Abstract
This chapter concerns dynamics, bifurcations and synchronization properties of von Bertalanffy’s functions, a new class of continuous one-dimensional maps, which was first studied in [22]. This family of unimodal maps is proportional to the right hand side of von Bertalanffy’s growth equation. We provide sufficient conditions for the occurrence of stability, period doubling, chaos and non admissibility of von Bertalanffy’s dynamics. These dynamics are dependent on the variation of the intrinsic growth rate of the individual weight, which is given by r = r(K, W ∞ ), where K is von Bertalanffy’s growth rate constant and W ∞ is the asymptotic weight. A central point of our investigation is the study of bifurcations structure for this class of functions, on the two-dimensional parameter space (K, W ∞ ). Another important approach in this work is the study of synchronization phenomena of von Bertalanffy’s models in some types of networks: paths, grids and lattices. We study the synchronization level when the local dynamics vary and the topology of the network is fixed. This variation is expressed by the Lyapunov exponents, as a function of the intrinsic growth rate r. Moreover, we present some results about the evolution of the network synchronizability, as the number of nodes increases, keeping fixed the local dynamics, in some types of networks: paths, grids and lattices. We also discuss the evolution of the network synchronizability as the number of edges increases. To support our results, we present numerical simulations for these types of networks.
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Acknowledgements
Research partially sponsored by national funds through the Fundação Nacional para a Ciência e Tecnologia, Portugal – FCT, under the project PEst-OE/MAT/UI0006/2014, CEAUL, CIMA and ISEL. The authors are grateful to Prof. Danièle Fournier-Prunaret for having made the image of Fig. 2. The authors are grateful to the anonymous referees for a careful checking of the details and for helpful comments that improved this work. 
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Rocha, J.L., Aleixo, S.M., Caneco, A. (2015). Chaotic Dynamics and Synchronization of von Bertalanffy’s Growth Models. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Dynamics, Games and Science. CIM Series in Mathematical Sciences, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-16118-1_30
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DOI: https://doi.org/10.1007/978-3-319-16118-1_30
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