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Universality in Systems with Power-Law Memory and Fractional Dynamics

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Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of families of nonlinear dynamical systems converging to regular systems in the case of an integer power-law memory or an integer order of derivatives/differences. The examples considered in this review include the logistic family of maps (converging in the case of the first order difference to the regular logistic map), the universal family of maps, and the standard family of maps (the latter two converging, in the case of the second difference, to the regular universal and standard maps). Correspondingly, the phenomenon of transition to chaos through a period doubling cascade of bifurcations in regular nonlinear systems, known as “universality”, can be extended to fractional maps, which are maps with power-/asymptotically power-law memory. The new features of universality, including cascades of bifurcations on single trajectories, which appear in fractional (with memory) nonlinear dynamical systems are the main subject of this review.

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Acknowledgements

The author expresses his gratitude to R. Cole and R. V. Kohn, for the opportunity to complete this work at the Courant Institute. The author is grateful to the organizers of the 6th International Conference on Nonlinear Science and Complexity in Sao Jose dos Campos, Brazil, for financial support. The author acknowledges continuing support from Yeshiva University.

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Authors and Affiliations

  1. Department of Physics, Stern College at Yeshiva University, 245 Lexington Ave, New York, NY, 10016, USA

    Mark Edelman

  2. Courant Institute of Mathematical Sciences, New York University 251 Mercer St, New York, NY, 10012, USA

    Mark Edelman

  3. Department of Mathematics, BCC, CUNY 2155 University Avenue, Bronx, NY, 10453, USA

    Mark Edelman

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  1. Mark Edelman
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Correspondence to Mark Edelman .

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Editors and Affiliations

  1. Stern College for Women, Yeshiva University, New York, New York, USA

    Mark Edelman

  2. INPE/LAC, Instituto Nacional de Pesquisas Espaciai, São José dos Campos, São Paulo, Brazil

    Elbert E. N. Macau

  3. Departamento de Fisica, Universidad Rey Juan Carlos, Móstoles, Madrid, Spain

    Miguel A. F. Sanjuan

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Edelman, M. (2018). Universality in Systems with Power-Law Memory and Fractional Dynamics. In: Edelman, M., Macau, E., Sanjuan, M. (eds) Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-68109-2_8

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