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Nonintegrability, chaos, and complexity

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  • McCauley, Joseph L.

Abstract

Two-dimensional driven-dissipative flows are generally integrable via a conservation law that is singular at equilibria. Nonintegrable dynamical systems are confined to n⩾3 dimensions. Even driven-dissipative deterministic dynamical systems that are critical, chaotic or complex have n − 1 local time-independent conservation laws that can be used to simplify the geometric picture of the flow over as many consecutive time intervals as one likes. Those conservation laws generally have either branch cuts, phase singularities, or both. The consequence of the existence of singular conservation laws for experimental data analysis, and also for the search for scale-invariant critical states via uncontrolled approximations in deterministic dynamical systems, is discussed. Finally, the expectation of ubiquity of scaling laws and universality classes in dynamics is contrasted with the possibility that the most interesting dynamics in nature may be nonscaling, nonuniversal, and to some degree computationally complex.

Suggested Citation

  • McCauley, Joseph L., 1997. "Nonintegrability, chaos, and complexity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 237(3), pages 387-404.
  • Handle: RePEc:eee:phsmap:v:237:y:1997:i:3:p:387-404
    DOI: 10.1016/S0378-4371(96)00408-6
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    Cited by:

    1. Pavlos, G.P. & Karakatsanis, L.P. & Xenakis, M.N., 2012. "Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma, Part one: Sunspot dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6287-6319.

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