Skip to main content
SpringerLink
Log in
Book cover

Trajectory Analysis in Health Care pp 117–130Cite as

Chaos Theory and Sensitive Dependence on Initial Conditions

  • Chapter
  • First Online:

  • 786 Accesses

Abstract

Henri Poincare introduced researchers to the concept of sensitive dependence on initial conditions. Edward Lorenz mapped trajectory changes for atmospheric systems to demonstrate these ubiquitous effects. As a result, long-term patterns can be roughly predicted, but never exactly. The Lyapunov exponent λ represents a measure of chaos, at the same time representing change in entropy δ2S, the inverse of coherence length ξ−1, as well as Kolmogorov entropy and fractal dimensions. The human body has positive and negative feedback processes that self-regulate. Nevertheless, perturbations to system trajectories can alter the periodicities of trajectories over time and even plunge some systems into chaos. Health interventions are designed to restore stability either by maintaining stable desirable conditions or by phase shifting undesirable conditions.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Abbreviations

Λ:

Lyapunov exponent

ξ:

Coherence length

LDL:

Low density lipoprotein cholesterol

S:

Entropy

References

  • Benincà, E., Ballantine, B., Ellner, S. P., & Huisman, J. (2015). Species fluctuations sustained by a cyclic succession at the edge of chaos. Proceedings of the National Academy of Sciences of the United States of America, 112(20), 6389–6394.

    Article  PubMed  PubMed Central  Google Scholar 

  • Christian, J. J., & Davis, D. E. (1956). The relationship between adrenal weight and population status of urban Norway rats. Journal of Mammalogy, 37(4), 475–486.

    Article  Google Scholar 

  • Christian, J. J., & Davis, D. E. (1964). Endocrines, behavior, and population. Science, 146(3651), 1550–1560.

    Article  CAS  PubMed  Google Scholar 

  • Christian, J. J., Flyger, V., & Davis, D. E. (1960). Factors in the mass mortality of a herd of Sika deer, Cervus nippon. Chesapeake Science, 1(2), 79–95.

    Article  Google Scholar 

  • Darwin, C. (1859 [1985]). The origin of species by means of natural selection. London: Penguin.

    Google Scholar 

  • Darwin, C. (1907). Journal of researches into the natural history and geology of the countries visited during the voyage round the world of HMS beagle. London: John Murray.

    Google Scholar 

  • Davis, S., Trapman, P., Leirs, H., Begon, M., & Heesterbeek, J. A. P. (2008). The abundance threshold for plague as a critical percolation phenomenon. Nature, 454, 634–637.

    Article  CAS  PubMed  Google Scholar 

  • Dawes, R. M. (1988). Rational choice in an uncertain world: The psychology of judgment and decision making. San Diego: Harcourt Brace Jovanovich.

    Google Scholar 

  • Dobson, A. (2000). Raccoon rabies in space and time. Proceedings of the National Academy of Sciences USA, 97(26), 14041–14043.

    Article  CAS  Google Scholar 

  • Eigen, M. (2002). Error catastrophe and antiviral strategy. Proceedings of the National Academy of Sciences USA, 99(21), 13374–13376.

    Article  CAS  Google Scholar 

  • Feder, J. (1988). Fractals. New York: Plenum Press.

    Book  Google Scholar 

  • Flockhart, D.T.T., Wassenaar, L.I., Martin, T.G., Hobson, K.A., Wunder, M.B., & Norris, D.R. (2013). Tracking multi-generational colonization of the breeding grounds by monarch butterflies in eastern North America. Proceedings of the Royal Society B, 280, 20131087. http://dx.doi.org/10.1098/rspb.2013.1087.

  • Gammaitoni, L., Hänggi, P., Jung, P., & Marchesoni, F. (1998). Stochastic resonance. Reviews of Modern Physics, 70(1), 223–287.

    Article  CAS  Google Scholar 

  • Glass, L., & Mackey, M. C. (1988). From clocks to chaos: The rhythms of life. Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Gleick, J. (1987). Chaos: Making a new science. New York: Viking.

    Google Scholar 

  • Hall, E. T. (1969). The hidden dimension. New York: Doubleday & Company/Anchor.

    Google Scholar 

  • Hastie, R., & Dawes, R. M. (2009). Rational choice in an uncertain world: The psychology of judgment and decision making (2nd ed.). Thousand Oaks, CA: SAGE.

    Google Scholar 

  • Hollar, D. W. (1992). Nonlinear maps and chaos. In F. N. Magill & T. A. Tombrello (Eds.), Magill’s survey of science: Physical science (pp. 1556–1563). Pasadena, CA: Salem Press.

    Google Scholar 

  • Hollar, D. W. (2009). Risk for intentional violent death associated with HLA genotypes: A preliminary survey of deceased American organ donors. Genetica, 137(3), 253–264.

    Article  CAS  Google Scholar 

  • Hotamisligil, G. S. (2006). Inflammation and metabolic disorders. Nature, 444, 860–867.

    Article  CAS  PubMed  Google Scholar 

  • Lavialle, C., Cornelis, G., Dupressoir, A., Esnault, C., Heidmann, O., Vernochet, C., et al. (2013). Paleovirology of ‘syncytins’, retroviral env genes exapted for a role in placentation. Philosophical Transactions of the Royal Society B, 368, 20120507. http://dx.doi.org/10.1098/rstb.2012.0507.

    Article  Google Scholar 

  • Lee, K. A. (2006). Linking immune defenses and life history at the levels of the individual and the species. Integrative and Comparative Biology, 46(6), 1000–1015.

    Article  CAS  Google Scholar 

  • Li, T.-Y., & Yorke, J. A. (1975). Period three implies chaos. The American Mathematical Monthly, 82(10), 985–992.

    Article  Google Scholar 

  • Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130–141.

    Article  Google Scholar 

  • MacArthur, R. H., & Wilson, E. O. (1963). An equilibrium theory of insular zoogeography. Evolution, 17(4), 373–387.

    Article  Google Scholar 

  • Oberhauser, K. S., & Solensky, M. J. (2004). The monarch butterfly: Biology and conservation. Ithaca, NY: Cornell University Press.

    Google Scholar 

  • Real, L. A., Henderson, J. C., Biek, R., Snaman, J., Jack, T. L., Childs, J. E., et al. (2005). Unifying the spatial population dynamics and molecular evolution of epidemic rabies virus. Proceedings of the National Academy of Sciences USA, 102(34), 12107–12111.

    Article  CAS  Google Scholar 

  • Ricklefs, R. E., & Finch, C. E. (1995). Aging: A natural history. New York: Scientific American Library.

    Google Scholar 

  • Robbins, J. R., Skrzypczynska, K. M., Zeldovich, V. B., Kapidzic, M., & Bakardjiev, A. I. (2010). Placental syncytiotrophoblast constitutes a major barrier to vertical transmission of Listeria monocytogenes. PLoS Pathogens, 6(1), e1000732. http://dx.doi.org/10.1371/journal.ppat.1000732.

    Article  PubMed  PubMed Central  Google Scholar 

  • Ruelle, D. (1989). Chaotic evolution and strange attractors. New York: Cambridge University Press.

    Book  Google Scholar 

  • Saltzman, B. (1962). Finite amplitude free convection as an initial value problem – I. Journal of the Atmospheric Sciences, 19, 329–341.

    Article  Google Scholar 

  • Simberloff, D. S., & Wilson, E. O. (1969). Experimental zoogeography of islands: The colonization of empty islands. Ecology, 50(2), 278–296.

    Article  Google Scholar 

  • Solhkhah, R., Olds, J., & Englund, D. W. (1999). To change the patient or the patient’s world: The suicide attempt of a teased 12-year-old girl. Harvard Review of Psychiatry, 7(2), 102–108.

    Article  CAS  PubMed  Google Scholar 

  • Swets, J. A., Dawes, R. M., & Monahan, J. (2000). Psychological science can improve diagnostic decisions. Psychological Science in the Public Interest, 1(1), 1–26.

    Article  CAS  PubMed  Google Scholar 

  • Thom, R. (1972). Structural stability and morphogenesis. New York: W.A. Benjamin/Westview.

    Google Scholar 

  • Tregenza, T., Simmons, L. W., Wedell, N., & Zuk, M. (2006). Female preference for male courtship song and its role as a signal of immune function and condition. Animal Behaviour, 72, 809–818.

    Article  Google Scholar 

  • van de Walle, A., Hong, Q., Kadkhodaei, S., & Sun, R. (2014). The free energy of mechanically unstable phases. Nature Communications, 6, 7559. doi:10.1038/ncomms8559.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Health Administration, Pfeiffer University, Morrisville, North Carolina, USA

    David W. Hollar

Authors
  1. David W. Hollar
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Cite this chapter

Hollar, D.W. (2018). Chaos Theory and Sensitive Dependence on Initial Conditions. In: Trajectory Analysis in Health Care. Springer, Cham. https://doi.org/10.1007/978-3-319-59626-6_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-59626-6_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59625-9

  • Online ISBN: 978-3-319-59626-6

  • eBook Packages: MedicineMedicine (R0)

Publish with us

Policies and ethics

Search

Navigation