Discontinuity, Nonlinearity, and Complexity
Fractional Maps and Fractional Attractors. Part I: αFamilies of Maps
Discontinuity, Nonlinearity, and Complexity 1(4) (2012) 305324  DOI:10.5890/DNC.2012.07.003
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., NY 10012, USA
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Abstract
In this paper we present a uniform way to derive families of maps from the corresponding differential equations describing systems which experience periodic kicks. The families depend on a single parameter  the order of a differential equation α > 0. We investigate general properties of such families and how they vary with the increase in α which represents increase in the space dimension and the memory of a system (increase in the weight of the earlier states). To demonstrate general properties of the α families we use examples from physics (Standard α Family of Maps) and population biology (Logistic α Family of Maps). We show that with the increase in α systems demonstrate more complex and chaotic behavior.
Acknowledgments
The author expresses his gratitude to V.E. Tarasov for the useful remarks, to E. Hameiri and H. Weitzner for the opportunity to complete this work at the Courant Institute and Yeshiva University for the financial support.
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