Skip to main content

Advertisement

Springer Nature Link
Log in

Application of chaos measures to a simplified boids flocking model

  • Published:
Swarm Intelligence Aims and scope Submit manuscript

Abstract

Swarm intelligence systems are characterised by the emergence of structure without explicit external control, the coexistence of many stable states and the existence of state transitions with significant change of the system dynamics. Despite these characteristics being shared with the dynamics of chaotic systems, little is known of the potential relationship between swarm and chaotic systems. If indeed there is a relationship between the two, it may be possible to complement the current subjective and quantitative measures used for identifying swarming dynamics with the objective measures used in chaotic systems. This study shows that a simplified version of the original Reynolds’ ‘boids’ (short for ‘birdoid’) model and two degraded versions of the simplified model display flocking dynamics based on current measures of group and order. The simplified version of the original boids model and one of the degraded forms also show chaotic dynamics using three proposed measures of information flow, complexity and sensitivity to initial conditions. A novel \(\hbox {Chaos}_\mathrm{composite}\) measure is introduced which combines all three measures into a single measure and shows promise for characterising flocking dynamics and potentially swarming dynamics. The measures proposed in this study, therefore, may have potential for predicting and controlling the behaviour of swarm intelligence systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. An attractor is defined as a set of states of a dynamic physical system towards which that system tends to evolve, regardless of the starting conditions of the system.

References

  • Aoki, I. (1982). A simulation study on the schooling mechanism in fish. Bulletin of the Japanese Society of Scientific Fisheries, 48, 119–131.

    Article  Google Scholar 

  • Azahar, M. A. B. M., Sunar, M. S., Daman, D., & Bade, A. (2008). Survey on real-time crowds simulation. In Technologies for E-learning and digital entertainment (pp. 573–580). New York: Springer.

  • Bellaachia, A., & Bari, A. (2012). Flock by leader: A novel machine learning biologically inspired clustering algorithm. In Advances in Swarm intelligence, volume 7332 of lecture notes in computer science (pp. 117–126). New York: Springer.

  • Bialek, W., Cavagna, A., Giardina, I., Mora, T., Silvestri, E., Viale, M., et al. (2012). Statistical mechanics for natural flocks of birds. Proceedings of the National Academy of Sciences, 109(13), 4786–4791.

    Article  Google Scholar 

  • Brambilla, M., Ferrante, E., Birattari, M., & Dorigo, M. (2013). Swarm robotics: A review from the swarm engineering perspective. Swarm Intelligence, 7(1), 1–41.

    Article  Google Scholar 

  • Cavagna, A., Cimarelli, A., Giardina, I., Parisi, G., Santagati, R., Stefanini, F., et al. (2010). Scale-free correlations in starling flocks. Proceedings of the National Academy of Sciences, 107(26), 11865–11870.

    Article  Google Scholar 

  • Chaté, H., Ginelli, F., Grégoire, G., Peruani, F., & Raynaud, F. (2008). Modeling collective motion: Variations on the vicsek model. The European Physical Journal B-Condensed Matter and Complex Systems, 64(3), 451–456.

    Article  Google Scholar 

  • Crowther, B., & Riviere, X. (2003). Flocking of autonomous unmanned air vehicles. Aeronautical Journal, 107(1068), 99–109.

    Google Scholar 

  • Czirók, A., Barabási, A.-L., & Vicsek, T. (1999). Collective motion of self-propelled particles: Kinetic phase transition in one dimension. Physical Review Letters, 82(1), 209.

    Article  Google Scholar 

  • Das, S., Halder, U., & Maity, D. (2012). Chaotic dynamics in social foraging swarms an analysis. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 42(4), 1288–1293.

  • DeCoster, G. P., & Mitchell, D. W. (1991). The efficacy of the correlation dimension technique in detecting determinism in small samples. Journal of Statistical Computation and Simulation, 39(4), 221–229.

    Article  Google Scholar 

  • Ding, M., Grebogi, C., Ott, E., Sauer, T., & Yorke, J. A. (1993). Plateau onset for correlation dimension: When does it occur? Physical Review Letters, 70(25), 3872.

    Article  Google Scholar 

  • Ennis, C., Peters, C., & O’Sullivan, C. (2008). Perceptual evaluation of position and orientation context rules for pedestrian formations. In Proceedings of the 5th symposium on applied perception in graphics and visualization (pp. 75–82). New York: ACM.

  • Ferrante, E., Turgut, A. E., Stranieri, A., Pinciroli, C., Birattari, M., & Dorigo, M. (2014). A self-adaptive communication strategy for flocking in stationary and non-stationary environments. Natural Computing, 13(2), 225–245.

    Article  MathSciNet  Google Scholar 

  • Fine, B. T., & Shell, D. A. (2013). Unifying microscopic flocking motion models for virtual, robotic, and biological flock members. Autonomous Robots, 35(2–3), 195–219.

    Article  Google Scholar 

  • Grassberger, P., & Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica D, 9, 189–208.

    Article  MATH  MathSciNet  Google Scholar 

  • Hamilton, W. D. (1971). Geometry for the selfish herd. Journal of Theoretical Biology, 31(2), 295–311.

    Article  Google Scholar 

  • Itoh, M., & Chua, L. O. (2007). Boids control of chaos. International Journal of Bifurcation and Chaos, 17(02), 427–444.

    Article  MATH  MathSciNet  Google Scholar 

  • Jadbabaie, A., Lin, J., & Morse, A. S. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. In IEEE Transactions on Automatic Control, 48, 988–1001.

    Article  MathSciNet  Google Scholar 

  • Khandoker, A. H., Palaniswami, M., & Begg, R. K. (2008). A comparative study on approximate entropy measure and poincaré plot indexes of minimum foot clearance variability in the elderly during walking. Journal of Neuroengineering and Rehabilitation, 5(1), 1–10.

    Article  Google Scholar 

  • Kugiumtzis, D., & Tsimpiris, A. (2010). Measures of analysis of time series (mats): A matlab toolkit for computation of multiple measures on time series data bases. arXiv preprint arXiv:1002.1940.

  • Lee, K. (2012). Fast approximate entropy. http://www.mathworks.com.au/matlabcentral/fileexchange/32427-fast-approximate-entropy/content/ApEn.m.

  • Liu, W., & Winfield, A. F. (2010). Modeling and optimization of adaptive foraging in swarm robotic systems. The International Journal of Robotics Research, 29(14), 1743–1760.

    Article  Google Scholar 

  • Parker, C. (2010). Boids psuedocode. http://www.kfish.org/boids/pseudocode.html.

  • Pincus, S. M. (1991). Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences, 88(6), 2297–2301.

    Article  MATH  MathSciNet  Google Scholar 

  • Reynolds, C. W. (1987). Flocks, herds and schools: A distributed behavioral model. In ACM SIGGRAPH computer graphics (Vol. 21, pp. 25–34). New York: ACM.

  • Ridsdale, G., & Calvert, T. (1990). Animating microworlds from scripts and relational constraints. In Computer animation (Vol. 90, pp. 107–118). New York: Springer.

  • Strömbom, D. (2011). Collective motion from local attraction. Journal of Theoretical Biology, 283(1), 145–151.

    Article  MathSciNet  Google Scholar 

  • Strömbom, D. (2013). Attraction based models of collective motion. Uppsala Dissertations in Mathematics, 82, 9–35.

    Google Scholar 

  • Tanner, H. G., Jadbabaie, A., & Pappas, G. J. (2003). Stable flocking of mobile agents, part 1: Fixed topology. In 42nd IEEE conference on decision and control, 2003 (Vol. 2, pp. 2010–2015).

  • Tanner, H. G., Jadbabaie, A., & Pappas, G. J. (2007). Flocking in fixed and switching networks. IEEE, 52, 863–868.

    MathSciNet  Google Scholar 

  • Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., & Shochet, O. (1995). Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75(6), 1226.

    Article  Google Scholar 

  • Vicsek, T., & Zafeiris, A. (2012). Collective motion. Physics Reports, 517(3), 71–140.

    Article  Google Scholar 

  • Wright, W. A., Smith, R. E., Danek, M., & Greenway, P. (2001). A generalisable measure of self-organisation and emergence. In Artificial neural networks—ICANN 2001 (pp. 857–864). New York:Springer.

  • Wu, Y., Su, J., Tang, H., & Tianfield, H. (2011). Analysis of the emergence in swarm model based on largest lyapunov exponent. Mathematical Problems in Engineering, 2011, 1–21.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Engineering and Information Technology, The University of New South Wales, Northcott Drive, Campbell, Canberra, ACT, 2600, Australia

    John Harvey, Kathryn Merrick & Hussein A. Abbass

Authors
  1. John Harvey
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Kathryn Merrick
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Hussein A. Abbass
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to John Harvey.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 231 KB)

Supplementary material 2 (pdf 2878 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Harvey, J., Merrick, K. & Abbass, H.A. Application of chaos measures to a simplified boids flocking model. Swarm Intell 9, 23–41 (2015). https://doi.org/10.1007/s11721-015-0103-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11721-015-0103-0

Keywords