## 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].

## Words and Reality and Change: What Is a Fluctuation?

Ludwig Boltzmann who died in 1906 was a giant in the history of physics.

His name is associated with various fields like statistical mechanics, entropy and so on.

A standard physics overview book called Introducing Quantum Theory (2007, Icon/Totem Books) shows a “cartoon” of Boltzmann which says, “I also introduced the controversial notion of fluctuations.” (page 25)

In common parlance, some common synonyms of fluctuate are oscillate, sway, swing, undulate, vibrate and waver. While all these words mean “to move from one direction to its opposite,” fluctuate suggests (sort of) constant irregular changes of level, intensity or value. Pulses and some pulsations suggest themselves as related.

Expressions like “Boltzmann brains” refer to this great physicist Boltzmann and you can find this notion described here: “Boltzmann Brain.”

Notice that the word “fluctuation” occurs four times in one of the paragraphs of the article “Boltzmann Brain,” as you can see:

“In 1931, astronomer Arthur Eddington pointed out that, because a large fluctuation is exponentially less probable than a small fluctuation, observers in Boltzmann universes will be vastly outnumbered by observers in smaller fluctuations. Physicist Richard Feynman published a similar counterargument within his widely read 1964 Feynman Lectures on Physics. By 2004, physicists had pushed Eddington’s observation to its logical conclusion: the most numerous observers in an eternity of thermal fluctuations would be minimal “Boltzmann brains” popping up in an otherwise featureless universe.”

You may remember perhaps you’ve also heard the term, perhaps on a PBS Nova episode on quantum fluctuation.

In the classic history of science book, The Merely Personal by Dr. Jeremy Bernstein (Ivan Dee, Chicago, 2001), one encounters the word fluctuation all over:

“This uniform density of matter …and fluctuations from the average are what would produce the unwanted instability.”

“So Einstein chose the cosmological constant…” (page 83 of Bernstein’s book)

Suppose we allow our minds to be restless and turn to economics to “change the lens” we are using to look at the world, since lens-changing is one of the pillars of Meta Intelligence.

What do we see?

In 1927, Keynes’s professor Arthur Cecil Pigou (died in 1959) published the famous work, Industrial Fluctuations.

In 1915, twelve years earlier, the famous Sir Dennis Holme Robertson (died in 1963) published A Study of Industrial Fluctuation.

The word fluctuation seems to be migrating to or resonating in economics.

The larger point (i.e., the Meta Intelligent one): is the use of this word a linguistic accident or fashion or is something basic being discovered about how some “things” “jump around” in the world?

Is the world seen as more “jumpy” or has it become more jumpy due to global integration or disintegration or in going to the deeper levels of physics with the replacement of a Newtonian world by an Einsteinian one?

The phenomena of change—call it “change-ology” whooshes up in front of us and a Meta Intelligent student of the world would immediately ponder fluctuations versus blips versus oscillations versus jumps and saltations (used in biology) and so on. What about pulsations? Gyrations?

This immediately places in front of you the question of the relationship of languages (words, numbers, images) to events.

The point is not to nail down some final answer. Our task here is not to delve into fields like physics or economics or whatever but to notice the very terms we are using across fields and in daily life (i.e., stock price fluctuations).

Notice, say, how the next blog post on oil price dynamics begins:

“Our oil price decomposition, reported weekly, examines what’s behind recent fluctuations in oil prices…”

The real point is to keep pondering and “sniffing” (i.e., Meta Intelligence), since MI is an awareness quest before all.