MULTIPLE SOLUTIONS FOR HAMILTONIAN SYSTEMS WITH PERIODIC NONLINEARITY AND STRONG RESONANCE

I. Introduction In this paper we are looking for solutions of the following Hamiltonian system of second order: where x= （x1, x2） and V satisfies （V. 1） V: R×R2→R is a C1-function, 1-periodic In t, （V.2） V is periodic in x1 with the period T>0, （V. 3） V→O, Vx→O as |x2|→∞, uniformly in （t, x1）.

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.

Different optical properties in different periodic slot cavity geometrical morphologies

In this paper,optical properties of two-dimensional periodic annular slot cavity arrays in hexagonal close-packing on a silica substrate are theoretically characterized by finite difference time domain（FDTD） simulation method.By simulating reflectance spectra,electric field distribution,and charge distribution,we confirm that multiple cylindrical surface plasmon resonances can be excited in annular inclined slot cavities by linearly polarized light,in which the four reflectance dips are attributed to Fabry–Perot cavity resonances in the coaxial cavity.A coaxial waveguide mode TE11 will exist in these annular cavities,and the wavelengths of these reflectance dips are effectively tailored by changing the geometrical pattern of slot cavity and the dielectric materials filled in the cavities.These resonant wavelengths are localized in annular cavities with large electric field enhancement and dissipate gradually due to metal loss.The formation of an absorption peak can be explained from the aspect of phase matching conditions.We observed that the proposed structure can be tuned over the broad spectral range of 600–4000 nm by changing the outer and inner radii of the annular gaps,gap surface topography.Meanwhile,different lengths of the cavity may cause the shift of resonance dips.Also,we study the field enhancement at different vertical locations of the slit.In addition,dielectric materials filling in the annular gaps will result in a shift of the resonance wavelengths,which make the annular cavities good candidates for refractive index sensors.The refractive index sensitivity of annular cavities can also be tuned by the geometry size and the media around the cavity.Annular cavities with novel applications can be implied as surface enhanced Raman spectra substrates,refractive index sensors,nano-lasers,and optical trappers.

Analytical and numerical studies for Seiches in a closed basin with bottom friction

A standing wave oscillation in a closed basin,known as a seiche,could cause destruction when its period matches the period of another wave generated by external forces such as wind,quakes,or abrupt changes in atmospheric pressure.It is due to the resonance phenomena that allow waves to have higher amplitude and greater energy,resulting in damages around the area.One condition that might restrict the resonance from occurring is when the bottom friction is present.Therefore,a modified mathematical model based on the shallow water equations will be used in this paper to investigate resonance phenomena in closed basins and to analyze the effects of bottom friction on the phenomena.The study will be conducted for several closed basin shapes.The model will be solved analytically and numerically in order to determine the natural resonant period of the basin,which is the period that can generate a resonance.The computational scheme proposed to solve the model is developed using the staggered grid finite volume method.The numerical scheme will be validated by comparing its results with the analytical solutions.As a result of the comparison,a rather excellent compatibility between the two results is achieved.Furthermore,the impacts that the friction coefficient has on the resonance phenomena are evaluated.It is observed that in the prevention of resonances,the bottom friction provides the best performance in the rectangular type while functioning the least efficient in the triangular basin.In addition,non-linearity effect as one of other factors that provide wave restriction is also considered and studied to compare its effect with the bottom friction effect on preventing resonance.

The phenomenon of stochastic resonance in a single-mode laser system with an input periodical signal

Using the linear approximation method, we study a single-mode laser system driven by colored pump noise and quantum noise with coupling between the real and imaginary parts when the laser is operated well above threshold. The steady state mean intensity fluctuation C(0) and signal-to-noise ratio (SNR) are calculated. It is found that there is a maximum in SNR when there is a minimum in the fluctuation of laser system if the coupling coefficient between real and imaginary parts of the quantum noise equals zero.

Analytical and numerical studies for seiches in a closed basin with various geometric shapes of longitudinal section

Seiches are long-period standing waves with a unique period called a natural resonant period,during which the phenomenon of resonance occurs.The occurrence of resonance in coastal areas can cause destruction to surrounding natural and man-made structures.By determining the resonant period of a given basin,we can pinpoint the conditions that allow waves to achieve resonance.In this study,a mathematical model is developed from the shallow water equations to examine seiches and resonances in various types of closed basin.The developed model is solved analytically using the separation of variables method to determine the seiches’fundamental resonant periods.Comparisons between the analytical solutions and experimental measurements for resonant periods are also provided.It is shown that the analytical resonant period confirms the experimental data for closed basin of various geometric profiles.Using a finite volume method on a staggered grid,the model is solved numerically to simulate the wave profile when resonance phenomenon occurs.Through those numerical simulations,we also obtain the fundamental resonant period for each basin which agrees with the derived analytical period.

Dynamic Analysis of a Single-Column Platform

In this paper, a single-column structure used as well-head platform is studied. The loads of wave and current exerted on the single-column will be greatly reduced, therefore the cost of the structure will be decreased. The advantages of the single-column structure compared with ordinary jacket structure are explained. A dynamic analysis of this type of structure is made and some problems related to dynamic analysis are solved. In order to check the reliability of computation theory and programme, model tests have been carried out. However, as space is limited, the conclusion of tests will be introduced in another paper. Therefore, this type of structure is applicable for proctical engineering.

Multi-scale variability of water discharge and sediment load into the Bohai Sea from 1950 to 2011

This paper examines the changes in the time series of water discharge and sediment load of the Yellow River into the Bohai Sea. To determine the characteristics of abrupt changes and multi-scale periods of water discharge and sediment load, data from Lijin station were analyzed, and the resonance periods were then calculated. The Mann-Kendall test, order clustering, power-spectrum, and wavelet analysis were used to observe water discharge and sediment load into the sea over the last 62 years. The most significant abrupt change in water discharge into the sea occurred in 1985, and an abrupt change in sediment load happened in the same year. Significant decreases of 64.6% and 73.8% were observed in water discharge and sediment load, respectively, before 1985. More significant abrupt changes in water discharge and sediment load were observed in 1968 and 1996. The characteristics of water discharge and sediment load into the Bohai Sea show periodic oscillation at inter-annual and decadal scales. The main periods of water discharge are 9.14 years and 3.05 years, whereas the main periods of sediment load are 10.67 years, 4.27 years, and 2.78 years. The significant resonance periods between water discharge and sediment load are observed at the following temporal scales： 2.86 years, 4.44 years, and 13.33 years. Water discharge and sediment load started to decrease after 1970 and has decreased significantly since 1985 for several reasons. Firstly, the precipitation of the Yellow River drainage area has reduced since 1970. Secondly, large-scale human activities, such as the building of reservoirs and floodgates, have increased. Thirdly, water and soil conservation have taken effect since 1985.

Low-threshold wavelength-tunable ultraviolet vertical-cavity surface-emitting lasers from 376 to 409 nm

Optically pumped wavelength-tunable vertical-cavity surface-emitting lasers(VCSELs)operating in the ultraviolet A(UVA)spectrum were demonstrated.The VCSELs feature double dielectric distributed brag reflectors and a wedge-shaped cavity fabricated using the substrate transfer technique and laser lift off,resulting in a graded cavity length in one device.A resonant period gain structure is used in the InGaN/GaN multi-quantum well active region to enhance the coupling between the cavity mode field and the active layers.The optical field inside the cavity is modulated by the cavity length;thus,tunable lasing at different wavelengths is realized at different points of a single VCSEL chip.The lasing wavelength extends from 376 to 409 nm,covering most of the UVA band below the band gap of GaN.The threshold pumping power density of the UVA VCSELs at different wavelengths ranges from 383 to 466 kW/cm^(2),which is among the lowest values for ultraviolet(UV)VCSELs.This study is promising for the development of small-footprint,power-efficient UV light sources.

Topology and analysis of an N-stage cascaded polymer eight-port microring optical router with 7N channel wavelengths

Eight-port optical routers are widely used in cluster-mesh photonic networks-on-chip(No C). By using 24 groups of cross-coupling two-ring resonators, a 1-stage 8-port polymer optical router is proposed, which can optically route 7 channel wavelength data streams along definite path in two-dimensional(2D) plane. Under the selected 7 channel wavelengths, the insertion losses along all routing paths are within 0.02-0.58 d B, the maximum crosstalk of all routing operations is less than-39 d B, and the device footprint size is about 0.79 mm2. Then, a universal novel structure and routing scheme of N-stage cascaded 8-port optical router are presented, which contains 7N channel wavelengths. Because of the good scalability in wavelength, this device shows potential application of wideband signal routing in optical No C.

Two Kinds Equal Frequency Circuits to Achieve Locally Resonant Band Gap of a Circular Plate Attached Alternately by Piezoelectric Unimorphs

A circular thin plate is proposed for vibration attenuation,which is attached alternately by annular piezoelectric unimorphs with resonant shunt circuits.Two kinds of equal frequency resonant shunt circuits are designed to achieve an integrated locally resonant（LR）band gap（BG） with a much smaller transmission factor：（1） the structure is arrayed periodically while the resonant shunt circuits are aperiodic;（2） the resonant shunt circuits are periodic while the structure is aperiodic.The transmission factor curve is calculated,which is validated by the finite element method.Dependences of the LR BG performance upon the geometric and electric parameters are also analyzed.