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| In the study of [[dynamical systems]] the term '''Feigenbaum function''' has been used to describe two different functions introduced by the physicist [[Mitchell Feigenbaum]]:
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| * the solution to the [[Feigenbaum-Cvitanović]] functional equation; and
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| * the scaling function that described the covers of the [[attractor]] of the [[logistic map]]
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| == Functional equation==
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| The functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. The functional equation is the mathematical expression of the [[Universality (dynamical systems)|universality]] of period doubling. The equation is used to specify a function ''g'' and a parameter ''λ'' by the relation
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| :<math> g(x) = \frac{1}{-\lambda} g( g(\lambda x ) ) </math>
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| with the boundary conditions
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| * ''g''(0) = 1,
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| * ''g''′(0) = 0, and
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| * ''g''′′(0) < 0
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| For a particular form of solution with a quadratic dependence of the solution
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| near x=0, the inverse ''1/λ=2.5029...'' is one of the [[Feigenbaum constant]]s.
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| ==Scaling function==
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| The Feigenbaum scaling function provides a complete description of the [[attractor]] of the [[logistic map]] at the end of the period-doubling cascade. The attractor is a [[Cantor set]] set, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size ''d<sub>n</sub>''. For a fixed ''d<sub>n</sub>'' the set of segments forms a cover ''Δ<sub>n</sub>'' of the attractor. The ratio of segments from two consecutive covers, ''Δ<sub>n</sub>'' and ''Δ<sub>n+1</sub>'' can be arranged to approximate a function ''σ'', the Feigenbaum scaling function.
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| ==See also==
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| * [[Logistic map]]
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| * [[Presentation function]]
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| ==References==
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| * {{MathWorld|urlname=FeigenbaumFunction|title=Feigenbaum Function}}
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| * {{cite journal
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| |journal= Journal of Statistical Physics
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| |year=1978
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| |title=Quanitative universality for a class of nonlinear transformations
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| |last1=Feigenbaum
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| |first1=M.
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| |volume=19
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| |issue=1
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| |pages=25–52
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| |doi=10.1007/BF01020332
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| |bibcode=1978JSP....19...25F
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| |mr= 0501179
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| }}
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| * {{cite journal
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| |journal = Journal of Statistical Physics
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| |year = 1979
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| |title = The universal metric properties of non-linear transformations
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| |pages = 669–706
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| |last1=Feigenbaum
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| |first1=M.
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| |volume = 21
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| |issue=6
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| |doi = 10.1007/BF01107909
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| |bibcode=1979JSP....21..669F
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| |mr = 0555919
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| }}
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| * {{cite journal
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| |journal = Communications in Mathematical Physics
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| |year = 1980
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| |title = The transition to aperiodic behavior in turbulent systems
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| |pages = 65–86
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| |first1=Mitchell J.
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| |last1=Feigenbaum
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| |volume = 77
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| |issue =1
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| |bibcode=1980CMaPh..77...65F
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| |doi=10.1007/BF01205039
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| }}
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| * {{cite journal
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| |last1=Epstein
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| |first1=H.
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| |first2=J.
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| |last2=Lascoux
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| |title=Analyticity properties of the Feigenbaum Function
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| |journal=Commun. Math. Phys.
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| |volume=81
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| |issue=3
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| |doi=10.1007/BF01209078
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| |pages=437–453
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| |bibcode=1981CMaPh..81..437E
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| |year=1981
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| }}
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| * {{cite journal
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| |first1=Mitchell J.
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| |last1=Feigenbaum
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| |title=Universal Behavior in Nonlinear Systems
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| |journal=Physica
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| |volume=7D
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| |year=1983
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| |pages=16–39
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| |doi=10.1016/0167-2789(83)90112-4
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| |bibcode = 1983PhyD....7...16F }} Bound as ''Order in Chaos, Proceedings of the International Conference on Order and Chaos held at the Center for Nonlinear Studies, Los Alamos, New Mexico 87545,USA 24–28 May 1982'', Eds. David Campbell, Harvey Rose; North-Holland Amsterdam ISBN 0-444-86727-9.
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| * {{cite journal
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| |doi=10.1090/S0273-0979-1982-15008-X
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| |first1=Oscar E.
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| |last1= Lanford III
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| |title=A computer-assisted proof of the Feigenbaum conjectures
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| |journal=Bull. Am. Math. Soc.
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| |volume=6
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| |issue=3
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| |pages=427–434
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| |year=1982
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| |mr=0648529
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| }}
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| * {{cite journal
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| |first1=M.
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| |last1=Campanino
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| |first2=H.
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| |last2=Epstein
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| |first3=D.
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| |last3=Ruelle
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| |title=On Feigenbaums functional equation <math>g\circ g(\lambda x)+\lambda g(x)=0</math>
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| |journal=Topology
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| |volume=21
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| |issue=2
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| |pages=125–129
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| |doi=10.1016/0040-9383(82)90001-5
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| |year=1982
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| |mr=0641996
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| }}
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| * {{cite journal
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| |first1=Oscar E.
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| |last1=Lanford III
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| |title=A shorter proof of the existence of the Feigenbaum fixed point
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| |journal=Commun. Math. Phys.
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| |volume=96
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| |issue=4
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| |pages=521–538
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| |doi=10.1007/BF01212533
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| |bibcode=1984CMaPh..96..521L
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| |year=1984
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| }}
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| * {{cite journal
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| |first1=H.
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| |last1=Epstein
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| |title=New proofs of the existence of the Feigenbaum functions
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| |journal=Commun. Math. Phys.
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| |volume=106
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| |issue=3
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| |doi=10.1007/BF01207254
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| |pages=395–426
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| |bibcode=1986CMaPh.106..395E
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| |year=1986
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| }}
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| * {{cite journal
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| |first1=Jean-Pierre
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| |last1=Eckmann | author1-link = Jean-Pierre Eckmann
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| |first2=Peter
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| |last2=Wittwer
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| |title=A complete proof of the Feigenbaum Conjectures
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| |journal=J. Stat. Phys.
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| |volume=46
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| |issue=3/4
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| |page=455
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| |year=1987
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| |bibcode=1987JSP....46..455E
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| |doi=10.1007/BF01013368
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| |mr=0883539
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| }}
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| * {{cite journal
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| |first1=John
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| |last1=Stephenson
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| |first2=Yong
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| |last2=Wang
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| |title=Relationships between the solutions of Feigenbaum's equation
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| |journal=Appl. Math. Lett.
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| |volume=4
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| |issue=3
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| |pages=37–39
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| |doi=10.1016/0893-9659(91)90031-P
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| |year=1991
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| |mr=1101871
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| }}
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| * {{cite journal
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| |first1=John
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| |last1=Stephenson
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| |first2=Yong
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| |last2=Wang
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| |title=Relationships between eigenfunctions associated with solutions of Feigenbaum's equation
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| |journal=Appl. Math. Lett.
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| |volume=4
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| |issue=3
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| |pages=53–56;
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| |doi=10.1016/0893-9659(91)90035-T
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| |year=1991
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| |mr=1101875
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| }}
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| * {{cite journal
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| |first1=Keith
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| |last1=Briggs
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| |title=A precise calculation of the Feigenbaum constants
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| |journal=Math. Comp.
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| |volume=57
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| |issue=195
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| |pages=435–439
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| |doi=10.1090/S0025-5718-1991-1079009-6
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| |year=1991
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| |mr=1079009
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| |bibcode = 1991MaCom..57..435B }}
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| *{{ cite journal
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| |first1=Alexei V.
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| |last1=Tsygvintsev
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| |first2=Ben D.
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| |last2=Mestel
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| |first3=Andrew H.
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| |last3=Obaldestin
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| |title=Continued fractions and solutions of the Feigenbaum-Cvitanović equation
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| |journal=C. R. Acad. Sci. Paris, Ser. I
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| |volume=334
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| |issue=8
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| |pages=683–688
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| |doi=10.1016/S1631-073X(02)02330-0
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| |year=2002
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| }}
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| * {{cite arxiv
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| |first1=Richard J.
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| |last1=Mathar
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| |title=Chebyshev series representation of Feigenbaum's period-doubling function
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| |eprint=1008.4608
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| |year=2010
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| |class=math.DS
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| }}
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| * {{Cite journal
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| |first1=V. P.
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| |last1=Varin
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| |title=Spectral properties of the period-doubling operator
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| |journal=KIAM Preprint
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| |volume=9
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| |year=2011
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| |url=http://www.keldysh.ru/preprint.asp?id=2011-9&lg=e
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| }}
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| [[Category:Chaos theory]]
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| [[Category:Dynamical systems]]
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Hi there. Let me begin by introducing the writer, her title is Myrtle Cleary. I am a meter reader but I strategy on altering it. North Dakota is her beginning place but she will have to move 1 day or an additional. Doing ceramics is what her family and her enjoy.
my web page :: std home test