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(21)
A computerassisted proof of the Feigenbaum conjectures
ITÉRATIONS ...
Feigenbaum universality and the thermodynamic formalism
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Quantitative universality for a class of nonlinear transformations
Quantitative universality for a class of nonlinear transformations,10.1007/BF01020332,Journal of Statistical Physics,Mitchell J. Feigenbaum
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Quantitative universality for a class of nonlinear transformations
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Citations: 565
)
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Mitchell J. Feigenbaum
A large class of recursion relationsxn + 1 = ?f(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum $$\bar x$$ . With $$f(\bar x)  f(x) \sim \left {x  \bar x} \right^z (for\left {x  \bar x} \right$$ sufficiently small),z > 1, the universal details depend only uponz. In particular, the
local structure
of highorder stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratioa (a = 2.5029078750957... forz = 2). This structure is determined by a universal functiong*(x), where the 2nth iterate off,f(n), converges locally toang*(anx) for largen. For the class off's considered, there exists a?n such that a 2npoint stable
limit cycle
including $$\bar x$$ exists;?8 ?n R~dn (d = 4.669201609103... forz = 2). The numbersa andd have been computationally determined for a range ofz through their definitions, for a variety off's for eachz. We present a recursive mechanism that explains these results by determiningg* as the fixedpoint (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.
Journal:
Journal of Statistical Physics  J STATIST PHYS
, vol. 19, no. 1, pp. 2552, 1978
DOI:
10.1007/BF01020332
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Citation Context
(146)
...Feigenbaum proved that this route is universal for dissipative systems possessing a quadratic nonlinearity and derived the scaling behaviour for successive perioddoubling bifurcations
2
,
3
...
Dmitry V. Savin
,
et al.
The selfoscillating system with compensated dissipation – the dynamic...
...Already relatively simple systems have interactions, nonlinear dynamics, and sensitivity that lead to chaos, strange attractors, and catastrophes that make a good prediction hard to find (Lorenz
1963
; Ruelle and Takens
1971
; Thom
1972
; Nicolis and Prigogine
1977
; Feigenbaum
1978
; Haken
1981
)...
Ilkka Tuomi
.
Foresight in an unpredictable world
...66920 [
42
]...
Gianmarco Pizza
,
et al.
Chaotic dynamics in premixed hydrogen/air channel flow combustion
...Renormalization theory of onedimensional dynamical systems was introduced by Feigenbaum in [
7
] to explain some universal phenomena in the rates of convergence from some sequences of points in the bifurcation diagram of the quadratic family...
Nuno Franco
,
et al.
Symbolic dynamics and renormalization of nonautonomous k periodic dyn...
... This effect is similar to perioddoubling bifurcations in dynamical system
...
Fabian Brau
,
et al.
Multiplelengthscale elastic instability mimics parametric resonance ...
References
(1)
On Finite Limit Sets for Transformations on the Unit Interval
(
Citations: 137
)
M. L. Stein
,
P. R. Stein
Journal:
Journal of Combinatorial Theory  JCT
, vol. 15, no. 1, pp. 2544, 1973
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Citations
(565)
NONLINEAR AND COMPLEX DYNAMICS IN ECONOMICS
(
Citations: 4
)
William A. Barnett
,
Alfredo Medio
,
Apostolos Serletis
Published in 2012.
The selfoscillating system with compensated dissipation – the dynamics of the approximate discrete map
Dmitry V. Savin
,
Alexey V. Savin
,
Alexander P. Kuznetsov
,
Sergey P. Kuznetsov
,
Ulrike Feudel
Journal:
Dynamical Systemsan International Journal  DYN SYST
, vol. aheadofp, no. aheadofp, pp. 113, 2012
Foresight in an unpredictable world
Ilkka Tuomi
Journal:
Technology Analysis & Strategic Management  TECHNOL ANAL STRATEG MANAGE
, vol. 24, no. 8, pp. 735751, 2012
A numerical study of infinitely renormalizable areapreserving maps
Denis Gaidashev
,
Tomas Johnson
Journal:
Dynamical Systemsan International Journal  DYN SYST
, vol. aheadofp, no. aheadofp, pp. 119, 2012
Chaotic dynamics in premixed hydrogen/air channel flow combustion
Gianmarco Pizza
,
Christos E. Frouzakis
,
John Mantzaras
Journal:
Combustion Theory and Modelling  COMBUST THEORY MODEL
, vol. 16, no. 2, pp. 275299, 2012