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  • Letter

Extreme fractal dimension at periodicity cascades in parameter spaces

Carlos E. P. Abreu

Joelson D. V. Hermes

Diogo Ricardo da Costa

Everton S. Medeiros

Rene O. Medrano-T*

  • Departamento de Física, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, UNESP, 13506-900, Campus Rio Claro, São Paulo, Brazil and Instituto Federal de Educação, Ciência e Tecnologia do Sul de Minas Gerais, IFSULDEMINAS, 37417-158, Campus Três Corações, Minas Gerais, Brazil

  • Instituto Federal de Educação, Ciência e Tecnologia do Sul de Minas Gerais, IFSULDEMINAS, 37576-000, Campus Inconfidentes, Minas Gerais, Brazil

  • Departamento de Física, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, UNESP, 13506-900, Campus Rio Claro, São Paulo, Brazil

  • Departamento de Física, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, UNESP, 13506-900, Campus Rio Claro, São Paulo, Brazil and Institute for Chemistry and Biology of the Marine Environment, Carl von Ossietzky University Oldenburg, 26111 Oldenburg, Germany

  • Departamento de Física, Universidade Federal de São Paulo, UNIFESP, 09913-030, Campus Diadema, São Paulo, Brazil and Departamento de Física, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, UNESP, 13506-900, Campus Rio Claro, São Paulo, Brazil

  • *Contact author: rene.medrano@unifesp.br
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Phys. Rev. E 110, L032201 – Published 11 September, 2024

DOI: https://doi.org/10.1103/PhysRevE.110.L032201

Abstract

In the parameter spaces of nonlinear dynamical systems, we investigate the boundaries between periodicity and chaos and unveil the existence of fractal sets characterized by a singular fractal dimension that deviates greatly from the fractal sets in their vicinity. This extreme fractal dimension stands out from the typical value previously considered universal for these parameter boundaries. We show that such singular fractal sets dwell along parameter curves, called extreme curves, that intersect periodicity cascades at their centers of stability across all scales of parameter spaces. The results reported here are generally demonstrated for the class of one-dimensional maps with at least two control parameters. Generalizations to other classes of systems are possible.

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Phys. Rev. E 110, L032201– Published 11 September, 2024
  • Received 16 January 2024
  • Accepted 12 August 2024
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