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Renormalization structure of the transition to chaos

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Il Nuovo Cimento A (1965-1970)

Summary

In this paper we establish, in parallel with QFT and statistical mechanics, a completely similar structure lying behind the study of transition of dynamical systems to chaos. We give the transformationsT m, which are a scale transformation supplemented by related changes of the parameters, in order to achieve invariance. Then we show that theT m, obey the composition ruleT mTk==T m+k ,m⩾0. We thus find that a discrete semi-group structure exists, to exactly the same extent as it exists in statistical mechanics. The domain of validity is established like the UV domain of validity of QFT, or the long-range domain in statistical mechanics.

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References

  1. Wilson K. andKogut J.,Phys. Rep. C,12 (1974) 76;Wilson K.,Phys. Rev. B,4 (1971) 3174, 3184;Kadanoff L.,Physics,2 (1966) 263;Feigenbaum M.,J. Stat. Phys.,19 (1978) 25;21 (1979) 669.

    Article  ADS  Google Scholar 

  2. Metropolis N., Stein M. andStein P.,J. Combinatorial Theory,15 (1973) 25;Coullet P. andTresser C.,C. R. Acad. Sci. Paris,287 (1978) 577;Collet P. andEckmann J. P., IHES/P/78-250;Derrida B., Gervais A. andPomeau Y.,J. Phys. A,12 (1979) 269;May R. H.,Nature,261 (1976) 459.

    Article  MATH  MathSciNet  Google Scholar 

  3. Collet P. andEckmann J. P., IHES/P/78-250;Iterated Maps on the Interval (Birkhäuser, Basel) 1980.

  4. Feigenbaum M.,J. Stat. Phys.,19 (1978) 25;21 (1979) 669.

    Article  MathSciNet  ADS  Google Scholar 

  5. Wegner F. J.,Phase Transitions and Critical Phenomena, edited byC. Domb andM. S. Green, Vol.6 (Academic Press) 1976, p. 8.

  6. See ref. [2] and, for instance;Chirikov B.,Renormalization Group 1986, edited byD. V. Shirkov (World Scientific) 1988, p. 221;Sinaï Ya.,Renormalization Group 1986, edited byD. V. Shirkov (World Scientific) 1988, p. 251.

    Article  MathSciNet  Google Scholar 

  7. Bogoljubov N. N. andShirkov D. V.,Introduction to the Theory of Quantized Fields (Interscience Pub., New York, N.Y.) 1959, Chapt. 8;Dokl Akad. Nauk SSSR,103 (1955) 203, 391;Shirkov D. V.,Sov. Phys. Dokl.,27 (1982) 197.

    Google Scholar 

  8. Peterman A. andZichichi A., to be published.

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Authors and Affiliations

  1. CERN, CH-1211, Geneva 23, Switzerland

    A. Peterman & A. Zichichi

  2. CPT Luminy, CNRS, Marseille, France

    A. Peterman

  3. Dipartimento di Fisica, Università di Bologna, Italy

    A. Zichichi

  4. INFN, Sezione di Bologna, Italy

    A. Zichichi

Authors
  1. A. Peterman
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  2. A. Zichichi
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Peterman, A., Zichichi, A. Renormalization structure of the transition to chaos. Nuov Cim A 109, 341–355 (1996). https://doi.org/10.1007/BF02731019

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  • DOI: https://doi.org/10.1007/BF02731019

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