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Gotowa bibliografia na temat „Feigenbaum renormalization”

Autor: Grafiati

Data publikacji: 2 czerwca 2025

Data aktualizacji: 22 czerwca 2025

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Artykuły w czasopismach na temat "Feigenbaum renormalization"

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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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 show that in regions near critical points having periodic codes, the infinitely intricate topography of the parameter plane reveals a property of self-similarity.

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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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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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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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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: that is, a proof that does not use generalizations of interval arithmetics to functional spaces — but rather relies on interval arithmetics on real numbers only to estimate otherwise explicit expressions. The proof relies on several instances of the Contraction Mapping Principle, which is, again, verified via mild computer assistance.

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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 - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

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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 signal. It is shown that a bifurcation takes place in the RG equation, and the behavior at the onset of chaos may be described by either Feigenbaum or non-Feigenbaum fixed point solutions. The results of numerical simulations are presented to illustrate the scaling properties of the dynamics forced by the fractal signal.

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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 scaling ratios depending on the Feigenbaum constant. Leading order predictions are shown to agree reasonably well with experimental data.

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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 renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space ofC2+αunimodal maps, forα>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space ofC2unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to geta prioribounds. In this smoother class, calledC2+∣⋅∣, the failure of hyperbolicity is tamer than inC2. Things get much worse with just a bit less smoothness thanC2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space ofC1+Lipunimodal maps has infinite topological entropy.

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Rozprawy doktorskie na temat "Feigenbaum renormalization"

Sendrowski, Janek. "Feigenbaum Scaling." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-96635.

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In this thesis I hope to provide a clear and concise introduction to Feigenbaum scaling accessible to undergraduate students. This is accompanied by a description of how to obtain numerical results by various means. A more intricate approach drawing from renormalization theory as well as a short consideration of some of the topological properties will also be presented. I was furthermore trying to put great emphasis on diagrams throughout the text to make the contents more comprehensible and intuitive.

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Książki na temat "Feigenbaum renormalization"

Rajeev, S. G. Fluid Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.001.0001.

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Starting with a review of vector fields and their integral curves, the book presents the basic equations of the subject: Euler and Navier–Stokes. Some solutions are studied next: ideal flows using conformal transformations, viscous flows such as Couette and Stokes flow around a sphere, shocks in the Burgers equation. Prandtl’s boundary layer theory and the Blasius solution are presented. Rayleigh–Taylor instability is studied in analogy with the inverted pendulum, with a digression on Kapitza’s stabilization. The possibility of transients in a linearly stable system with a non-normal operator is studied using an example by Trefethen et al. The integrable models (KdV, Hasimoto’s vortex soliton) and their hamiltonian formalism are studied. Delving into deeper mathematics, geodesics on Lie groups are studied: first using the Lie algebra and then using Milnor’s approach to the curvature of the Lie group. Arnold’s deep idea that Euler’s equations are the geodesic equations on the diffeomorphism group is then explained and its curvature calculated. The next three chapters are an introduction to numerical methods: spectral methods based on Chebychev functions for ODEs, their application by Orszag to solve the Orr–Sommerfeld equation, finite difference methods for elementary PDEs, the Magnus formula and its application to geometric integrators for ODEs. Two appendices give an introduction to dynamical systems: Arnold’s cat map, homoclinic points, Smale’s horse shoe, Hausdorff dimension of the invariant set, Aref ’s example of chaotic advection. The last appendix introduces renormalization: Ising model on a Cayley tree and Feigenbaum’s theory of period doubling.

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Części książek na temat "Feigenbaum renormalization"

Kuznetsov, S. P. "Generalization of the Feigenbaum-Kadanoff-Shenker Renormalization and Critical Phenomena Associated with the Golden Mean Quasiperiodicity." In Synchronization: Theory and Application. Springer Netherlands, 2003. http://dx.doi.org/10.1007/978-94-010-0217-2_5.

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Moreira, Carlos Gustavo, and Daniel Smania. "Metric stability for random walks (with applications in renormalization theory)." In Frontiers in Complex Dynamics, edited by Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland. Princeton University Press, 2014. http://dx.doi.org/10.23943/princeton/9780691159294.003.0013.

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This chapter considers the stability of metric (measure-theoretic) properties of dynamical systems. A well-known example is that of (C²) expanding maps on the circle; this class is structurally stable, and all such maps have an absolutely continuous and ergodic invariant probability satisfying certain decay of correlations estimates. In particular, in the measure theoretic sense, most of the orbits are dense in the phase space. The chapter uses the idea of random walk, which describes transitions between various dynamical scales, to prove a surprising rigidity result: the conjugacy between two unimodal maps of the same degree with Feigenbaum or wild attractors is absolutely continuous.

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