Birds and frogs in mathematics and physics

Dyson, Freeman J.

doi:10.3367/ufne.0180.201008f.0859

Cited by 17 publications

(1 citation statement)

References 25 publications

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“…If f is a continuous map on an interval, there is a remarkable theorem proved by Sharkovskii [1] and later independently by Li and Yorke, [2] said to be the only rigorous theorem in chaos. [3] This theorem asserts that if 3-cycle exists in some domain of such a map, there exist all other cycles in said domain. Thus the existence of 3-cycle according to Li and Yorke implies chaos itself in that domain.…”

Section: Introduction To One-dimensional Continuous Maps On An Intervalmentioning

confidence: 99%

Solving for the Fixed Points of 3-Cycle in the Logistic Map and Toward Realizing Chaos by the Theorems of Sharkovskii and Li—Yorke

Lee

2014

Commun. Theor. Phys.

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Sharkovskii proved that, for continuous maps on intervals, the existence of 3-cycle implies the existence of all others. Li and Yorke proved that 3-cycle implies chaos. To establish a domain of uncountable cycles in the logistic map and to understand chaos in it, the fixed points of 3-cycle are obtained analytically by solving a sextic equation. At one parametric value, a fixed-point spectrum, resulted from the Sharkovskii limit, helps to realize chaos in the sense of Li and Yorke.

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Section: Introduction To One-dimensional Continuous Maps On An Intervalmentioning

confidence: 99%

Solving for the Fixed Points of 3-Cycle in the Logistic Map and Toward Realizing Chaos by the Theorems of Sharkovskii and Li—Yorke

Lee

2014

Commun. Theor. Phys.

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Spectral statistics and scattering resonances of complex primes arrays

2018

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We introduce a novel class of aperiodic arrays of electric dipoles generated from the distribution of prime numbers in complex quadratic fields (Eisenstein and Gaussian primes) as well as quaternion primes (Hurwitz and Lifschitz primes), and study the nature of their scattering resonances using the vectorial Green's matrix method. In these systems we demonstrate several distinctive spectral properties, such as the absence of level repulsion in the strongly scattering regime, critical statistics of level spacings, and the existence of critical modes, which are extended fractal modes with long lifetimes not supported by either random or periodic systems. Moreover, we show that one can predict important physical properties, such as the existence spectral gaps, by analyzing the eigenvalue distribution of the Green's matrix of the arrays in the complex plane. Our results unveil the importance of aperiodic correlations in prime number arrays for the engineering of novel gapped photonic media that support far richer mode localization and spectral properties compared to usual periodic and random media.

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How in the 20th century physicists, chemists and biologists answered the question: what is life?

Реутов

Schechter

2010

Phys.-Usp.

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Birds and frogs in mathematics and physics

Cited by 17 publications

References 25 publications

Solving for the Fixed Points of 3-Cycle in the Logistic Map and Toward Realizing Chaos by the Theorems of Sharkovskii and Li—Yorke

Solving for the Fixed Points of 3-Cycle in the Logistic Map and Toward Realizing Chaos by the Theorems of Sharkovskii and Li—Yorke

Spectral statistics and scattering resonances of complex primes arrays

How in the 20th century physicists, chemists and biologists answered the question: what is life?

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