Quantitative universality for a class of nonlinear transformations

Feigenbaum, Mitchell J.

doi:10.1007/bf01020332

Cited by 3,261 publications

(1,207 citation statements)

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“…3 ≈ µ [Feigenbaum, 1978], the time signal complexification by intermittence in the neighbourhood of 82 . 3 ≈ µ [Pomeau & Manneville, 1980] and the random-like dynamics for 4 = µ .…”

Section: The Logistic Model For One Isolated Speciesmentioning

confidence: 99%

Indirect Allee effect, bistability and chaotic oscillations in a predator–prey discrete model of logistic type

López‐Ruiz¹,

Fournier-Prunaret²

2005

Chaos, Solitons & Fractals

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A cubic discrete coupled logistic equation is proposed to model the predator-prey problem. The coupling depends on the population size of both species and on a positive constant λ , which could depend on the prey reproduction rate and on the predator hunting strategy. Different dynamical regimes are obtained when λ is modified. For small λ , the species become extinct. For a bigger λ , the preys survive but the predators extinguish. Only when the prey population reaches a critical value then predators can coexist with preys. For increasing λ , a bistable regime appears where the populations apart of being stabilized in fixed quantities can present periodic, quasiperiodic and chaotic oscillations. Finally, bistability is lost and the system settles down in a steady state, or, for the biggest permitted λ , in an invariant curve. We also present the basins for the different regimes.The use of the critical curves lets us determine the influence of the zones with different number of first rank preimages in the bifurcation mechanisms of those basins.

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“…3 ≈ µ [Feigenbaum, 1978], the time signal complexification by intermittence in the neighbourhood of 82 . 3 ≈ µ [Pomeau & Manneville, 1980] and the random-like dynamics for 4 = µ .…”

Section: The Logistic Model For One Isolated Speciesmentioning

confidence: 99%

Indirect Allee effect, bistability and chaotic oscillations in a predator–prey discrete model of logistic type

López‐Ruiz¹,

Fournier-Prunaret²

2005

Chaos, Solitons & Fractals

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“…Two independent proofs (Campanino et al [1,2], Lanford [4]) have been given for the existence of an even analytic solution to the Feigenbaum-Cvitanovi6 functional equation [3] g(x)= --~g (g(--2x)), g(O) = 1 (1.1) with g"(O) < O; 2 ---g(l)>O.…”

Section: Introductionmentioning

confidence: 99%

A shorter proof of the existence of the Feigenbaum fixed point

Lanford

1984

Commun.Math. Phys.

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“…It turns out that differences of a function which is chaotic at a point show a near doubling of size with increasing order of differences. Connections with work of Feigenbaum [2] seem to be of possible interest.…”

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confidence: 99%

“…In fact observation of false period doubling in computations with Feigenbaum's function (fh: fh(x) = bx(l -x), x G [0,1]; cf. [2]) led to the idea of connecting [5][6][7] with the study of chaos.…”

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confidence: 99%