Chaotic Fractal Movement Variability

A healthy, natural, and adaptive movement has fractal dynamics. As soon as intention or external disturbances kick in, this dynamic changes, and the movement becomes more artificial and less adaptive. Is it possible to modulate fractal dynamics on purpose, and what would be the practical and aesthetic implications?

 

Natural Movement, Fractals, and Chaos

Studies have shown that our body is swaying gently when we are standing still in a calm resting state, and that swaying has fractal properties [1] [2]. Constant fractal fluctuations characterize the healthy variability that allows adaptation to environmental change [3].

Image from Yamada (1995) showing the chaotic variability of a still body’s small natural movements.

Fractal (in this context) means that the movement dynamics over time has a self-repetitive shape, which will be similar at any scale: whether we observe the movement for only a short time or for much longer. There will be many small-amplitude movements and a few, but a significant number of big-amplitude ones. In plain terms, it means that somebody whose movement is fractal will repeat the fluctuation patterns both on a very small scale (e.g., hands moving gently) and also on a big scale (e.g., higher amplitude, faster movement). This dynamics is also referred to as “chaotic” or exhibiting the 1/f (power-law) pink-noise dynamics.

If we look at a signal with chaotic fractal dynamics, it may look unpredictable, but there is a certain pattern that emerges over time. This pattern follows a power-law: small frequent changes are followed by a few big ones (hence, the “1/f” law — the higher the frequency, the lower the amplitude.

Another, more intuitive understanding of the fractals is provided when we look at the image below that shows romanesco and apples. The sizes of the romanesco heads are all different, but if we plot them by frequency we will see that they follow a power-law: the bigger is the head, the less frequently it appears. On the other side, the apples may also all have different shapes but their deviations from the mean are more or less the same:

The different apple sizes will oscillate around a mean. While the romanesco heads distribution will follow a power law: the average does not really exist, we have many heads that are small in size, but also a significant number of bigger ones. Image from Hardstone et al (2012) Detrended Fluctuation Analysis

The movement itself may appear diverse and noisy, but it is not random because there’s a relationship between the short-term and the long-term fluctuations. There will be many small changes, and their accumulation will be followed by an occasional big change, which, in turn, will either shift the system to a new state or bring it back to the low-amplitude fluctuation mode. That’s why a system that is fractal is also adaptable: it can accommodate an incoming impulse and change accordingly.

Another state is pure randomness and is called “white noise”, as it has no particular state or color. The proportion of small and big fluctuations is more or less the same, no matter what scale (time period) we take. The short-term and the long-term do not communicate, which means the system does not have any memory and the past actions carry no influence on the future ones. A 0 will be followed by 1, a series of fluctuations in one direction will be followed by a series of fluctuations in the other—a sort of ideal equilibrium, maximum entropy, which is incompatible with life.

Different types of noise: A (top) – random white noise (no correlation between the frequency and the amplitude — all the different fluctuations have a similar chance of appearing). C – pink noise (power-law) – the lower is the power, the higher is the frequency (and vice versa). E – brown noise – the correlation is even stronger but not fractal: big changes (phase shifts) happen quite often (in other words, the process has external influence).

Interestingly enough, researchers have shown that when the body is tense or in an anxious state, there will be “whitening” of the signal from its natural “pink noise” fractal dynamics [2]. So as the body becomes less adaptable and more tense, it becomes less fractal and more random (white noise). The amplitude of the big fluctuations decreases while their frequency rises. The body-mind system loses its memory; the various influences cancel each other out; any incoming impulse with a clear intention will have a direct and predictable effect unless absorbed by the white noise entropy.

Fractal dynamics is not only found in body movement. It is also present in speech [4], locomotion in complex environments [5], foraging patterns of animals [6], as well as the mood shifts, cognitive dynamics [7], heartbeat, and the dynamics of the brain [12]. Moreover, fractal dynamics is considered to be a sign of a healthy, adaptive state of the body. It is used to detect heart pathologies, for example.

Generally, every “natural” process developed by evolution seems to have fractal properties, so that’s why it may be interesting to emulate and modulate this dynamics, to make the movement more “natural” and adaptive in its most basic form. But how do the fractals and chaos emerge?

 

Where do the Chaos and Fractals Come From

There are multiple reasons for the emergence of chaos and fractal properties in a dynamic system.

The consensus is that chaotic fractal behavior is an indicator of

  • the existence of memory in a system
  • several underlying interacting processes which co-exist in the state of “winnerless competition” (i.e., breathing, heartbeat, walking), where none overtakes the others.
  • adaptive behavior influenced by external factors and instabilities [8]

The behavior becomes less fractal when intentionality or external obstacles appear. For instance, a human gait naturally has fractal properties. However, as soon as pathologies or obstacles emerge, the fractal quality is lost [9] [10].

Winnerless competition (unlike winner-takes-all) adds an extra force to the system which will then create a temporary attractor that may compete with the other forces. When that third force loses, the ball falls down. When it is strong, the ball keeps hanging between them.

Therefore, chaotic (or fractal, 1/f “pink noise”) dynamics represents an adaptive system, which is influenced by several underlying interacting processes. Its unpredictability may look random, but it has long-range correlations: small perturbations will eventually affect the emergence of the bigger ones and vice versa. The intrinsic variability of such a system makes it possible for it to respond to an external change in a quick and efficient way. Thus, chaos may be an evolutionary strategy acquired by a system because it allows it to survive and to develop within a changing environment.

Interestingly, studies have shown [11] that the complex patterns arising in nature are a result of sustained instability (see this beautiful video for more information). Phenomena as diverse as snowflakes, riverbeds, spiral growth patterns arise due to a system going through instabilities over a long period of time, forming self-similar fractal structures. While this is a result of instability, it is also a testimony to the fact that the system, through its adaptivity, has managed to overcome this instability and to settle into a non-equilibrium stable state.

A river bed that is fractal because it is self-repetitive on every scale. If you zoom in, you will see a similar pattern. The principle stays intact. Photo by Stuart Rankin https://www.flickr.com/photos/24354425@N03/

River bed close up. The same principles of organization. Photo by Stuart Rankin.

 

Now that we know the main qualities of chaotic dynamics, the reasons for its emergence, and its underlying processes, let’s see how we could modulate this dynamics through the movement using the physical body.

 

Modulating Fractality and Natural Movement

Many different practices aspire to move in a “natural” way. Therapeutic approaches, such as the Alexander Technique and Feldenkreis method, Noguchi Taiso gymnastics, various forms of dance, and martial arts talk about the “natural” movement as the ultimate objective.

As our definition of the “natural” is often subjective, in the previous sections, we have demonstrated one of the possible objective interpretations of this term. The main premise of our approach is to acknowledge the fact that when we do “nothing”, our body naturally moves in a way that has fractal dynamics. We have identified that “fractal” in this context means self-similar movement, which is self-similar across different scales. We have also shown that the purpose of this fractality is to overcome instability and to adapt to a changing environment. It is not only useful, however, but it is also beautiful, so it is interesting both from an ethical and aesthetic perspective. How do we move in a fractal way then?

In the most simple terms, the fractal movement will look like a succession of many small-amplitude movements with a few but a significant number of occasional big-amplitude changes. The strength of the amplitude may be expressed through the speed but also through the size of the movement, its position in space, orientation, even the length of pauses between the movements. There may be different systems of coordinates, and each offers its own fractal perspective.

 

 

In order to identify the movement’s fractality in pure mathematical terms, we adopted an approach used in medical science called Detrended Fluctuation Analysis (DFA) [13]. This approach generates an Alpha exponent score for a time-series (originally — the variations between the heartbeats happening over a period of time). The value of the score shows whether the movement is random, has positive / negative auto-correlation, shows fractal properties, or is complex.

A person or a group of people can, therefore, send their movement data (via a sensor) to a processing algorithm, which will analyze the resulting time series using the DFA approach. It will then deliver the Alpha exponent, which can be used by the participants to adjust or alter their movement quality, depending on their intentions.

For instance, if the objective is to stay highly adaptive, “natural”, and physically versatile, they will try to score the Alpha component close to 1 (which has the fractal property). If, on the other hand, they have stayed in the fractal zone for a long enough time and would like to explore mathematical randomness — what it feels like and looks like — they will aim for the Alpha exponent of 0.5 (random noise).

Overall, there are 4 different general states, which can be identified using the Alpha exponent calculated using the Detrended Fluctuation Analysis approach:

  • random — meaning the movements are fluctuating around an average
    (0 < alpha < 0.6, ideal: alpha = 0.5)
  • regular, auto-correlated — meaning that the longer we look, the more likely we are to see large fluctuations
    (0.6 < alpha < 0.85)
  • fractal, scale-free — the deviations of fluctuations on a small scale have the same pattern as in the long-scale
    (0.85 < alpha < 1.15, ideal: alpha = 1)
  • complex, non-stationary — meaning the movement quality changes drastically
    (alpha > 1.15 ~ 1.50)

The movement itself will have a different quality depending on the score.

For instance, we have observed during the EightOS sessions that during the “normal” everyday actions or dance, the DFA Alpha exponent stays at about 0.7-0.8, indicating a self-correlated regular, organized activity.

The accelerometer reading of a “regular” movement. Spikes of activity tend to be clustered, there is a correlation and “memory” in that system in that we can expect that the movement will be repeated if it’s already happened just before.

 

 

However, when the movement becomes more repetitive (e.g., stomping, dancing to techno, repetitive prayer movement), the Alpha exponent is closer to 0.5, thus indicating more of a random dynamics, which really means the absence of self-correlation (it is simply 1 or 0, up or down, movement or stillness, oscillating around an average value —  the difference between the movements is not very pronounced).

The accelerometer profile of an oscillatory movement has random properties: it oscillates around a mean and its deviations from the average have a normal distribution in that most of the time most movements will look more or less the same and the deviation will not be that big.

 

 

Big phase shifts and changes in the quality of movement will score about 1.2 – 1.5 Alpha exponent.

The movement which has fractal variability, where a succession of short-range movements (or small, slow changes) is followed by a few big oscillations and where this same pattern can be seen both in a short and in a long time period, will have the DFA Alpha exponent score close to 1 and indicate fractal dynamics. Interestingly, this same dynamics is shown when a person is standing still in a calm, resting state with no tension of agitation, referring us back to the relation between the “natural” and the “fractal” we talked about earlier. Once we start moving on purpose, it is much harder to get the fractal dynamics, however, it has high variability and its variability is of a very special kind.

The accelerometer reading of a fractal movement shows that the periods of low-power activity are followed by the spikes of high-power moves. Those can be seen both on a very short time scale and on the long one (hence the fractal — self-repetitive pattern).

 

 

Here is how this movement looks in real life:

 

If you would like to learn more about how the DFA algorithm works from a mathematical perspective, please, see the Appendix below. We will now demonstrate how DFA is integrated into EightOS body-mind operating system software and how it is used to modulate fractal dynamics during our sessions.

 

EightOS Movement Recommender System

In the frame of our study of fractal dynamics, we have developed a special movement approach, which we implement during the live physical sessions using the live feedback from the algorithm.

The open-source EightOS Movement Recommender (on GitHub) analyzes the time series of human movements and detects its fractality using the detrended fractal analysis algorithm (DFA).

There can be 4 states:

  • random,
  • organized,
  • fractal,
  • complex.

We assume that it can be interesting to go through each of these states while staying in every one of them for a long enough time.

Based on that data, we will see what the current state is and what it has been. If it’s been too regular for a long time, the algorithm will recommend bringing in some fractal variability or repetitiveness. If it has been too fractal, the algorithm will recommend rendering it a bit more regular, repetitive, or introduce a phase-shift. If it has been too regular, we’ll propose to disrupt it again.

As it has been pointed out above, healthy dynamics has fractal properties, while pathologies, tensions, or environmental obstacles that are not dealt with in an adaptive way, will often make the signal less fractal and more random. Therefore, our approach helps us detect the emergence of pathology in a movement and to either bring the dynamics back to adaptable, or, on the contrary, explore the pathology, observe how it feels, how it looks, and how it moves.

The recommendation can be made using an instruction, a symbol generated by the software, with a physical, sonic, or visual cue. The person can then “read” the signal and interpret it in their own way: either to “understand” what is happening and incorporate this understanding into what they are doing, to self-reflect, get an aesthetic experience, or to simply change what they’re doing to reach for another state.

We follow this EightOS variability schema:

In this schema, we start from disrupted singularities and a certain level of predictability. Then we go into less predictability while we’re starting to make connections. Then the system becomes predictable again (3) but in the sense of long-range-correlations (fractal dynamics). Finally, we go to stage 4 (phase-shift) and then to 1 again and start over.

Technically, one of the possible implementations is made using Xsens DOT sensors, which are attached to the participants who are taking part in a physical movement session. A modified version of Xsens DOT Server (which we call EightOS Fractal Control) is used to record the accelerator signal from xSens DOT sensors (logging it to a CSV file) then sending it via the OSC signal to the specified port/host. The OSC is then forwarded to a musical instrument (via Usine) to give a direct sonic feedback on movement.

Our software will also apply Detrended Fluctuation Analysis to the signal and calculate its related Alpha exponent, which shows how fractal the signal is.

This is calculated both for each sensor and for the group of sensors to know the individual state of each sensor and the common state for all the sensors.

The Alpha exponent at

  • around 0.5 indicates random white noise (sensor in an idle state, all cumulative micro-movements tend to zero),
  • between 0.65 and 0.85 is a positively correlated signal, meaning the movement has “memory”: if a movement with a certain acceleration is made, it is likely it will appear again.
  • between 0.85 and 1.15 the signal has fractal (self-similar, scale-free) properties, which means
  • more than 1.15 is a complex signal where the bigger is the scale, the higher are the deviations and they are not scale-free.

EightOS Fractal Control system will then provide a recommendation based on the number of states that occurred during a period of time (e.g., the last 2 minutes). If it’s been too much in the “regular” zone, it will recommend that the participants shift into the fractal dynamics. If it’s stayed too long in the fractal dynamics, the recommendation will propose to move to the random one.

EightOS system providing  feedback on the different sensors attached to the different parts of the body.

In EightOS, we aim to go through various Alpha exponents in a movement session so that both the individuals and the group can experience all the different states (randomness, positive correlation, fractality, and disruption).

It relates to the variability principle and is a part of our ongoing research on the aesthetic of change and its ethical implications.

 

Appendix: Detrended Fluctuation Analysis

It may be interesting to look closer at how the Detrended Fluctuation Analysis approach [13] works as it sheds more light on the phenomenon of fractality.

The DFA algorithm does the following:
– take a discrete time-series data
– detrend it and normalize it (so the values are positive and the trends are removed) — we only have fluctuations left
– calculate the sum of the square of the deviations for the different time scales: the short ones (e.g. fraction of a second) and the long ones (e.g. several seconds)
– build the log / log graph where on the X axis we have the size of the scale, on the Y axis we have the sums of the fluctuations
– plot the values obtained and calculate the slope of the line — this will be the alpha exponent.

In simple terms, Alpha exponent shows how much the accumulation of fluctuations in movement increases as we increase the time period we look at. If the increase in the fluctuations as we increase the time scale, the Alpha exponent is zero, meaning that they will always revert to the mean. If we increase the scale 4 times but the number of fluctuations only increases 2 times, then alpha-exponent is 0.5 and we have a random process. In a random process the fluctuations accumulate, but the absence of occasional high-amplitude oscillations reduces the alpha component. Now, if we bring in an occasional high-amplitude fluctuation, we will move towards a positively correlated process: the more we move, the more fluctuations we accumulate. When the number of those high-frequency oscillations is high enough, in fact, exactly at the right “sweet-spot,” we will reach fractality. This means that no matter what scale we look at, the pattern of fluctuation will be the same (so it’s self-similar across different scales or “scale-free”). Introduce more change, and we look at a situation where big time scales will yield a very high number of changes, so the movement becomes fractal (it is not similar across the scales anymore).

 

References

[1] Yamada, N. (1995). Chaotic swaying of the upright posture. Human movement science.

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[3] Rickles D , Hawe P, Shiell A. (2007) A simple guide to chaos and complexity. Journal of Epidemilogoy & Community Health. 61:933–937.

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[5] Matthey, L., Righetti, L., & Ijspeert, A. J. (2008). Experimental study of limit cycle and chaotic controllers for the locomotion of centipede robots. In 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems (pp. 1860–1865). Nice: IEEE.

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[10] J. M. Hausdorff, P. L. Purdon, C. K. Peng, Z. Ladin, J. Y. Wei, and A. L. Goldberger (1996) Fractal dynamics of human gait: stability of long-range correlations in stride interval fluctuations. Journal of Applied Physiology, 80:5, 1448-1457

[11] Rabinovich M, Trubetskov, (2012). Oscillations and Waves.

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[13] Hardstone, R., Poil, S.-S., Schiavone, G., Jansen, R., Nikulin, V., Mansvelder, H., & Linkenkaer-Hansen, K. (2012). Detrended Fluctuation Analysis: A Scale-Free View on Neuronal Oscillations. Frontiers in Physiology, 3, 450. https://doi.org/10.3389/fphys.2012.00450