Max-Product Weierstrass Type Functions

A 3-D invertible map generating a 4-loop solenoid attractor is presented. The attractor of the "standard" 4-solenoid turns out to be "fat" - that is, fractal dimensionality exceeds topological dimensionality by more than unity. If a fourth variable that is weakly damped is added to this 3-variable map, a new type of attractor is generated. In a cross section it is a continuous nowhere-differentiable function over a 2-D domain which is a 2-D Cantor set. Such an object can be called a "singular-continuous 2-D Weierstrass function". The latter can be "superfat" (dimension gap larger than 2). Realistic systems possessing attractors of this new type are possible.

Fractal and Hausdorff dimensions for systems of iterative functional equations

Fractals can model many classes of time-series data. The fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales. The covering methods are a class of efficient approaches to measure the fractal dimension of an arbitrary fractal signal by creating multiscale covers around the signal’s graph. In this paper we develop a general method that uses multiscale morphological operations with varying structuring elements to unify and extend the theory and digital implementations of covering methods. It is theoretically established that, for the fractal dimension computation, covering one-dimensional signals with planar sets is equivalent to morphologically transforming the signal by one-dimensional functions, which reduces the computational complexity from quadratic in the signal’s length to linear. Then a morphological covering algorithm is developed and applied to discrete-time signals synthesized from Weierstrass functions, fractal interpolation functions, and fractional Brownian motion. Further, for deterministic parametric fractals depending on a single parameter related to their dimension, we develop an optimization method that starts from an initial estimate and iteratively con- verges to the true fractal dimension by searching in the parameter space and minimizing a distance between the original signal and all such signals from the same class. Experimental results are also provided to demonstrate the good performance of the developed methods.

Wind speed modelling using Weierstrass function fitted by a genetic algorithm

The fundamental relationship between self‐similar, that is, fractal, arrays and their ability to generate radiation patterns which possess fractal features is examined in this paper. The theoretical foundation and design procedures are developed for using fractal arrays to synthesize fractal radiation patterns having certain desired characteristics. A family of functions, known as generalized Weierstrass functions, are shown to play a pivotal role in the theory of fractal radiation pattern synthesis. These functions are everywhere continuous but nowhere differentiable and exhibit fractal behavior at all scales. It will be demonstrated that the array factor for a nonuniformly but symmetrically spaced linear array can be expressed in terms of a Weierstrass partial sum (band‐limited Weierstrass function) for an appropriate choice of array element spacings and excitations. The resulting fractal radiation patterns from these arrays possess structure over a finite range of scales. This range of scales can be controlled by the number of elements in the array. For a fixed array geometry, the fractal dimension of the radiation pattern may be varied by changing the array current distribution. A general and highly flexible synthesis technique is introduced which is based on the theory of Fourier‐Weierstrass expansions. One of the appealing attributes of this synthesis technique is that it provides the freedom to select an appropriate generating function, in addition to the dimension, for a desired fractal radiation pattern. It is shown that this synthesis procedure results in fractal arrays which are composed of a sequence of self‐similar uniformly spaced linear subarrays. Finally, a synthesis technique for application to continuous line sources is presented which also makes use of Fourier‐Weierstrass expansions.

Complexity modeling and analysis of chaos and other fluctuating phenomena

Determination of fractal dimension and prefactor of agglomerates with irregular structure

Three-dimensional flows are addressed in porous media with resurgences. These media are characterized by a double structure, i.e., a continuous porous medium and capillaries with impermeable walls which relate distant points of the continuous medium. Spatially periodic harmonic functions with singularities at given points are constructed by means of the Berdichevskij functions; these functions extend to three dimensions the classical Weierstrass functions. The solution of the same medium with two vertices is obtained in two and three dimensions and thoroughly compared and discussed.

Particle growth by Brownian coagulation at high concentration in the continuum regime is investigated by solving the Langevin dynamics (LD) equations for each particle trajectory of polydisperse suspensions. By monitoring the LD attainment of the self-preserving size distribution (SPSD), it is shown that the classic Smoluchowski collision frequency function is accurate for dilute particle volume fractions, phis, below 0.1%. At higher phis, coagulation is about 4 and 10 times faster than for the classic theory at phis = 10 and 20%, respectively. For complete particle coalescence upon collision, SPSDs develop even in highly concentrated suspensions (up to phis = 35%), as with dilute ones, but are broadened with increasing phis. At high particle concentration, an overall coagulation rate is proposed that reduces to the classic one at low concentration. Detailed collision frequency functions are also obtained at various phis values. Fractal-like agglomerates undergoing coagulation at constant fractal dimension attain an SPSD only temporarily because their effective volume fraction continuously increases, approaching gelation in the absence of restructuring or fragmentation.

Starting in the last two decades of 20th Century, fractal geometry and underlying analysis became a part of material sciences. From ore distribution in Earth layers to nanotechnology, from biomedical to energy applications, fractal descriptors of physicochemical properties of matter are inevitable part of scientific research and presentations. Fractal and spectral dimensions replace the usual (geometric) dimension, multifractal distributions appear instead convention statistical ones, Weierstrass functions replace standard algebraic or trigonometric polynomials. In the fractal frame of mind, description, construction or reconstruction of fairly complicated and uneven objects such as ceramic grains, ferromagnetic domains or pores labyrinth are as easy as working with ideal Euclidean objects like circles and squares. The aim of this work is to point out the most important issues where powder materials technology, like mass density, particle density, electric charge density, energy density etc., intersects with the notion of fractal dimension and power law equations and other fractal constructions and corresponding applications.

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect gaps in essential spectra and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these long-standing problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the solvability complexity index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale’s comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications.

Let [Formula: see text] be a [Formula: see text] expanding map of the circle and let [Formula: see text] be a [Formula: see text] function. Consider the twisted cohomological equation [Formula: see text] which has a unique bounded solution [Formula: see text]. We show that [Formula: see text] is either [Formula: see text] or continuous but nowhere differentiable. If [Formula: see text] is nowhere differentiable then the Newton quotients of [Formula: see text], after an appropriated normalization, converges in distribution (with respect to the unique absolutely continuous invariant probability of [Formula: see text]) to the normal distribution. In particular, [Formula: see text] is not a Lipschitz continuous function on any subset with positive Lebesgue measure.

Fractal basin boundary constructed by superposition of the Weierstrass function

This is a revision (and partial retraction) of my previous abstarct. Let $\lambda(X)$ denote Lebesgue measure. If $X\subseteq [0,1]$ and $r \in (0,1)$ then the $r$-Hausdorff capacity of $X$ is denoted by $H^r(X)$ and is defined to be the infimum of all $\sum_{i=0}^\infty \lambda(I_i)^r$ where $\{I_i\}_{i\in\omega}$ is a cover of $X$ by intervals. The $r$ Hausdorff capacity has the same null sets as the $r$-Hausdorff measure which is familiar from the theory of fractal dimension. It is shown that, given $r < 1$, it is possible to enlarge a model of set theory, $V$, by a generic extension $V[G]$ so that the reals of $V$ have Lebesgue measure zero but still have positive $r$-Hausdorff capacity.

This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sections contain some very general results in nonuniform hyperbolic theory. We consider (f, μ), where f is an arbitrary dynamical system and μ is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical quantities of interest, including entropy, fractal dimension and rates of escape. The second half of this review focuses on observable chaos, characterized by positive Lyapunov exponents on positive Lebesgue measure sets. Much attention is given to SRB measures, a very special kind of invariant measures that offer a way to understand observable chaos in dissipative systems. Paradoxical as it may seem, given a concrete system, it is generally impossible to determine with mathematical certainty if it has observable chaos unless strong geometric conditions are satisfied; case studies will be discussed. The final section is on noisy or stochastically perturbed systems, for which we present a dynamical picture simpler than that for purely deterministic systems. In this short review, we have elected to limit ourselves to finite-dimensional systems and to discrete time. The phase space, which is assumed to be or a Riemannian manifold, is denoted by M throughout. The Lebesgue or the Riemannian measure on M is denoted by , and the dynamics are generated by iterating a self-map of M, written f: M↺. For flows, the reviewed results are applicable to time-t maps and Poincaré return maps to cross-sections.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.