Monday, December 29, 2014

Peloton hysteresis revisited

In analyzing peloton dynamics, an area ripe for exploration involves the mechanics of peloton density as it stretches and compresses. Speed and power output changes are obvious underlying mechanisms for these oscillations, but there are other more subtle dynamics at play. Among these subtleties are lags or delays in the transitions between density oscillations which are characteristic of hysteresis effects.

There is more than one type of hysteresis, but the general category of hysteresis I consider involves a lag between system input and output [1], meaning there is an asymmetry between a system trajectory, such as acceleration, and its complementary trajectory, such as deceleration. The following description, made in the context of vehicle traffic [2], well describes hysteretic behavior in pelotons:

"The dynamics of traffic flow result in the hysteresis phenomenon. This consists of a generally retarded behavior of vehicle platoons after emerging from a disturbance compared to the behavior of the same vehicles approaching the disturbance."

In a 2010 conference paper "Hysteresis in Competitive Bicycle Pelotons" [3] I proposed three different types of hysteresis in pelotons. Below I re-state and re-categorize them, hopefully with better clarity. In the 2010 paper I referred to "flow" to describe changing peloton density to be consistent with vehicle traffic models, but it is more intuitive to refer to compression (increase in peloton density) and stretching (peloton lengthening).

Types of peloton hysteresis

I.    Where a peloton decelerates relatively rapidly with corresponding rapid compression, followed by a proportionately longer acceleration time and peloton stretching. There are two sub-types:
  • A: occurs with corresponding changes in speed and power output preceding the disturbance and after it, such as slowing before a corner and accelerating out of it;
  • B: occurs with corresponding changes in speed but where power output may be approximately retained, such as when peloton compression and deceleration precede a climb and stretching occurs during the climb, but power output is roughly constant before and during the climb.
II. Where a peloton accelerates rapidly with corresponding stretching effect, followed by increased stretching  even as speeds decrease.

III. Inversely to II, where a peloton decelerates rapidly with corresponding compression, following by an increase in speed where compression either continues to increase for a short period or is temporarily retained even as the peloton accelerates.

For the purpose of this post, I am primarily concerned with types II and III.

Figures 1 & 2 below are re-scaled images from my 2010 hysteresis paper. Figures 3 & 4 show data from indoor velodrome races I videoed a couple of years ago.  Figures 1 - 4 involve speed data that is generated by the front rider in the group, which speed is applied to the entire group.

In Fig 1, the circled area and arrows show a period of decreasing speed following a rapid acceleration with a corresponding increase in stretching. In Fig 2, the arrows show the direction of the hysteresis curve as speed drops even as the peloton continues to stretch. The short arrow pointing down and slightly to the right (Fig 2 speed-stretch curve) is the hysteretic period of deceleration accompanies by increasing stretch. This direction in the speed-stretch trajectory characterizes Type II hysteresis. This indicates a lag or delay in the system's return to a compressed state as deceleration occurs. If the system dynamics were symmetrical, one would expect compression to accompany deceleration immediately, but we can see that is not necessarily the case. This delay is explainable by cyclists' collective fatigue and a recovery period following high acceleration. I refer to this fatigue induced delay in density oscillation as competitive hysteresis. This is the defining feature of type II peloton hysteresis.

Figure 1 independent speed and stretch curves
Figure 2 speed-stretch curve

Figures 3 & 4 below show another example of Type II hysteresis from a 3km indoor mass-start race (red arrows). Increasing stretch follows a double-whammy acceleration (the two sharp spikes preceding the arrows). Following the first of these accelerations to about 48km/h, there is deceleration and corresponding compression, as one would expect. After the second acceleration "whammy" there is the characteristic Type II down-and-to-the-right trajectory in system dynamics, as shown in Figure 4 (red arrow), indicating deceleration accompanied by increased stretching.

Figures 3& 4 also show an example of Type III hysteresis (green). Following soon after the Type II period, there is a period of increasing speed and increasing compression (Fig 3, green arrows), characterized by an up-and-to-the-left trajectory (Fig 4, green arrow) in the speed-stretch curve. This may occur when there is an acceleration from cyclists toward the rear of the peloton who pass riders ahead. In this case the hysteretic lag is in the return to a stretched state.

Type III hysteresis is also a precursor dynamic to the convective phase discussed in earlier papers [e.g. 5].

Figure 3 speed of front rider, and stretch 

Figure 4 speed-stretch

Using the model in [4], I've sought to identify occurrences of Types II and III hysteresis in simulated peloton dynamics.

Similar to the speed data in Figures 1 - 4, Figures 5 & 6 show the speed of the front rider versus peloton density. Speed of the front rider was coded to randomly fluctuate every 60 ticks (equivalent to 60s) within a narrow range, hence the "square" appearance of the speed data, as well as in the speed-stretch curve.  In Fig 6, generally wherever the curve proceeds at an angle it indicates a hysteretic delay. Using Fig 6, it is difficult to trace the direction of the curve which would tell us the hysteretic Type, although we can roughly eyeball regions of both Types II and III in Fig 5.  I've noted two periods of stable front-rider speed versus increased stretching, which appear to be Type II periods (although difficult to know given the speed curve is stable in those regions).

In contrast, Figures 7 & 8 show the mean speeds of all the riders versus peloton density, which is a level of detail that provides a clearer indication of whether there is acceleration or deceleration occurring. These aggregate speeds are easy to trace in simulations, but are a level of detail I have not obtained for any empirical data. This level of detail could be obtained empirically if speed data was obtained individually for each rider and averaged over the course of the race. This sort of data is desirable.

In Fig 7 there are arrows in about the same locations as for Fig 5, and the opposing direction of the aggregate speed curve do suggest that these regions are indeed hysteretic periods, of Type II. While I have not noted a Type III occurrence, the Fig 8 speed-stretch curve winds in all directions rather like the trajectory of a fruit fly, so it seems the simulation captures a fairly continuous but non-linear hysteresis trajectory. A speed-stretch curve of the moving averages of the mean speeds and the stretch might produce visually more apparent looping patterns.

Figure 5 speed of front rider, and stretch

Figure 6 speed-stretch

Figure 7 mean speeds of all riders, and stretch 

Figure 8 speed-stretch 

Where else might this be applied? 

Regarding other types of flocking systems, similar system lags could be tested by accelerating groups to maximal sustainable outputs for specific periods, followed by deceleration periods, in much the same way as demonstrated here for pelotons.

Here I get highly speculative, but I wonder if there are similar kinds of competitive hysteresis processes in bacterial colonies, or perhaps cancer growths, that might reveal vulnerabilities in those systems. I do not know enough about either of those systems even to propose an experimental protocol to test such a hypothesis. It does seem worth looking into further, however, if it has not already been done.  

It further occurs to me that, having mentioned fruitflies, when we look at 3-dimensional fruit fly trajectories, perhaps there is a way to decompose those trajectories into 2-dimensional speed and stretch data. I don't have a good sense of how this would be done at this point, however, and am simply putting the idea out there.

If nothing else, we gain insight into peloton dynamics specifically, and may gain understanding of general evolutionary mechanisms for group formation in a variety of systems and how these systems behave.  

2. Treiterer, J., Myers, J.A. 1974. The Hysteresis Phenomenon in Traffic Flow. In: Buckley, D.J. (ed.), Proceedings of the 6th International Symposium on Transportation and Traffic Theory: 13-38
3. Complex Adaptive Systems —Resilience, Robustness, and Evolvability: Papers from the AAAI Fall Symposium (FS-10-03)
4. Trenchard, H., Ratamero, E., Richardson, A., Perc, M. 2015. A deceleration model for peloton dynamics and group sorting. Applied Mathematics and Computation 251, 24–34
5. Trenchard, H.Richardson, A.Ratamero, E.  Perc, M. 2014. Collective behavior and the identification of phases in bicycle pelotons. Physica A: Statistical Mechanics and its Applications, 405 . pp. 92-103. ISSN 0378-4371

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