Fractal Boundaries Research Articles

Fractal subsystem symmetries, anomalies, boundaries, and effective field theory

Fractal subsystem symmetries, anomalies, boundaries, and effective field theory

Intersecting the Twin Dragon with rational lines

The Knuth Twin Dragon is a compact subset of the plane with fractal boundary of Hausdorff dimension s = (\log \lambda)/(\log \sqrt{2}) , \lambda^{3} = \lambda^{2} + 2 . Although the intersection with a generic line has Hausdorff dimension s-1 , we prove that this does not occur for lines with rational parameters. We further describe the intersection of the Twin Dragon with the two diagonals as well as with various axis parallel lines.

On the Katugampola fractional integral and dimensional analysis of the fractal basin boundary for a random dynamical system

In this paper, we mainly investigate the geometric based relationship between the Katugampola fractional calculus and a Weierstrass-type function whose graph can be characterized as a fractal basin boundary for a random dynamical system. Using the potential-theoretic approach with some classical analytical tools, we have derived some kinds of fractal dimensions of the graph of the Katugampola fractional integral of this fractal function. It has been shown that there is a linear relationship between the order of the Katugampola fractional integral and the fractal dimension of the graph of this generalized Weierstrass function. Numerical results have also been provided to corroborate such linear connection.

The fluctuational transition mechanism of non-hyperbolic chaotic invariant sets

In order to reveal the general escape mechanism of non-hyperbolic chaotic invariant sets under both weak noise limit and finite noise intensity, the simplest example of Hénon map, which represents the stretching and folding of the phase space, is taken to study the escape mechanism under two kinds of global bifurcation: the fractal boundary crisis and the attractor contact crisis. In this paper, we revealed the general exit mechanism by analyzing the escape paths derived from the shooting method and Monte Carlo simulation. Finally, to further demonstrate universality, an example of high-dimensional differential dynamical system, namely the transient chaos Duffing oscillator, is examined, which underscores the main idea of this paper analyzing specific deterministic structures not only enhances the comprehensibility of the escape process, but also allows for predictions of general escape behavior under weak noise intensity.

Fractal Numerical Investigation of Mixed Convective Prandtl-Eyring Nanofluid Flow with Space and Temperature-Dependent Heat Source

An explicit computational scheme is proposed for solving fractal time-dependent partial differential equations (PDEs). The scheme is a three-stage scheme constructed using the fractal Taylor series. The fractal time order of the scheme is three. The scheme also ensures stability. The approach is utilized to model the time-varying boundary layer flow of a non-Newtonian fluid over both stationary and oscillating surfaces, taking into account the influence of heat generation that depends on both space and temperature. The continuity equation of the considered incompressible fluid is discretized by first-order backward difference formulas, whereas the dimensionless Navier–Stokes equation, energy, and equation for nanoparticle volume fraction are discretized by the proposed scheme in fractal time. The effect of different parameters involved in the velocity, temperature, and nanoparticle volume fraction are displayed graphically. The velocity profile rises as the parameter I grows. We primarily apply this computational approach to analyze a non-Newtonian fluid’s fractal time-dependent boundary layer flow over flat and oscillatory sheets. Considering spatial and temperature-dependent heat generation is a crucial factor that introduces additional complexity to the analysis. The continuity equation for the incompressible fluid is discretized using first-order backward difference formulas. On the other hand, the dimensionless Navier–Stokes equation, energy equation, and the equation governing nanoparticle volume fraction are discretized using the proposed fractal time-dependent scheme.

Reduction procedure for obtaining solutions of the scalar additive Jump problem and Riemann boundary value problem in vectorial Clifford analysis

In this paper, we study existence of solutions to the scalar additive Jump problem and the Riemann boundary value problems in the context of vectorial Clifford analysis on domains with fractal boundaries. A reduction procedure is applied with great effectiveness to find the solution of the problems.

Morphological analysis for thermodynamics of cavitation collapse near fractal solid wall

A fractal geometric boundary with natural wall features is introduced into a hybrid lattice-Boltzmann-method (LBM) multiphase model. The physical model of cavitation bubble collapse near the irregular geometric wall is established to study the thermodynamic characteristics of the bubble collapse. Due to the lack of periodicity, symmetry, spatial uniformity and obvious correlation in the LBM simulation of the bubble collapse near the fractal wall, the morphological analysis based on Minkowski functional is introduced into the thermodynamic investigation of cavitation bubble so as to analyze and obtain the effective information. The results show that the Minkowski functional method can employed to study the temperature information in complex physical fields hierarchically and quantitatively. The high/low temperature region of the cavitation flow is explored, and thermal effect between irregular and fractal geometric wall and cavitation bubble can be revealed. It illustrates that LBM and morphological analysis complement each other, and morphological analysis can also be used as an optional and potential tool in research field of complex multiphase flows.

Study of fractality nature in VO2 films and its influence on metal-insulator phase transition

The mechanisms underlying the origin of fractal shape of inclusions of a new phase in VO2 films during metal-insulator phase transition are discussed. The obtained results show that hysteresis of the temperature dependence of resistance R(T) significantly depends on the film morphology and texture. Moreover, some fractal features are observed. To determine the fractal dimension D of the structural elements of the studied films from their images, different fractal analysis approaches were preliminary compared and discussed. As a result of the film image treatments, the boundaries of the structural elements were found to have fractal dimensions of 1.3 to 1.5 or higher and to correlate with the shape of R(T). The fractal boundaries indicate the dominant role of elastic stress on the phase transition of films, which is confirmed by numerical modeling. Based on these results, an analytical model is proposed that relates the free energy of a film to the fractal dimension of its constituents. Depending on the ratio of the elastic and interface specific energies, the position of the free energy minimum F corresponds to a certain fractal dimensionality D. A small interface energy leads to a higher fractal dimension making the initial phase more stable. This conclusion explains well all the effects observed experimentally in VO2. The obtained results provide a better understanding of the influence of structure and morphology on other properties of the studied films.

Lagrangian descriptors: The shearless curve and the shearless attractor.

Hamiltonian systems with a nonmonotonic frequency profile are called nontwist. One of the key properties of such systems, depending on adjustable parameters, is the presence of a robust transport barrier in the phase space called the shearless curve, which becomes the equally robust shearless attractor when dissipation is introduced. We consider the standard nontwist map with and without dissipation. We derive analytical expressions for the Lagrangian descriptor (LD) for the unperturbed map and show how they are related to the rotation number profile. We show how the LDs can reconstruct finite segments of the invariant manifolds for the perturbed map. In the conservative case, we demonstrate how the LDs distinguish the chaotic seas from regular structures. The LDs also provide a remarkable tool to identify when the shearless curve is destroyed: we present a fractal boundary, in the parameter space, for the existence or not of the shearless torus. In the dissipative case, we show how the LDs can be used to localize point attractors and the shearless attractor and distinguish their basins of attraction.

Micro- and Nano-Roughness Separation Based on Fractal Analysis.

When describing the tribological behaviour of technical surfaces, the need for full-length scale microtopographic characterization often arises. The self-affine of surfaces and the characterisation of self-affine using a fractal dimension and its implantation into tribological models are commonly used. The goal of our present work was to determine the frequency range of fractal behaviour of surfaces by analysing the microtopographic measurements of an anodised aluminium brake plunger. We also wanted to know if bifractal and multifractal behaviour can be detected in real machine parts. As a result, we developed a new methodology for determining the fractal range boundaries to separate the nano- and micro-roughness. To reach our goals, we used an atomic force microscope (AFM) and a stylus instrument to obtain measurements in a wide frequency range (19 nm-3 mm). Power spectral density (PSD)-based fractal evaluation found that the examined surface could not be characterised by a single fractal dimension. A new method capable of separating nano- and micro-roughness has been developed for investigating multifractal behaviour. The presented procedure separates nano- and micro-roughness based on the geometric characteristics of surfaces. In this way, it becomes possible to specifically examine the relationship between the micro-geometry that can be measured in each wavelength range and the effects of cutting technology and the material structure that creates them.

VARIATIONAL FORMULATIONS FOR A COUPLED FRACTAL–FRACTIONAL KdV SYSTEM

Every shallow-water wave propagates along a fractal boundary, and its mathematical model cannot be precisely represented by integer dimensions. In this study, we investigate a coupled fractal–fractional KdV system moving along an irregular boundary within the framework of variational theory, which is commonly employed to derive governing equations. However, not every fractal–fractional differential equation can be formulated using variational principles. The semi-inverse method proves to be challenging in finding an appropriate variational principle for nonlinear problems and eliminating extraneous components from the studied model. We consider the coupled fractal–fractional KdV system with arbitrary coefficients and establish its variational formulation to unveil the remarkable insights into the energy structure of the model and interrelationships among coefficients. Encouraging results are obtained for this coupled KdV system.

A FRACTAL-FRACTIONAL TSUNAMI MODEL CONSIDERING NEAR-SHORE FRACTAL BOUNDARY

Every fluid problem is greatly affected by its boundary conditions, especially the near-shore seabed could produce an irrevocable harm when a tsunami wave is approaching, and a real-life mathematical model could stave off the worst effect. This paper assumes that the unsmooth seabed is a fractal surface, and fractal-fractional governing equations are established according to physical laws in the fractal space. The geometrical potential theory is used to explain the force produced by the wave surface, and Kong-He friction law is applied to further figuring out the local and memory properties of the friction along the fractal boundary. This paper aims at studying tsunami waves in a fractal space, rendering a reliable mathematical model for both prediction of the tsunami motion and the coastal protection.

Limits to predictability of the asymptotic state of the Atlantic Meridional Overturning Circulation in a conceptual climate model

Anticipating critical transitions in the Earth system is of great societal relevance, yet there may be intrinsic limitations to their predictability. For instance, the asymptotic state of a dynamical system possessing multiple chaotic attractors depends sensitively on the initial condition in the proximity of a fractal basin boundary. Here, we approach the problem of final-state sensitivity of the Atlantic Meridional Overturning Circulation (AMOC) using a conceptual climate model, composed of a slow bistable ocean coupled to a fast chaotic atmosphere. First, we explore the occurrence of long chaotic transients in the monostable regime, which can mask a loss of stability near bifurcations. In the bistable regime, we explicitly construct the chaotic saddle using the edge tracking technique. We quantify the final-state sensitivity through the maximum Lyapunov exponent and the lifetime of the saddle and find that the system exhibits a fractal basin boundary with almost full phase space dimension, implying vanishing predictability of the second kind near the basin boundary. Our results demonstrate the usefulness of studying non-attracting chaotic sets in the context of predicting climatic tipping points, and provide guidance for the interpretation of critical transitions in higher-dimensional climate models.

Fractal basins as a mechanism for the nimble brain

An interesting feature of the brain is its ability to respond to disparate sensory signals from the environment in unique ways depending on the environmental context or current brain state. In dynamical systems, this is an example of multi-stability, the ability to switch between multiple stable states corresponding to specific patterns of brain activity/connectivity. In this article, we describe chimera states, which are patterns consisting of mixed synchrony and incoherence, in a brain-inspired dynamical systems model composed of a network with weak individual interactions and chaotic/periodic local dynamics. We illustrate the mechanism using synthetic time series interacting on a realistic anatomical brain network derived from human diffusion tensor imaging. We introduce the so-called vector pattern state (VPS) as an efficient way of identifying chimera states and mapping basin structures. Clustering similar VPSs for different initial conditions, we show that coexisting attractors of such states reveal intricately “mingled” fractal basin boundaries that are immediately reachable. This could explain the nimble brain’s ability to rapidly switch patterns between coexisting attractors.

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

AbstractWe study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in , , with respect to subordinate killed Brownian motions via ‐stable subordinators and establish the asymptotic behavior of the spectral heat content as for the full range of . Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when , where is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.

Dirichlet problem for electromagnetic fields in chiral media with fractal boundary

The main goal of this work is to formulate and solve the Dirichlet problem for time-harmonic electromagnetic fields in case of chiral media with fractal boundary. To this end, a reformulation of the Maxwell system in quaternionic form is taken as a starting point. The Whitney extension theorem as well as some results from fractal geometry are used.

Coordination Analysis between the Development of Urban Built-Up Areas and Urban Environmental Factors through Remote Sensing of Nighttime Lights: A Case Study in Nanjing, China

The fast development of urban built-up areas in China is causing many problems, such as pollution, congestion, etc. How to effectively evaluate the coordination between urban areas and environmental problems has been attracting many scholars worldwide. This paper intends to discover this “secretary” through investigating the built-up areas and their accompanied economic and environmental factors over almost 30 years (1992 to 2020) in Nanjing, China. DMSP/OLS nighttime lights images from 1992 to 2013 and the NPP/VIIRS nighttime lights images from 2012 to 2022 are used for extraction of built-up areas. A spatiotemporal evolution model is established to evaluate whether the built-up areas have developed in coordination and the relationship between urban built-up areas and various factors, including compactness, the fractal dimension, boundary and shape changes, exhaust emissions, and the production of general industrial solid waste, which was further investigated to ascertain whether there was coordination or not. The investigated results discovered that Nanjing’s built-up areas had maintained continuous growth from 1992 to 2020, with the compactness of built-up areas gradually decreasing from 0.42 to 0.23 and the built-up differentiation dimension changing from 1.31 to 1.39, demonstrating that built-up areas had gradually moved from a loose pattern to a compact pattern and from irregular development to balanced development in all directions. The macro model of the coordination index change trend is 0.847 from 1995 to 2020, which indicates that the coordination between urban built-up areas of development and their environments has been improving; however, the reduction in urban green space, the increase in waste emissions, and the increased production of general industrial solid waste has raised questions regarding sustainable development.

Imaginary scators quadratic mapping in 1+2D dynamic space

The elliptic scator algebra quadratic iteration is evaluated in 1+2 dimensions in dynamic space. There exists a non divergent K set in the scator three dimensional space with a highly complex boundary J=∂K. Two and three dimensional renderings of the sets in dynamic space are published for the first time. The sets exhibit a rich fractal boundary in all three directions. Some of the salient features of the sets can be described in terms of square nilpotent iterations. The Julia and filled in Julia sets are identically reproduced at two perpendicular planes where only one non-vanishing hypercomplex director component is present. The fixed points of K in S1+2 with real constant c, can be obtained from the roots of a scator quadratic polynomial equation. In S1+2 there can be, in addition to the usual complex roots, hypercomplex roots that give rise to four additional fixed points. The inverse orbits of the hypercomplex roots reveal a rich complex structure, that allow for the evaluation of an infinite set of points in ∂K. The ∂K set of the origin is equal to the unit magnitude scator surface, named a cusphere. The J set exhibits self similar structures in 3D at different scales, typical of fractal phenomena. The ix set, is the three dimensional equivalent of the M-set in three dimensions. It is conjectured that the ix-set with some restrictions, is the set of parameters where the J set is connected.

Nonlinear oscillations of a cylindrical shell with a circumferentially discontinuous elastic base

This work investigates the linear and nonlinear vibrations of a functionally graded (FG) transversally excited cylindrical shell and in contact along its length with a circumferentially discontinuous elastic base, a configuration frequently found in practical applications. The elastic medium is modeled as a Winkler foundation. The adopted distribution of the FG material along the thickness is similar to that of a sandwich shell, with one material predominating in the core and the other on both faces. First a finite element analysis (FEA) is performed to obtain the natural frequencies and vibration modes of the FG cylindrical shell and a parametric analysis is conducted to study the influence of the contact region and foundation stiffness on natural frequencies and mode shapes. Even though the problem is a symmetric, the cylindrical shell displays a sequence of symmetric and antisymmetric vibrations modes and these modes may interact due to nonlinear coupling. For much geometry the lowest vibration modes have very close natural frequencies, leading to internal resonances, which influence strongly the nonlinear shell response. To obtain a reliable reduced order model (ROM) for the nonlinear dynamic analysis, Donnell’s nonlinear shallow shell theory is considered. To discretize the nonlinear equations of motion by the Galerkin method, a consistent modal solution is obtained considering nonlinear couplings and interactions. Here, first the displacements are consistently derived from a perturbation procedure proposed in previous papers. To start the perturbation procedure, a seed solution is necessary for the transversal displacement field. For this, first the linear vibration modes obtained from the finite element analysis are expanded in Fourier series in terms of the circumferential coordinate. Then, the essential modes to be used in the ROM are identified using a mathematical procedure that combines Fourier transform and Parseval theorem. The proposed modal solution for the transversal displacements is also used to obtain the modal expansions for the axial and circumferential displacements and their modal amplitudes are determined as a function of the modal amplitudes of the transversal displacements, thus reducing even further the number of unknows. These expansions are finally used to discretize the nonlinear equation of motion in the transversal direction and the resulting equations of motion are solved numerically using continuation software AUTO. A geometry displaying multiple internal resonances is selected and resonance curves, time-responses and phase-portraits are obtained to analyze the nonlinear dynamics of the shell. Depending on the excitation, different bifurcation scenarios are obtained, with important changes in stability and resonant peaks due to modal interaction. Competition between coexisting stable attractors is investigated using basins of attraction, which exhibit a complex topology, often with fractal boundaries.

A Novel Method for the Calculation of Oil–Water Relative Permeability for Tight Oil Reservoirs by Considering Nonlinear Seepage Characteristics

In view of the lack of clear physical significance of the parameters of the traditional nonlinear seepage models and the difficulty of obtaining accurate experimental measurements of the two-phase relative permeability curve, a nonlinear seepage model of a tight reservoir is established on the basis of fractal theory and boundary layer theory. The results show that the proposed model can comprehensively reflect the effects of reservoir matrix physical properties, reservoir fluid physical properties, wettability, and displacement-pressure gradient on the single-phase and two-phase nonlinear seepage characteristics of tight reservoirs. Furthermore, the introduction of the permeability loss factor makes the two-phase relative permeability model more representative of the morphological characteristics of the actual relative permeability curve and avoids the disadvantage that the relative permeability at the end point of the wetting phase has in the traditional model. Finally, by taking the tight core of Changqing Oilfield as an example, a sensitivity analysis of the proposed model is conducted, which proves the practical application of this model. The proposed model provides a convenient theoretical method for the accurate characterization of nonlinear seepage characteristics of tight reservoirs and is of great significance to the numerical simulation, productivity evaluation, and optimization of tight reservoirs.