Devil's Staircase Research Articles

Quantized Spiral Tip Motion in Excitable Systems with Periodic Heterogeneities

Meandering spiral tips in homogeneous reaction-diffusion systems are characterized by two generically incommensurate radii and frequencies. Here, we create periodic perturbations in space to induce a transition to commensurate radii and frequencies that exhibit a devil's staircase. The plateaus of the staircase correspond to pinned or complex periodic orbits of the spiral tip.

BIFURCATIONS AND CHAOS IN TWO-CELL CELLULAR NEURAL NETWORKS WITH PERIODIC INPUTS

This study investigates bifurcations and chaos in two-cell Cellular Neural Networks (CNN) with periodic inputs. Without the inputs, the time periodic solutions are obtained for template A=[r,p,s] with p>1, r>p-1 and -s>p-1. The number of periodic solutions can be proven to be no more than two in exterior regions. The input is b sin 2πt/T with period T>0 and amplitude b>0. The typical trajectories Γ(b,T,A) and their ω-limit set ω(b,T,A) vary with b, T and A are also considered. The asymptotic limit cycles Λ∞(T,A) with period T of Γ(b,T,A) are obtained as b→∞. When [Formula: see text] (given in (67)), Λ∞and -Λ∞can be separated. The onset of chaos can be induced by crises of ω(b,T,A) and -ω(b,T,A) for suitable T and b. The ratio [Formula: see text], of largest amplitude a1(b) except for T-mode and amplitude of the T-mode of the Fast Fourier Transform (FFT) of Γ(b,T,A), can be used to compare the strength of sustained periodic cycle Λ0(A) and the inputs. When [Formula: see text], Λ0(A) dominates and the attractor ω(b,T,A) is either a quasi-periodic or a periodic. Moreover, the range b of the window of periodic cycles constitutes a devil's staircase. When [Formula: see text], finitely many chaotic regions and window regions exist and interweave with each other. In each window, the basic periodic cycle can be identified. A sequence of period-doubling is observed to the left of the basic periodic cycle and a quasi-periodic region is observed to the right of it. For large b, the input dominates, ω(b,T,A) becomes simpler, from quasi-periodic to periodic as b increases.

A Kinetics Driven Commensurate–Incommensurate Transition

We study the steady-state structure of an interface in an Ising system on a square lattice with a nonuniform external field. This structure exhibits a commensurate–incommensurate transition driven by the velocity of the interface. The nonuniform field has a profile with a fixed shape which is designed to stabilize a flat interface, and is translated with velocity ve . For small velocities, the interface is stuck to the profile and is rippled with a periodicity which may be either commensurate or incommensurate with the lattice parameter of the square lattice. For a general orientation of the profile, the local slope of the interface locks into one of infinitely many rational directions, producing a devil's staircase structure. These lock-in or commensurate structures disappear as ve increases through a kinetics driven commensurate–incommensurate transition. For large ve , the interface becomes detached from the field profile and coarsens with Kardar–Parisi–Zhang (KPZ) exponents. The complete phase diagram and the multifractal spectrum corresponding to these structures have been obtained numerically together with several analytic results concerning the dynamics of the rippled phases. Our work has technological implications in crystal growth and the production of surfaces with desired surface morphologies.

A CHARACTERISTIC OF THE TRANSITION TO CHAOS VIA TYPE V INTERMITTENCY

In the transition to chaos via type V intermittency, when a periodic motion loses stability, there may be no intermediate higher-period orbits displayed before chaos or a sequential such orbits that form a staircase. The staircase may take a form of a conventional devil's staircase, a multiple devil's staircase or a seemingly irregular form. This article briefly presents typical examples for these situations.

Quantum criticality and deconfinement in phase transitions between valence bond solids

We consider spin-half quantum antiferromagnets in two spatial dimensions in the quantum limit, where the spins are in a valence bond solid (VBS) phase. The transitions between two such VBS phases is studied. In some cases, an interesting second order transition controlled by a fixed line with varying critical exponents if found. A specific example is provided by an antiferromagnetically coupled bilayer system on the honeycomb lattice where a continuous quantum phase transition can generically exist between two VBS phases. Furthermore, these critical points are deconfined, in the sense that gapped spin-1/2 spinon excitations emerge right at the transition. The low energy physics of this critical point (upto marginally irrelevant interactions) contains just a free quadratically dispersing `photon'. The phase structure on one side of this continuous transition is very intricate consisting of a series of infinitely closely spaced further transitions in a `devil's staircase' form. Analogies with previous examples of deconfined quantum criticality are emphasized. Closely related transitions in single layer systems are explored. These are second order only at some multicritical points. The solvable Rokshar-Kivelson point of quantum dimer models of single layer systems is found to correspond to a very special non-generic multicritical point

Bipartite Rokhsar–Kivelson points and Cantor deconfinement

Quantum dimer models on bipartite lattices exhibit Rokhsar--Kivelson points with exactly known critical ground states and deconfined spinons. We examine generic, weak perturbations around these points. In $d=2+1$ we find a first-order transition between a ``plaquette'' valence bond crystal and a region with a devil's staircase of commensurate and incommensurate valence bond crystals. In the part of the phase diagram where the staircase is incomplete, the incommensurate states exhibit a gapless photon and deconfined spinons on a set of finite measure, almost but not quite a deconfined phase in a compact $U(1)$ gauge theory in $d=2+1$! In $d=3+1$, we find a continuous transition between the $U(1)$ resonating valence bond phase and a deconfined staggered valence bond crystal. In an appendix, we comment on analogous phenomena in quantum vertex models, most notably the existence of a continuous transition on the triangular lattice in $d=2+1$.

Self-organization at finite temperatures of the devil’s staircase in Pb/Si(111)

The ``devil's staircase'' (DS) is one of the outstanding phase diagrams in physics because it shows a high degree of self-organization driven by a repulsive long-range adatom interaction. An infinite number of phases are predicted to exist which are built hierarchically with simple combinatorial rules. It was found with diffraction and scanning tunneling microscopy that a DS exists in Pb/Si (111) in the range $1.2l\ensuremath{\theta}l1.33$ monolayers, surprisingly at $\ensuremath{\sim}120\mathrm{K},$ and with phases which extend spatially over macroscopic distances $(\ensuremath{\sim}0.5\mathrm{mm}).$ The extraordinary amount of atom rearrangement necessary for these phases to form indicates an unexpected degree of self-organization at low temperatures.

ENTROPY OF MAPS WITH HORIZONTAL GAPS

We study the behavior of topological entropy in one-parameter families of interval maps obtained from a continuous map f by truncating it at the level depending on the parameter. When f is piecewise monotone, the entropy function has the devil's staircase structure.

Statistically locked-in transport through periodic potential landscapes.

Classical particles driven through periodically modulated potential energy landscapes are predicted to follow a devil's staircase hierarchy of commensurate trajectories depending on the orientation of the driving force. Recent experiments on colloidal spheres flowing through arrays of optical traps do indeed reveal such a hierarchy, but not with the predicted structure. The microscopic trajectories, moreover, appear to be random, with commensurability emerging only in a statistical sense. We introduce an idealized model for periodically modulated transport in the presence of randomness that captures both the structure and statistics of such statistically locked-in states.

Numerical modelization of disordered media via fractional calculus

In this paper, the framework for the mechanics of solids, deformable over fractal subsets, is outlined. Anomalous mechanical quantities with fractal dimensions are introduced, i.e., the fractal stress [σ ∗] , the fractal strain [ε ∗] and the fractal work of deformation W ∗ . By means of the local fractional operators, the static and kinematic equations are obtained, and the principle of virtual work for fractal media is demonstrated. Afterwards, from the definition of the fractal elastic potential φ ∗ , the linear elastic constitutive relation is derived. The direct formulation of the elastic problem is obtained in terms of the fractional Lamé operators and of the equivalence equations at the boundary. The variational form of the elastic problem is also obtained, through minimization of the total potential energy. Finally, discretization of the fractal medium is proposed, in the spirit of the Ritz–Galerkin approach, and a finite element formulation is obtained by means of devil's staircase interpolating splines.

Nonlinear Dynamics: Integrability, Chaos and Patterns

When the editorial office of Journal of Physics A: Mathematical and General of the Institute of Physics Publishing asked me to review a book on nonlinear dynamics I experienced an undeniable apprehension. Indeed, the domain is a rapidly expanding one and writing a book aiming at a certain degree of completeness looks like an almost impossible task. My uneasiness abated somewhat when I saw the names of the authors, two well-known specialists of the nonlinear domain, but it was only when I held the book in my hands that I felt really reassured. The book is not just a review of the recent (and less so) findings on nonlinear systems. It is also a textbook. The authors set out to provide a detailed, step by step, introduction to the domain of nonlinearity and its various subdomains: chaos, integrability and pattern formation (although this last topic is treated with far less detail than the other two). The public they have in mind is obviously that of university students, graduate or undergraduate, who are interested in nonlinear phenomena. I suspect that a non-negligible portion of readers will be people who have to teach topics which figure among those included in the book: they will find this monograph an excellent companion to their course. The book is written in a pedagogical way, with a profusion of examples, detailed explanations and clear diagrams. The point of view is that of a physicist, which to my eyes is a major advantage. The mathematical formulation remains simple and perfectly intelligible. Thus the reader is not bogged down by fancy mathematical formalism, which would have discouraged the less experienced ones. A host of exercises accompanies every chapter. This will give the novice the occasion to develop his/her problem-solving skills and acquire competence in the use of nonlinear techniques. Some exercises are quite straightforward, like `verify the relation 14.81'. Others are less so, such as `prepare a write-up on a) frequency-locking and b) devil's staircase'. I do not quite grasp the usefulness of such project-like exercises. Projects must be assigned by the person who indeed teaches the course. There are things that I really like a lot in this book. For instance, the section on `chaos in nonlinear electronic circuits' is particularly interesting. It offers a simple and rather inexpensive way to visualize chaos in the laboratory. The closing section of the book devoted to technological applications of nonlinear dynamics is also quite useful. The fact that the treatment remains rather elementary, based on review articles and monographs rather than research articles, adds to the intelligibility of the chapter, which will certainly prove stimulating to many a student. Of course, not everything can be perfect, and a 600-page book is bound to have some weak points. I find the treatment of quantum chaos rather sketchy and that of chaotic scattering even more so. Also, while the authors are aware of the importance of complex time in integrability, they do not attempt an explanation of the fundamental puzzle: `why, while the physical time is par excellence real, do we need a complex time in order to study the long-time behaviour of dynamical systems?'. Also the book devotes just four pages to integrable discrete systems. Given the tremendous development of this domain over the past decade, this short presentation is not doing justice to the subject. (However as the present reviewer is editing Springer Lecture Notes in Physics on precisely `Integrable Discrete Systems', to appear in early 2004, he would be the last one to complain about the absence of more details on the matter in the present book.) To sum it up, the monograph of Lakshmanan and Rajasekar is a book written by physicists and for physicists. It will be of interest to both the experienced practitioner and to the uninitiated. Its main quality resides in its thorough, pedagogical approach to the matter. Moreover the relaxed, not too formal, style makes for easy reading. Given that I am writing this review just a few days before Christmas I cannot help thinking that this book could be a nice present for a physicist.

One-sided multifractal analysis and points of non-differentiability of devils staircases

We examine the multifractal spectra of one-sided local dimensions of Ahlfors regular measures on . This brings into a natural context a curious property that has been observed in a number of instances, namely that the Hausdorff dimension of the set of points of non-differentiability of a self-affine ‘devil's staircase’ function is the square of the dimension of the set of points of increase.

Controllable step motors and rectifiers of magnetic flux quanta using periodic arrays of asymmetric pinning defects

We study the transport of vortices in superconductors with regular arrays of asymmetric pinning wells when applying an alternating electrical current. The asymmetric traps are modelled by the superposition of two interpenetrating square lattices of weak and strong pinning centers with separation smaller than the lattice constant. We show that this system can induce a net rectifying or diode effect for the vortex motion, including collective step-motor-type dynamics, where many vortices move forward a controlled and exact number of pin-lattice spacings at each cycle of the ac driving force. This system exhibits a remarkable net dc response with striking sawtooth-type oscillations. The net dc voltage response ${V}_{\mathrm{dc}}$ of the ac-driven vortices versus both the half period P and the amplitude ${F}_{L}$ of the ``square wave'' ac drive has been detailed in the present work. The influence of the equilibrium thermal noise, the shift between the two pinning sublattices, the degree of translational and orientational disorder, and the size of the simulation system on the ${V}_{\mathrm{dc}}$ response of the vortex motion at ac drive has also been addressed. Devil staircase and Arnold's tongue structures are revealed. We also analytically derive all the key features of our numerical results. This system provides a very controllable stepmotor for the control of collective motion. Our results apply mutatis mutandis to arrays of Josephson junctions, colloidal systems with optical traps, Wigner crystals, and any system with repelling movable objects that can be pinned by a lattice of traps.

"Devil's staircase" in Pb/Si(111) ordered phases.

With scanning tunneling microscopy we have found that ordered phases in Pb/Si(111) are one of the best examples of the "devil's staircase" phase diagram. Phases within a narrow coverage range (1.2<theta<1.3 monolayers) are constructed with the rules similar to the ones found in theoretical models.

Quantum phase transition in the Frenkel-Kontorova chain: from pinned instanton glass to sliding phonon gas.

We study analytically and numerically the one-dimensional quantum Frenkel-Kontorova chain in the regime where the classical model is located in the pinned phase characterized by the gaped phonon excitations and devil's staircase. By extensive quantum Monte Carlo simulations, we show that for the effective Planck constant Planck smaller than the critical value Planck(c) the quantum chain is in the pinned instanton glass phase. In this phase, the elementary excitations have two branches: phonons, separated from zero energy by a finite gap, and instantons that have an exponentially small excitation energy. At Planck = Planck(c) the quantum phase transition takes place and for Planck > Planck(c) the pinned instanton glass is transformed into the sliding phonon gas with gapless phonon excitations. This transition is accompanied by the divergence of the spatial correlation length and appearance of sliding modes at Planck > Planck(c).

Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback

Nonlinear dynamics of a semiconductor laser with delayed negative optoelectronic feedback are studied both numerically and experimentally. Mappings of the dynamic states and bifurcation diagrams are compared between a delayed negative optoelectronic feedback system and a delayed positive optoelectronic feedback system. Both systems follow a quasiperiodic route to chaos, where regular pulsing, quasiperiodic pulsing, and chaotic pulsing states are observed. Frequency-locked pulsing states are also found in a delayed negative optoelectronic feedback system, but not in a delayed positive optoelectronic feedback system. These frequency-locked pulsing states are experimentally observed to exhibit a harmonic frequency-locking phenomenon, where the pulsing frequency is locked to a harmonic of the delay loop frequency instead of the delay loop frequency itself. The rotation numbers of these frequency-locked pulsing states show a Devil's staircase structure.

Intercellular Communication Via Intracellular Calcium Oscillations

In this letter, we present the results of a simple model for intercellular communication via calcium oscillations, motivated in part by a recent experimental study. The model describes two cells (a “donor” and “sensor”) whose intracellular dynamics involve a calcium-induced, calcium release process. The cells are coupled by assuming that the input of the sensor cell is proportional to the output of the donor cell. As one varies the frequency of calcium oscillations of the donor cell, the sensor cell passes through a sequence of N : M phase-locked regimes and exhibits a “Devil's staircase” behavior. Such a phase-locked response has been seen experimentally in pulsatile stimulation of single cells. We also study a stochastic version of the coupled two-cell model. We find that phase locking holds for realistic choices for the cell volume.

DEVIL'S STAIRCASE OF GAP MAPS

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

Devil's staircase in kinetically limited growth.

The devil's staircase is a term used to describe surface or an equilibrium phase diagram in which various ordered facets or phases are infinitely closely packed as a function of some model parameter. A classic example is a one-dimensional Ising model [P. Bak and R. Bruinsma, Phys. Rev. Lett. 49, 249 (1982)] wherein long-range and short-range forces compete, and the periodicity of the gaps between minority species covers all rational values. In many physical cases, crystal growth proceeds by adding surface layers that have the lowest energy, but are then frozen in place. The emerging layered structure is not the thermodynamic ground state, but is uniquely defined by the growth kinetics. It is shown that for such a system, the grown structure tends to the equilibrium ground state via a devil's staircase traversing an infinity of intermediate phases. It would be extremely difficult to deduce the simple growth law based on measurement made on such a grown structure.

Non-differentiability of devil's staircases and dimensions of subsets of Moran sets

Let C be the homogeneous Cantor set invariant for x→ax and x→1−a+ax. It has been shown by Darst that the Hausdorff dimension of the set of non-differentiability points of the distribution function of uniform measure on C equals (dimHC)2 = (log 2/log a)2. In this paper we generalize the essential ingredient of the proof of this result. Let Ω = {0, 1, …, r}. Let F be a Moran set associated with {0 &lt; ai &lt; 1, i ∈ Ω} and Ωw = Ω×Ω×⃛. Let ø be the associated coding map from Ωe onto F. Fix a non-empty set Γ ⊆ Ω with Γ≠Ø and let z(σ, n) denote the position of the nth occurrence of the elements of Γ in σ ∈ Ωw.