Elliptic Perturbation Research Articles

A broadband circularly polarized antenna for millimeter-wave applications

Abstract This paper presents a broadband circularly polarized (CP) antenna array for millimeter-wave applications, and the antenna array has the advantages of wide impedance bandwidth (IBW), novel CP design, and low profile. The antenna unit consists of a two-layer substrate and two pairs of magnetoelectric dipoles. Stepped microstrip lines coupled by rectangular slits form a feeder network for easy integration. The axial ratio bandwidth (ARBW) is extended because a pair of parasitic patches is loaded and an elliptical perturbation is added. The simulation results show that the antenna has an ARBW of 18.6% (26.4–31.9 GHz) and an IBW of 45.5% (20.6–32.7 GHz), with a gain greater than 7.11 dBic in the IBW. To improve the gain of the antenna, a 2 × 2 antenna array is designed, fabricated, and measured. The measured results show that the array has an ARBW of 16.6% (26.42–31.21 GHz), an IBW of 41.6% (22.28–33.97 GHz), a peak gain of 13.89 dBic in the IBW, the cross-polarization levels in the xoz-plane and yoz-plane are above 20 dB, and a radiation efficiency greater than 89%.

Development of an Elliptical Perturbation System that provides unexpected perturbations during elliptical walking (the EPES system)

Background‘Perturbation-based balance training’ (PBBT) is a training method that was developed to improve balance reactive responses to unexpected balance loss. This training method is more effective in reducing fall rates than traditional balance training methods. Many PBBTs are performed during standing or treadmill walking which targeted specifically step reactive responses, we however, aimed to develop and build a mechatronic system that can provide unexpected perturbation during elliptical walking the Elliptical Perturbation System (the EPES system), with the aim of improving specifically the trunk and upper limbs balance reactive control.MethodsThis paper describes the development, and building of the EPES system, using a stationary Elliptical Exercise device, which allows training of trunk and upper limbs balance reactive responses in older adults.ResultsThe EPES system provides 3-dimensional small, controlled, and unpredictable sudden perturbations during stationary elliptical walking. We developed software that can identify a trainee’s trunk and arms reactive balance responses using a stereo camera. After identifying an effective trunk and arms reactive balance response, the software controls the EPES system motors to return the system to its horizontal baseline position after the perturbation. The system thus provides closed-loop feedback for a person’s counterbalancing trunk and arm responses, helping to implement implicit motor learning for the trainee. The pilot results show that the EPES software can successfully identify balance reactive responses among participants who are exposed to a sudden unexpected perturbation during elliptical walking on the EPES system.ConclusionsEPES trigger reactive balance responses involving counter-rotation action of body segments and simultaneously evoke arms, and trunk reactive response, thus reactive training effects should be expected.

Asymptotic linearized inversion for acoustic variable-tilt transversely isotropic media through elliptic perturbation analysis

The linearized inverse scattering problem for acoustic variable-tilt transversely isotropic (TTI) media is studied. In depicting the acoustic anisotropy behavior of the TTI medium with the P-wave normal moveout velocity (NMO) Vn, density ρ, and anisotropic parameters δ and η, a single-scattering procedure of NMO pressure wavefields is developed through elliptical perturbation analysis. By high-frequency approximation, the singly scattered pressure field is naturally related with an acoustic TTI generalized Radon transform (GRT), in which the TTI scattering patterns are explicitly expressed. We asymptotically invert for the multiple material parameters of an acoustic TTI medium via constructing the GRT backprojection operator. The effectiveness of the proposed approach is demonstrated on the layered model, and its superiority over the traditional acoustic isotropic and VTI GRT inversion methods is illustrated on the overthrust model.

A circularly polarized low‐profile magnetoelectric dipole antenna

AbstractIn this article, we proposed a novel circularly polarized magnetoelectric (ME) dipole antenna with low profile, which is excited by substrate integrated waveguide aperture‐coupled feeding structure with a butterfly‐shaped slot. The ME dipole antenna is formed of a couple of oblong metallic patches with square corner cutting off and elliptic perturbation acting as electric dipole, and two couples of vertical metalized via holes working as magnetic dipole. The profile of the antenna is 0.122λ0. The antenna achieves a fractional bandwidth of 32.6% for |S11| smaller than −10 dB from 22.6 to 31.3 GHz and the 3‐dB axial‐ratio bandwidth (BW) of 12.8% from 22.6 to 25.7 GHz. Moreover, the gain is maintained over 6 dB for a large portion of the operational BW. The measured boresight gain is 7.8 dB.

Realization of a dual mode filter in ridge gap waveguide technology

AbstractA proof of concept of a dual mode filter based on ridge gap waveguide cavity is presented by introducing perturbation to one of the first two degenerate modes. Rectangular, circular, and elliptical shaped perturbations are investigated. The one with elliptical perturbation was fabricated and tested. Good agreement between measured and simulation results was observed.

Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment

We study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on $\mathbb{Z} ^d$. Standard conditions for ballisticity and the central limit theorem require ellipticity, and are typically non-local. We use oriented percolation and martingale arguments to find non-trivial local conditions for ballisticity and an annealed invariance principle in the non-elliptic setting. The use of percolation allows certain non-elliptic models to be treated even though ballisticity has not been proved for elliptic perturbations of these models.

Riesz transforms through reverse Hölder and Poincaré inequalities

We study the boundedness of Riesz transforms in $$L^p$$ for $$p>2$$ on a doubling metric measure space endowed with a gradient operator and an injective, $$\omega $$ -accretive operator L satisfying Davies–Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Holder inequality, the Riesz transform is always bounded on $$L^p$$ for p in some interval $$[2,2+\epsilon )$$ , and that $$L^p$$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $$L^q$$ for $$q \in [2,p)$$ . This improves results of Auscher et al. (Ann Sci Ecole Norm Sup 37(4):911–957, 2004) and Auscher and Coulhon (Ann Scuola Norm Sup Pisa 4:531–555, 2005), where the stronger assumption of a Poincare inequality and the assumption $$e^{-tL}(1)=1$$ were made. The Poincare inequality assumption is also weakened in the setting of a sectorial operator L. In the last section, we study elliptic perturbations of Riesz transforms.

Three-dimensional instabilities and optimal perturbations of a counter-rotating vortex pair in stratified flows

This paper investigates the three-dimensional instabilities and the optimal perturbations on a pair of horizontal counter-rotating Lamb-Oseen vortices in a vertically stably stratified flow. Two-dimensional (2D) simulations are first performed, showing that while the dipole moves vertically against the stratification the vortex parameters: the radius a∗, the separation distance b∗, and the circulation Γ∗ are solely function of the time rescaled by the Brunt-Väisälä frequency N, independently of the Froude number, when the Reynolds number is large enough. Here, the Froude number is Fr = W0/Nb0 with W0 the initial advection velocity of the dipole and b0 the initial separation distance between the two vortices. For weak and moderate stratifications (large Fr), the stratification acts on a long time scale compared to the advection time of the dipole implying that the 2D flow can be considered as quasi-steady. In that case, when three dimensional instabilities are added, a linear stability analysis of this 2D flow at different instants retrieves the instability peaks corresponding to the Crow instability for the long wavelengths and to the elliptic instability for the short wavelengths showing that the dynamics is almost unaffected by buoyancy effects. The Crow and elliptic instabilities scale with the instantaneous dipole parameters showing in particular that stratification promotes instability by reducing the distance b∗ between vortices [K. K. Nomura et al., “Short-wavelength instability and decay of a vortex pair in stratified fluid,” J. Fluid Mech. 553, 283-322 (2006)]. For strong stratifications (Froude numbers of order unity or smaller), the quasi-steady approximation is not valid, and the question of stability should be formulated in a different way, by, for example, searching for the transient growth of the energy of perturbation that may be computed for steady or unsteady base flow. Then, for each time horizon τ, we should determine the critical perturbation leading to the largest energy growth by the time τ. Presently, we compute the optimal perturbations at two time horizons τ = 4 and τ = 10 dimensionalized by 2πb02/Γ0 with a direct-adjoint technique which takes into account the evolution of the base flow. In the homogeneous case, this technique allows to investigate the effect of the weak unsteadiness of the flow due to viscous diffusion which induces a growth of the vortex core radius a∗. Both Crow and elliptic instabilities are retrieved in the optimal response and in the energy gain curves. Even if very slow, the viscous diffusion is found to increase the gain of the antisymmetric elliptic perturbation compared to the symmetric one. When the fluid is stratified, peaks at small wavenumber and at wavenumber of the order of the vortex core size are found for all Froude numbers with optimal responses strongly resembling, respectively, the Crow and the elliptic modes with optimal gains corresponding to mean growth rates larger than in the homogeneous case for both modes. However, as the strength of stratification increases (Froude numbers smaller than 2), optimal perturbations start departing from their homogeneous counterpart with large perturbation in the wake of the dipole associated with density effects.

A Perturbation Procedure for Limit Cycle and Heteroclinic Connection Analysis of Certain Self-Excited Oscillator

A perturbation procedure, in which the elliptic perturbation method and the hyperbolic perturbation method are applied, is presented for predicting heteroclinic connection of limit cycle or self-excited ocsillator. The limit cycle can be analytically constructed first by the elliptic perturbation method after Hopf bifurcation, in which the amplitude of limit cycle can be controlled by the modulus of elliptic functions. The heteroclinic trajectories, which are formed by the heteroclinic connection of limit cycle, can also be constructed by similar perturbation procedure but adopting the hyperbolic functions instead of elliptic functions. And the criterion of heteroclinic connection is given in the perturbation procedure. A typical self-excited oscillator is studied in detail to assess the present method.

Analytical methods for an elliptic singular perturbation problem in a circle

We consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain.

Rows of optical vortices from elliptically perturbing a high-order beam

An optical vortex (phase singularity) with a high topological strength resides on the axis of a high-order light beam. The breakup of this vortex under elliptic perturbation into a straight row of unit-strength vortices is described. This behavior is studied in helical Ince-Gauss beams and astigmatic, generalized Hermite-Laguerre-Gauss beams, which are perturbations of Laguerre-Gauss beams. Approximations of these beams are derived for small perturbations, in which a neighborhood of the axis can be approximated by a polynomial in the complex plane: a Chebyshev polynomial for Ince-Gauss beams, and a Hermite polynomial for astigmatic beams.

On cooperative instabilities of parallel vortex pairs

We present a combined analytical and numerical study of the instabilities of a pair of parallel unequal-strength vortices. We extend the analyses of a vortex in an external strain field (Crow, AIAA J. vol. 8, 1970, p. 2172; Widnall et al., J. Fluid Mech. vol. 66, 1974, p. 35) to include the orbital motion of the vortex pair. For counter-rotating pairs, the classic Crow-type periodic displacement perturbations are unstable for all vortex strength ratios, with fastest-growing wavelengths several times the vortex spacing. For co-rotating pairs, the orbital motion acts to suppress instability due to displacement perturbations. Instabilities in this case arise for elliptic perturbations at wavelengths that scale with the vortex core size. We also examine the influence of a second vortex pair by extending Crouch's (J. Fluid Mech. vol. 350, 1997, p. 311) analysis. Numerical results from a spectral initial-value code with subgrid-scale modelling agree with the growth rates from the theoretical models. Computations reveal the nonlinear evolution at late times, including wrapping and ring-rejection behaviour observed in experiments. A pair of co-rotating Gaussian vortices perturbed by noise develops elliptic instabilities, leading to the formation of vorticity bridges between the two vortices. The bridging is a prelude to vortex merger. Analytic, computational and experimental results agree well at circulation Reynolds numbers of order .

A small elliptic perturbation of a backward-forward parabolic problem with applications to stochastic models

We consider an elliptic PDE in two variables. As one parameter approaches zero, this PDE collapses to a parabolic one, that is forward parabolic in a part of the domain and backward parabolic in the remainder. Such problems arise naturally in various stochastic models, such as fluid models for data-handling systems and Markov-modulated queues. We employ singular perturbation methods to study the problem for small values of the parameter.

Reply to “Comment on ‘Cylindrical phase of block copolymers: Stability of circular configuration to elliptical distortions and thin film morphologies’ ”

The result of Pereira [Phys. Rev. E 63, 061809 (2001)] does not contradict previous work, but rather complements it. Both this work and previous work show there is no reason to suspect that a circular interface in a hexagonal Wigner-Seitz cell is stable to small perturbations, whether they are hexagonal or elliptical. The elliptical perturbation is on the downward energy pathway from a perfectly circular interface to a hexagonally modulated interface. Finally, my thin film conclusions are shown to be valid using an independent method.

Mapping properties of generalized Fourier transforms and applications to Kirchhoff equations

We prove the differentiability of generalized Fourier transforms associated with a self-adjoint and strictly elliptic perturbation A of the Laplacian with variable coefficients in an exterior domain, using results on the spectral differentiability of the resolvent of A. Moreover we show that differentiable functions with bounded support and vanishing near the origin are mapped by the generalized Fourier transform into polynomially weighted L 2-spaces. As an application of the generalized Fourier transform and exploiting the previous results, we deal with equations of Kirchhoff type. We will not only show the global (in t) existence and uniqueness of solutions for a class of small data, but also an assertion on its time asymptotic behavior. In addition, we obtain amplified results for Schrodinger operators $A = - \Delta + V$ .

Self‐consistent Response of a Galactic Disk to an Elliptical Perturbation Halo Potential

We calculate the self-consistent response of an axisymmetric galactic disk perturbed by an elliptical halo potential of harmonic number m = 2 and obtain the net disk ellipticity. Such a potential is commonly expected to arise due to galactic tidal encounter and also during the galaxy formation process. The self-gravitational potential corresponding to the self-consistent, nonaxisymmetric density response of the disk is obtained by inversion of the Poisson equation for a thin disk. This response potential is shown to oppose the perturbation potential because physically the disk self-gravity resists the imposed potential. This results in a reduction in the net ellipticity of the perturbation halo potential in the disk plane. The reduction factor denoting this decrease is independent of the strength of the perturbation potential and has a typical minimum value of ~0.75-0.9 for a wide range of galaxy parameters. The reduction is most important at 1.4 exponential disk scale lengths and is progressively less so at higher radii. For the solar neighborhood region of the Galaxy, the reduction factor is 0.8. Beyond twice the Holmberg radius, the reduction is negligible, and there the disk asymmetry in the atomic hydrogen gas traces the true ellipticity of the halo potential. The reduction is negligible at all radii for higher harmonics (m ≥ 3) of the halo potential. On correcting for the negative disk response, the true ellipticity of the halo potential for a typical spiral galaxy is shown to be higher by ~20% than the typical halo ellipticity of ≤0.1 deduced in the literature from observations of isophotal or kinematical asymmetry of disks.

Inviscid damping of asymmetries on a two-dimensional vortex

The inviscid damping of an asymmetric perturbation on a two-dimensional circular vortex is examined theoretically, and with an electron plasma experiment. In the experiment, an elliptical perturbation is created by an external impulse. After the impulse, the ellipticity (quadrupole moment) of the vortex exhibits an early stage of exponential decay. The measured decay rate is in good agreement with theory, in which the perturbation is governed by the linearized Euler equations. Often, the exponential decay of ellipticity is slow compared to a vortex rotation period, due to the excitation of a quasimode. A quasimode is a vorticity perturbation that behaves like a single azimuthally propagating wave, which is weakly damped by a resonant interaction with corotating fluid. Analytically, the quasimode appears as a wave packet of undamped continuum modes, with a sharply peaked frequency spectrum, and it decays through interference as the modes disperse. When the exponential decay rate of ellipticity is comparable to the vortex rotation frequency, the vorticity perturbation does not resemble a quasimode; rather, it is rapidly dominated by spiral filaments. Over longer times, linear theory predicts algebraic decay of ellipticity; however, nonlinear oscillations of ellipticity emerge in the experiment before a transition to algebraic decay would occur.

Electrohydrodynamic stability of a liquid column under cross fields: Application to continuous flow electrophoresis

An infinite, conductive cylindrical column of electrolyte lying inside another electrolyte flattens when it is submitted to a transverse continuous electric field. This “ribbon” effect can be canceled by a second oscillatory field whose direction is perpendicular to both the column axis and to the first electric field. The linear stability of the equilibrium with respect to elliptic perturbations of the cross section is studied, and it is shown that the stability of the circular shape depends on the sign of a discriminating function of the conductivity and permittivity ratios. Possible applications of the results to continuous flow electrophoresis are discussed: Even for unstable situations the growth rate of disturbances is reduced by the application of a second field so that we expect the efficiency of the separation to be improved.

ANALYTICAL METHODS FOR SOLVING STRONGLY NON-LINEAR DIFFERENTIAL EQUATIONS

In this paper various approximate analytical methods for obtaining solutions for strongly non-linear differential equations in a complex function are developed. The methods are based on the solution of the generating differential equation with a cubic complex term. The method of harmonic balance, the method of Krylov–Bogoliubov and the elliptic perturbation method are adopted for solving strongly non-linear differential equations in a complex function. Three examples are analyzed. They describe the vibrations of a rotor with non-linearity.

PERIODIC SOLUTIONS OF STRONGLY QUADRATIC NON-LINEAR OSCILLATORS BY THE ELLIPTIC PERTURBATION METHOD

The elliptic perturbation method is applied to the study of the periodic solutions of strongly quadratic non-linear oscillators of the formx+c1x+c]2x2=ϵf(x,x), in which the Jacobian elliptic functions are employed. The generalized van der Pol equation withf(x,x)=μ0+μ1x−μ2x2is studied in detail. Comparisons are made with the solutions obtained by using the Lindstedt–Poincaré method and Runge–Kutta method to show the efficiency of the present method.