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Symbolic Dynamics and Big Bang Bifurcation in Weibull-Gompertz-Fréchet’s Growth Models |
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PP: 2377-2388 |
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Author(s) |
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J. Leonel Rocha,
Abdel-Kaddous Taha,
Danièle Fournier-Prunaret,
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Abstract |
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| In this paper, motivated by the interest and relevance of the study of tumor growth models, a central point of our investigation
is the study of the chaotic dynamics and the bifurcation structure of Weibull-Gompertz-Fr´echet’s functions: a class of continuousdefined
one-dimensional maps. Using symbolic dynamics techniques and iteration theory, we established that depending on the
properties of this class of functions in a neighborhood of a bifurcation point PBB, in a two-dimensional parameter space, there exists
an order regarding how the infinite number of periodic orbits are born: the Sharkovsky ordering. Consequently, the corresponding
symbolic sequences follow the usual unimodal kneading sequences in the topological ordered tree. We verified that under some
sufficient conditions, Weibull-Gompertz-Fr´echet’s functions have a particular bifurcation structure: a big bang bifurcation point PBB.
This fractal bifurcations structure is of the so-called “box-within-a-box” type, associated to a boxe W¯ 1, where an infinite number of
bifurcation curves issues from. This analysis is done making use of fold and flip bifurcation curves and symbolic dynamics techniques.
The present paper is an original contribution in the framework of the big bang bifurcation analysis for continuous maps. |
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