×

Chaos in first-order partial difference equations. (English) Zbl 1144.39002

The author considers the following 2D first-order partial difference equation \[ x(n+1, m)=f(x(n, m), x(n, m+1)). \] Since the dynamical behaviors of partial difference equations will be helpful for understanding the dynamical behaviors of the corresponding partial differential equation, the author changes the partial difference equations into certain ordinary difference equations, and establishes several criteria of chaos.

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.2307/2324899 · Zbl 0758.58019
[2] Block L.S., Lecture Notes in Mathematics 1513 (1992)
[3] DOI: 10.1016/j.chaos.2004.11.058 · Zbl 1071.37018
[4] Collet P., Progress in Physics 1 (1980)
[5] Devaney R.L., An Introduction to Chaotic Dynamical Systems (1989) · Zbl 0695.58002
[6] DOI: 10.1017/S0004972700010157 · Zbl 0577.54041
[7] Kelley W.G., Difference Equations: An Introduction with Applications (1991)
[8] DOI: 10.2307/2589144 · Zbl 0992.37029
[9] DOI: 10.2307/2318254 · Zbl 0351.92021
[10] Liang, W., Shi, Y. and Zhang, C., 2006, Chaotification for a class of first-order partial difference equations, submitted for publication. · Zbl 1147.37325
[11] DOI: 10.2307/2691012 · Zbl 1008.37014
[12] DOI: 10.1126/science.252.5003.226
[13] Robinson C., Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2. ed. (1999) · Zbl 0914.58021
[14] DOI: 10.1109/81.257286
[15] DOI: 10.1016/j.chaos.2004.02.015 · Zbl 1067.37047
[16] Shi Y., Science in China, Series A: Mathematics, Chinese Version 34 pp 595– (2005)
[17] DOI: 10.1142/S021812740601629X · Zbl 1185.37084
[18] Shi Y., Dynamics of Continuous, Discrete, and Impulsive Systems, Series B: Applications and Algorithms 14 pp 174– (2007)
[19] Shi, Y. and Yu, P., 2006, Chaos induced by regular snap-back repellers. Journal of Mathematical Analysis and Applications, in press. · Zbl 1131.37023
[20] DOI: 10.1080/1023619021000054006 · Zbl 1012.39004
[21] DOI: 10.1137/S0036144500376649 · Zbl 1049.37027
[22] Wiggins S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (1990) · Zbl 0701.58001
[23] Zhou Z., Symbolic Dynamics (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.