Discontinuity, Nonlinearity, and Complexity
Universal Behavior of the Convergence to the Stationary State for a Tangent Bifurcation in the Logistic Map
Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 6370  DOI:10.5890/DNC.2020.03.005
Joelson D. V. Hermes$^{1}$, Flávio Heleno Graciano$^{2}$, Edson D. Leonel$^{3}$
$^{1}$ Instituto Federal de Educação Ciência e Tecnologia do Sul de Minas Gerais, Praç Tiradentes, 416  37576000, Centro, Inconfidentes, MG, Brazil
$^{2}$ Instituto Federal de Educação Ciência e Tecnologia do Sul de Minas Gerais, Avenida Maria da Conceição Santos, 900  37550970, Parque Real, Pouso Alegre, MG, Brazil
$^{3}$ Departamento de Física, UNESP  Univ Estadual Paulista, Av. 24A, 1515, Bela Vista, 13506900, Rio Claro, SP  Brazil
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Abstract
The scaling formalism is applied to understand and describe the evolution towards the equilibrium at and near at a tangent bifurcation in the logistic map. At the bifurcation the convergence to the steady state is described by a homogeneous function leading to a set of critical exponents. Near the bifurcation the convergence is rather exponential whose relaxation time is given by a power law. We use two different approaches to obtain the critical exponents: (1) a phenomenological investigation based on three scaling hypotheses leading to a scaling law relating three critical exponents and; (2) an approximation that transforms the recurrence equations in a differential equation which is solved under appropriate conditions given analytically the scaling exponents. The numerical results give support for the theoretical approach.
Acknowledgments
Instituto Federal de Educação, Ciência e Tecnologia do Sul de Minas Gerais, IFSULDEMINAS  Campus Inconfidentes. EDL thanks to CNPq, FUNDUNESP and FAPESP (2017/144142), Brazilian agencies.
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