---

Prog. Theor. Phys. Vol. 97 No. 2 (1997) pp. 179-200

The Renormalization-Group Method Applied to Asymptotic Analysis of Vector Fields

Teiji Kunihiro

Faculty of Science and Technology, Ryukoku University, Ohtsu 520-21

(Received February 19, 1996)

Abstract:

The renormalization group method of Goldenfeld, Oono and their collaborators is applied to the asymptotic analysis of vector fields. The method is formulated on the basis of the theory of envelopes, as was done for scalar fields. This formulation completes the discussion of the previous work for scalar equations. It is shown in a generic way that the method applied to equations with a bifurcation leads to the Landau-Stuart and (time-dependent) Ginzburg-Landau equations. It is confirmed that this method is actually a powerful theory for the reduction of dynamics as is the reductive perturbation method. Some examples for ordinary differential equations, such as the forced Duffing, the Lotka-Volterra and the Lorenz equations, are worked out in this method: The time evolution of the solution of the Lotka-Volterra equation is given explicitly, while the center manifolds of the Lorenz equation are constructed in a simple way using the RG method.

URL : http://ptp.ipap.jp/link?PTP/97/179/
DOI : 10.1143/PTP.97.179

[ Full Text PDF : FREE ACCESS (838K) ] Citation: Plain Text BiBTeX EndNote(RIS) RSS

---

References:

1. E. C. G. Stuckelberg and A. Petermann, Helv. Phys. Acta 26 (1953), 499.
   M. Gell-Mann and F. E. Low, Phys. Rev. 95 (1953), 1300[APS].
   See also for the significance of the latter paper, K. Wilson, Phys. Rev. D3 (1971), 1818, [APS]and S. Weinberg, in Asymptotic Realms of Physics, ed. A. H. Guth et al. (MIT Press, 1983).
2. As review articles, J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1989);
   D. V. Shirkov, hep-th/9602024[e-print arXiv].
3. N. Goldenfeld, O. Martin and Y. Oono, J. Sci. Comp. 4 (1989), 4.
   N. Goldenfeld, O. Martin, Y. Oono and F. Liu, Phys. Rev. Lett. 64 (1990), 1361[APS].
   N. D. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, Reading, Mass., 1992).
   L. Y. Chen, N. Goldenfeld, Y. Oono and G. Paquette, Physica A204 (1994), 111.
   G. Paquette, L. Y. Chen, N. Goldenfeld and Y. Oono, Phys. Rev. Lett. bf 72 (1994), 76.
   L. Y. Chen, N. Goldenfeld and Y. Oono, Phys. Rev. Lett. 73 (1994), 1311[APS].
4. L. Y. Chen, N. Goldenfeld and Y. Oono, Phys. Rev. E54 (1996), 376[APS].
   See also R. Graham, Phys. Rev. Lett. 76 (1996), 2185[APS].
5. T. Kunihiro, Prog. Theor. Phys. 94 (1995), 503[PTP]; 95 (1996), 835E.
6. T. Kunihiro, Jpn. J. Ind. Appl. Math 14 (1997), 51.
7. R. Courant and D. Hilbert, Methods of Mathematical Pysics, vol. 2 (Interscience Publishers, N. Y., 1962).
8. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer-Verlag 1984); this monograph gives an excellent account of the solvability condition in the reductive perturbation method. See also P. Manneville, Dissipative Structures and Weak Turbulence (Academic Press, INC, 1990).
9. T. Kunihiro, talk presented at the JPS meeting at Yamaguchi University (October 4, 1996); see also T. Kunihiro and J. Matsukidaira, in preparation.
10. See for example, J. Guckenheimer and P. Holmes, Nonlinear Oscillators, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).
11. A. J. Lotka, J. Amer. Chem. Soc. 42 (1920), 1595.
    V. Volterra, Theorie mathematique de lalutte pour lavie (Gauthier-Villars, Paris, 1931).
    See also, L. E. Reichl, A Modern Course in Statistical Physics, Chapt. 17 (Univ. of Texas Press, 1980).
12. E. N. Lorenz, J. Atmos. Sci. 20 (1963), 130.
13. L. D. Landau and E. M. Lifscitz, Fluid Mechanics, § 27 (Pergamon, London, 1959).
    J. T. Stuart, J. Fluid Mech. 4 (1958), 1[CrossRef].
14. J. S. Frame, J. Theor. Biol. 43 (1974), 73.
15. P. Glansdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability, and Fluctuations (Wiley, London, 1971).
16. T. Kunihiro, unpublished (1995).
17. Y. Kuramoto, Prog. Theor. Phys. Suppl. No. 99 (1989), 244[PTP]; Bussei Kenkyu (Kyoto) 49 (1987), 299.
18. V. N. Bogaevski and A. Povner, Algebraic Methods in Nonlinear Perturbation Theory (Springer-Verlag, N. Y., 1991).

Citing Article(s) :

1. Progress of Theoretical Physics Vol. 102 No. 3 (1999) pp. 471-497 :

   Lie-Group Approach to Perturbative Renormalization Group Method

   Shin-itiro Goto, Yuji Masutomi and Kazuhiro Nozaki

2. Progress of Theoretical Physics Vol. 115 No. 2 (2006) pp. 251-258 :

   Analytical Expression for Low-Dimensional Resonance Islands in a 4-Dimensional Symplectic Map

   Shin-itiro Goto

3. Progress of Theoretical Physics Vol. 118 No. 2 (2007) pp. 211-227 :

   Renormalization Reductions for Systems with Delay

   Shin-itiro Goto

4. Progress of Theoretical Physics Vol. 122 No. 4 (2009) pp. 881-910 :

   Dynamical Density Fluctuations around QCD Critical Point Based on Dissipative Relativistic Fluid Dynamics

   Yuki Minami and Teiji Kunihiro

5. Progress of Theoretical Physics Vol. 125 No. 5 (2011) pp. 871-878 :

   Renormalization-Group for Amplitude Equations in Cellular Pattern Formation with and without Conservation Law

   Yasuhiro Shiwa

6. Progress of Theoretical Physics Vol. 126 No. 5 (2011) pp. 761-809 :

   First-Principles Derivation of Stable First-Order Generic-Frame Relativistic Dissipative Hydrodynamic Equations from Kinetic Theory by Renormalization-Group Method

   Kyosuke Tsumura and Teiji Kunihiro

7. Progress of Theoretical Physics Supplement No.131 (1998) pp. 459-470 :

   Renormalization-Group Resummation of a Divergent Series of the Perturbative Wave Functions of Quantum Systems

   Teiji Kunihiro

8. Progress of Theoretical Physics Supplement No.195 (2012) pp. 19-28 :

   New Forms of Non-Relativistic and Relativistic Hydrodynamic Equations as Derived by the Renormalization-Group Method

   Kyosuke Tsumura and Teiji Kunihiro

---

[PTP HOME] [BROWSE] [SEARCH] [SUBSCRIPTION] [SUBMISSION] [REGISTRATION]

Copyright © Progress of Theoretical Physics 2013 All rights reserved.