Discontinuity, Nonlinearity, and Complexity
Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α Families of Maps
Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 391402  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 αfamily 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 factoriallaw memory which is asymptotically powerlaw 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 powerlaw 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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