Persistence of the Feigenbaum Attractor in One-Parameter Families

Consider a map ψ₀ of class C ^r for large r of a manifold of dimension n greater than or equal to 2 having a Feigenbaum attractor. We prove that any such ψ₀ is a point of a local codimension-one manifold of C ^r transformations also exhibiting Feigenbaum attractors. In particular, the attractor persists when perturbing a one-parameter family transversal to that manifold at ψ₀. We also construct such a transversal family for any given ψ₀, and apply this construction to prove a conjecture by J. Palis stating that a map exhibiting a Feigenbaum attractor can be perturbed to obtain homoclinic tangencies. Received: 4 August 1998 / Accepted: 11 May 1999     

On the Hausdorff dimension of fractal attractors

We consider such mappingsx _n+1=F(x_n) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the Hausdorff dimension of the attractor is universally equal toD=0.538 ...     

Labyrinthine pathways towards supercycle attractors in unimodal maps

As 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 hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated with the family of supercycles. iv) This map is given in closed form by the same kind of q-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is final-stage ultra-fast dynamics towards the attractor, with a sensitivity to initial conditions which decreases as an exponential of an exponential of time. We discuss the relevance of these properties to the comprehension of the discrete scale-invariance features, and to the identification of the statistical-mechanical framework present at the period-doubling transition to chaos. This is the first of three studies (the other two are quoted in the text) which together lead to a definite conclusion about the applicability of q-statistics to the dynamics associated to the Feigenbaum attractor.      

Temporal scaling at Feigenbaum points and nonextensive thermodynamics

We show that recent claims for the nonstationary behavior of the logistic map at the Feigenbaum point based on nonextensive thermodynamics are either incorrect or can be easily deduced from well-known properties of the Feigenbaum attractor. In particular, there is no generalized Pesin identity for this system, the existing attempts at proofs being based on misconceptions about basic notions of ergodic theory. In deriving several new scaling laws of the Feigenbaum attractor, thorough use is made of its detailed structure, but there is no obvious connection to nonextensive thermodynamics.     

Intermittency at critical transitions and aging dynamics at the onset of chaos

We recall that at both the intermittency transitions and the Feigenbaum attractor, in unimodal maps of non-linearity of order ζ > 1, the dynamics rigorously obeys the Tsallis statistics. We account for theq-indices and the generalized Lyapunov coefficients λ^q that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the onset of chaos with the appearance of a special value for the entropic indexq. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.     

Numerical study of the oscillatory convergence to the attractor at the edge of chaos

This paper compares three different types of “onset of chaos” in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three window; the transition to chaos in the generalized logistic with inflection 1/2 (x_n+1 = 1-μx_n^1/2), in which the main bifurcation cascade, as well as the bifurcations generated by the periodic windows in the chaotic region, collapse in a single point. The occupation number and the Tsallis entropy are studied. The different regimes of convergence to the attractor, starting from two kinds of far-from-equilibrium initial conditions, are distinguished by the presence or absence of log-log oscillations, by different power-law scalings and by a gap in the saturation levels. We show that the escort distribution implicit in the Tsallis entropy may tune the log-log oscillations or the crossover times.     

Incidence of nonextensive thermodynamics in temporal scaling at Feigenbaum points

Recently, in [Phys. Rev. Lett. 95 (2005) 140601], Grassberger addresses the interesting issue of the applicability of q-statistics to the renowned Feigenbaum attractor. He concludes there is no genuine connection between the dynamics at the critical attractor and the generalized statistics and argues against its usefulness and correctness. Yet, several points are not in line with our current knowledge, nor are his interpretations. We refer here only to the dynamics on the attractor to point out that a correct reading of recent developments invalidates his basic claim.     

Critical Slowing Down in One-Dimensional Maps and Beyond

This is a brief review on critical slowing down near the Feigenbaum period-doubling bifurcation points and its consequences. The slowing down of numerical convergence leads to an “operational” fractal dimension D=2/3 at a finite order bifurcation point. There is a cross-over to D ₀=0.538... when the order goes to infinity, i.e., to the Feigenbaum accumulation point. The problem of whether there exists a “super-scaling” for the dimension spectrum D _q ^W that does not depend on the primitive word W underlying the period-n-tupling sequence seems to remain open     

Bifurcation phenomena and processes in the flow formed in a three-dimensional phase space by a dynamic system with square-law nonlinearity

The basic laws of operation of a dynamic system with square-law nonlinearity and three-dimensional phase space are studied analytically, numerically, and experimentally. Results of analytical investigations of the stability of special points in the system and of numerical and full-scale experiments that indicate the existence of a sequence of bifurcation phenomena described by the Feigenbaum scenario are presented. The existence of two critical values of the control parameter, the first of which characterizes the first Hopf bifurcation and the second describes the destruction of motion at the expense of confluence of the chaotic attractor with the vicinity of the special repeller-type point, is proved. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 78–86, September, 2006.     

Route to and from the NMR chaos in diamagnets

The route to and from the chaos via period doubling bifurcations in nuclear spin system with dipole-dipole interactions is investigated. The transition points are found. It is shown that route from the chaos proceeds according the Feigenbaum scenario. Received 19 August 1998 and Received in final form 15 December 1998     

Ghost-matter mixing and feigenbaum universality in string theory

Branelike vertex operators, defining backgrounds with ghost-matter mixing in Neveu-Schwarz-Ramond superstring theory, play an important role in a world-sheet formulation of D branes and M theory, being creation operators for extended objects in the second quantized formalism. We show that the dilaton beta function in ghost-matter mixing backgrounds becomes stochastic. The renormalization group (RG) equations in ghost-matter mixing backgrounds lead to non-Markovian Fokker-Planck equations whose solutions describe superstrings in curved spacetimes with branelike metrics. We show that the Feigenbaum universality constant δ=4.669 ..., describing transitions from order to chaos in a huge variety of dynamical systems, appears analytically in these RG equations. We find that the appearance of this constant is related to the scaling of relative spacetime curvatures at fixed points of the RG flow. In this picture, the fixed points correspond to the period doubling of Feigenbaum iterational schemes.

     

Role of multistability in the transition to chaotic phase synchronization

In this paper we describe the transition to phase synchronization for systems of coupled nonlinear oscillators that individually follow the Feigenbaum route to chaos. A nested structure of phase synchronized regions of different attractor families is observed. With this structure, the transition to nonsynchronous behavior is determined by the loss of stability for the most stable synchronous mode. It is shown that the appearance of hyperchaos and the transition from lag synchronization to phase synchronization are related to the merging of chaotic attractors from different families. Numerical examples using Rossler systems and model maps are given. (c) 1999 American Institute of Physics.     

Further scaling properties for circle maps

It is shown that certain iterations of (k–1)tuples of commuting invertible circle maps whose rotation numbers are algebraic of degree k, show very similar scaling properties to those found by Feigenbaum et al. in the case k=2.     

超混沌吸引子的翼倍增方案

通过对一个五维超混沌系统施加平移变换、镜像映射和滞回切换操作,将多翼混沌吸引子结构由双翼倍增为四翼.施加n-1次相似操作可以得到2ⁿ翼的超混沌吸引子.设计了一个简单的电路实现吸引子翼数量的倍增.该方法在保留了系统原有超混沌特性的基础上,增加了吸引子的拓扑结构复杂性,使之更适合保密通信等领域的应用. 关键词： 多涡卷吸引子 多翼吸引子 超混沌系统     

Black hole attractors and the entropy function in four‐ and five‐dimensional N = 2 supergravity

In this overview of selected aspects of the black hole attractor mechanism, after introducing the necessary foundations, we examine the relationship between two ways to describe the attractor phenomenon in four‐dimensional N = 2 supergravity: the entropy function and the black hole potential. We also exemplify their practical application to finding solutions to the attractor equations for a conifold prepotential. Next we describe an extension of the original definition of the entropy function to a class of rotating black holes in five‐dimensional N = 2 supergravity based on cubic polynomials, exploiting a connection between four‐ and five‐dimensional black holes. This link allows further the derivation of five‐dimensional first‐order differential flow equations governing the profile of the fields from infinity to the event horizon and construction of non‐supersymmetric interpolating solutions in four dimensions by dimensional reduction. Finally, since four‐dimensional extremal black holes in N = 2 supergravity can be viewed as certain two‐dimensional string compactifications with fluxes, we discuss implications of the conifold example in the context of the entropic principle, which postulates as a probability measure on the space of these string compactifications the exponentiated entropy of the corresponding black holes.     

On the Feigenbaum-Cvitanović equation in the theory of chaotic behaviour

We propose an analytic perturbative approach for the determination of the Feigenbaum-Cvitanović function and the universal parameterα occurring in the Feigenbaum scenario of period doubling for approach to chaotic behaviour. We apply the method to the caseZ=2 whereZ is the order of the unique local maximum of the nonlinear map. Our third order approximation givesα=2.5000 as compared to “exact” numerical valueα=2.5029 ... We also obtain a reasonably accurate value of the Feigenbaum-Cvitanović function.     

On the existence of non‐supersymmetric black hole attractors for two‐parameter Calabi‐Yau's and attractor equations

We look for possible nonsupersymmetric black hole attractor solutions for type II compactification on (the mirror of) CY₃(2,128) expressed as a degree‐12 hypersurface in WCP ⁴[1,1,2,2,6]. In the process, (a) for points away from the conifold locus, we show that the existence of a non‐supersymmetric attractor along with a consistent choice of fluxes and extremum values of the complex structure moduli, could be connected to the existence of an elliptic curve fibered over C ⁸ which may also be “arithmetic” (in some cases, it is possible to interpret the extremization conditions for the black‐hole superpotential as an endomorphism involving complex multiplication of an arithmetic elliptic curve), and (b) for points near the conifold locus, we show that existence of non‐supersymmetric black‐hole attractors corresponds to a version of A₁‐singularity in the space Image( Z ⁶→ R ²/ Z ₂ (↪ R ³)) fibered over the complex structure moduli space. The (derivatives of the) effective black hole potential can be thought of as a real (integer) projection in a suitable coordinate patch of the Veronese map: CP ⁵→ CP ²⁰, fibered over the complex structure moduli space. We also discuss application of Kallosh's attractor equations (which are equivalent to the extremization of the effective black‐hole potential) for nonsupersymmetric attractors and show that (a) for points away from the conifold locus, the attractor equations demand that the attractor solutions be independent of one of the two complex structure moduli, and (b) for points near the conifold locus, the attractor equations imply switching off of one of the six components of the fluxes. Both these features are more obvious using the attractor equations than the extremization of the black hole potential.     

Higher derivative correction to Kaluza–Klein black hole solution

We investigate the attractor mechanism in a Kaluza–Klein black hole solution in the presence of higher derivative terms. In particular, we discuss the attractor behavior of static black holes by using the effective potential approach as well as the entropy function formalism. We consider different higher derivative terms with a general coupling to the moduli field. For the R ² theory, we use an effective potential approach, looking for solutions which are analytic near the horizon and showing that they exist and enjoy attractor behavior. The attractor point is determined by extremization of the modified effective potential at the horizon. We study the effect of the general higher derivative corrections of R ⁿ terms. Using the entropy function we define the modified effective potential and we find the conditions to have the attractor solution. In particular for a single charged Kaluza–Klein black hole solution we show that a higher derivative correction dresses the singularity for an appropriate coupling, and we can find the attractor solution.     

The Maximum Principle and the Global Attractor for the Dissipative 2D Quasi-Geostrophic Equations

The long time behavior of the solutions to the two dimensional dissipative quasi-geostrophic equations is studied. We obtain a new positivity lemma which improves a previous version of A. Cordoba and D. Cordoba [10] and [11]. As an application of the new positivity lemma, we obtain the new maximum principle, i.e. the decay of the solution in L^p for any p [2,+) when f is zero. As a second application of the new positivity lemma, for the sub-critical dissipative case with the existence of the global attractor for the solutions in the space H^s for any s>2(1–) is proved for the case when the time independent f is non-zero. Therefore, the global attractor is infinitely smooth if f is. This significantly improves the previous result of Berselli [2] which proves the existence of an attractor in some weak sense. For the case =1, the global attractor exists in H^s for any s0 and the estimate of the Hausdorff and fractal dimensions of the global attractor is also available.Acknowledgement The author thanks Prof. P. Constantin for encouragement and kind help for his research on the subject of 2D QG equations, Prof. J. Wu for useful conversation and Prof. A. Cordorba for providing preprints. This work was started while the author visited IPAM at UCLA with an IPAM fellowship. The hospitality and support of IPAM is gratefully acknowledged. This work is partially supported by the Oklahoma State University new faculty start-up fund and the Deans Incentive Grant.