Feigenbaum Map Research Articles

Limit drift for complex Feigenbaum mappings

AbstractWe 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 $\infty $ . We also prove the convergence of the drifts to a finite limit, which can be expressed purely in terms of the limiting tower, which corresponds to a Feigenbaum map with a flat critical point.

A Novel Image-Encryption Scheme Based on a Non-Linear Cross-Coupled Hyperchaotic System with the Dynamic Correlation of Plaintext Pixels.

Based on a logistic map and Feigenbaum map, we proposed a logistic Feigenbaum non-linear cross-coupled hyperchaotic map (LF-NCHM) model. Experimental verification showed that the system is a hyperchaotic system. Compared with the existing cross-coupled mapping, LF-NCHM demonstrated a wider hyperchaotic range, better ergodicity and richer dynamic behavior. A hyperchaotic sequence with the same number of image pixels was generated by LF-NCHM, and a novel image-encryption algorithm with permutation that is dynamically related to plaintext pixels was proposed. In the scrambling stage, the position of the first scrambled pixel was related to the sum of the plaintext pixel values, and the positions of the remaining scrambled pixels were related to the pixel values after the previous scrambling. The scrambling operation also had a certain diffusion effect. In the diffusion phase, using the same chaotic sequence as in the scrambling stage increased the usage rate of the hyperchaotic sequence and improved the calculation efficiency of the algorithm. A large number of experimental simulations and cryptanalyses were performed, and the results proved that the algorithm had outstanding security and extremely high encryption efficiency. In addition, LF-NCHM could effectively resist statistical analysis attacks, differential attacks and chosen-plaintext attacks.

Oscillating sequences, MMA and MMLS flows and Sarnak’s conjecture

We define anoscillating sequence, an important example of which is generated by the Möbius function in number theory. We also define aminimally mean attractable (MMA) flowand aminimally mean-L-stable (MMLS) flow. One of the main results is that any oscillating sequence is linearly disjoint from all MMA and MMLS flows. In particular, this confirms Sarnak’s conjecture for all MMA and MMLS flows. We provide several examples of flows that are MMA and MMLS. These examples include flows defined by all$p$-adic polynomials of integral coefficients, all$p$-adic rational maps with good reduction, all automorphisms of the$2$-torus with zero topological entropy, all diagonalizable affine maps of the$2$-torus with zero topological entropy, all Feigenbaum maps, and all orientation-preserving circle homeomorphisms.

Poly-time computability of the Feigenbaum Julia set

We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.

On the attractors of Feigenbaum maps

A solution of the Feigenbaum functional equation is called a Feigenbaum map. We investigate the likely limit set (i.e. the maximal attractor in the sense of Milnor) of a non-unimodal Feigenbaum map, prove that it is a minimal set that attracts almost all

On the Attractor of the Product System

We investigate the attractor of the product system and mainly prove that the likely limit set of the product system equals the product of one of each factor system for n compact systems with solenoid attractors. Specially, this holds for the product map of Feigenbaum maps. Furthermore, we deduce that the Hausdorff dimension of the likely limit set of the product map for Feigenbaum maps is the sum of one of each factor map.

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let Tg: [−1, 1] → [−1, 1] be the Feigenbaum map. It is well known that Tg has a Cantor-type attractor F and a unique invariant measure µ0 supported on F. The corresponding unitary operator (Ugφ)(x) = φ(g(x)) has pure point spectrum consisting of eigenvalues λn,r, n ≥ 1, 0 ≤ r ≤ 2n−1 − 1 with eigenfunctions er(n)(x). Suppose that f ∈ C1([−1, 1]), f′ is absolutely continuous on [−1, 1] and f″ ∈ Lp([−1, 1], dµ0), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for Tg we prove that Sn(f) ∼ 2−nqn, as n → ∞, where the constant q ∈ (0, 1) does not depend on f.

Measure of the Julia set of the Feigenbaum map with infinite criticality

AbstractWe consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

A NOTE ON THE UNIMODAL FEIGENBAUM'S MAPS

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 map but not periodic ones.

Multifractality of the Feigenbaum Attractor and Fractional Derivatives

It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities $f(\alpha)$. This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions (Frisch and Matsumoto (J. Stat. Phys. 108:1181, 2002)). The relation between the thermodynamic approach (Vul, Sinai and Khanin (Russian Math. Surveys 39:1, 1984)) and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations.

A Markov partition for the Feigenbaum dynamics

We build a Markov partition of the Feigenbaum dynamical plane, and a system of ‘external rays’, which are invariant under rescaling. Applications to geometry of the Feigenbaum map are given.

Wreath products along the period-doubling route to chaos

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.

Homoclinic tangencies near cascades of period doubling bifurcations

We consider perturbations of the Feigenbaum map in n dimensions. In the analytic topology we prove that the maps that are accumulated by period doubling bifurcations are approximable with homoclinic tangencies. We also develop a n-dimensional Feigenbaum theory in the Cr topology, for r large enough. We apply this theory to extend the result of approximation with homoclinic tangencies for Cr maps.

Symbolic dynamics, entropy and complexity of the feigenbaum map at the accumulation point

This paper aims to make further contributions to the exploration of the symbolic dynamics generated by the logistic map at Feigenbaum accumulation point. In particular we are interested in the grammar of these sequences; completing earlier studies we study here arbitrary partitions also. Our main aim is the investigation of the special grammars which characterize the long‐range correlations between letters. Considering these sequences as standard examples of a complex system, we introduce and discuss a complexity function derived from the conditional entropies. Further we discuss local predictabilities.

When Scott is weak on the top

We construct an approximating chain of simple valuations on the upper space of a compact metric space whose lub is a given probability measure on the metric space. We show that whenever a separable metric space is homeomorphic to a Gδ subset of an ω-continuous dcpo equipped with its Scott topology, the space of probability measures of the metric space equipped with the weak topology is homeomorphic with a subset of the maximal elements of the probabilistic power domain of the ω-continuous dcpo. Given an effective approximation of a probability measure by an increasing chain of normalised valuations on the upper space of a compact metric space, we show that the expected value of any Hölder continuous function on the space can be obtained up to any given accuracy. We present a novel application in computing integrals in dynamical systems. We obtain an algorithm to compute the expected value of any Hölder continuous function with respect to the unique invariant measure of the Feigenbaum map in the periodic doubling route to chaos.

Period-doubling bifurcations in the presence of colored noise.

We study the effects of colored noise on period-doubling bifurcations. Using the Feigenbaum map as a model, the technique of cumulant equations is applied to analyze the bifurcation behavior. We find that the universal properties of the period-doubling sequences are preserved in the case of colored noise. Moreover, the resonancelike response of the period-doubling cascade to the colored noise forcing is observed, while the noise correlation time is varied. This response is reflected in both power spectra and bifurcation diagrams.

Random perturbations of the Feigenbaum map

In this paper we study some geometric properties of the invariant measure mu ' of the in randomly perturbed Feigenbaum map. We give a rigorous analysis of how the support of mu ' changes as in varies.

The Fibonacci Unimodal Map

This paper will study topological, geometrical and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan. It turns out that the situation can be understood completely and is of quite regular nature. In particular, any Fibonacci map (with negative Schwarzian and non-degenerate critical point) has an absolutely continuous invariant measure (so, we deal with a ``regular'' type of chaotic dynamics). It turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the Feigenbaum map: its Hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter.

The Fibonacci unimodal map

The recurrence of the critical orbit appeared in the work of Branner and Hubbard on complex cubic polynomials [BH, §12] and in Yoccoz's work [Yl, Y2] on quadratic ones, as the worst pattern of recurrence. On the other hand, a real quadratic map was suggested by Hofbauer and Keller [HK] as a possible candidate for a map having a attractor (that is, a set A which is the w-limit set for Lebesgue almost every orbit but is strictly smaller than the w-limit set for a generic orbit). The w-limit set of the critical orbit in [HK] possesses all known topological properties of wild attractors (compare [BL2]). In fact, we will see below that the quadratic map does not have a wild attractor; however, the corresponding question for a map with a degenerate critical point remains open. Actually, the first indication of the map appeared in the numerical work of Tsuda [T], related to the Belousova-Zhabotinskii reaction, and also in numerical work of Shibayama [Sh] (more precisely, they studied the sequence of Fibonacci bifurcations creating the map). This paper will study topological, geometrical, and measure-theoretical properties of the real map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan [S]. It turns out that the situation can be understood completely and is of quite nature. In particular, any map (with negative Schwarzian and nondegenerate critical point) has an absolutely continuous invariant measure (so, we deal with a regular type of chaotic dynamics). It turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the Feigenbaum map: its Hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter. Branner and Hubbard introduce the concept of a in order to describe recurrence of critical orbits. Their Fibonacci tableau is a basic example, which corresponds to one particularly close and pattern of recurrence. If a complex quadratic map z 1-+ z2 + c realizes this tableau, then the

Computing S(α) using symbolic Monte Carlo

We describe a method for carrying out a computation of the entropy function of the invariant set of a particular class of maps. We illustrate the method for the case of the quadratic and the Feigenbaum map. Elementary consequences of the symbolic dynamics associated with a map f lead to an algorithm that is a novel application of the Monte Carlo method. To sample, we use the fact that the salient features of the thermodynamic properties of the map are captured by its symbolic dynamics. In the symbolic domain, samples are chosen by manipulating character strings, rather than by performing floating-point calculations. Therefore, exact symbolic answers can be obtained by purely combinatorial methods. These symbolic answers then provide a recipe for performing high-precision numerical computations.