Tutorials and ReviewsNo Access

CHAOTIC BEHAVIOR OF INTERVAL MAPS AND TOTAL VARIATIONS OF ITERATES

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

Supported in part by Texas A&M University Interdisciplinary Research Initiative IRI 99-22, a TITF initiative, and a DARPA grant.

Search for more papers by this author

TINGWEN HUANG

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

Search for more papers by this author

, and

YU HUANG

Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275, P. R. China

Supported in part by the National Natural Science Foundation of P. R. China and the Natural Science Foundation of Guangdong Province.

Search for more papers by this author

https://doi.org/10.1142/S0218127404010540Cited by:39 (Source: Crossref)

Abstract

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fⁿ will display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fⁿ on subintervals. There are four distinctive cases of the growth of total variations of fⁿ as n→∞:

(i) the total variations of fⁿ on I remain bounded;

(ii) they grow unbounded, but not exponentially with respect to n;

(iii) they grow with an exponential rate with respect to n;

(iv) they grow unbounded on every subinterval of I.

We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fⁿ are more extreme, the more complex is the behavior of the system.

Keywords:

References

R. L. Ader, A. G. Konheim and M. H. McAndrew, Trans. Amer. Math. Soc. 114, 309 (1965). Crossref, Web of Science, Google Scholar
M. Artin and B. Mazur, Ann. Math. 81, 82 (1965). Crossref, Web of Science, Google Scholar
J. Bankset al., Amer. Math. Monthly 99, 332 (1992). Crossref, Web of Science, Google Scholar
L. Block, Proc. Amer. Math. Soc. 72, 576 (1978). Crossref, Web of Science, Google Scholar
L. Block, Trans. Amer. Math. Soc. 254, 391 (1979). Web of Science, Google Scholar
L. Block and W. A. Coppel , Dynamics in One Dimension , Lecture Notes in Mathematics 1513 ( Springer-Verlag , NY, Heidelberg, Berlin , 1992 ) . Crossref, Google Scholar
R. Bowen, Global Analysis, Proc. Symp. Pure Math., Amer. Math. Soc. 14, 23 (1970). Crossref, Google Scholar
R. Bowen, Trans. Amer. Math. Soc. 153, 401 (1971). Crossref, Web of Science, Google Scholar
Bowen R. and J. Franks, Topology 15, 337 (1976). Web of Science, Google Scholar
R. Brown and L. O. Chua, Int. J. Bifurcation and Chaos 6, 219 (1996). Link, Web of Science, Google Scholar
L. Cesari , Asymptotic Behavior and Stability Problems in Ordinary Differential Equations ( Springer-Verlag , NY, Heidelberg, Berlin , 1959 ) . Google Scholar
G. Chenet al., Control of Nonlinear Distributed Parameter Systems, Marcel Dekker Lectures Notes on Pure & Appl. Math., eds. G. Chenet al. (Dekker, NY, 2001) pp. 15–43. Crossref, Google Scholar
G. Chen, S. B. Hsu and T. Huang, Int. J. Bifurcation and Chaos 12, 965 (2002). Link, Web of Science, Google Scholar
R. L. Devaney , An Introduction to Chaotic Dynamical Systems ( Addison-Wesley , NY , 1989 ) . Google Scholar
G. Edgar , Measure, Topology, and Fractal Geometry ( Springer-Verlag , NY, Berlin, Heidelberg , 1990 ) . Crossref, Google Scholar
K. Falconer , Fractal Geometry ( Wiley , NY , 1990 ) . Google Scholar
P. Hartman , Ordinary Differential Equations ( Wiley , NY , 1964 ) . Google Scholar
E. Hewitt and K. Stromberg , Real and Abstract Analysis ( Springer-Verlag , NY , 1965 ) . Crossref, Google Scholar
Huang, T. [2002] "Chaotic vibration of the wave equation studied through the unbounded growth of total variation," Ph.D. Dissertation, Department of Mathematics, Texas A&M University, College Station, Texas, U.S.A . Google Scholar
Y. Huang, Int. J. Bifurcation and Chaos (2003). Google Scholar
A. Katok and B. Hasselblatt , Introduction to the Modern Theory of Dynamical Systems ( Cambridge University Press , Cambridge, U.K. , 1995 ) . Crossref, Google Scholar
Juang, J. & Hsieh, S. F. [2001] "Interval maps, total variation and chaos," preprint . Google Scholar
T. Li and L. Yorke, Amer. Math. Monthly 82, 985 (1975). Crossref, Web of Science, Google Scholar
Liapunov, A. [1907] "Probléme general de la stabilite du mouvement," Ann. Fac. Sci. Univ. Toulouse 9, 203–475, reproduced [1947] Ann. Math. Study 17, Princeton . Google Scholar
R. Mane , Ergodic Theory and Differential Dynamics ( Springer-Verlag , NY, Heidelberg, Berlin , 1987 ) . Crossref, Google Scholar
J. Milnor and W. Thurston, On Iterates Maps of the Interval, Lecture Notes in Mathematics 1342 (Springer-Verlag, NY, 1988) pp. 465–563. Crossref, Google Scholar
M. Misiurewicz and W. Szlenk, Studia Math 67, 45 (1980). Crossref, Web of Science, Google Scholar
Z. Nitecki, Proc. Amer. Math. Soc. 85, 451 (1982). Crossref, Web of Science, Google Scholar
C. Robinson , Dynamical Systems, Stability, Symbolic Dynamics and Chaos , 2nd edn. ( CRC Press , Boca Raton, FL , 1999 ) . Google Scholar
Rothschild, J. [1971] "On the computation of topological entropy," thesis, CUNY, New York, U.S.A . Google Scholar
D. Ruelle , Chaotic Evolution and Strange Attractors ( Cambridge University Press , Cambridge, U.K. , 1989 ) . Crossref, Google Scholar
P. Stefan, Commun. Math. Phys. 53, 237 (1977). Web of Science, Google Scholar
P. Walters , An Introduction to Ergodic Theory ( Springer-Verlag , NY, Heidelberg, Berlin , 1982 ) . Crossref, Google Scholar
S. Wiggins , Introduction to Applied Nonlinear Dynamical Systems and Chaos ( Springer-Verlag , NY , 1990 ) . Crossref, Google Scholar
Z. L. Zhou, Chinese Sci. Bull. 31, 1 (1986). Crossref, Google Scholar
Z. L. Zhou, Chinese Quart. J. Math. 3, 42 (1988). Web of Science, Google Scholar
Z. L. Zhou , Symbolic Dynamics ( Shanghai Scientific and Technological Education Publishing House , Shanghai, China , 1997 ) . Google Scholar