Fractal Boundaries Research Articles

Computing eigenvalues of the Laplacian on rough domains

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

Characterization methods for natural fractures distribution in shale and tight reservoirs

Aperture distribution and spatial arrangement are fundamental characteristics of natural fractures. Since it is impossible to directly measure the aperture of all existing fractures, a power law (fractal model) has been commonly used to describe fracture apertures, which has also been extended to estimate fracture spacing and macrofractures density. Although a power law (fractal model) has commonly been used to describe fracture apertures, data points in small and large fracture size ranges tend to deviate from this distribution. In this study, a Variable Shape Distribution (VSD) method was derived from a fractal theory to model fracture size distribution. The VSD method was compared with seven other commonly used models. Eleven fracture datasets collected from five countries on four continents were regressed using these methods. The scale-free methods of the VSD and power law models showed more advantages than the scale-dependent methods. The VSD method was found to best fit all variable aperture size distribution patterns, with an average coefficient of determination of 0.99. The average fractal dimension for all samples calculated by the VSD method (1.36) was slightly higher than that of the power law (1.06). This study is unique in showing that the VSD method can solve a boundary characterization problem that has been widely neglected in previous studies. With the assumption of an even fracture distribution, the VSD method was further applied to estimate average fracture spacing, predict a threshold fracture length that controls the linkage of fractures, and predict fractures outside of the measurement range. This study not only proposes an advanced fractal method to model aperture size distributions in shale and tight rocks but also helps to better understand the importance of fractal dimensions and boundary characterizations.

Reconstruction wedges in AdS/CFT with boundary fractallike structures

In this work, we show the robustness of uberholography and its associated quantum error correcting code against the breakdown of entanglement wedge in the presence of highly entropic mixed states in the bulk. We show that for Cantor-set-like erasure in the boundary in $AdS_3/CFT_2$, the code distance is independent of the mixed-state entropy in the bulk in the $m\rightarrow\infty$ limit. We also show that for a Sierpinski triangle shaped boundary subregion with fractal boundary erasures in $AdS_4/CFT_3$, bulk reconstruction is possible in the presence of highly entropic mixed states in the bulk in the large $m$ regime.

Coexistence of triple, quadruple attractors and Wada basin boundaries in a predator–prey model with additional food for predators

In biological control programs, the provision of additional food is regarded as one of the most useful tools for species conservation and pest management. In a system, providing additional food for predators can distract the predators from the overconsumption of prey (short term) or enhance the predation rate (long term). In the present article, we consider a predator–prey model with Holling type-II functional response by incorporating additional food for the predator species in a discrete-time setup. Here, we intend to explore the impact of additional food on the growth of prey as well as the overall dynamics of the underlying system. We analyze the system dynamics by varying two control parameters, viz, the biomass of additional food and the intrinsic growth rate of prey species simultaneously with the help of Lyapunov exponent and isoperiodic diagrams. Through the simultaneous variation of two parameters, very rich and complex dynamical behaviors of a simple deterministic system can come into view with the presence of structurally stable periodic patterns in the transitional and chaotic zones, which is not looked on only by the variability of one parameter. In this study, we notice the presence of infinite families of different organized periodic structures, like Arnold tongues, shrimp-shaped domains, a new kind of ‘double fishhook’ structures, etc. in the quasiperiodic and chaotic zones of the parameter planes with various kinds of period-adding phenomena. The study also discloses the transition to chaos via shrimp-induced period-bubbling process. We observe the coexistence of double and triple attractors in several different sets of attractors. The most fascinating finding in this study is the coexistence of quadruple attractors, one of the rarest phenomena in ecological systems. The basin boundaries of all these coexisting double, triple, and quadruple attractors are of a very complex nature, they are either fractal basin boundaries or Wada basin boundaries.

Chaotic saddles and interior crises in a dissipative nontwist system.

We consider a dissipative version of the standard nontwist map. Nontwist systems present a robust transport barrier, called the shearless curve, that becomes the shearless attractor when dissipation is introduced. This attractor can be regular or chaotic depending on the control parameters. Chaotic attractors can undergo sudden and qualitative changes as a parameter is varied. These changes are called crises, and at an interior crisis the attractor suddenly expands. Chaotic saddles are nonattracting chaotic sets that play a fundamental role in the dynamics of nonlinear systems; they are responsible for chaotic transients, fractal basin boundaries, and chaotic scattering, and they mediate interior crises. In this work we discuss the creation of chaotic saddles in a dissipative nontwist system and the interior crises they generate. We show how the presence of two saddles increases the transient times and we analyze the phenomenon of crisis induced intermittency.

Charging up boosted black holes

Contrary to a prevailing assumption that black holes would swiftly discharge, we argue that black holes can charge preferentially when boosted through an ambient magnetic field. Though the details are very different, the preference for charge is related to the precipitation of the Wald charge on a spinning black hole in an ambient magnetic field. The gravitoelectrodynamics upstage naive arguments about screening electric fields---in vacuum---in determining the value of the charge accrued. Charged test particles, which build up the black hole charge, exhibit chaotic behavior as evidenced by fractal basin boundaries between dynamical regions. Charged, boosted black holes will generate their own electromagnetic fields and thereby their own luminous signatures, even if they are initially bare. We therefore add boosted black holes to the growing list of potentially observable black hole signatures, alongside black hole batteries and black hole pulsars. The implications should be relevant for supermassive black holes that are boosted relative to a galactic magnetic field as well as black holes merging with magnetized neutron stars.

Unpredictability and basin entropy

The basin entropy is a simple idea that aims to measure the the final state unpredictability of multistable systems. Since 2016, the basin entropy has been widely used in different contexts of physics, from cold atoms to galactic dynamics. Furthermore, it has provided a natural framework to study basins of attraction in nonlinear dynamics and new criteria for the detection of fractal boundaries. In this article, we describe the concept as well as fundamental applications. In addition, we provide our perspective on the future challenges of applying the basin entropy idea to understanding complex systems.

Estimates for integrals of derivatives of $n$-valent functions and geometric properties of domains

A number of questions concerning the behaviour of double integrals of the moduli of the derivatives of bounded $n$-valent functions and, in particular, of rational functions of fixed degree $n$ are considered. For domains with rectifiable boundaries the sharp order of growth of such integral means is found in its dependence on $n$. Upper bounds for domains with fractal boundaries are obtained, which depend on the Minkowski dimension of the boundary of the domain. In certain cases these bounds are shown to be close to sharp ones. Lower bounds in terms of the integral means spectra of conformal mappings are also found. These inequalities refine Dolzhenko's classical results (1966) and some recent results due to the authors. Bibliography: 32 titles.

Advances in Singular and Degenerate PDEs

This review was written with a dual purpose; on the one hand, to collect all the topics dealt with during the conference “Advances in Singular and Degenerate PDEs”. For this reason, in the first part of this work, the abstracts of the lectures during the workshop are shown. On the other hand, as well as the workshop, this work is a way to celebrate the career of Professor Maria Agostina Vivaldi. Professor Vivaldi’s long career and her work are hence highlighted in this work; moreover, some of the more recent results obtained by Professor Vivaldi about problems involving p-Laplace-type operators in fractal and pre-fractal boundary domains are here illustrated and discussed. In conclusion, some new recent outcomes and new perspectives are outlined.

PERIODIC WAVE STRUCTURE OF THE FRACTAL GENERALIZED FOURTH-ORDER BOUSSINESQ EQUATION TRAVELING ALONG THE NON-SMOOTH BOUNDARY

In this study, we present a fractal generalized fourth-order Boussinesq equation which can describe the shallow water waves with the non-smooth boundary (such as the fractal boundary). Aided by the semi-inverse method, we establish its variational principle, which is proved to have a strong minimum condition via the He–Weierstrass theorem. Then, two powerful approaches namely the variational method (VM) and energy balance theory (EBT) are utilized to search for the periodic wave solutions. As expected, the results obtained by the two methods are almost the same. Furthermore, the impact of the fractal orders on the periodic wave structure is illustrated via the 3D plot and 2D curve. The results of this paper are expected to provide a reference for the study of periodic wave theory in fractal space.

FRACTAL BOUNDARY LAYER AND ITS BASIC PROPERTIES

In this paper, the fractal calculus is introduced to study a non-smooth boundary layer of a viscous fluid, and a fractal-fractional modification of the Blasius equation is suggested and solved analytically. The results show that the non-smooth boundary might lead to smaller friction, this can explain well the lotus effect, the waving sand dune and Fangzhu’s water collection. The fractal boundary layer theory has opened the path for a new way to optimal design of a high moving surface with the minimal friction.

Nonautonomous semilinear Wentzell problems in fractal domains

We study a nonautonomous semilinear parabolic problem with a dynamical boundary condition in an irregular domain with fractal boundary. Local existence, uniqueness and regularity results for the mild solution are established via a family of evolution operators. A sufficient condition on the initial datum for global existence is given.

A metasurface enabled dual band antenna exhibiting dual polarizations with reduced beam squint and cross polarization

A compact, dual-band metasurface-based microstrip patch antenna with both linear and circular polarization (CP) for multiband communication is reported in this work. The ingredients in the antenna design consist of a combination of metasurface and metamaterial inspired structures that are demonstrated as an effective technique to correct the beam squint problem and reduce the cross-polarized radiation, usually experienced by thick substrate-based broadband CP antennas. A fractal boundary metasurface is introduced to achieve CP over a small frequency band. Eight Electric-LC resonators are symmetrically placed in the same plane of the radiating patch to widen the CP bandwidth, correct the beam squint, reduce the cross-polarized radiation, and realize a subwavelength resonance of λ0/5.15 order. The measured impedance bandwidths (|S11| < −10 dB) of the proposed antenna are 7.8% (3.7–4.0 GHz) and 27.58% (5.05–6.65 GHz) in the lower and upper-frequency band, respectively. The measured 3-dB axial ratio bandwidth (ARBW) is 24.8% (5070–6510 MHz) at 5.8 GHz. The fabricated prototype has an overall size of 55 × 55 × 4.08 mm (0.696λ0 × 0.696λ0 × 0.051λ0 at 3.8 GHz or 1.063λ0 × 1.063λ0 × 0.078λ0 at 5.8 GHz) with a realized peak gain of 4 dBi in the lower band and 7.1 dBic in the upper band. The measured results are consistent with the simulation results.

Design of C band Bandpass Filter using Fractal based Symmetrical Ring Resonator

Symmetrical ring resonator metamaterial along with fractal boundary is proposed for Band Pass Filter (BPF) design in this paper. A combination of symmetrical ring resonators and vias is used for designing a bandpass filter. Bandpass filter with low insertion loss, better fractional bandwidth even at higher frequencies is achieved by using moore fractal applied symmetrical ring metamaterial resonators along the microstrip transmission line. The operating frequency range of the simulated filter is in the C-band between 5.47 GHz - 6 GHz having fractional bandwidth (FBW) of 9.25% and with a minimum insertion loss of 1.2 dB. Application of moore fractal to the above implementation improved the bandwidth of the filter. Fractal applied symmetrical ring resonator simulated filter operates in the C-band between 7.15 GHz - 8.15 GHz having FBW of 13%, with a minimum insertion loss of 1 dB. The proposed filter is simulated, fabricated and S-parameters are measured using network analyzer N5222A. S-parameters results of fractal applied symmetrical ring resonator filter realized from simulations match closely with those from measurements results performed on prototypes but with a small shift in a frequency range. The measured filter operates in 6.95 GHz - 7.8 GHz having FBW of 11.5%, with a minimum insertion loss of 0.4 dB.

Numerical treatment of fractal boundary value problems for heat conduction in polar bear with spatial variation of thermal conductivity

A high-resolution compact discretization scheme for the numerical approximation of two-point nonlinear fractal boundary value problems is presented to study the stationary anomalous diffusion process. Hausdorff derivative is applied to derive the models in fractal media. The proposed scheme solves the nonlinear fractal model and achieves an accuracy of order four by employing only three mesh points in a stencil and consumes short computing time. Numerical simulations with heat conduction in polar bear, convection–diffusion, boundary layer, Bessel’s and Burgers equation in a fractal medium are carried out to illustrate the utility of the scheme and their numerical rate of convergence.

Dynamics in the biparametric spaces of a three-species food chain model with vigilance

In nature, vigilance is a common anti-predator strategy prey individuals employ to protect themselves from a possible attack. It gives them sufficient time to take cover or flee. Yet this seemingly profitable strategy, which lowers the number of successful predatory attacks, has a significant impact on the prey's growth rate. Researchers attribute this to reduced foraging time. In this article, we study a three-species food chain model in which both the basal prey and middle predator show active vigilance against their respective predators. Apart from the positivity, boundedness, and local stability analysis of the equilibrium points of the model, we find the conditions for global stability and the occurrence of Hopf bifurcation around the coexisting equilibrium point. Bifurcation diagrams and their corresponding Lyapunov exponent diagrams (largest and second-largest) illustrate diverse dynamical scenarios ranging from stable coexistence to periodic to chaotic oscillations. The isospike diagrams and their associated largest Lyapunov exponent diagrams in the biparametric space of basal prey's and middle predator's vigilance reveal a broader picture of the effects of different levels of vigilance on the dynamics of the system. A great deal of attention has been paid to unwrapping the complex structural beauties like shrimp-shaped periodic structures inside the chaotic sea and multistability between several sets of attractors in the overlapping zones of periodic windows. We observe that multiple coexisting attractors possess fractal basin boundaries. We also explore the dynamics of the system in two other biparametric spaces. The density variation maps at the end show the effect of vigilance on the equilibrium density of all three species. Our results suggest that vigilance promotes regularity and coexistence, but too high levels of vigilance can cause species extinction. For better visualization of the system dynamics, we provide a few high-resolution animations of the phase portraits and the basins of attraction in the supplementary material.

Noise-induced behavioral change driven by transient chaos

We study behavioral change in the context of a stochastic, non-linear consumption model with preference adjusting, interdependent agents. Changes in long-run consumption behavior are modelled as noise induced transitions between coexisting attractors. A particular case of multistability is considered: two fixed points, whose immediate basins have smooth boundaries, coexist with a periodic attractor, with a fractal immediate basin boundary. If a trajectory leaves an immediate basin, it enters a set of complexly intertwined basins for which final state uncertainty prevails. The standard approach to predicting transition events rooted in the stochastic sensitivity function technique due to Mil'shtein and Ryashko (1995) does not apply since the required exponentially stable attractor, for which a confidence region could be constructed, does not exist. To solve the prediction problem we propose a heuristic based on the idea that a vague manifestation of a non-attracting chaotic set (chaotic repellor) - could serve as a surrogate for an attractor. A representation of the surrogate is generated via an algorithm for generating the boundary of an absorbing area due to Mira et al. (1996). Then a confidence domain for the surrogate is generated using the approach due to Bashkirtseva and Ryashko (2019). The intersections between this confidence region and the immediate basins of the coexisting attractors can then be used to make predictions about transition events. Preliminary assessments show that the heuristic indeed explains the transition probabilities observed in numerical experiments.

The occurrence of riddled basins and blowout bifurcations in a parametric nonlinear system

In this paper, a two parameters family Fβ1,β2 of maps of the plane having two different invariant subspaces is studied. We observe that, our model exhibits two chaotic attractors Ai, i=0,1, lying in these invariant subspaces and identify the parameters at which Ai has a locally riddled basin of attraction or becomes a chaotic saddle. Then, the occurrence of riddled basin in the global sense is investigated in an open region of β1β2-plane. We semi-conjugate our system to a random walk model and define a fractal boundary which separates the basins of attraction of the two chaotic attractors, then we describe riddled basin in detail. We show that the model undergoes a sequence of bifurcations: “a blowout bifurcation”, “a bifurcation to normal repulsion” and “a bifurcation to emergence a new chaotic attractor with an intermingled basin”. Numerical simulations are presented graphically to confirm the validity of our results.

Coriolis coupling in a Hénon–Heiles system

We study the impact of a Coriolis term in the dynamics of a perturbed Hénon–Heiles Hamiltonian. The strength of the Coriolis coupling is measured by a frequency ω that regulates two different regimes. If (in scaled units) ω1, the minimum becomes a dynamically stable maximum, and there appears an open region of stable motion around that maximum. We restrict our study to ω∈[0,1) because we find richer dynamics in that interval. Poincaré surfaces of section are used to show how the strength of the Coriolis coupling controls the size of the trapping area. While, for ω=0, most of the orbits escape, for ω≈1 most of the orbits remain trapped. The transition from one situation to the other one reveals complex resonant structures giving rise to a chaotic sticky region of long living orbits. The study of the escape basins reveals that they evolve from a complex structure, with fractal boundaries, to basins with smooth boundaries. Explicit computation of the evolution of the basin entropy confirms this fact. The escape probability as a function of ω is also calculated. Both the evolution of the escape probability and the entropy are not monotonic and exhibit intricate and complex dynamics for intermediate values of ω.

Mixed boundary valued problems for linear and nonlinear wave equations in domains with fractal boundaries

The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in a large natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in \(\mathbb {R}^2\) or \(\mathbb {R}^3\) we also validate the approximation of the solution of the Westervelt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.