Graphical exploration of the connectivity sets of alternated Julia sets; M, the set of disconnected alternated Julia sets

Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in [1], on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in [1], the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness) which characterizes the standard Julia sets. Via computer graphics, we unveil these connectivity domains which are four-dimensional fractals. The computer graphics results show here, without substituting the proof but serving as a research guide, that for the alternated Julia sets, the Mandelbrot set consists of the set of all parameter values, for which each alternated Julia set is not only connected, but also disconnected.Comment: 7 figure

A self-similar field theory for 1D linear elastic continua and self-similar diffusion problem

This paper is devoted to the analysis of some fundamental problems of linear elasticity in 1D continua with self-similar interparticle interactions. We introduce a self-similar continuous field approach where the self-similarity is reflected by equations of motion which are spatially non-local convolutions with power-function kernels (fractional integrals). We obtain closed-form expressions for the static displacement Green's function due to a unit $\delta$-force. In the dynamic framework we derive the solution of the {\it Cauchy problem} and the retarded Green's function. We deduce the distribution of a self-similar variant of diffusion problem with L\'evi-stable distributions as solutions with infinite mean fluctuations describing the statistics L\'evi-flights. We deduce a hierarchy of solutions for the self-similar Poisson's equation which we call "self-similar potentials". These non-local singular potentials are in a sense self-similar analogues to the 1D-Dirac's $\delta$-function. The approach can be the starting point to tackle a variety of scale invariant interdisciplinary problems

Power-law fluctuations in phase-separated lipid membranes

URL:http://link.aps.org/doi/10.1103/PhysRevE.60.7354 DOI:10.1103/PhysRevE.60.7354The spatial structure of three binary lipid mixtures, prepared as multilamellar vesicles, was studied by small-angle neutron scattering. In the fluid-gel coexistence region, large- scale concentration fluctuations appear which scatter like surface fractals for small acyl- chain mismatch and like mass fractals for large mismatch over about one decade of length. The transition is highly discontinuous: The fractal dimension of the boundary between the gel and fluid drops from 2.7 to 1.7, the gel fraction in the fluctuations drops from about 0.5 to 0.07, and the gel domains interlamellar correlation drops from strong to weak. We interpret the fluctuations as long-lived descendants of the incipient two-phase equilibrium state and the transition as due to changes in the gel rigidity and phase diagram.We gratefully acknowledge financial support from the Deutsch Forschungsgemeinschaft and the Petroleum Research Fund, administered by the American Chemical Society

Quantum Mechanics on Laakso Spaces

We first review the spectrum of the Laplacian operator on a general Laakso Space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally, we compute the spectrum for the Laplacian and its multiplicities when certain regions of a Laakso space are compressed or stretched and calculate the Casimir force experienced by two uncharged conducting plates by imposing physically relevant boundary conditions and then analytically regularizing the result. Lastly, we derive a general formula for the spectral zeta function and its derivative for Laakso spaces with strict self-similar structure before listing explicit spectral values for cases of interest.Comment: v2: Updated and revised text from preliminary report. v1:Preliminary Report from a Summer 2010 REU at the University of Connecticu

Integrated density of states for Poisson-Schr\"odinger perturbations of subordinate Brownian motions on the Sierpi\'nski gasket

We prove the existence of the integrated density of states for subordinate Brownian motions in presence of the Poissonian random potentials on the Sierpi\'nski gasket.Comment: 34 pages, 2 figure

Response of a Hodgkin-Huxley neuron to a high-frequency input

We study the response of a Hodgkin-Huxley neuron stimulated by a periodic sequence of conductance pulses arriving through the synapse in the high frequency regime. In addition to the usual excitation threshold there is a smooth crossover from the firing to the silent regime for increasing pulse amplitude $g_{syn}$. The amplitude of the voltage spikes decreases approximately linearly with $g_{syn}$. In some regions of parameter space the response is irregular, probably chaotic. In the chaotic regime between the mode-locked regions 3:1 and 2:1 near the lower excitation threshold the output interspike interval histogram (ISIH) undergoes a sharp transition. If the driving period is below the critical value, $T_i < T^*$, the output histogram contains only odd multiples of $T_i$. For $T_i > T^*$ even multiples of $T_i$ also appear in the histogram, starting from the largest values. Near $T^*$ the ISIH scales logarithmically on both sides of the transition. The coefficient of variation of ISIH has a cusp singularity at $T^*$. The average response period has a maximum slightly above $T^*$. Near the excitation threshold in the chaotic regime the average firing rate rises sublinearly from frequencies of order 1 Hz.Comment: 7 pages, 11 figure

Inversion Fractals and Iteration Processes in the Generation of Aesthetic Patterns

In this paper, we generalize the idea of star-shaped set inversion fractals using iterations known from fixed point theory. We also extend the iterations from real parameters to so-called q-system numbers and proposed the use of switching processes. All the proposed generalizations allowed us to obtain new and diverse fractal patterns that can be used, e.g., as textile and ceramics patterns. Moreover, we show that in the chaos game for iterated function systems - which is similar to the inversion fractals generation algorithm - the proposed generalizations do not give interesting results

Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles

In three-dimensional ideal magnetohydrodynamics, closed flux surfaces cannot maintain both rational rotational-transform and pressure gradients, as these features together produce unphysical, infinite currents. A proposed set of equilibria nullifies these currents by flattening the pressure on sufficiently wide intervals around each rational surface. Such rational surfaces exist at every scale, which characterizes the pressure profile as self-similar and thus fractal. The pressure profile is approximated numerically by considering a finite number of rational regions and analyzed mathematically by classifying the irrational numbers that support gradients into subsets. Applying these results to a given rotational-transform profile in cylindrical geometry, we find magnetic field and current density profiles compatible with the fractal pressure.Comment: 11 pages, 10 figure

Perturbation Mappings in Polynomiography

In the paper, a modification of rendering algorithm of polynomiograph is presented. Polynomiography is a method of visualization of complex polynomial root finding process and it has applications among other things in aesthetic pattern generation. The proposed modification is based on a perturbation mapping, which is added in the iteration process of the root finding method. The use of the perturbation mapping alters the shape of the polynomiograph, obtaining in this way new and diverse patterns. The results from the paper can further enrich the functionality of the existing polynomiography software

The Limit Cycles of Lienard Equations in the Weakly Nonlinear Regime

Li\'enard equations of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with $f(x)$ an even function, are considered in the weakly nonlinear regime ($\epsilon\to 0$). A perturbative algorithm for obtaining the number, amplitude and shape of the limit cycles of these systems is given. The validity of this algorithm is shown and several examples illustrating its application are given. In particular, an ${\mathcal O}(\epsilon^8)$ approximation for the amplitude of the van der Pol limit cycle is explicitly obtained.Comment: 17 text-pages, 1 table, 6 figure