Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations

Author: Koichiro Naito
Journal: Trans. Amer. Math. Soc. 354 (2002), 1137-1151
MSC (2000): Primary 11K60, 28A80, 35B15
Published electronically: September 21, 2001
MathSciNet review: 1867375
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we introduce recurrent dimensions of discrete dynamical systems and we give upper and lower bounds of the recurrent dimensions of the quasi-periodic orbits. We show that these bounds have different values according to the algebraic properties of the frequency and we investigate these dimensions of quasi-periodic trajectories given by solutions of a nonlinear PDE.

References [Enhancements On Off] (What's this?)

  • 1. V.Barbu and Th.Precupanu, Convexity and Optimization in Banach Spaces, Sijthoff & Noordhoff, 1978. MR 87k:49045
  • 2. P. Bergé, Y. Pomeau, C. Vidal, Order within chaos, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York; Hermann, Paris, 1986. MR 88f:58099
  • 3. H.Brezis, Opérateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973. MR 50:1060
  • 4. M.G.Crandall and A.Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972), 57-94. MR 45:9214
  • 5. C.M.Dafermos, Almost periodic processes and almost periodic solutions of evolution equations, in Dynamical Systems, Proc. Univ. Florida Inter. Symp." (A.R.Bednarek and L.Cesari, Eds.), 43-57, Academic Press, New York, 1977. MR 57:3545
  • 6. T.Kato, Nonlinear semi-groups and evolution equations J. Math. Soc. Japan 19 (1967), 508-520. MR 37:1820
  • 7. K.Naito, On the almost periodicity of solutions of a reaction diffusion system, J. Differential Equations 44 (1982), 9-20. MR 83f:35009
  • 8. -, Fractal dimensions of almost periodic attractors, Ergodic Theory and Dynamical Systems 16 (1996), 791-803. MR 97e:58156
  • 9. -, Dimension estimate of almost periodic attractors by simultaneous Diophantine approximation, J. Differential Equations, 141 (1997), 179-200. MR 98k:35215
  • 10. -, Correlation dimensions of quasi-periodic orbits with frequencies given by Roth numbers, Differential equations and applications (Chinju, 1998), 119-129, Nova Sci. Publ., Huntington, NY, 2000. MR 2001e:37038
  • 11. -, Correlation dimensions of quasi-periodic orbits with frequencies given by quasi Roth numbers, J. Korean Math. Soc. 37 (2000), 857-870. MR 2001h:37051
  • 12. -, Fractal dimensions of quasi-periodic orbits given by Weierstrass type functions, to appear in J. Dyn. Diff. Eq.
  • 13. Y. B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997. MR 99b:58003
  • 14. R.T.Rockafellar, Integrals which are convex functionals, I, Pacific J. Math. 24 (1968), 525-539. MR 38:4984
  • 15. -, Integrals which are convex functionals, II, Pacific J. Math. 39 (1971), 439-469. MR 46:9710
  • 16. W.M.Schmidt, Diophantine Approximation, Springer Lecture Notes in Math. 785, 1980. MR 81j:10038

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11K60, 28A80, 35B15

Retrieve articles in all journals with MSC (2000): 11K60, 28A80, 35B15

Additional Information

Koichiro Naito
Affiliation: Faculty of Engineering, Kumamoto University, Kurokami 2-39-1, Kumamoto, 860-8555, Japan

Keywords: Diophantine approximation, quasi-periodic solutions, fractal dimension
Received by editor(s): October 29, 2000
Received by editor(s) in revised form: May 9, 2001
Published electronically: September 21, 2001
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society