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Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 391--402 | DOI:10.5890/DNC.2015.11.003

M. Edelman

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

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

Department of Mathematics, BCC, CUNY, 2155 University Avenue, Bronx, New York 10453, USA

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Abstract

In this paper we extend the notion of an α-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic α- Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.

Acknowledgments

The author acknowledges support from the Joseph Alexander Foundation, Yeshiva University. The author expresses his gratitude to E. Hameiri, H. Weitzner, and G. Ben Arous for the opportunity to complete this work at the Courant Institute and to V. Donnelly for technical help.

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