On characterization of the solution set in case of generalized semiflow

Zdeněk Beran

Kybernetika (2009)

  • Volume: 45, Issue: 5, page 701-715
  • ISSN: 0023-5954

Abstract

top
In the paper, a possible characterization of a chaotic behavior for the generalized semiflows in finite time is presented. As a main result, it is proven that under specific conditions there is at least one trajectory of generalized semiflow, which lies inside an arbitrary covering of the solution set. The trajectory mutually connects each subset of the covering. A connection with symbolic dynamical systems is mentioned and a possible numerical method of analysis of dynamical behavior is outlined.

How to cite

top

Beran, Zdeněk. "On characterization of the solution set in case of generalized semiflow." Kybernetika 45.5 (2009): 701-715. <http://eudml.org/doc/37694>.

@article{Beran2009,
abstract = {In the paper, a possible characterization of a chaotic behavior for the generalized semiflows in finite time is presented. As a main result, it is proven that under specific conditions there is at least one trajectory of generalized semiflow, which lies inside an arbitrary covering of the solution set. The trajectory mutually connects each subset of the covering. A connection with symbolic dynamical systems is mentioned and a possible numerical method of analysis of dynamical behavior is outlined.},
author = {Beran, Zdeněk},
journal = {Kybernetika},
keywords = {generalized semiflow; chaos; symbolic dynamics; chaos; generalized semiflow; symbolic dynamics},
language = {eng},
number = {5},
pages = {701-715},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On characterization of the solution set in case of generalized semiflow},
url = {http://eudml.org/doc/37694},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Beran, Zdeněk
TI - On characterization of the solution set in case of generalized semiflow
JO - Kybernetika
PY - 2009
PB - Institute of Information Theory and Automation AS CR
VL - 45
IS - 5
SP - 701
EP - 715
AB - In the paper, a possible characterization of a chaotic behavior for the generalized semiflows in finite time is presented. As a main result, it is proven that under specific conditions there is at least one trajectory of generalized semiflow, which lies inside an arbitrary covering of the solution set. The trajectory mutually connects each subset of the covering. A connection with symbolic dynamical systems is mentioned and a possible numerical method of analysis of dynamical behavior is outlined.
LA - eng
KW - generalized semiflow; chaos; symbolic dynamics; chaos; generalized semiflow; symbolic dynamics
UR - http://eudml.org/doc/37694
ER -

References

top
  1. Differential inclusions, set-valued maps and viability theory, (Grundlagen Math. Wiss. 264.) Springer–Verlag, Berlin 1984. MR0755330
  2. Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations, Nonlinear Sci. 7 (1997), 475–502. Erratum, ibid 8 (1998), 233. Zbl0958.35101MR1462276
  3. Stability Theory of Dynamical Systems, Springer-Verlag, Berlin 1970. MR0289890
  4. Yet another chaotic attractor, Internat. J. Bifurcation Chaos 9 (1999), 1465–1466. MR1729683
  5. On a generalized Lorenz canonical form, Chaos, Solitons and Fractals 26 (2005), 1271–1276. MR2149315
  6. On a generalized Lorenz canonical form of chaotic systems, Internat. J. Bifurcation Chaos, in press. 
  7. Bilinear systems and chaos, Kybernetika 30 (1994), 403–424. 
  8. A generalized Lorenz system, Commun. Math. Phys. 60, (1978), 193–204. Zbl0387.76052MR0495037
  9. A Rigorous Numerical Method in Infinite Dimensions, PhD Thesis, Georgia Institute of Technology 2003. MR2620074
  10. Rigorous numerics for global dynamics: A study of the Swift–Hohenberg equation, SIAM J. Appl. Dynamical Systems 1 (2005), 4, 1–31. MR2136516
  11. Validated continuation for equilibria of PDFs, to appear in SIAM J. Numer. Anal. 2007. MR2338393
  12. [unknown], L. Dieci and T. Eirola. Numerical Dynamical Systems. School of Mathematics, Georgia Institute of Technology, Institute of Mathematics, Helsinky University of Technology 2005. 
  13. A strange, strange attractor, In: The Hopf Bifurcation and Applications (Applied Math. Sciences 19, J. Marsden and J. McCracken, eds.), Springer–Verlag, Berlin 1976, pp. 368–381. 
  14. Computational Homology, (Appl. Math. Sci. Series 157.) Springer–Verlag, New York 2004. MR2028588
  15. Introduction to the Modern Theory of Dynamic Systems, Cambridge University Press, Cambridge 1997. 
  16. Validated Continuation for Infinite Dimensional Problems, PhD Thesis, Georgia Institute of Technology 2007. MR2626583
  17. Deterministic nonperiodic flow, J. Atmospheric Sci. 20, (1963), 130–141. 
  18. A new chaotic attractor coined, Internat. J. Bifurcation Chaos 3 (2002), 12, 659–661. MR1894886
  19. Bridge the gap between the Lorenz system and the Chen system: Internat, J. Bifurcation Chaos 12 (2002), 12, 2917–2926. MR1956411
  20. A new chaotic system and beyond: the generalized Lorenz-like system, Internat. J. Bifurcation Chaos 5 (2004), 12, 1507–1537. MR2072347
  21. On attractors of multivalued semi-flows and differential inclusions, Set-Valued Analysis 6 (1998), 83–111. MR1631081
  22. Conley index theory: A brief introduction, In: Conley Index Theory, Banach Center Publication 1999. Zbl0946.37010MR1675403
  23. Conley index, mathematical computation, In: Handbook of Dynamical Systems, Berlin, Elsevier 2002. MR1901060
  24. Chaos in Lorenz equations: A computer assisted proof, Bull. Amer. Math. Soc. 32 (1995), 66–72. MR1276767
  25. Chaos in Lorenz equations: A computer assisted proof, Part II: Details. Math. Computation 67 (1998), 1023–1046. MR1459392
  26. Chaos in Lorenz Equations: A Computer Assisted Proof, Part III: Classical Parameter Values, Research supported by NSF 9805584 and by KBN, Grant 2 P03A 029 12. 
  27. Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars 1899. English translation: New Methods of Celestial Mechanics by D. Goroff, AIP Press. 
  28. On the nature of the turbulence, Commun. Math. Phys. 20 (1963), 167–192. MR0284067
  29. Some simple chaotic flows, Phys. Rev. E50 (1994), R647–R650. MR1381868
  30. Strange Attractors: Creating Patterns in Chaos, M&T Books, New York 1993. 
  31. Dynamical Systems and Numerical Analysis, Cambridge Univ. Press, Cambridge 1996. MR1402909
  32. Infinite-Dimensional Dynamical Systems, Springer, Berlin 1993. Zbl0871.35001MR1319878
  33. The structure of Lorenz attractors, Publ. Math. de l’I.H.É.S 50 (1979), 73–99. Zbl0484.58021MR0556583
  34. Newton’s method and symbolic dynamics, Proc. Amer. Math. Soc. 2 (1984), 91, 245–253. Zbl0554.65038MR0740179

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.