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Journal of Applied Nonlinear Dynamics 3(2) (2014) 115--130 | DOI:10.5890/JAND.2014.06.002

J.L. Rocha; S. Aleixo; A. Caneco

Instituto Superior de Engenharia de Lisboa - ISEL, ADM, CEAUL and CIMA-UE, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa, Portugal

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Abstract

In this work we establish new one-dimensional discrete dynamical systems: a family of unimodal maps that is proportional to the right hand side of Richards’ growth equation. We investigate in detail the bifurcation structure of Richards’ functions, on the twodimensional parameter space (β,r), where β is the shape parameter, related with the growth-retardation phenomena, and r is the intrinsic growth rate. Sufficient conditions are provided for the occurrence of extinction, stability, period doubling, chaos and non admissibility of Richards’ dynamics. We consider networks having in each node a Richards’ function. We prove some results about the synchronization level when fixing the network topology and changing the local dynamics expressed by the Lyapunov exponents, which depends on the (β,r) parameters. Moreover, we fix the local dynamics and prove results about the synchronization when the network topology change for some kind of networks. Finally, using numerical simulations, we compute the Lyapunov exponents to measure the system complexity, we obtain synchronization intervals of these networks, and we discuss the evolution of the synchronization level in terms of the parameters Richard’s function.

Acknowledgments

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-UE and ISEL. The authors are grateful to Profs. Dani`ele Fournier-Prunaret and Abdel-Kaddous Taha for constructive discussions and remarks.

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