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Siegel disk for complexified Hénon map

By O. B. Isaeva and S. P. Kuznetsov

Abstract

It is shown that critical phenomena associated with Siegel disk, intrinsic to 1D complex analytical maps, survives in 2D complex invertible dissipative Hénon map. Special numerical method of estimation of the Siegel disk scaling center position (for 1D maps it corresponds to extremum) for multi-dimensional invertible maps are developed. It is known that complexification of real 1D logistic map zn+1 = f(zn) = λ − z 2 n, (1) where λ, z ∈ C leads to the origination of the Mandelbrot set at the complex parameter λ plane [1] and a number of other accompanying phenomena. Opportunity for realization of the phenomena, characteristic for the dynamics of complex maps (Mandelbrot and Julia sets etc.) at the physical systems seems to be interesting problem [2, 3, 4, 5, 6]. In the context of this problem the following question is meaning: Does phenomena of dynamics of the 1D complex maps (like classic Mandelbrot map (1)) survive for the more realistic (from the point of view of possible physical applications) model – two-dimensional maps invertible in time. For example, more realistic model rather than logistic map, is the Hénon ma

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