Quasiperiodic motions in superquadratic time-periodic potentials

It is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time-periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most (in a precise sense) motions are in fact quasiperiodic. The class of potentials includes, as very particular examples, the exponential, polynomial and much more. This extends earlier results and gives an answer to a problem posed by Littlewood in the mid 1960's. Along the way machinery is developed for estimating the action-angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived.     

Trapping of quantum particles for a class of time-periodic potentials. A semi-classical approach

We study the time evolution for Schrödinger operators with time-periodic potentials when the classical equations of motion possess periodic orbits. We exhibit a class of time-periodic potentials such that for initial states suitably localized around these periodic orbits, then at the dominant order of the semi-classical approximation, the system is trapped forever at sufficiently large frequency. An estimation of the correction to the semi-classical approximation is given, which yields a minimum “trapping time” for these systems.     

Some comments on the Mindlin-Goodman method of solving vibrations problems with time dependent boundary conditions

Experimental evidence is presented for chaotic type non-periodic motions of a deterministic magnetoelastic oscillator. These motions are analogous to solutions in non-linear dynamic systems possessing what have been called “strange attractors”. In the experiments described below a ferromagnetic beam buckled between two magnets undergoes forced oscillations. Although the applied force is sinusoidal, nevertheless bounded, non-periodic, apparently chaotic motions result due to jumps between two or three stable equilibrium positions. A frequency analysis of the motion shows a broad spectrum of frequencies below the driving frequency. Also the distribution of zero crossing times shows a broad spectrum of times greater than the forcing period. The driving amplitude and frequency parameters required for these non-periodic motions are determined experimentally. A continuum model based on linear elastic and non-linear magnetic forces is developed and it is shown that this can be reduced to a single degree of freedom oscillator which exhibits chaotic solutions very similar to those observed experimentally. Thus, both experimental and theoretical evidence for the existence of a strange attractor in a deterministic dynamical system is presented.     

Model for modulated and chaotic waves in zero-Prandtl-number rotating convection

The effects of time-periodic forcing in a few-mode model for zero-Prandtl-number convection with rigid body rotation is investigated. The time-periodic modulation of the rotation rate about the vertical axis and gravity modulation are considered separately. In the presence of periodic variation of the rotation rate, the model shows modulated waves with a band of frequencies. The increase in the external forcing amplitude widens the frequency band of the modulated waves, which ultimately leads to temporally chaotic waves. The gravity modulation, on the other hand, with small frequencies, destroys the quasiperiodic waves at the onset and leads to chaos through intermittency. The spectral power density shows more power to a band of frequencies in the case of periodic modulation of the rotation rate. In the case of externally imposed vertical vibration, the spectral density has more power at lower frequencies. The two types of forcing show different routes to chaos.      

Ratchet motion and current reversal of coupled Brownian motors in pulsating symmetric potentials

In this study, we investigate the collective directed transport of coupled Brownian particles in spatially symmetric periodic potentials under time-periodic pulsating modulations. We find that the coupling between two particles can induce symmetry breaking and consequently collective directed motion. Moreover, the direction of motion can be reversed under certain conditions. The dependence of directed current on various parameters is systematically studied. reverse motion can be achieved by modulating the coupling free length and the phase shift of the pulsating potential. The dynamical mechanism of these transport properties is understood in terms of the effective-potential theory and the space-time transformation invariance. The directed transport of coupled Brownian motors can be manipulated and optimized by adjusting the coupling strength, pulsating frequency, or noise intensity.     

A model of one-dimensional metal (the dynamical solution)

A simple method of obtaining dynamical solutions of a model of one-dimensional metal is described. Using this method a new class of dynamical time-periodic solutions of breather type is obtained.I would like to express my gratitude to the referee for many useful comments and suggestions.     

Directed transport of a Brownian particle in a periodically tapered tube

The problem of the motion of a Brownian particle in a periodically tapered tube induced by a time-periodic longitudinal force with zero mean is considered. Under the action of this force, the particle is shown to drift in a direction opposite to the constant load force applied to it. Analytical solutions for the drift velocity, the stopping force (the load causing the effect to disappear), and the efficiency of converting the energy introduced by perturbations into directed motion have been obtained at a large amplitude of the driving force, when the effect being discussed is maximal. In the range of its applicability extending from zero to asymptotically large force switching frequencies (proportional to the amplitude of the driving force), these solutions are in good agreement with the results of Brownian dynamics simulations.     

Intraband discrete breathers in disordered nonlinear systems. I. Delocalization

The existence of Discrete Breathers or DBs (also called Intrinsic Localized Modes or ILMs) and multibreathers, is investigated in a simple one-dimensional chain of random anharmonic oscillators with quartic potentials coupled by springs. When the breather frequency is outside and above the linearized (phonon) spectrum, the existence theorems and numerical methods previously used in periodic nonlinear models for finding time-periodic and spatially localized solutions, hold identically in random nonlinear systems. These solutions are extraband discrete breathers (EDBs). When the frequencies penetrate inside the linearized spectrum, the existence theorems do not hold. Our numerical investigations demonstrate that the strict continuation of (localized) EDBs as intraband discrete breathers (IDBs) is impossible because of cascades of bifurcations generating many discontinuities. A detailed analysis of these bifurcations for small systems with increasing sizes, shows that only a relatively small subset of the spatially extended multibreathers can be strictly continued while their frequency varies inside the phonon spectrum. We propose an ansatz for finding the coding sequences of these solutions and continuing safely these multibreathers in finite systems of any size. This continuation ends at a lower limit frequency where the solution annihilates through a bifurcation with another multibreather. A smaller subset of these multibreather solutions can be continued to amplitude zero and become linear localized modes at this limit. Conversely, any linear localized mode can be continued when increasing its frequency as an extended multibreather. Extrapolation of these results to infinite systems yields the main conclusion of this first part which is that nonlinearity in disordered systems (with localized eigenmodes only) restores their capability of energy transportation by generating infinitely many spatially extended time-periodic solutions. This approach yields mainly spatially extended solutions, except sometimes at their bifurcation points. In the second part of this work, which is presented in our next article, we develop an accurate method for calculating in situ localized IDBs.     

Solution of the Hartree-Fock equations

We suggest to use the Newton iteration method for constructing a (locally unique) solution of the atomic and nuclear Hartree-Fock equations for an arbitrary number of particles. Our proposal is based on a theorem by Kantorovi and rests on the following points: 1) the two-body potential must satisfy a boundedness condition; 2) the zero-order approximation, used to start the iteration sequence, must satisfy certain conditions, to be proved numerically. Condition 1) holds, for instance, for all local potentials, defined by a bounded function and for a class of nonlocal potentials; it does not hold for local potentials, behaving as 1/r near the origin.This work has been supported in part by Istituto Nazionale di Fisica Nucleare (Sezione di Catania) and by Centro Siciliano di Fisica Nucleare e di Struttura della Materia (Catania).     

Dielectrophoresis and deformation of a liquid drop in a non-uniform, axisymmetric AC electric field

Analytical theory for the dielectrophoresis and deformation of a leaky dielectric drop, suspended in a leaky dielectric medium, subjected to non-uniform, axisymmetric Alternating Current (AC) fields is presented in the small deformation limit. The applied field is assumed to be a combination of a uniform part and a quadrupole component. The analysis shows that the magnitude and the sign of the steady and time-periodic dielectrophoretic velocity depend upon the frequency of the applied voltage. The frequency of oscillatory motion is twice that of the applied frequency and the phase lag is a consequence of charge dynamics. A deformed drop under non-uniform axisymmetric AC fields admits Legendre modes l = 2, 3, 4 . The deformation has a frequency-dependent steady and time-periodic parts due to charge and interface dynamics. The steady deformation can be zero at a certain critical frequency in leaky dielectric systems. The time-periodic deformation also has a frequency which is twice the frequency of the applied voltage. In perfect dielectric systems, unlike the steady state deformation which is a balance of Maxwell and curvature stresses, the time-periodic deformation additionally includes viscous stresses associated with the oscillatory shape changes of the drop. A consequence of this effect is a phase lag that is dependent on the charge and interface hydrodynamics and a lag of π/2 at high frequencies. It also results in vanishing amplitude of the oscillatory deformation at high frequencies. The study should lead to a better understanding of dielectrophoresis under non-uniform axisymmetric AC fields and better electrode design to affect drop breakup.     

Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems

Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtain simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved: if all moments of the initial distribution function are bounded by the corresponding moments of the MaxwellianA exp(−Bv ²), then all moments of the solution are bounded by the corresponding moments of the other MaxwellianA ₁ exp[−B ₁(t)v ²] for anyt > 0; moreoverB(t) = const for hard spheres. An estimate for a collision frequency is also obtained.     

Motion of spiral tip driven by local forcing in excitable media

We study the motion of a spiral wave controlled by a local periodic forcing imposed on a region around the spiral tip in an excitable medium. Three types of trajectories of spiral tip are observed： the epicycloid-like meandering, the resonant drift, and the hypocycloid-like meandering. The frequency of the spiral is sensitive to the local periodic forcing. The dependency of spiral frequency on the amplitude and size of local periodic forcing are presented. In addition, we show how the drift speed and direction are adjusted by the amplitude and phase of local periodic forcing, which is consistent with a theoretical analysis based on the weak deformation approximation.     

Non-linear vibrations of three-layer beams with viscoelastic cores I. Theory

Approximate equations of motion are developed for large amplitude motions of three-layer axially restrained unsymmetrical beams with viscoelastic cores. The external force consists of a constant plus an oscillatory term. The combination of this form of forcing and the large amplitude motions cause the beam to respond at multiples of the forcing frequency. This can lead to difficulties in the complex modulus approach to viscoelasticity. These are overcome here through use of hereditary integrals and their relationships with complex moduli. Theoretical results on the frequency response of clamped, symmetrical beams are compared with earlier experimental work. On the whole, reasonable agreement is found.     

On the Steady Compressible Navier–Stokes–Fourier System

We study the motion of the steady compressible heat conducting viscous fluid in a bounded three dimensional domain governed by the compressible Navier–Stokes–Fourier system. Our main result is the existence of a weak solution to these equations for arbitrarily large data. A key element of the proof is a special approximation of the original system guaranteeing pointwise uniform boundedness of the density as well as the positiveness of the temperature. Therefore the passage to the limit omits tedious technical tricks required by the standard theory. Basic estimates on the solutions are possible to obtain by a suitable choice of physically reasonable boundary conditions.     

Bifurcation structures of periodically forced oscillators

A theoretical investigation of bifurcation structures of periodically forced oscillators is presented. In the plane of forcing frequency and amplitude, subharmonic entrainment occurs in v-shaped (Arnol'd) tongues, or entrainment bands, for small forcing amplitudes. These tongues terminate at higher forcing amplitudes. Between these two limits, individual tongues fit together to form a global bifurcation structure. The regime in which the forcing amplitude is much smaller than the amplitude of the limit cycle is first examined. Using the method of multiple time scales, expressions for solutions on the invariant torus, widths of Arnol'd tongues, and Liapunov exponents of periodic orbits are derived. Next, the regime of moderate to large forcing amplitudes is examined through studying a periodically forced Hopf bifurcation. In this case the forcing amplitude and the amplitude of the limit cycle can be of the same order of magnitude. From a study of the normal forms for this case, it is shown how Arnol'd tongues terminate and how complicated bifurcation structures are associated with strong resonances. Aspects of model and experimental chemical systems that show some of the phenomena predicted from the above theoretical results are mentioned.     

Pattern transition of two-dimensional Faraday waves at an extremely shallow depth

In this paper, we experimentally investigate the pattern transition of two-dimensional Faraday waves at an extremely shallow depth in a Hele-Shaw cell. Several patterns of Faraday waves are observed, which have some significant differences in wave profile, wave height and wave length. It is found that, in a wide range of the forcing frequency f, there always exists a region of the acceleration amplitude A, in which there exist the so-called hysteretic jumps between different patterns of Faraday waves. All of these experimental observations could enrich our knowledges about the Faraday waves and would be helpful to the further theoretical studies on the related topic in future.     

Brownian motion in a solitary potential well in a bounded solid structure

The motion of a heavy Brownian particle in a low-dimensional bounded solid structure under the effect of a phonon’s excitation fluctuations is considered. Because of the finiteness of the system, the fluctuation spectrum has zero spectral density at zero frequency. The effect of this kind of noise, which is conditionally called “green” noise, is studied both analytically by using the averaging method and numerically on the basis of predictor-corrector algorithms. The effective potential is introduced, and its form is shown to govern the particle dynamics. Considering a Gaussian potential well (a trap) as an example, it is demonstrated that green noise leads to abrupt phase transitions in the system as a result of very small parameter variations (a catastrophe-type effect). The results are compared with the case of white noise in an unbounded structure. From numerical calculations, it follows that the boundedness of the structure, which changes the noise spectrum, favors a considerable increase in the lifetime of the particle in the trap.     

Spectral element method in time for rapidly actuated systems

In this paper, the spectral element (SE) method is applied in time to find the entire time-periodic or transient solution of time-dependent differential equations. The time-periodic solution is computed by enforcing periodicity of the element set. Of particular interest are periodic forcing functions possessing high frequency content. To maintain the spectral accuracy for such forcing functions, an h-refinement scheme is employed near the semi-discontinuity without increasing the number of degrees of freedom.Time discretization by spectral elements is applied initially to a standard form of a set of linear, first-order differential equations subject to harmonic excitation and an excitation admitting rapid variation. Other case studies include the application of the SE approach to parabolic and hyperbolic partial differential equations. The first-order form of these equations is obtained through semi-discretization using conventional finite-element, spectral element and finite-difference schemes. Element clustering (h-refinement) is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation. The convergence in time of the method is demonstrated. In some cases, machine precision is obtained with 25 degrees of freedom per cycle. Finally the method is applied to a weakly nonlinear problem with time-periodic solution to demonstrate its future applicability to the analysis of limit-cycle oscillations in aeroelastic systems.     

Minimax design of vibration absorbers for linear damped systems

This paper addresses the issue of design of a passive vibration absorber in the presence of uncertainties in the forcing frequency. A minimax problem is formulated to determine the parameters of a vibration absorber which minimize the maximum motion of the primary mass over the domain of the forcing frequency. The limiting solutions corresponding to the forcing frequency being unrestricted and to that where the forcing frequency is known exactly, are shown to match those available in the literature. The transition of the optimal vibration absorber parameters between the extreme two cases is presented and the solutions are generalized by permitting the mass ratio of the absorber mass and the primary mass to be design parameters. For the specific case where the primary system is undamped, detailed analysis is presented to determine the transition of the optimal vibration absorber parameters between three distinct domains of solutions.