Von Bertalanffy's dynamics under a polynomial correction: Allee effect and big bang bifurcation

In this work we consider new one-dimensional populational discrete dynamical systems in which the growth of the population is described by a family of von Bertalanffy's functions, as a dynamical approach to von Bertalanffy's growth equation. The purpose of introducing Allee effect in those models is satisfied under a correction factor of polynomial type. We study classes of von Bertalanffy's functions with different types of Allee effect: strong and weak Allee's functions. Dependent on the variation of four parameters, von Bertalanffy's functions also includes another class of important functions: functions with no Allee effect. The complex bifurcation structures of these von Bertalanffy's functions is investigated in detail. We verified that this family of functions has particular bifurcation structures: the big bang bifurcation of the so-called "box-within-a-box" type. The big bang bifurcation is associated to the asymptotic weight or carrying capacity. This work is a contribution to the study of the big bang bifurcation analysis for continuous maps and their relationship with explosion birth and extinction phenomena.info:eu-repo/semantics/publishedVersio

DYNAMICAL STUDY OF A SECOND ORDER DPCM TRANSMISSION SYSTEM MODELED BY A PIECE-WISE LINEAR FUNCTION

17 pagesInternational audienceThis paper analyses the behaviour of a second order DPCM (Differential Pulse Code Modulation) transmission system when the nonlinear characteristic of the quantizer is taken into consideration. In this way, qualitatively new properties of the DPCM system have been unravelled, which cannot be observed and explained if the nonlinearity of the quantizer is neglected. For the purposes of this study, a piece-wise linear nondifferentiable quantizer characteristic is considered. The resulting model of the DPCM is of the form of iteration equations (i.e. map), where the inverse iterate is not unique (i.e. noninvertible map). Therefore the mathematical theory of noninvertible maps is particularly suitable for this analysis, together with the more classic tools of Non Linear Dynamics. This study allowed us in addition to show from a theoretical point of view some new properties of nondifferentiable maps, in comparison with differentiable ones. After a short review of noninvertible maps, the presented methods and tools for noninvertible maps are applied to the DPCM system. An original algorithm for calculation of bifurcation curves for the DPCM map is proposed. Via the studies in the parameter and phase plane, different nonlinear phenomena such as the overlapping of bifurcation curves causing multistability, chaotic behaviour, or multiple basins with fractal boundary are pointed out. All observed phenomena show a very complex dynamical behaviour even in the constant input signal case, discussed here

Invariant Curves of Quadratic Maps of the Plane from the One-Parameter Family Containing the Trace Map

The rigorous proofs are given: (1) for the existence of the unbounded invariant curves, containing the fixed point – source (μ + 1; 1), of the maps from the one-parameter family Fμ(x,y) = (xy, (x − μ)2), μ ∈ [0, 2]; (2) for the birth of the closed invariant curve from the elliptic fixed point (μ − 1; 1) for μ = 3 / 2. Numerical results are presented for the main steps of the evolution of this invariant curve, when μ changes in the interval (3 / 2, 2)

Border Collision Bifurcations and route to chaos in a Two-dimensional piecewise linear map

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