1978 : The Lozi map

Christophe LETELLIER
JPG - 3.1 ko
René Lozi

M. Hénon defined a map from the plane to itself. The sequence of points obtained by iteration of the mapping either diverges or converges towards a chaotic attractor, which appears to be the product of a one-dimensional manifold by a Cantor set. René Lozi proposed another mapping for which the attractor seems to be more simple in that sense that it seems to be the product of parts of straight lines by a Cantor set  [1]. The mapping is

       x_{n+1} = 1 - a |x\n| + y_n \\[0.1cm]
       y_{n+1} = b x_n \, .

In his original paper, Lozi used the parameter values : a=1.7 and b=0.5 as used for producing Fig. 1. The chaotic nature of the attractor produced by this map was proved by Michal Misiurewicz [2]. A large review was recently published by Zeraoulia Elhadj [3]

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Fig. 1. Chaotic behavior solution to the Lozi map.


[1] R. Lozi, Un attracteur étrange du type attracteur de Hénon, Journal de Physique, 39 (8), C5-9 - C5-10, 1978.

[2] M. Misiurewicz, Strange attractors for the Lozi mappings, Annals of the New York Academy of Sciences, 357 (1), 348-358, 1980.

[3] Z. Elhadj, Lozi Mappings : Theory and Applications, CRC Press, 2013.

[4] Beautiful pictures produced by René’s son Here.

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