On a class of second-order impulsive boundary value problem at resonance

We consider the following impulsive boundary value problem, x″(t)=f(t,x,x′), t∈J\{t1,t2,…,tk}, Δx(ti)=Ii(x(ti),x′(ti)), Δx′(ti)=Ji(x(ti),x′(ti)), i=1,2,…,k, x(0)=(0), x′(1)=∑j=1m−2αjx′(ηj). By using the coincidence degree theory, a general theorem concerning the problem is given. Moreover, we get a concrete existence result which can be applied more conveniently than recent results. Our results extend some work concerning the usual m-point boundary value problem at resonance without impulses

Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks

In this paper a novel device aimed at controlling the mechanical vibrations of plates by means of a set of electrically-interconnected piezoelectric actuators is described. The actuators are embedded uniformly in the plate wherein they connect every node of an electric network to ground, thus playing the two-fold role of capacitive element in the electric network and of couple suppliers. A mathematical model is introduced to describe the propagation of electro-mechanical waves in the device; its validity is restricted to the case of wave-forms with wave-length greater than the dimension of the piezoelectric actuators used. A self-resonance criterion is established which assures the possibility of electro-mechanical energy exchange. Finally the problem of vibration control in simply supported and clamped plates is addressed; the optimal net-impedance is determined. The results indicate that the proposed device can improve the performances of piezoelectric actuationComment: 22 page

A long-lived horseshoe companion to the Earth

We present a dynamical investigation of a newly found asteroid, 2010 SO16, and the discovery that it is a horseshoe companion of the Earth. The object's absolute magnitude (H=20.7) makes this the largest object of its type known to-date. By carrying out numerical integrations of dynamical clones, we find that (a) its status as a horseshoe is secure given the current accuracy of its ephemeris, and (b) the time spent in horseshoe libration with the Earth is several times 10^5 yr, two orders of magnitude longer than determined for other horseshoe asteroids of the Earth. Further, using a model based on Hill's approximation to the three-body problem, we show that, apart from the low eccentricity which prevents close encounters with other planets or the Earth itself, its stability can be attributed to the value of its Jacobi constant far from the regime that allows transitions into other coorbital modes or escape from the resonance altogether. We provide evidence that the eventual escape of the asteroid from horseshoe libration is caused by the action of planetary secular perturbations and the stochastic evolution of the eccentricity. The questions of its origin and the existence of as-yet-undiscovered co-orbital companions of the Earth are discussed.Comment: Accepted in MNRAS; 6 pages, 3 figures, 2 table

Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics

This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of motion to the form, which is convenient for further use of analytical and numerical methods of analyses. Three different types of nonsmooth transformations are discussed as follows: positional coordinate transformation, state variables transformation, and temporal transformations. Illustrating examples are provided.Comment: 15 figure

Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions

In this paper, we focus on the existence and asymptotic analysis of positive solutions for a class of singular fractional differential equations subject to nonlocal boundary conditions. By constructing suitable upper and lower solutions and employing Schauder’s fixed point theorem, the conditions for the existence of positive solutions are established and the asymptotic analysis for the obtained solution is carried out. In our work, the nonlinear function involved in the equation not only contains fractional derivatives of unknown functions but also has a stronger singularity at some points of the time and space variables

Impulsive phase flare energy transport by large-scale Alfven waves and the electron acceleration problem

The impulsive phase of a solar flare marks the epoch of rapid conversion of energy stored in the pre-flare coronal magnetic field. Hard X-ray observations imply that a substantial fraction of flare energy released during the impulsive phase is converted to the kinetic energy of mildly relativistic electrons (10-100 keV). The liberation of the magnetic free energy can occur as the coronal magnetic field reconfigures and relaxes following reconnection. We investigate a scenario in which products of the reconfiguration - large-scale Alfven wave pulses - transport the energy and magnetic-field changes rapidly through the corona to the lower atmosphere. This offers two possibilities for electron acceleration. Firstly, in a coronal plasma with beta < m_e/m_p, the waves propagate as inertial Alfven waves. In the presence of strong spatial gradients, these generate field-aligned electric fields that can accelerate electrons to energies on the order of 10 keV and above, including by repeated interactions between electrons and wavefronts. Secondly, when they reflect and mode-convert in the chromosphere, a cascade to high wavenumbers may develop. This will also accelerate electrons by turbulence, in a medium with a locally high electron number density. This concept, which bridges MHD-based and particle-based views of a flare, provides an interpretation of the recently-observed rapid variations of the line-of-sight component of the photospheric magnetic field across the flare impulsive phase, and offers solutions to some perplexing flare problems, such as the flare "number problem" of finding and resupplying sufficient electrons to explain the impulsive-phase hard X-ray emission.Comment: 31 pages, 6 figure