Using kicked differential equations of motion with derivatives of noninteger orders, we obtain generalizations of the dissipative standard map. The main property of these generalized maps, which are called fractional maps, is long-term memory. The memory effect in the fractional maps means that their present state of evolution depends on all past states with special forms of weights. Already a small deviation of the order of derivative from the integer value corresponding to the regular dissipative standard map (small memory effects) leads to the qualitatively new behavior of the corresponding attractors. The fractional dissipative standard maps are used to demonstrate a new type of fractional attractors in the wide range of the fractional orders of derivatives.
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June 2010
Research Article|
June 21 2010
Fractional dissipative standard map
Vasily E. Tarasov;
Vasily E. Tarasov
1Courant Institute of Mathematical Sciences,
New York University
, 251 Mercer St., New York, New York 10012, USA
2Skobeltsyn Institute of Nuclear Physics,
Moscow State University
, Moscow 119991, Russia
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M. Edelman
M. Edelman
1Courant Institute of Mathematical Sciences,
New York University
, 251 Mercer St., New York, New York 10012, USA
3Department of Physics, Stern College,
Yeshiva University
, 245 Lexington Ave., New York, New York 10016, USA
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Chaos 20, 023127 (2010)
Article history
Received:
October 30 2009
Accepted:
May 12 2010
Citation
Vasily E. Tarasov, M. Edelman; Fractional dissipative standard map. Chaos 1 June 2010; 20 (2): 023127. https://doi.org/10.1063/1.3443235
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