Effects of a parametric perturbation in the Hassell mapping

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) ; Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) ; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) ; Processo FAPESP: 2005/56253-8 ; Processo FAPESP: 2008/57528-9 ; Processo FAPESP: 2012/23688-5 ; Processo FAPESP: 2014/18672-8 ; Processo FAPESP: 2015/22062-3 ; Processo FAPESP: 2017/14414-2 ; CNPq: 303707/2015-1 ; CNPq: 311105/2015-7 ; CNPq: 421254/2016-5 ; The convergence to the fixed point near at a transcritical bifurcation and the organization of the extreming curves for a parametric perturbed Hassell mapping are investigated. The evolution of the orbits towards the fixed point at the transcritical bifurcation is described using a phenomenological approach with the support of scaling hypotheses and homogeneous function hence leading to a scaling law related with three critical exponents. Near the bifurcation the decay to the fixed point is exponential with a relaxation time given by a power law. The extreming curves in the parameter space dictates the organization for the windows of periodicity, consequently demonstrating how the set of shrimp-like structures are organized.