Asymptotic Oscillation Research Articles

Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on $${\mathbb R^2}$$

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy solutions converge to a harmonic map as t goes to infinity (asymptotic stability), extending previous work down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m=3, involving (among other tools) a "normal form" for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schroedinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m=2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even "eternal oscillation".

Surf zone diffusivity on a rip‐channeled beach

Absolute and relative diffusivity are measured on a rip‐channeled beach using 30 position‐tracking drifters released in clusters (4–12 drifters) deployed on 7 days with different wave forcing and tidal elevations at Sand City, Monterey Bay, California. Diffusivity and dispersion were found to be larger on days with rip current flow patterns and larger waves. Rip currents cause material to diffuse quickly for t < 90 s in the cross shore (κxx = 5.4–6.1 m2/s) before decreasing to an asymptotic oscillation (κxx = 0.9–2.2 m2/s), while alongshore material diffusion is initially (t < 170 s) smaller than cross‐shore diffusion and asymptotes at a larger value (κyy = 2.8–3.8 m2/s). The cross‐ and alongshore absolute diffusivity modulate at ∼300 s corresponding to the average circulation time for a rip current. Two‐particle relative dispersion (Dp) grows like Dp2 ∼ t4/3 and the relative diffusivity (Kp) is scale dependent, Kp ∼ d0.2 (d is particle separation). Cluster relative diffusion (Ke) ranged from 1.0 to 4.5 m2/s and cluster relative dispersions (De) are significantly correlated with two‐particle relative dispersions (DS) [R2 > 0.9]. Two independent methods are used to measure the small‐scale turbulent diffusion contribution (kxy), which are found significantly correlated (R2 = 0.95) with each other and calculated surf zone wave breaking induced turbulent eddy viscosity. Here kxy is small relative to the total dispersion (Ke/kxy = 3–30), indicating that the shear flow is the primary mechanism responsible for dispersion in a rip current system.

Total Stability Properties Based on Fixed Point Theory for a Class of Hybrid Dynamic Systems

AbstractRobust stability results for nominally linear hybrid systems are obtained from total stability theorems for purely continuous-time and discrete-time systems by using the powerful tool of fixed point theory. The class of hybrid systems dealt consists, in general, of coupled continuous-time and digital systems subject to state perturbations whose nominal (i.e., unperturbed) parts are linear and, in general, time-varying. The obtained sufficient conditions on robust stability under a wide class of harmless perturbations are dependent on the values of the parameters defining the over-bounding functions of those perturbations. The weakness of the coupling dynamics in terms of norm among the analog and digital substates of the whole dynamic system guarantees the total stability provided that the corresponding uncoupled nominal subsystems are both exponentially stable. Fixed point stability theory is used for the proofs of stability. A generalization of that result is given for the case that sampling is not uniform. The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time. The existence of a fixed point for the sampled state-trajectory solution at sampling instants guarantees the existence of a fixed point of an extended auxiliary discrete system and the existence of a global asymptotic attractor of the solutions which is either a fixed point or a limit "Equation missing" globally stable asymptotic oscillation.

Asymptotic behavior and oscillation of functional differential equations

Asymptotic relations between the solutions of a linear autonomous functional differential equation and the solutions of the corresponding perturbed equation are established. In the scalar case, it is shown that the existence of a nonoscillatory solution of the perturbed equation often implies the existence of a real eigenvalue of the limiting equation. The proofs are based on a recent Perron type theorem for functional differential equations.

R modes of slowly pulsating B stars

Slowly pulsating B (SPB) stars are g-mode pulsators in main-sequence stages with mass ranging from M∼ 3 to ∼8 M⊙. In this paper, we examine pulsational stability of low-m r modes in SPB stars by calculating fully non-adiabatic oscillations of uniformly rotating stars, where m is an integer representing the azimuthal wavenumber around the rotation axis. r modes are rotationally induced, non-axisymmetric, oscillation modes, whose oscillation frequency strongly depends on the rotation frequency Ω of the star. They are conveniently classified by using two integer indices m and l′≥ ∼m∼ that define the asymptotic oscillation frequency 2mΩ/[l′(l′+ 1)] in the limit of Ω→ 0. We confirm Savonije's and Townsend's finding that low-m, high-radial order, odd r modes with l′=m in SPB stars are excited by the same iron opacity bump mechanism that excites low-frequency g modes of the variables, when the rotation frequency Ω is sufficiently high. No even r modes with low m are found to be pulsationally unstable. Since the surface pattern of the temperature perturbation of odd modes is antisymmetric about the equator of the star, observed photometric amplitudes caused by the unstable odd r modes with l′=m are strongly dependent on the inclination angle between the axis of rotation and the line of sight. Applying the wave-meanflow interaction formalism to non-adiabatic r modes in rapidly rotating SPB models, we find that because of the r φ component of the Reynolds stress and the radial transport of the eddy fluctuation of density in the rotating star, the surface rotation is accelerated by the forcing due to the low l′=m unstable r modes. We suggest that the amount of angular momentum redistribution in the surface region of the stars can be comparable to that needed to sustain decretion discs found in Be systems. We make a brief comparison between non-adiabatic r-mode calculations done with and without the traditional approximation. In the traditional approximation, the local horizontal component of the rotation vector Ω is ignored in the momentum conservation equation, which makes it possible to represent the angular dependence of an oscillation mode using a single Hough function. We find that the oscillation frequencies of low-m r modes computed with and without the traditional approximation are qualitatively in good agreement. We also find that the pulsational instability of r modes in the traditional approximation appears weaker than that without the approximation.

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form x n+1 = ∑ i=0 i≠j,j−1 k x n−i +x n−j+1 x n−j +1 ∑ i=0 k x n−i , j=1,2,…,k, where n ∈{0,1,…}, k ∈{1,2,…} and the initial values x − k , x − k +1 ,…, x 0 are positive real numbers. For these difference equations, we investigate the oscillatory behavior of the positive solutions and prove that the unique equilibrium x ̄ =1 is globally asymptotically stable.

Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form x n+1= ∑ i=0 i≠j,j−1 k x n−i+x n−j+1x n−j+1 ∑ i=0 k x n−i , j=1,2,…,k, where n∈{0,1,…}, k∈{1,2,…} and the initial values x − k , x − k+1 ,…, x 0 are positive real numbers. For these difference equations, we investigate the oscillatory behavior of the positive solutions and prove that the unique equilibrium x ̄ =1 is globally asymptotically stable.

Asymptotic Behavior and Oscillation of Delay Partial Difference Equations with Positive and Negative Coefficients

We obtain sufficient conditions for the oscillation of all solutions of the linear partial difference equations with positive and negative coefficients of the form A m-1,n + A m,n-1 - A mn + pA m+k - qA m+k' = 0, n+1 n+l' and A m-1,n + A m,n-1 - A mh + p mn A m+k - q mn A m+k' = 0, n+l n+l' where m,n = 0, 1,..., and k, k', l', l are nonnegative integers p, q ∈ (0, oo), and coefficients {q mn } and {p mn } are sequences of nonnegative real numbers. In this paper A mn = A m,n .

Time Correlations and Diffusion of a Conservative Forced Pendulum

The time correlations and diffusion of chaotic orbits in a periodically forced pendulum without friction are studied. The pendulum exhibits a Poincare section with period T at times t = jT (j =0 , 1, 2, ··· ). The time-correlation function C(t) ≡� p(t)p(0)� of the angular velocity p(t) oscillates with period T ,e ven ast →∞ , since the average quantities of the system have a periodicity with period T , due to the periodic external force. Studying the approach to asymptotic oscillation, we find that the time-correlation function C(t) exhibits an inverse power decay t −(β−1) (1 <β <2), where there exist islands of accelerator-mode tori. Then, it is also shown that the power spectrum Ip(ω )o fp(t) obeys an inverse power law ω −(2−β) for small frequency ω � 2π/T. We also calculate the mean square displacement σ 2 (n) of the angular variable qn ≡ q(nT ) on the Poincare section, and show that σ 2 (n) ∝ n ζ (ζ =3 − β) for n →∞ , leading to anomalous diffusion with 1 <ζ< 2.

Burst Oscillation Periods from 4U 1636−53: A Constraint on the Binary Doppler Modulation

The burst oscillations seen during type I X-ray bursts from low-mass X-ray binaries typically evolve in period toward an asymptotic limit that likely reflects the spin of the underlying neutron star. If the underlying period is stable enough, measurement of it at different orbital phases may allow a detection of the Doppler modulation caused by the motion of the neutron star with respect to the center of mass of the binary system. Testing this hypothesis requires enough X-ray bursts and an accurate optical ephemeris to determine the binary phases at which they occurred. We present here a study of the distribution of asymptotic burst oscillation periods for a sample of 26 bursts from 4U 1636-53 observed with the Rossi X-Ray Timing Explorer. The burst sample includes both archival and proprietary data and spans more than 4.5 yr. We also present new optical light curves of V801 Arae, the optical counterpart of 4U 1636-53, obtained during 1998-2001. We use these optical data to refine the binary period measured by Augusteijn et al. to 3.7931206(152) hr. We show that a subset of ~70% of the bursts form a tightly clustered distribution of asymptotic periods consistent with a period stability of ~1 × 10-4. The tightness of this distribution, made up of bursts spanning more than 4 yr in time, suggests that the underlying period is highly stable, with a time to change the period of ~3 × 104 yr. This is comparable to similar numbers derived for X-ray pulsars. We investigate the period and orbital phase data for our burst sample and show that it is consistent with binary motion of the neutron star, with vns sin i < 38 and 50 km s-1 at 90% and 99% confidence, respectively. We use this limit as well as previous radial velocity data to constrain the binary geometry and component masses in 4U 1636-53. Our results suggest that unless the neutron star is significantly more massive than 1.4 M☉, the secondary is unlikely to have a mass as large as 0.36 M☉, the mass estimated assuming that it is a main-sequence star that fills its Roche lobe. We show that a factor of ~3 increase in the number of bursts with asymptotic period measurements should allow a detection of the neutron star velocity.

The Long-Term Stability of Oscillations during Thermonuclear X-Ray Bursts: Constraining the Binary X-Ray Mass Function

We report on the long-term stability of the millisecond oscillations observed with the Rossi X-Ray Timing Explorer during thermonuclear X-ray bursts from the low-mass X-ray binaries (LMXBs) 4U 1728-34 and 4U 1636-53. We show that bursts from 4U 1728-34 spanning more than 1.5 yr have observed asymptotic oscillation periods that are within 0.2 μs of each other, well within the magnitude that could be produced by the orbital motion of the neutron star in a canonical LMXB. This stability implies a timescale to change the oscillation period of more than 23,000 yr in this system and suggests a highly stable process, such as stellar rotation, as the mechanism producing the oscillations. For 4U 1636-53, which has an orbital period of 3.8 hr, we show that the offsets in the asymptotic oscillation periods from three different bursts can be consistently interpreted as due to the orbital velocity of the neutron star with vsini/c ≈ 4.25 × 10−4. An updated optical ephemeris for the epoch of maximum light from V801 Arae would provide a strong test of this interpretation. We discuss the constraints on the X-ray mass function, which in principle can be derived using this technique.

Order and Oscillation in Delay Differential Systems

We consider asymptotic stability and oscillation of neutral delay differential systems. Using positivity and domination arguments we obtain sufficient conditions for oscillation. Furthermore, we prove some results concerning (strict) positivity of the solutions.

Asymptotic Behavior and Oscillation in Higher Order Nonlinear Differential Equations with Retarded Arguments

Asymptotic Behavior and Oscillation in Higher Order Nonlinear Differential Equations with Retarded Arguments

Asymptotic Periodicity, Monotonicity, and Oscillation of Solutions of Scalar Neutral Functional Differential Equations

We consider the periodic scalar neutral functional differential equation (d/dt)[x(t)−c(t)x(t−τ)]=−h(t,x(t))+h(t−σ,x(t−σ)), wherecis continuously differentiable,his increasing in its second argument, and bothcandhare 1-periodic in thet-variable. The two time-lags τ and σ are not required to be the same. It is shown that, under certain conditions, (i) the set of 1-periodic solutions is an ordered arc and each solution is convergent to a periodic solution; (ii) the asymptotic and oscillatory behaviors of each solution are completely classified in terms of the value of the first integral at the initial condition.

Asymptotic behavior and oscillation of difference equations of volterra type

The asymptotic behavior and oscillation of solutions of difference equation D my(n)+ ∑ s=0 n−1 a(n,s)f(s,y(s))=h(n) , are studied. Examples, which illustrate the importance of results, are included.

Asymptotic Behavior and Oscillation of Classes of Integrodifferential Equations

Asymptotic Behavior and Oscillation of Classes of Integrodifferential Equations

Asymptotic behaviour and oscillation of classes of integrodifferential equations

Under some conditions on the integrodifferential equations \[ y ¨ ( t ) + ∫ 0 t k ( t − s ) y ( s ) d s + φ ( t ) ∫ 0 t K ( t − s ) y ˙ ( s ) d s = f [ t , y ( t ) , y ˙ ( t ) , ∫ 0 t g ( t , s , y ( s ) , y ˙ ( s ) ) d s ] , t ≥ 0 , \ddot y\left ( t \right ) + \int _0^t {k\left ( {t - s} \right )y\left ( s \right )ds + \varphi \left ( t \right )} \int _0^t {K\left ( {t - s} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _0^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ]} ,\quad t \geq 0, \] , \[ y ¨ ( t ) + ∫ 1 t k ( t s ) y ( s ) 1 s d s + φ ( t ) ∫ 1 t K ( t s ) y ˙ ( s ) d s = f [ t , y ( t ) , y ˙ ( t ) , ∫ 1 t g ( t , s , y ( s ) , y ˙ ( s ) ) d s ] , t ≥ 1 , \ddot y\left ( t \right ) + \int _1^t {k\left ( {\frac {t}{s}} \right )y\left ( s \right )} \frac {1}{s}ds + \varphi \left ( t \right )\int _1^t {K\left ( {\frac {t}{s}} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _1^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ],\quad t \geq 1,} \] , the explicit asymptote of solutions is proved to be y ( t ) = A sin ⁡ ( ω t + δ ) y\left ( t \right ) = A\sin \left ( {\omega t + \delta } \right ) as t → ∞ t \to \infty . From this asymptote, the oscillatory behavior of the equations, the limit of the amplitudes, and the limit of the distance between consecutive zeros of the solutions are evident. Their definite values are also determined.

Visual fixation and tracking by flies: Mathematical properties of simple control systems

The study of the orientation behavior of flies requires the consideration of a few simple control systems for fixation and tracking. In this paper two such control systems are analyzed, in terms of the corresponding difference and differential equations. The first control system corrects with a delay ? the angular trajectory proportionally to the error angle ?; the second control system also corrects proportionally to the error angle ? but only when the absolute value of ? is increasing. The differential equations are $$\mathop \psi \limits^ \circ (t) = - \alpha '\psi (t - \varepsilon ) + A'$$ and $$\mathop \psi \limits^o (t) = - \alpha '\psi (t - \varepsilon ) \cdot u[\psi (t - \varepsilon )\mathop \psi \limits^o (t - \varepsilon )] + A'$$ u[ ] being the step function (u[x]=1 if x>0, otherwise u[x]=0). Under suitable restrictions on the parameters it is proved (a) that the difference equations $$x_{n + 1} = x_n - \alpha x_n + A$$ and $$x_{n + 2} = x_{n + 1} - \alpha x_{n + 1} \cdot u[x_{n + 1} \cdot (x_{n + 1} - x_n )] + A,$$ which can be associated to Eqs. (0) and (*), are "asymptotically equivalent" (for large n) if the time scale is "smoothed" over two time units and (b) that the second equation, with 0<?<2, always converge to a set of oscillating solutions of period 2 for arbitrary initial conditions. Numerical simulations show that the delay-differential equations behave in a similar way. We have also demonstrated with computer simulations that both control systems can satisfactorily predict the 3-D trajectory of a fly chasing another fly. The main biological implications of the analysis are: (1) The two control systems are practically equivalent descriptions of the fly's control of flight on a "coarse" time scale (2 times the fly's delay), consistently with an earlier more general derivation of Eq. (0) (Poggio and Reichardt, 1973). (2) On a fine time scale the second control system is characterized by an asymptotic oscillation with period twice the fly's delay. It is conjectured that a wide range of control systems of the same general type must have a similar oscillatory behavior. Finally, we predict the existence of asymptotic oscillations in the angular trajectory and the torque of tracking flies (if a control system of the second type is involved to a significant extent). Such oscillations should have a basic period of twice the effective reaction delay, and should be best detectable outside the binocular region. Closed loop experiments and the analysis of free flight trajectories may provide critical tests of this prediction.

Asymptotic oscillation hypothesis

The observed persistence of threshold effects into the Regge asymptotic region of single-particle inclusive cross sections can be accommodated in a natural fashion by complex Regge poles which, if factorizable, will produce asymptotic damped oscillations in a wide variety of other observable quantities. It is shown that a single pair of such poles, the imaginary part of whose position in the complex J plane is of the order ± 1, can qualitatively explain various anomalies that have been observed in high energy phenomena. Predictions are made for future measurements.

Asymptotic oscillation results for solutions to first-order nonlinear differential-difference equations of advanced type

Asymptotic oscillation results for solutions to first-order nonlinear differential-difference equations of advanced type