Fractal Boundaries Research Articles

Design and analysis of m‐segment fractal boundary antennas

AbstractThis article investigates the resonant behavior of a novel family of fractal boundary antennas at the fundamental mode of operation. The miniaturization patterns over iterations as well as over the number of segments on the boundary have been studied by simulating the fractal antennas in High Frequency Structure Simulator (HFSS). The antennas are fed by a 50‐Ω coaxial probe, which is placed at the best position on the patch, with impedance matching and S11 < −10 dB. Analysis of the resonant frequency with respect to the square‐shaped fractal generator resulted in curve‐fit expressions that vary with a single variable of either iteration or number of segments on the boundary. The derived equations are independent from the substrate thickness. They are useful to design miniature patch antennas within a specified area in order to resonate at a desired frequency by simply changing the boundary or the fractal iteration.

Miniaturised microstrip patch design based on highly capacitive defected ground structure with fractal boundary for X‐band microwave communications

This article proposes a new class of miniaturised microstrip patch design for wireless communications at 10 GHz. Antenna miniaturisation is achieved here by loading a highly capacitive modified Minkowski fractal (type-2) defected ground structure (MFDGS-II) exactly beneath the center of the radiating patch. The proposed methodology involves sensitivity analysis to select best DGS configuration. The resonant frequency corresponding to the patch is reduced from 16.832 GHz to 10 GHz incorporating MFDGS-II without any change in the physical size of the antenna. This enables a patch size reduction of as high as 68% and an overall volumetric reduction of 84% along with an improvement in bandwidth and efficiency of the antenna. A prototype of the proposed antenna is fabricated, and its performance parameters are measured. An equivalent transmission-line model is presented to explain the theoretical background behind the shift in the resonant frequency of the antenna. Our antenna prototype with a lower patch size of 0.20 λ 0 × 0.15 λ 0 , exhibits an impedance bandwidth of 270 MHz, a realised gain of 3.2 dBi with radiation efficiency of 98% centered at 10 GHz. The proposed compact antenna has the potential to meet the practical requirements for portable X-band wireless sensor applications.

A Bimonogenic Cauchy Transform on Higher Order Lipschitz Classes

We prove that a Cauchy transform naturally arising for bimonogenic Clifford algebra valued functions theory behaves invariant on the higher order Lipschitz classes and obtain a Plemelj–Privalov theorem for the related singular integral transform. Moreover, we obtain an upper bound for the norm of such an operator in the pathological case of fractal boundaries.

Fine regularity for elliptic and parabolic anisotropic Robin problems with variable exponents

We investigate a class of quasi-linear elliptic and parabolic anisotropic problems with variable exponents over a general class of bounded non-smooth domains, which may include non-Lipschitz domains, such as domains with fractal boundary and rough domains. We obtain solvability and global regularity results for both the elliptic and parabolic Robin problem. Some a priori estimates, as well as fine properties for the corresponding nonlinear semigroups, are established. As a consequence, we generalize the global regularity theory for the Robin problem over non-smooth domains by extending it for the first time to the variable exponent case, and furthermore, to the anisotropic variable exponent case.

Global analysis of crises in shape memory thin plate system

The unique global properties of shape memory alloy are mainly derived from the martensite phase transition and its inverse, which result from the change of temperature and external load. In this paper, the global characteristics of shape memory alloy thin plate system are analyzed with the temperature and harmonic excitation amplitude as control parameters. Based on the method of Poincare map, the complex crisis phenomenon of the system including the sudden change in number, size and type of attractors can be observed through the global multivalued bifurcation diagram. However, the specific crisis type is not clear, it is necessary to be analyzed from the global viewpoint. By computing the global diagram with the composite cell coordinate system method which constructs a composite cell state space by multistage division of the continuous phase space, the attractors, saddles and basins of attraction of the system can be obtained more accurately. The vivid evolutionary processes of the crisis phenomena of the system are illustrated, and it can be found that the system presents a complex global structure with amplitude and temperature changing. There exist two kinds of crises: one is the boundary crisis resulting from the collision between a chaotic/periodic attractor and a chaotic saddle within the basin boundary, which causes the attractor to vanish, and the other is the merging crisis caused by the collision of two or more attractors with the chaotic saddle within the basin boundary where a new chaotic attractor appears. When multiple attractors coexist in the system, the basin boundary may be smooth or fractal, and for any point at boundary, its small open neighborhood always has a nonempty intersection with three or more basins, which is known as Wada basin boundary. It is difficult to predict the dynamic behavior of the system accurately due to the fractal, the Wada-Wada, Wada-fractal and fractal-Wada basin boundary metamorphoses which can be observed along with the variation of temperature and amplitude through the composite cell coordinate system method, which owns a unique advantage in depicting basin boundary. Furthermore, the Wada property is displayed more clearly by refining specified region. The results of this paper provide a theoretical analysis tool for adjusting the dynamic response of shape memory alloy thin plate system and optimizing the deformation and vibration control of mechanical equipment through controlling temperature and excitation intensity.

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

Influence of transient escape and added load noise on the dynamic integrity of multistable systems

Concepts and tools of global dynamics play a meaningful role in enhancing the engineering design and safety of multistable structural systems in a dynamic environment. Basins of attraction topology can vary meaningfully as a function of system parameters, and the basin boundaries can be smooth or fractal. Fractal boundaries have important practical consequences as they lead to uncertainty with respect to the outcome of a given initial condition, thus reducing the system safety, which can be quantified through different dynamic integrity measures and their variation with a control parameter. In the present paper two key issues influencing the integrity measures and basins’ erosion profiles are addressed: transient escape and added random noise. Transient escape is a significant issue in many structural systems since it may lead to unwanted stresses and strains or exceed some safe design requirements in terms of maximum displacements and velocities. In turn, the random noise may increase the uncertainty of the systems, thus decreasing their practical safety and escape load. As a case study, the bistable von Mises truss is considered and the effects of transient escape and added random noise on the global dynamics of the harmonically excited truss are investigated in detail, clarifying their influence on the system safety as evaluated with different integrity measures.

Boundaries of synchronization in oscillator networks

We analyze the final state sensitivity of nonlocal networks with respect to initial conditions of their units. By changing the initial conditions of a single network unit, we perturb an initially synchronized state. Depending on the perturbation strength, we observe the existence of two possible network long-term states: (i) The network neutralizes the perturbation effects and returns to its synchronized configuration. (ii) The perturbation leads the network to an alternative desynchronized state. By computing uncertainty exponents of a two-dimensional cross section of the state space, we find the existence of fractal basin boundaries separating synchronized solutions from desynchronized ones. We attribute these features to an unstable chaotic set in which trajectories persist for times indefinitely long in the network.

Experimental behaviour of an acoustic wave in a medium comprising self-similar (fractal) objects

After the pioneering work of one of the authors (Sapoval et al. J. Acoust. Soc. Am. 104, 1997), the study of acoustic propagation over fractal boundaries came up against the difficulty of experimenting with self-similar objects immersed in two- and three-dimensional space. The idea of one of us to use the capabilities offered by three-dimensional printing has made it possible to overcome this difficulty and produce numerous objects, as small or large as desired, in large numbers. We will present the experimental results obtained for acoustic propagation, reflection coefficient, attenuation and trapping in the case of an half fractal octahedron, isolated as well as in the case of a flat paving consisting of an important number of these objects.

Compact Wideband Microstrip Patch Antenna based on High Impedance Surface

A compact single probe feed asymmetrical semicircular fractal boundary patch antenna based on HIS (high impedance surface) is proposed for wide bandwidth at Wi-Fi band. Circular polarization operation can be obtained by embedding semi-circle fractal curves along the edges of the square patch antenna. The 10-dB return loss bandwidth is 15.13% (2.32GHz-2.70GHz). The 3-dB axial ratio bandwidth is 4.11% (2.38GHz-2.48GHz). The close relationship between simulation results and measured results establishes the antenna usefulness.

Asymptotics for quasilinear obstacle problems in bad domains

We study two obstacle problems involving the p-Laplace operator in domains with n-th pre-fractal and fractal boundary. We perform asymptotic analysis for \begin{document} $p \to \infty $ \end{document} and \begin{document} $n \to \infty $ \end{document} .

On Nevanlinna domains with fractal boundaries

On Nevanlinna domains with fractal boundaries

Revealing the Newton–Raphson basins of convergence in the circular pseudo-Newtonian Sitnikov problem

In this paper we numerically explore the convergence properties of the pseudo-Newtonian circular restricted problem of three and four primary bodies. The classical Newton–Raphson iterative scheme is used for revealing the basins of convergence and their respective fractal basin boundaries on the complex plane. A thorough and systematic analysis is conducted in an attempt to determine the influence of the transition parameter on the convergence properties of the system. Additionally, the roots (numerical attractors) of the system and the basin entropy of the convergence diagrams are monitored as a function of the transition parameter, thus allowing us to extract useful conclusions. The probability distributions, as well as the distributions of the required number of iterations are also correlated with the corresponding basins of convergence.

Design & Performance Analysis of Symmetric & Asymmetric Fractal Boundary Microstrip Antenna for Wireless Application

Design & Performance Analysis of Symmetric & Asymmetric Fractal Boundary Microstrip Antenna for Wireless Application

On the First Eigenvalue of the Degenerate $$\varvec{p}$$ p -Laplace Operator in Non-convex Domains

In this paper we obtain lower estimates of the first non-trivial eigenvalues of the degenerate p-Laplace operator, $$p>2$$ , in a large class of non-convex domains. This study is based on applications of the geometric theory of composition operators on Sobolev spaces that permits us to estimate constants of the Poincare–Sobolev inequalities. On this base we obtain lower estimates of the first non-trivial eigenvalues for Ahlfors-type domains (i.e. quasidiscs). This class of domains includes some snowflake-type domains with fractal boundaries.

Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary

This paper uses a connection between bounded remainder sets in Rd and cut-and-project sets in R together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some lattice in order to construct several bounded remainder sets with fractal boundary. Moreover it is shown that there are cut-and-project sets being not bounded distance equivalent to each other even if they are locally indistinguishable, more precisely: even if they are contained in the same hull.

Comparing the basins of attraction for several methods in the circular Sitnikov problem with spheroid primaries

The circular Sitnikov problem, where the two primary bodies are prolate or oblate spheroids, is numerically investigated. In particular, the basins of convergence on the complex plane are revealed by using a large collection of numerical methods of several order. We consider four cases, regarding the value of the oblateness coefficient which determines the nature of the roots (attractors) of the system. For all cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distribution of the iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical computations indicate that most of the iterative schemes provide relatively similar convergence structures on the complex plane. However, there are some numerical methods for which the corresponding basins of attraction are extremely complicated with highly fractal basin boundaries. Moreover, it is proved that the efficiency strongly varies between the numerical methods.

Stochastic Resonance and Safe Basin of Single-Walled Carbon Nanotubes with Strongly Nonlinear Stiffness under Random Magnetic Field.

In this paper, a kind of single-walled carbon nanotube nonlinear model is developed and the strongly nonlinear dynamic characteristics of such carbon nanotubes subjected to random magnetic field are studied. The nonlocal effect of the microstructure is considered based on Eringen’s differential constitutive model. The natural frequency of the strongly nonlinear dynamic system is obtained by the energy function method, the drift coefficient and the diffusion coefficient are verified. The stationary probability density function of the system dynamic response is given and the fractal boundary of the safe basin is provided. Theoretical analysis and numerical simulation show that stochastic resonance occurs when varying the random magnetic field intensity. The boundary of safe basin has fractal characteristics and the area of safe basin decreases when the intensity of the magnetic field permeability increases.

Analysis of global properties for dynamical systems by a modified digraph cell mapping method

A modified procedure is proposed in this paper to solve the limitations of digraph cell mapping method in accurately analyzing the global properties of dynamical system, such as fractal basin boundaries. Firstly a rough cell structure is applied to the generalized cell mapping (GCM) method to obtain the general global properties of dynamical systems with digraph algorithm, and then a procedure based on the composite cell coordinate system method is proposed to increase the calculation accuracy. In order to further increase the calculation speed, a simple and feasible parallel strategy is applied during the creation process of one-step transition probability matrix for the GCM method. Meanwhile, the memory consumption can be greatly reduced by storing the one-step transition probability matrix as finite separate data files. The accurate global properties of two examples, a nonlinear oscillator with fractal basin structure and the Lorenz system, are demonstrated to show the effectiveness of our proposed improvement strategy.

Conversion time to heat of the vortices injected only initially is much longer than the ones in a well-developed equilibrium turbulence: A possible relevance of the super-tsunami caused by the 3.11 earthquake off the Pacific coast of Tohoku to the appearance of a prominent SST anomaly in the northern pacific ocean in the summer of 2011

In an equilibrium steady turbulence where vortices of a large size are constantly injected, the vortices are known to decay successively to the smaller eddies by splitting with a time constant of the order of the turnover time of each eddy. We studied in this paper how long the decay time is for a transient turbulence in which large eddies are injected only initially into an thermal equilibrium fluid system or into a fluid system of a fractal boundary with an average constant flow. We showed that the decay time of vortices in a transient turbulence is much longer than the turnover time of the eddies because of much weaker perturbation to the transient vortices compared to eddies in a well-developed equilibrium turbulence. It was suggested that the sea surface temperature anomaly observed in the northern pacific ocean in August, 2011 may be related to the large tsunami attack on east coast of Tohoku Japan caused by the 3.11 earth quakes of magnitude 9.0.