Small-amplitude Period Oscillation Research Articles

Small-amplitude nonlinear periodic oscillations in a suspension bridge system

This paper deals with a coupled nonlinear beam-wave system, proposed by Lazer and McKenna, modeling a suspension bridge. By constructing a suitable Nash–Moser-type iteration scheme, the existence of small-amplitude nonlinear periodic oscillations, i.e., the existence of small-amplitude periodic solutions for the suspension bridge system, is obtained, and so is their local uniqueness.

Fluid-structure interaction of a rolling restrained body of revolution at high angles of attack

The current work investigates numerically rolling instabilities of a free-to-roll slender rigid-body of revolution placed in a wind tunnel at a high angle of attack. The resistance to the roll moment is represented by a linear torsion spring and equivalent linear damping representing friction in the bearings of a simulated wind tunnel model. The body is subjected to a three-dimensional, compressible, laminar flow. The full Navier-Stokes equations are solved using the second-order implicit finite difference Beam-Warming scheme, adapted to a curvilinear coordinate system, whereas the coupled structural second order equation of motion for roll is solved by a fourth-order Runge-Kutta method. The body consists of a 3.5-diameter tangent ogive forebody with a 7.0-diameter long cylindrical afterbody extending aft of the nose-body junction to x/D = 10.5. We describe in detail the investigation of three angles of attack 20°, 40°, and 65°, at a Reynolds number of 30 000 (based on body diameter) and a Mach number of 0.2. Three distinct configurations are investigated as follows: a fixed body, a free-to-roll body with a weak torsion spring, and a free-to-roll body with a strong torsion spring. For each angle of attack the free-to-roll configuration portrays a distinct and different behavior pattern, including bi-stable limit-cycle oscillations. The bifurcation structure incorporates both large and small amplitude periodic roll oscillations where the latter lose their periodicity with increasing stiffness of the restraining spring culminating with distinct quasiperiodic oscillations. We note that removal of an applied upstream disturbance for a restrained body does not change the magnitude or complexity of the oscillations or of the flow patterns along the body. Depending on structure characteristics and flow conditions even a small rolling moment coefficient at the relatively low angle of attack of 20° may lead to large amplitude resonant roll oscillations.

Effect of delay on a predator–prey model with parasitic infection

In the paper an eco-epidemic system with delay and parasitic infection in the prey is investigated. The conditions for asymptotic stability of steady states are derived and the length of the delay preserving the stability is also estimated. Further, the criterion for existence of Hopf-type small amplitude periodic oscillations of the predator and prey biomass is derived. Numerical results indicate that the delay does not affect the stability of the system in the process but makes all populations oscillate more intensively. In addition, the results show that the recovery makes the levels of the infected prey and the predator become lower but makes the sound prey higher in limit time.

The energetics of flow through a rapidly oscillating tube. Part 2. Application to an elliptical tube

In Part 1 of this work, we derived general asymptotic results for the three-dimensional flow field and energy fluxes for flow within a tube whose walls perform prescribed small-amplitude periodic oscillations of high frequency and large axial wavelength. In the current paper, we illustrate how these results can be applied to the case of flow through a finite-length axially non-uniform tube of elliptical cross-section – a model of flow in a Starling resistor. The results of numerical simulations for three model problems (an axially uniform tube under pressure–flux and pressure–pressure boundary conditions and an axially non-uniform tube with prescribed flux) with prescribed wall motion are compared with the theoretical predictions made in Part 1, each showing excellent agreement. When upstream and downstream pressures are prescribed, we show how the mean flux adjusts slowly under the action of Reynolds stresses using a multiple-scale analysis. We test the asymptotic expressions obtained for the mean energy transfer E from the flow to the wall over a period of the motion. In particular, the critical point at which E = 0 is predicted accurately: this point corresponds to energetically neutral oscillations, the condition which is relevant to the onset of global instability in the Starling resistor.

First R and I Lights and Their Photometric Analyses of GSC 02393–00680

Abstract We obtained complete $R$ and $I$ light curves of GSC 02393$-$00680 in 2008 and analyzed them with the 2003 version of the W–D code. It has been shown that GSC 02393$-$00680 is a W-type shallow contact binary system with a high mass ratio, $q =$ 1.600, and a degree of contact factor, $f =$ 5.0% ($\pm$1.3%). It will be a good example to check up on the TRO theory. A period investigation based on all available data suggests that the system has a small-amplitude period oscillation ($A_3 =$ 0$.^{\rm d}$0030; $T_3 =$ 1.92 years). This may indicate that it has a moderate-mass close third body, which is similar to XY Leo.

An Orbital Period Investigation of the Solar-Type Overcontact Binary V700 Cygni

Orbital period changes of the solar-type contact binary V700 Cyg were investigated based on one new eclipse time combined with the others collected from the literature. A cyclic oscillation, with a period of 39.1 yr and an amplitude of 0.$^{\rm d}$0197, was found to be superimposed on a secular period increase at a rate of $dP/dt$$=$$+$2.8 $\times$ 10$^{-8}{\rm d}$ yr$^{-1}$. Weak evidence indicates that there exists another small-amplitude period oscillation. The long-term period increase is in agreement with the conclusion that high-mass ratio W-type overcontact binaries usually show period increasing. This suggests that it may be in the stage controlled by thermal relaxation oscillations. Cyclic period oscillation can be explained as being the result of a light-travel time effect via the presence of an additional body. On the other hand, since both components of V700 Cyg are late-type stars, the period oscillation may be evidence of magnetic activity cycles of the two component stars.

An Orbital Period Investigation of the Semi-Detached Near-Contact Binary DD Monocerotis

The variations of the orbital period of the near-contact binary system DD Monocerotis were analyzed based on three new CCD times of light minimum together with others compiled from the literature. The general trend of the ($O-C$)$_1$ curve indicates a continuous period increase at a rate of $dP/dt$$=$$+$1.40 $\times$ 10$^{-7}$d yr$^{-1}$. Meanwhile, by using all available photoelectric and CCD data, a small-amplitude period oscillation ($A$$=$ 0.0039 d; $T=$ 18.8 yr) was found to be superimposed on the period increase. The continuous period increase is explained by mass transfer from the less-massive component to the more-massive one ($dM_{2}/dt$$=$$+$6.99 $\times$ 10$^{-8}M_{\odot}$yr$^{-1}$), which is consistent with the semi-detached configuration of the binary with a lobe-filling secondary discovered by Qian et al. (1997, A&AS, 125, 475). We showed that the timescale of the period change is close to the thermal time scale of the secondary, suggesting that DD Monocerotis is undergoing a case A mass transfer. The small-amplitude period oscillation reveals the presence of an extremely cool stellar companion ($M_3$sin${i^{\prime}}$$=$ 0.13$M_{\odot}$) in the system. Accumulation of photoelectric and CCD data will discover small-amplitude cyclic changes in many close binary stars in the future.

Complex Period Variations of the Neglected W UMa-type Binary System NY Lyrae

AbstractOrbital-period variations of the neglected W UMa-type binary star, NY Lyr, were analyzed based on two newly determined eclipse times together with the others compiled from the literature. A cyclic oscillation with a period of 82.1 yr and an amplitude of 0.0247 d was discovered to be superimposed on a continuous period increase (dP/dt = +1.33 × 10−7 d yr−1). After the long-term period increase and the large-amplitude cyclic oscillation were removed from the O–C diagram, the residuals suggest that there is another small-amplitude period oscillation (A4 = 0.0053 d, P4 = 19.4 years) in the orbital period changes. As in the cases of AH Cnc and AD Cnc, both the continuous period increase and the two cyclic period oscillations make NY Lyr an interesting system to study in the future. In order to understand the evolutionary state of the binary system, new photometric and spectroscopic observations and a careful investigation on those data are needed.

Effect of delay on nutrient cycling in phytoplankton–zooplankton interactions in estuarine system

Phytoplankton–zooplankton interactions with delayed nutrient cycling from senescence and mortality of phytoplankton in Hooghly-Matla estuarine system are considered in this present study. Average numbers of zooplankton interacting with phytoplankton follow Poisson distribution (random zooplankton grazing). The condition for asymptotic stability of steady state is derived. The length of the delay preserving the stability is also estimated. The criterion for existence of Hopf-type small amplitude periodic oscillations of phytoplankton and zooplankton biomass is derived. Delay in the decomposition process of dead phytoplankton from senescence and other mortalities do not affect the stability of the system. All analytical results are interpreted ecologically.

AD Cancri: A Shallow Contact Solar‐Type Eclipsing Binary and Evidence for a Dwarf Third Component and a 16 Year Magnetic Cycle

CCD photometric observations of AD Cancri obtained from 2000 March 7 to 2004 December 20 are presented. Variations of the light levels at the primary minimum and both maxima are found. Uniform solutions of four sets of photometric data were derived by using the Wilson-Devinney method. The solutions suggest that AD Cancri is a shallow W-type contact binary (f 8.3% +/- 1.3%) with a highmass ratio of 1/q = 0.770 +/- 0.002. The long-term variation of the light curve is explained by variable dark-spot models of the more massive component star with a possible 17 yr cycle. Our 13 times of light minimum over 5 years, including others collected from the literature, have been used for the period study. The complex period changes can be sorted into a long-term period increase at rate of dP/dt +(4.94 +/- 0.16) x 10(-7) days yr(-1), a 16.2 yr periodic component (A(3) = 0.0155 days), and a very small amplitude period oscillation (A(4) = 0.0051 days, P-4 = 6.6 yr). The existence of third light may indicate that there is a tertiary component in the binary system. Solving the four-band light curves of Samec & Bookmyer, it is found that the contribution of the tertiary component to the total light of the triple system increases with wavelength, which suggests that it is very cool and may be a very red main-sequence star. The small-amplitude period oscillation may be caused by the light-time effect of the cool tertiary component (M-3 similar to 0.41 M-circle dot). The 16.2 yr periodic component in the orbital period and the 17 yr cyclic activity of the dark spot on the more massive component both may reveal that the more massive component displays solar-type magnetic activity with a cycle length of about 16 yr.

CCD Photometric Study of the Contact Binary TX Cnc in the Young Open Cluster NGC 2632

Abstract TX Cnc is a member of the young open cluster NGC2632. In the present paper, four CCD epochs of light minimum and a complete $V$ light curve of TX Cnc are presented. A period investigation based on all available photoelectric or CCD data showed that it is superimposed on a long-term increase ($dP/dt=+3.97\times{10^{-8}}$d yr$^{-1}$), and weak evidence suggests that it includes a small-amplitude period oscillation ($A_3=0\rlap{.}^{\rm d} 0028$; $T_3=26.6$yr). The light curves in the $V$ band obtained in 2004 were analyzed with the 2003 version of the W-D code. It was shown that TX Cnc is an overcontact binary system with a degree of contact factor $f=24.8\% (\pm0.9\%)$. The absolute parameters of the system were calculated: $M_1=1.319\pm0.007M_{\odot}, M_2=0.600\pm0.01M_{\odot}$; $R_1=1.28\pm0.19R_{\odot}, R_2=0.91\pm0.13R_{\odot}$. TX Cnc may be on the TRO-controlled stage of the evolutionary scheme proposed by Qian (2001a, b, 2003a), and may contain an invisible tertiary component ($m_3\approx0.097M_{\odot}$). If this is true, the tertiary component has played an important role in the formation and evolution of TX Cnc by removing angular momentum from the central system (Pribulla & Rucinski 2006). In this way the contact binary configuration can be formed in the shortlife time of a young open cluster via AML.

Deep, Low Mass Ratio Overcontact Binary Systems. VI. AH Cancri in the Old Open Cluster M67

CCD photometric light curves in the B and V bands obtained in 2001 and in the V band obtained in 2002 of AH Cnc in the old open cluster M67 are presented. It is shown that AH Cnc is a total-eclipsing binary and its light curves correspond to a typical A type according to Binnendijk's classification. The variations of the light curve around the primary minimum and second maximum were found. Our nine epochs of light minimum monitored from 2001 to 2005, including others collected from the literature, were used to create the first study of the period changes of the binary system. A cyclic oscillation with a period of 36.5 yr and an amplitude of 0.0237 days was discovered to be superposed on a continuous period increase (dP/dt = 3: 99; 10(-7) days yr(-1)). Weak evidence indicates that there exists another small-amplitude period oscillation (A(4) = 0: 0035 days, P-4 = 7.75 yr). The symmetric light curves in the B and V bands obtained in 2001 were analyzed with the 2003 version of the Wilson-Devinney code. It is confirmed that AH Cnc is a deep overcontact binary system with a high degree of overcontact f = 58.5%+/- 4.5% and a low mass ratio of q = 0.1682 +/- 0.0012. The existence of the third light and the cyclic period oscillation both may suggest that AH Cnc is a triple system containing an unseen third body. The tertiary component may have played an important role in the origin of the overcontact binary star by removing angular momentum from the central system, which would cause it to have a short initial orbital period and thus evolve into an overcontact configuration by angular momentum loss. The long-term period increase can be interpreted as a mass transfer from the less massive component to the more massive one. As the orbital period increases, the decrease of the mass ratio will cause it finally to evolve into a single rapid-rotating star when the system meets the more familiar criterion that the orbital angular momentum be less than 3 times the total spin angular momentum. Therefore, AH Cnc may be a progenitor of the blue straggler stars in M67.

Simplest piston problem. II. Inelastic collisions

We study the dynamics of three particles in a finite interval, in which two light particles are separated by a heavy "piston," with elastic collisions between particles but inelastic collisions between the light particles and the interval ends. A symmetry breaking occurs in which the piston migrates near one end of the interval and performs small-amplitude periodic oscillations on a logarithmic time scale. The properties of this dissipative limit cycle can be understood simply in terms of a effective restitution coefficient picture. Many dynamical features of the three-particle system closely resemble those of the many-body inelastic piston problem.

YY Geminorum: A Very Late Type Close Binary with Possible Magnetic Stellar Wind

The O-C curve of the very late type close binary (dMle+dMle)YY Geminorum is formed and analyzed based on all available times of minimum light. It is found that the orbital period shows a secular decrease with rate dP/dt = -1.08x10(-8) days yr(-1). There is weak evidence indicating that a small-amplitude period oscillation is superposed on the period decrease. YY Gem is a well-detached eclipsing binary containing two very active components. The secular period decrease may suggest that the system is undergoing secular mass and angular momentum loss via a magnetic stellar wind. If the small-amplitude oscillation is real, it can be explained either by magnetic activity cycles or by the presence of a giant planet or brown dwarf.

Stability and oscillations in a resource-based model of two interacting species with nutrient cycling

In the present study we consider a resource-based model of autotroph–herbivore interaction with nutrient cycling in which the amount of autotroph biomass consumed by the herbivore is assumed to follow a Holling type-II function. We have derived the criteria for existence of a unique feasible steady state of the system. It is shown that there is a threshold value of the external nutrient input at which the system may exhibit Hopf type small amplitude periodic oscillations and above which large amplitude periodic oscillations occur. Finally, we have verified the analytical results through computer simulations.

Bifurcations of numerically simulated thermocapillary flows in axially symmetric float zones

Two-dimensional time-dependent flow in an axially symmetric liquid bridge of Si at zero g with flat ends at the melting temperature θM of Si driven by motions of the free, lateral surface is studied numerically. The system is energized by a ring heater H at temperature θH>θM that surrounds the midcircle m of the bridge. The radiation from H creates a ∇σS(δS is surface temperature), which in turn creates a ∇σ (σ is surface tension) that drives oscillations of its free surface and thermocapillary flow in the bridge. For aspect ratios A of (1)/(2) , (2)/(3) , and 1 it is found that as the relevant control parameter Δθ≡θH−θM is increased, the flow in the bridge bifurcates differently depending on A. For A= (1)/(2) and (2)/(3) the flow bifurcates from steady state to small-amplitude periodic oscillations, breaking symmetry, to large-amplitude, period-tripled ones, to their subharmonics, then becomes aperiodic, and finally chaotic. Toroidal cells in the bridge oscillate to-and-fro along its axis as a compound pendulum, and they change in shape, number, and strength of rotation over the period of an oscillation. For A=1, the bifurcation is from steady state to asymmetric oscillations steadily growing in amplitude as Δθ increases and then directly to semiperiodic ones and thence to chaos. Agreement between the present results and recent experimental and numerical ones, are also described.