A Physical Approach to Chaos

  • Hazime Mori
  • Yoshiki Kuramoto


The term ‘chaos’ is used in reference to unstable, aperiodic motion in dynamical systems and also to the state of a system which exhibits such motion. Almost any nonequilibrium open system will, when some bifurcation parameter characterizing the system is made sufficiently large, display chaotic behavior. It can be said that chaos is Nature’s universal dynamical form. Chaos is characterized by the coexistence of an infinite number of unstable periodic orbits that determine the form and structure of the chaotic behavior exhibited by any given system. In this chapter we consider the problem of identifying the descriptive signature of such a set of orbits, and we establish the point of view from which we will elucidate the nature of chaos.


Periodic Orbit Invariant Manifold Unstable Manifold Chaotic Attractor Fluid Particle 
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  1. Baker, G.L., Gollub, J.P. (1990) Chaotic Dynamics, An Introduction. Cambridge University Press, New YorkGoogle Scholar
  2. Berge, P., Pomeau, Y., Vidal, C. (1984) Order Within Chaos. Hermann, Paris Eckmann, J.-P., Ruelle, D. (1985) Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656Google Scholar
  3. Eckmann, J.-P., Ruelle, D. (1985) Ergodic theory of chaos and strange atractors. Rev. Mod. Phys. 57, 617–656MathSciNetADSCrossRefGoogle Scholar
  4. Fujisaka, H. (1983) Statistical dynamics generated by fluctuations of local Lyapunov exponents. Prog. Theor. Phys. 70, 1264–1275MathSciNetADSMATHCrossRefGoogle Scholar
  5. Guckenheimer, J., Holmes, P. (1986) Nonlinear Oscillations, Dynamical Systems, and Bifurcations and Vector Fields, 2nd edn. Springer, Berlin, HeidelbergGoogle Scholar
  6. Morita, T., Hata, H., Mori, H., Horita, T., Tomita, K. (1988) Spatial and temporal scaling properties of strange attractors and their representations by unstable periodic orbits. Prog. Theor. Phys. 79, 296–312MathSciNetADSCrossRefGoogle Scholar
  7. Oseledec, V.I. (1968) A multiplicative ergodic theorem Liapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231MathSciNetGoogle Scholar
  8. Ottino, J.M. (1989) The kinematics of mixing: stretching, chaos and trans¬port. Cambridge University Press, New YorkGoogle Scholar
  9. Rasband, S.N. (1990) Chaotic Dynamics of Nonlinear Systems. Wiley, New YorkGoogle Scholar
  10. Sinai, Y.G. (1981) Randomness in non-random systems. Nature No. 3, 72–80Google Scholar
  11. Takahashi, Y.,(1980) Chaos, periodic points and entropy. Butsuri 35,149–161 (Japanese)Google Scholar
  12. Takahashi, Y., Oono, Y. (1984) Towards the statistical mechanics of chaos. Prog. Theor. Phys. 71, 851–854MathSciNetADSMATHCrossRefGoogle Scholar
  13. Wiggins, S. (1992) Chaotic transport in dynamical systems. Springer, Berlin, HeidelbergGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Hazime Mori
    • 1
    • 2
  • Yoshiki Kuramoto
    • 3
  1. 1.Kyushu UniversityJapan
  2. 2.Higashi-ku FukuokaJapan
  3. 3.Graduate School of SciencesKyoto UniversityKyotoJapan

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