Symbolic dynamics and big bang bifurcation in Weibull-Gompertz-Fréchet's growth models

Abstract

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-Frechet's functions: a class o f continuous- defined one-dimensional maps. Using symbolic dynamics tech niques 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-Frechet's func tions have a particular bifurcation structure: a big bang bi furcation point PBB. This fractal bifurcations structure is of the so-called bo x-within-a-box type, associated to a boxe ¯ Ω1, where an infinite number of bifurcation curves issues from. This analysis is done making use of fold and flip bifurcation curves and symbolic dynamic s techniques. The present paper is an original contribution in the framework of the big bang bifurcation analysis for continuous maps.