.

The paper is available as a pre-print here [archived PDF].

## Education and the Long-Term: Automation As Example

### The American Revolution: Pages From a Negro Worker’s Notebook

##### Chapter 2: The Challenge of Automation

“Since 1955 and the advent of automation, overtime has been detrimental to the workers. Again and again workers have been faced with the decision to work overtime or not to work overtime, and the decision has usually been: ‘To hell with those out of work. Let’s get the dollar while the dollar is gettable.’ The amazing thing is that this has nothing to do with the backwardness of these workers. Not only can they run production and think for themselves, but they sense and feel the changes in conditions way in advance of those who are supposed to be responsible for their welfare. But with all these abilities there is one big organic weakness. Over and over again workers in various shops and industries, faced with a critical issue, only divide and become disunited, even though they are well aware that they are being unprincipled and weakening their own cause as workers. Since the advent of automation there has not been any serious sentiment for striking, particularly if the strike was going to come at the expense of material things that the workers already had in their possession, like cars, refrigerators, TV sets, etc. They were not ready to make any serious sacrifices of these; they would rather sacrifice the issue. Between the personal things and the issue, they have chosen the personal. Most American workers have geared themselves to a standard of living that is based on a five-day week plus—either in the form of overtime or another job, part or full time. And any time this standard of living is threatened, it is a personal crisis, which means that more and more decisions are being personalized and individualized rather than collectivized and socialized.”

(The American Revolution: Pages From a Negro Worker’s Notebook, James Boggs, Monthly Review Press, 1963, page 33)

As far back as 1963, with President John Kennedy in office, James Boggs (a Detroit autoworker) was already quite aware of automation and its challenges.

A “meta-intelligent” education means we learn from any sources available including “angry pamphlets” without worrying about the ideological blinders or fireworks because our desire is not to engage in polemics but to “extract signals” from a noisy world.

Chapter 2 of James Boggs’s pamphlet is called “The Challenge of Automation” and begins: “Since 1955 and the advent of automation, overtime has been detrimental to the workers…”

This immediately tells you that automation is a very long-run historical trend and should be seen in a larger sweep with history as your searchlight.

Indeed the famous German classic The Weavers by Gerhart Hauptmann is about machines as a threat to employment:

The Weavers (German: Die Weber, Silesian German: De Waber) is a play written by the German playwright Gerhart Hauptmann in 1892. The play sympathetically portrays a group of Silesian weavers who staged an uprising during the 1840s due to their concerns about the Industrial Revolution and replacement by machines and automation.

In 1927, it was adapted into a German silent film The Weavers, directed by Frederic Zelnik and starring Paul Wegener.

A Broadway version of The Weavers was staged in 1915–1916.

To dismiss all such movements and revolts as Luddite-like is not useful since it sweeps legitimate problems under the rug.

This includes Ernst Toller’s classic The Machine Wreckers (German: Die Maschinenstürmer). Two of his early plays were produced in this period: The Machine Wreckers (1922), whose opening night in 1937 he attended, and No More Peace, produced in 1937 by the Federal Theatre Project and presented in New York City in 1938.

All of these critiques of machines and automation are part of a long-term historical overview of machines and jobs and in our time, robotics and AI, etc which should be analyzed as a trajectory and arc where “machine wreckers” à la Hauptmann or Toller are understood empathetically and realistically and not dismissed as vandals.