Elliptic Islands Research Articles

Abundance of elliptic isles at conservative bifurcations

We prove a conservative analogue of a theorem of Newhouse on the abundance of wild hyperbolic sets: Arbitrarily close to any area preserving map with a homoclinic tangency there are maps with 'wild' hyperbolic sets having persistent homoclinic tangencies. Furthermore, for a residual subset in an open set of nearby systems, these basic sets are accumulated by periodic elliptic islands.

Size and shape effects of patterned polycrystalline islands

Patterned polycrystalline cobalt islands with different sizes and shapes have been simulated by a micromagnetic simulator. An elliptic shape is found to be preferred for patterned islands to form a uniform single domain demagnetised state and have sharper squareness compared with square islands. Therefore, in a single-bit-per-island recording system, the elliptic islands could lead to a stable state and a predictable overwrite process and act independently even in a closely packed arrays.

On a possible mechanism of anomalous diffusion by Rossby waves

We study the advection of passive tracers by traveling plane Rossby waves of finite amplitude. In distinction with previous studies the nonlinearity of the wave field is taken into account in the first order of perturbation theory by considering the Lagrangian transport by resonant wave triads. Using the waves’ phases as new dynamical variables we reduce the problem to the study of a specific one-and-a-half degree of freedom Hamiltonian system with nonharmonic modulation. By using a symplectic integrator we study this system numerically and find an interesting series of bifurcations of its phase portrait as the nonlinearity increases. As is standard in the systems of this type we commonly see a chaotic sea with elliptic islands in the phase space, which means that in the physical space the resonant triads give rise to chaotic mixing and ballistic transport, respectively. The relevance of these results to the transport properties of β-plane turbulence is discussed.

Elliptic islands appearing in near-ergodic flows

It is proved that periodic and homoclinic trajectories which are tangent to the boundary of any scattering (ergodic) billiard produce elliptic islands in the `nearby' Hamiltonian flows i.e. in a family of two-degrees-of-freedom smooth Hamiltonian flows which converge to the singular billiard flow smoothly where the billiard flow is smooth and continuously where it is continuous. Such Hamiltonians exist; indeed, sufficient conditions are supplied, and thus it is proved that a large class of smooth Hamiltonians converges to billiard flows in this manner. These results imply that ergodicity may be lost in the physical setting, where smooth Hamiltonians which are arbitrarily close to the ergodic billiards, arise.

Higher-order narrow-tube quantizations of quasienergies

The narrow-tube quasienergy quantization allows an identification of part of the spectrum due to stable elliptic islands of the period map. With its transparent connection to the classical dynamics it also sets an absolute scale to the understanding of the Brillouin zone spectrum of quasienergies. In combination with a recent time-dependent normal-form expansion we show that the applicability and accuracy of the narrow-tube quantization is improved. Its analytic form helps us understand the quasiregular motion centred near periodic responses. We numerically consider harmonic and subharmonic elliptic islands of the weakly nonlinear Duffing oscillator.

Ray splitting in a class of chaotic triangular step billiards

For a class T of triangular step billiards (TSBs) we prove analytically the absence of elliptic islands in phase space. There is numerical evidence that TSBs are ergodic, sensitive, and mixing. Thus TSBs are chaotic, although their Kolmogorov-Sinai entropy is zero. We study the quantum implications of ray splitting (RS) in TSBs. The signature of non-Newtonian periodic RS orbits is identified in the Fourier transform of the TSB level density. The RS correction of the Weyl formula is tested in the TSB context. In contrast to the rich structure of split circle wave functions, TSB wave functions, except for the emergence of short-range correlations in the form of scarlets, appear featureless and homogeneous.

Elliptic islands in generalized Sinai billiards

AbstractWe find necessary conditions for a generalized Sinai billiard to be ergodic, independent of the geometry of the disks; i.e. if the conditions are not satisfied, then we can always construct special geometries for which the system is not ergodic. We prove non-ergodicity by constructing an elliptic periodic orbit and showing that this orbit satisfies the non-degeneracy condition of KAM theory. By a combination of theory and numerical calculation involving a computer algebra system (Mathematica), we show that the first Birkhoff invariant is non-zero. We apply this result to produce non-ergodic Hamiltonian flows that are generated by smooth potentials of finite range (repelling potentials and Lennard–Jones potentials) and by geodesic flows on the torus.

An in-situ laser-light scattering study of the development of surface topography during GaAs and In xGa 1 − xAs chemical beam epitaxy

The development of surface topography during the annealing of epi-ready (001) GaAs wafers and the subsequent CBE growth of GaAs and In x Ga 1 − x As layers were investigated using laser-light scattering (LLS) from a rotating substrate. The change in scattered light intensity during the pre-growth anneal is usually attributed to oxide desorption. However, we have found that a wide range of behaviour is possible depending on the growth history of the chamber. We suggest that this is due to the desorption of previously deposited species from the cryopanel as a result of radiant heating from the substrate, leading to contamination of the wafer surface. Procedures to minimise this contamination are outlined. LLS was used to study the changes in surface topography during growth, particularly the development of elliptical islands during the growth of GaAs and the onset of the “cross-hatch” pattern during the growth of InGaAs. It has recently been established that the ridges of the cross-hatch pattern are associated with dislocations which lie parallel to the surface. The development of these ridges parallel to 〈110〉 is related to the critical thickness of the layer. Interference oscillations in scattered light intensity during the growth of multilayer structures are also shown to be observable using LLS.

Tomography of (2, 1) and (3, 2) magnetic island structures on Tokamak Fusion Test Reactor

High-resolution electron cyclotron emission (ECE) image reconstruction has been used to observe (m,n)=(2,1) and (3, 2) island structures on Tokamak Fusion Test Reactor [Plasma Phys. Controlled. Fusion 33, 1509 (1991)], where m and n are the poloidal and the toroidal mode number, respectively. The observed island structure is compared with other diagnostics, such as soft x-ray tomography and magnetic measurements. A cold elliptic island is observed after lithium pellet injection. Evidence for the enhancement of the heat transfer due to the island is observed. A relaxation phenomenon due to the m=2 mode is newly observed in Ohmic plasmas.

Semiclassical narrow-tube quantizations of time-periodic Hamiltonian systems

An approximate semiclassical quantization procedure for time-periodic Hamiltonians is formulated, which applies to motion near stable periodic orbits of the system, corresponding to elliptic islands of the period map. The theory involves canonical transformations and the use of a particular dynamical Lewis invariant. The basic assumption in the narrow-tube approximation is the localization of the quantal wavefunction inside the elliptic islands.

Plenty of elliptic islands for the standard family of area preserving maps

For the Standard Map, a well-known family of conservative diffeomorphisms on the torus, we construct large basic sets which fill in the torus as the parameter runs to ∞. Then we prove that, for a residual set of large parameters, these basic sets are accumulated by elliptic periodic islands. We also show that there exists a k0 > 0 and a dense set of parameters in [k0, ∞) for which the standard map exhibits homoclinic tangencies.

Numerical aspects of wave forces on an artificial island

Abstract The wave forces on an artificial island standing on the sea‐bed are investigated by the proposed hybrid method. A boundary integral equation is developed for the inner fluid domain encompassing the arbitrarily shaped island, while an eigenfunction expansion is used to represent a simple velocity field in the outer fluid domain. Applying the natural boundary condition on the matching surface, the wave force and moment on the wetted surface of the island are found according to the calculated velocity potentials. A gravity step circular dock is used as a test case to confirm the present computation. As a practical application of the present method, numerical results for an elliptical artificial island are presented.

Scanning Tunneling Microscopy Study of GaAs Step Structures on Vicinal Substrate Grown by Metalorganic Chemical Vapor Deposition

Scanning tunneling microscopy (STM) is used to study the step structure of metalorganic chemical vapor deposition (MOCVD)-grown GaAs on a vicinal substrate. We passivated the as-grown surfaces with As in a vacuum chamber connected to the MOCVD system in order to protect it from oxidation during its transfer to the STM system. The undulation amplitude of the monolayer steps grown on a surface misoriented in the [110] direction is about twice that of these grown on a surface misoriented in the [1̄10] direction. On a substrate at misorientation angles equal to or less than 7.0°, surfaces show a monolayer step staircase when the growth temperature (T g)>650° C; step bunching occurs and the vicinal surface decomposed into (001) terraces and facets on which steps bunch when 575° C≤T g≤650° C; and elliptical two-dimensional islands form between the monolayer steps because of a decrease of Ga diffusion coefficient when T g<575° C.

Locally most robust circles and boundary circles for area-preserving maps

What is the rotation number of the last rotational invariant circle to break in a family of area-preserving maps? And what are the rotation numbers of the outermost invariant circles around elliptic islands? Under the hypothesis that the breakup of invariant circles of arbitrary rotation number for a significant class of area-preserving maps is governed by a critical set of a renormalization operator with certain properties, the authors show that for maps in this universality class the rotation number of locally most robust circles is always noble, and that the rotation number of every boundary circle lies on the stable manifold of a certain Cantor set of numbers of constant type under the Gauss map.

Coastal-trapped waves in elliptical lakes and around elliptical islands

Abstract The propagation of low-frequency waves, trapped around the shelves of lakes and islands, is investigated. A two-layer model with a simple step topography is used to study the combined effects of depth changes and stratification. The lake or island boundary and corresponding shelf break are modelled as confocal ellipses and analytical solutions for the barotropic streamfunction, interface displacement and dispersion relation are found. Results show the presence of a topographically modified internal Kelvin wave and a stratification-modified topographic Rossby wave. The model is applied to the small, elongated northern basin of Lake Lugano. Although the Rossby wave frequency is close to its barotropic value the two mode 1 dispersion curves “kiss” for very weak stratification with important effects on the wave structures and velocities. The velocity field of the observed 74th oscillation more closely resembles that of an internal Kelvin wave than a topographic Rossby wave.

Dynamics of quadratic correspondences

The author investigates the dynamics of maps z to z' determined by implicit complex quadratic equations g (z,z')=0. Such maps are 2-valued and have a 2-valued inverse. They may be considered as a generalisation of both quadratic maps and two-generator Kleinian groups. Under iteration they exhibit not only properties typical of these two classes, but also features of Hamiltonian systems. Maps of this type with a reflection symmetry can be restricted to a half plane, where they are homeomorphisms except for 'zip' discontinuities. When additional symmetries are imposed they have hierarchies of elliptic island chains and isolated invariant semicircle. Restricted to other domains the same maps have 'Julia sets', which appear to vary continuously with the domain.

Nearshore Circulation Around a Elliptical Island

Nearshore Circulation Around a Elliptical Island

An analogue model study of EM induction for an island near bay and cape coastlines

Laboratory electric and magnetic model measurements were carried out to examine how the addition of a nearby elliptical island, to an otherwise complex continental coastline involving bays and capes, affects electromagnetic induction in coastal regions. Irregularities in a coastline in the form of bays and capes lead to significant spatial field variations in coastal regions due to variations in density of induced current flowing in the ocean caused by current deflection or channelling by the coastal irregularities. The addition of a nearby island, which for the appropriate field polarization can result in current channelling through the strait, greatly influences these field spatial variations. For E-polarization, the case of the model electric field of the inducing source parallel to the strait, and for short period (4.2 min studied here) variations, the effect on the fields, of the model elliptical island in the shallow coastal ocean, at points on the nearby continental coastline, was to increase by factors as large as two to three the H y amplitudes at the centres of the bay and the cape, and the E x amplitude at each bay, cape and straight coastline in the region of the strait, as compared with the values for coastline models without an offshore island. For H-polarization, the case of the electric field of the source perpendicular to the strait, the effect of the island was to increase by a factor of two to three the H z coastline values at the edge of the bay, and to decrease by a similar factor the value at the edge of the cape. Further, the spatial variations in the fields for both polarizations, over the island and over the strait were strongly influenced by current channelling through the strait. Thus any interpretation of field station measurements on such an island, on the sea floor in the strait, or on the continent in coastal areas near an island, should take into account the effects due to current concentrations in the strait particularly at short periods.

A piecewise linear model for the zones of instability of an area-preserving map

In this note we study the global behavior of the piecewise linear area-preserving transformation x 1 = 1 − y 0 + | x 0|, y 1 = x 0, of the plane. We show that there are infinitely many invariant polygons surrounding an elliptic fixed point. The regions between these invariant polygons serve as models for the “zones of instability” in the corresponding smooth case. For our model we show that some of these annular zones contain only finitely many elliptic islands. The map is hyperbolic on the complement of these islands and hence exhibits stochastic behavior in this region. Unstable periodic points are dense in this region.

Development of the embryonic chick otic placode. I. Light microscopic analysis.

AbstractAt the 7–8 somite stage of embryonic chick development (29‐31 hours of incubation), a slightly elliptical island of thickened ectoderm appears laterally on either side of the most distal constricture of the rhombencephalon at the level of the anterior intestinal portal. The appearance and extent of this auditory placode is precisely correlated with the subjacent accumulation of neural crest cells. By 33 hours of incubation, there is a distinct depression in the developing otic placode, and by 40 to 45 hours, the placode is visibly invaginated, forming an epithelial vesicle or otocyst. Carefully staged embryos were serially sectioned, and the area underlying the developing otic placode was traced with a planimeter. It was found that placode size (area 60,000 μm2) is nearly unchanged from 30 to 42 hours of development. During this time interval, the placode cells first become columnar, show nuclear orientation, and then pseudostratify. The increase in placode cell number during this time interval is not likely to be the result of localized, accelerated cell division: the population doubling time of placode cells is eight and one‐half hours and the mitotic index of 2.5% is similar to that of cells in an equivalent area of adjacent, non‐placode forming head ectoderm. A model of otic placode formation is proposed which suggests that by 30 hours of development, a discrete population of placode forming cells is segregated from head ectoderm. Subsequent epithelial pseudostratification results from accumulation of this dividing population within the limits of the placode.