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Artykuły w czasopismach na temat „Feigenbaum renormalization”

Autor: Grafiati

Data publikacji: 2 czerwca 2025

Data aktualizacji: 19 lipca 2025

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1

Kuznetsov, Sergey. "Period-doubling for complex cubic map." Izvestiya VUZ. Applied Nonlinear Dynamics 4, no. 4 (1996): 3–12. https://doi.org/10.18500/0869-6632-1996-4-4-3-12.

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Scaling properties are reported for period-doubling cascade in complex cubic map z -> c-z^3. Renormalization group analysis is developed and the associated complex solution of the Feigenbaum - Cvitanovic equation is obtained numerically.

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2

KUZNETSOV, A. P., S. P. KUZNETSOV, I. R. SATAEV, and L. O. CHUA. "TWO-PARAMETER STUDY OF TRANSITION TO CHAOS IN CHUA'S CIRCUIT: RENORMALIZATION GROUP, UNIVERSALITY AND SCALING." International Journal of Bifurcation and Chaos 03, no. 04 (1993): 943–62. http://dx.doi.org/10.1142/s0218127493000799.

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A complex fine structure in the topography of regions of different dynamical behavior near the onset of chaos is investigated in a parameter plane of the 1D Chua's map, which describes approximately the dynamics of Chua's circuit. Besides piecewise-smooth Feigenbaum critical lines, the boundary of chaos contains an infinite set of codimension-2 critical points, which may be coded by itineraries on a binary tree. Renormalization group analysis is applied which is a generalization of Feigenbaum's theory for codimension-2 critical points. Multicolor high-resolution maps of the parameter plane sho

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3

BARBARO, G. "FORMAL SOLUTIONS OF THE CVITANOVIC–FEIGENBAUM EQUATION." International Journal of Bifurcation and Chaos 17, no. 09 (2007): 3275–80. http://dx.doi.org/10.1142/s0218127407019020.

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The so-called renormalization group equation or Cvitanovic–Feigenbaum (CF) equation arises in the universal scaling theory of iterated maps. In this Letter, a set of formal analytic solutions of this equation is obtained by assuming that the solutions have a particular functional form.

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4

SMITH, J. D. H. "Wreath products along the period-doubling route to chaos." Ergodic Theory and Dynamical Systems 19, no. 6 (1999): 1617–36. http://dx.doi.org/10.1017/s0143385799151927.

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The wreath-product construction is used to give a complete combinatorial description of the dynamics of period-doubling quadratic maps leading to the Feigenbaum map. An explicit description of the action on periodic points uses the Thue–Morse sequence. In particular, a wreath-product construction of this sequence is given. The combinatorial renormalization operator on the period-doubling family of maps is invertible.

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5

Delbourgo, R., and BG Kenny. "Universal Features of Tangent Bifurcation." Australian Journal of Physics 38, no. 1 (1985): 1. http://dx.doi.org/10.1071/ph850001.

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We exhibit certain universal characteristics of limit cycles pertaining to one-dimensional maps in the 'chaotic' region beyond the point of accumulation connected with period doubling. Universal, Feigenbaum-type numbers emerge for different sequences, such as triplication. More significantly we have established the existence of different classes of universal functions which satisfy the same renormalization group equations, with the same parameters, as the appropriate accumulation point is reached.

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6

GAIDASHEV, DENIS G. "PERIOD DOUBLING RENORMALIZATION FOR AREA-PRESERVING MAPS AND MILD COMPUTER ASSISTANCE IN CONTRACTION MAPPING PRINCIPLE." International Journal of Bifurcation and Chaos 21, no. 11 (2011): 3217–30. http://dx.doi.org/10.1142/s0218127411030477.

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A universal period doubling cascade analogous to the famous Feigenbaum–Coullet–Tresser period doubling has been observed in area-preserving maps of ℝ2. The existence of the "universal" map with orbits of all binary periods has been proved via a renormalization approach in [Eckmann et al., 1984] and [Gaidashev et al., 2011]. These proofs use "hard" computer assistance.In this paper, we attempt to reduce computer assistance in the argument, and present a mild computer aided proof of the analyticity and compactness of the renormalization operator in a neighborhood of a renormalization fixed point

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7

Kozlovski, Oleg, and Sebastian van Strien. "Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws." Communications in Mathematical Physics 379, no. 1 (2020): 103–43. http://dx.doi.org/10.1007/s00220-020-03835-9.

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Abstract We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x

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8

Kuznetsov, Aleksandr, Sergey Kuznetsov, and Igor Sataev. "Fractal signal and dynamics of periodic-doubling systems." Izvestiya VUZ. Applied Nonlinear Dynamics 3, no. 5 (1995): 64–87. https://doi.org/10.18500/0869-6632-1995-3-5-64-87.

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The model of fractal signal having a phase portrait in a form of two-scale Cantor set provides a possibility to describe many real signals generating by dynamical systems at the onset of chaos and to treat them in a unified way. The points in the parameter plane of the fractal signal are outlined, which correspond to these real types of dynamical behavior. Simple electronic circuit admitting experimental realization is suggested, that generates the fractal signal with tunable parameters. Renormalization group analysis is developed for the case of period-doubling system forced by the fractal si

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9

GOLDFAIN, ERVIN. "FEIGENBAUM ATTRACTOR AND THE GENERATION STRUCTURE OF PARTICLE PHYSICS." International Journal of Bifurcation and Chaos 18, no. 03 (2008): 891–96. http://dx.doi.org/10.1142/s0218127408020756.

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The standard model (SM) for high-energy physics describes fundamental interactions between subatomic particles down to a distance scale on the order of 10-18 m. Despite its widespread acceptance, SM operates with a large number of arbitrary parameters whose physical origin is presently unknown. Our work suggests that the generation structure of at least some SM parameters stems from the chaotic regime of renormalization group flow. Invoking the universal route to chaos in systems of nonlinear differential equations, we argue that the hierarchical pattern of parameters amounts to a series of sc

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10

CHANDRAMOULI, V. V. M. S., M. MARTENS, W. DE MELO, and C. P. TRESSER. "Chaotic period doubling." Ergodic Theory and Dynamical Systems 29, no. 2 (2009): 381–418. http://dx.doi.org/10.1017/s0143385708080371.

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AbstractThe period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormal

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11

FRAME, MICHAEL, and DAVID PEAK. "METRIC UNIVERSALITY OF ORDER IN ONE-DIMENSIONAL DYNAMICS." International Journal of Bifurcation and Chaos 03, no. 03 (1993): 567–72. http://dx.doi.org/10.1142/s0218127493000453.

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The orbit of the critical point of a nonlinear dynamical system defines a family of functions in the parameter space of the system. For unimodal maps a renormalization makes these functions indistinguishable over a wide range of parameter values. The universal representation of these functions leads directly to a number of interesting results: (1) the positions in the parameter space of the windows of order; (2) the sizes of the windows of order; (3) measures of distortion in the window structure; and (4) various generalized Feigenbaum numbers. We explicitly discuss the examples of the quadrat

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12

KUZNETSOV, A. P., S. P. KUZNETSOV, and I. R. SATAEV. "BICRITICAL DYNAMICS OF PERIOD-DOUBLING SYSTEMS WITH UNIDIRECTIONAL COUPLING." International Journal of Bifurcation and Chaos 01, no. 04 (1991): 839–48. http://dx.doi.org/10.1142/s0218127491000610.

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The simplest case of bicritical behavior arises in a system of two logistic maps with unidirectional coupling in the point of a parameter plane where lines of transition to chaos in both subsystems meet. We develop a renormalization group analysis of the bicriticality and find the corresponding fixed point universal function and constants featuring the scaling properties of the second system while the first one is in the Feigenbaum critical state. Fractal properties of the bicritical attractor and its quantitative characteristics (σ-functions, f(α)-spectra, generalized dimensions) are consider

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13

Jalnine, Aleksej, and Sergey Kuznetsov. "Universality and scaling in the circle map under external periodic driving." Izvestiya VUZ. Applied Nonlinear Dynamics 10, no. 6 (2003): 3–15. http://dx.doi.org/10.18500/0869-6632-2002-10-6-3-15.

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We investigate scaling properties of а neighborhood of the «golden mean» critical point in the circle map in presence of external periodic forcing upon the system. We consider such perturbations of the fixed point of the Feigenbaum-Kadanoff-Shenker renormalization group equation, which are initiated by periodic forcing. We show that, depending upon the frequency of external forcing, two types of scaling behavior can be observed. The first (P-type) 18 associated with periodic repetition of the structures of dynamical regimes in parameter space under subsequent scaling transformations. For the s

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14

Kuznetsov, S. P. "A renormalization-group analysis for two unidirectionally coupled Feigenbaum systems at the hyperchaos threshold." Radiophysics and Quantum Electronics 33, no. 7 (1990): 578–81. http://dx.doi.org/10.1007/bf01048488.

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15

Kuznetsov, A. P., S. P. Kuznetsov, and I. R. Sataev. "Influence of a fractal signal on a Feigenbaum system and bifurcation in renormalization group equations." Radiophysics and Quantum Electronics 34, no. 6 (1991): 556–63. http://dx.doi.org/10.1007/bf01039580.

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16

GAIDASHEV, DENIS, and HANS KOCH. "Period doubling in area-preserving maps: an associated one-dimensional problem." Ergodic Theory and Dynamical Systems 31, no. 4 (2010): 1193–228. http://dx.doi.org/10.1017/s0143385710000283.

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AbstractIt has been observed that the famous Feigenbaum–Coullet–Tresser period-doubling universality has a counterpart for area-preserving maps of ℝ2. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in Eckmannet al[Existence of a fixed point of the doubling transformation for area-preserving maps of the plane.Phys. Rev. A 26(1) (1982), 720–722; A computer-assisted proof of universality for area-preserving maps.Mem. Amer. Math. Soc. 47(1984), 1–121]. As is the case with all non-trivial universality pr

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17

HUANG, GUIFENG, LIDONG WANG, and GONGFU LIAO. "A NOTE ON THE UNIMODAL FEIGENBAUM'S MAPS." International Journal of Modern Physics B 23, no. 14 (2009): 3101–11. http://dx.doi.org/10.1142/s0217979209052649.

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We mainly investigate the likely limit sets and the kneading sequences of unimodal Feigenbaum's maps (Feigenbaum's map can be regarded as the fixed point of the renormalization operator [Formula: see text], where λ is to be determined). First, we estimate the Hausdorff dimension of the likely limit set for the unimodal Feigenbaum's map and then for every decimal s ∈ (0, 1), we construct a unimodal Feigenbaum's map which has a likely limit set with Hausdorff dimension s. Second, we prove that the kneading sequences of unimodal Feigenbaum's maps are uniformly almost periodic points of the shift

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18

Kuznetsov, Aleksandr, Sergey Kuznetsov, and Igor Sataev. "Critical dynamics for one-dimensional maps. Part II. Two-parametre transition to chaos." Izvestiya VUZ. Applied Nonlinear Dynamics 1, no. 3 (1993): 17–35. https://doi.org/10.18500/0869-6632-1993-1-3-17-35.

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Generalization of Feigenbaum’s method is considered with respect to the two—parametre transition to chaos in one—dimensional maps. The approximate and exact renormalization group analyses are developed. Illustrations of scaling are presented and physical examples are discussed.

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19

Ivank’ov, N. Yu, and S. P. Kuznetsov. "Different types of scaling in the dynamics of period–doubling maps under external periodic driving." Discrete Dynamics in Nature and Society 5, no. 3 (2000): 223–32. http://dx.doi.org/10.1155/s1026022600000546.

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Based on the renormalization group approach developed by Kuznetsov and Pikovsky (Phys. Lett., A140, 1989, 166) several types of scaling are discussed, which can be observed in a neighborhood of Feigenbaum’s critical point at small amplitudes of the driving. The type of scaling behavior depends on a structure of binary representation of the frequency parameter:F-scaling (Feigenbaum’s) for finite binary fractions,P- andQ-scaling (periodic and quasiperiodic) for periodic binary fractions, andS-scaling (statistical) for non-periodic binary fractions. All types of scaling are illustrated by paramet

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20

Kuznetsov, Aleksandr, and Sergey Kuznetsov. "Critical dynamics for one-dimensional maps part 1: Feigenbaum's scenario." Izvestiya VUZ. Applied Nonlinear Dynamics 1, no. 1 (1993): 15–33. https://doi.org/10.18500/0869-6632-1993-1-1-15-33.

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A review of main results is given, concerning the Feigenbaum's scenario in the context of critical phenomena theory. Computer-generated illustrations of scaling are presented. Approximate renormalization group (RG) analysis is considered, allowing to obtain RG transformation in an explicit form. Examples of nonlinear systems are discussed, demonstrating this type of critical behaviour.

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21

KUZNETSOV, A. P., S. P. KUZNETSOV, I. R. SATAEV, and L. O. CHUA. "SELF-SIMILARITY AND UNIVERSALITY IN CHUA'S CIRCUIT VIA THE APPROXIMATE CHUA'S 1-D MAP." Journal of Circuits, Systems and Computers 03, no. 02 (1993): 431–40. http://dx.doi.org/10.1142/s0218126693000265.

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In this paper we investigate the features of the transition to chaos in a one-dimensional Chua's map which describes approximately the Chua's circuit. These features arise from the nonunimodality of this map. We show that there exists a variety of types of critical points, which are characterized by a universal self-similar topography in a neighborhood of each critical point in the parameter plane. Such universalities are associated with various cycles of Feigenbaum's renormalization group equation.

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22

Moon, Ki-Jung, and Sang Don Choi. "Reducible expansions and related sharp crossovers in Feigenbaum’s renormalization field." Chaos: An Interdisciplinary Journal of Nonlinear Science 18, no. 2 (2008): 023104. http://dx.doi.org/10.1063/1.2902826.

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23

MESEGUER, A., F. MARQUÈS, and J. SÁNCHEZ. "FEIGENBAUM’S UNIVERSALITY IN A LOW-DIMENSIONAL FLUID MODEL." International Journal of Bifurcation and Chaos 06, no. 08 (1996): 1587–94. http://dx.doi.org/10.1142/s0218127496000953.

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We present a low-dimensional truncated model for a viscous fluid contained in a two-dimensional square box, obtained by truncating a dynamical system of amplitudes for the velocity field. This low-dimensional model exhibits a route to chaos via a period doubling cascade (Feigenbaum’s Scenario). In order to compute with high accuracy the period doubling, a numerical method based on the first order variational equations and a Poincaré map has been developed. This methodology can also be applied to the analysis of bifurcations of periodic orbits in low-dimensional ordinary differential equations.

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24

Coppersmith, S. N. "A simpler derivation of Feigenbaum’s renormalization group equation for the period-doubling bifurcation sequence." American Journal of Physics 67, no. 1 (1999): 52–54. http://dx.doi.org/10.1119/1.19190.

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25

Moon, Ki-Jung. "Erratum: “Reducible expansions and related sharp crossovers in Feigenbaum’s renormalization field” [Chaos 18, 023104 (2008)]." Chaos: An Interdisciplinary Journal of Nonlinear Science 20, no. 4 (2010): 049902. http://dx.doi.org/10.1063/1.3530128.

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26

Zhang, Yuhan, Jianyong Qiao, and Junyang Gao. "Feigenbaum Julia Sets Concerning Renormalization Transformation." Frontiers of Mathematics, September 15, 2024. http://dx.doi.org/10.1007/s11464-022-0118-y.

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27

LEVIN, GENADI, and GRZEGORZ ŚWIA̧TEK. "Limit drift for complex Feigenbaum mappings." Ergodic Theory and Dynamical Systems, September 28, 2020, 1–52. http://dx.doi.org/10.1017/etds.2020.53.

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Abstract We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order $\ell $ of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to $0$ under the dynamics of the tower for corresponding $\ell $ . That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when $\ell $ tends to $\i

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28

Moyano, Luis, D. Silva, and A. Robledo. "Labyrinthine pathways towards supercycle attractors in unimodal maps." Open Physics 7, no. 3 (2009). http://dx.doi.org/10.2478/s11534-009-0065-1.

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AbstractAs an important preceding step for the demonstration of an uncharacteristic (q-deformed) statisticalmechanical structure in the dynamics of the Feigenbaum attractor we uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the pre-images of the repellor, display hierarch

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29

Ziep, Otto. "Fractal Universe and Atoms." May 17, 2025. https://doi.org/10.36347/sjpms.2025.v12i04.002.

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Fractal universes and atoms are assigned to k-components or stable orbiting laps of simplest cycles of elliptic invariants. Cosmological redshift, expansion of the universe, origin of cosmic rays, cosmic microwave background, quantum entanglement and the cosmological constant problem are resolvable easily by fractal universes of bifurcating spacetime. Quantum entanglement is explainable by a highly correlated pseudo-congruent k-component in bifurcating spacetime. A one-dimensional complex contour around nontrivial zeros of zeta and L- functions is capable to create a zero-energy universe- acti

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