On the complete chaotic transformations that preserve symmetric invariant densities. (English) Zbl 1152.37325

Huang, Weihong

Chaos Solitons Fractals 38, No. 4, 1065-1074 (2008).

Summary: A transformation \(f:[0,1]\rightarrow [0,1]\) is said to be complete chaotic if it is (i) ergodic with respect to the Lebesgue measure and (ii) chaotic in the probabilistic sense, that is, an absolutely continuous invariant density \(\varphi \) is preserved. The characteristics of the complete chaotic transformations that preserve symmetric invariant densities, that is, \(\varphi (x)=\varphi (1-x)\), for all \(x \in [0,1]\), are explored. It is found that such transformations are “invariant” with both horizontal and vertical mirroring operations in the sense that the transformations resulted do not only remain to be chaotic but also preserve an identical invariant density. Numerical examples and computer simulations are consistent with theoretical findings.

MSC:

37E05	Dynamical systems involving maps of the interval
28D05	Measure-preserving transformations

References:

[1]	Lorenz, E.N., The essence of chaos, (1993), University of Washington · Zbl 0835.58001
[2]	Boyarsky, A.; Góra, P., Laws of chaos: invariant measures and dynamical systems in one dimension, (1994), Addison-Wesley Reading, MA
[3]	Grosjean, G.C., The invariant density for a class of discreet-time dynamical systems involving an arbitrary monotonic function operator and an integer parameter, J math phy, 28, 1265-1274, (1987) · Zbl 0632.60030
[4]	Grossmann, S.; Thomae, S., Invariant distributions and stationary correlation functions of one-dimensional discrete transformation, Z naturforschung, 32a, 1353-1363, (1977)
[5]	Ershov, S.H.; Malinetskii, G.G., The solution of the inverse problem for the frobenius – perron equation, USSR comput maths math phys, 28, 136-141, (1988) · Zbl 0694.39001
[6]	Pingel, D.; Schmelcher, P.; Diakonos, F.K., Theory and examples of the inverse frobenius – perron problem for complete chaotic dynamical systems, Chaos, 9, 357-366, (1999) · Zbl 0982.37024
[7]	Gyögyi, G.; Szépfalusy, P., Fully developed chaotic 1-d dynamical systems, Z phys B-condens matter, 55, 179-186, (1984)
[8]	Huang, W., On complete chaotic maps with tent-map-like structures, Chaos, solitons & fractals, 24, 287-299, (2005) · Zbl 1064.37029
[9]	Huang, W., Characterizing chaotic transformations that generate uniform invariant density, Chaos, solitons & fractals, 25-2, 449-460, (2005) · Zbl 1101.37024
[10]	Huang, W., Constructing an opposite map to a specified chaotic map, Nonlinearity, 18, 1375-1391, (2005) · Zbl 1125.37313
[11]	Huang, W., Constructing complete chaotic maps with reciprocal structures, Discrete dyn nature soc, 3, 357-372, (2005) · Zbl 1134.37336
[12]	Huang W. Constructing chaotic transformations with closed functional forms, Discrete Dyn Nature Soc Sci, forthcoming. · Zbl 1130.37363
[13]	May, R.M., Simple mathematical model with very complicated dynamics, Nature, 26, 457-467, (1976) · Zbl 1369.37088