Stochastic Perturbations of Periodic Orbits with Sliding

Vector fields that are discontinuous on codimension-one surfaces are known as Filippov systems and can have attracting periodic orbits involving segments that are contained on a discontinuity surface of the vector field. In this paper we consider the addition of small noise to a general Filippov system and study the resulting stochastic dynamics near such a periodic orbit. Since a straight-forward asymptotic expansion in terms of the noise amplitude is not possible due to the presence of discontinuity surfaces, in order to quantitatively determine the basic statistical properties of the dynamics, we treat different parts of the periodic orbit separately. Dynamics distant from discontinuity surfaces is analyzed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analyzed through stochastic averaging methods. The influence of noise on points at which the periodic orbit escapes a discontinuity surface is determined by zooming into the transition point. We combine the results to quantitatively determine the effect of noise on the oscillation time for a three-dimensional canonical model of relay control. For some parameter values of this model, small noise induces a significantly large reduction in the average oscillation time. By interpreting our results geometrically, we are able to identify four features of the relay control system that contribute to this phenomenon.Comment: 44 pages, 9 figures, submitted to: J Nonlin. Sc

Propositional Dynamic Logic for Message-Passing Systems

We examine a bidirectional propositional dynamic logic (PDL) for finite and infinite message sequence charts (MSCs) extending LTL and TLC-. By this kind of multi-modal logic we can express properties both in the entire future and in the past of an event. Path expressions strengthen the classical until operator of temporal logic. For every formula defining an MSC language, we construct a communicating finite-state machine (CFM) accepting the same language. The CFM obtained has size exponential in the size of the formula. This synthesis problem is solved in full generality, i.e., also for MSCs with unbounded channels. The model checking problem for CFMs and HMSCs turns out to be in PSPACE for existentially bounded MSCs. Finally, we show that, for PDL with intersection, the semantics of a formula cannot be captured by a CFM anymore

Field evaluation of entomopathogenic nematodes against orchard pests

Survival of pest in micro-plot trials (container studies) or field plot trials was monitored after exposure to commercially used EPN strains. Experimental plots were artificially infested with pest larvae that naturally burrowed into the soil for diapause. Either larval mortality or adult emergence, was assessed to estimate the control effect of the EPN treatment. Here we present preliminary results from three ongoing projects

The Injector Layout of BERLinPro

BERLinPro is an Energy Recovery Linac Project running since 2011 at the HZB in Berlin. A conceptual design report has been published in 2012 [1]. One of the key components of the project is the 100 mA superconducting RF photocathode gun under development at the HZB since 2010. Starting in 2016 the injector will go into operation, providing 6.6 MeV electrons with an emittance well below 1mm mrad and bunches shorter than 5 ps. In 2017 the 50 MeV linac will be set up and full recirculation is planned for 2018. The injector design has been finalized and is described in detail in this paper. Emphasis is further laid on beam dynamics aspects and performance simulations of two different gun cavitie

Stochastic Regular Grazing Bifurcations

A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincare map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude, epsilson [e]. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms, or, if there exists an attracting period-n solution when e = 0, be well approximated by the sum of n Gaussian densities centred about each point of the deterministic solution, and scaled by 1/n, for sufficiently small e > 0. We explain the irregular forms and calculate the covariance matrices associated with the Gaussian approximations in terms of the parameters of the map. Close to the grazing bifurcation the size of the invariant density may be proportional to the square-root of e, as a consequence of a square-root singularity in the map. Sequences of transitions between different dynamical regimes that occur as the primary bifurcation parameter is varied have not been described previously.Comment: Submitted to: SIAM J. Appl. Dyn. Sy

Improvements of the Set Up and Procedures for Beam Energy Measurements at BESSY II

With a 7 T wiggler in operation all attempts to detect the resonant depolarization of the electron spins were unsuccessful at BESSY II. This was attributed to the severely reduced degree of spin polarization of the electrons moving in the alternating fields of the strong wiggler which on the other hand nearly doubles the radiation loss per turn. The key to a clear detection of the depolarization is the improvement of the sensitivity of the polarimeter, based on the spin dependent Touschek scattering cross section and the more effective and thus full depolarization of the beam. With these improvements the high precision beam energy determination can again be performed in parallel to the normal user operation and without any noticeable perturbations to the bea

Mean Field Effects for Counterpropagating Traveling Wave Solutions of Reaction-Diffusion Systems

In many problems, e.g., in combustion or solidification, one observes traveling waves that propagate with constant velocity and shape in the x direction, say, are independent of y and z and describe transitions between two equilibrium states, e.g., the burned and the unburned reactants. As parameters of the system are varied, these traveling waves can become unstable and give rise to waves having additional structure, such as traveling waves in the y and z directions, which can themselves be subject to instabilities as parameters are further varied. To investigate this scenario we consider a system of reaction-diffusion equations with a traveling wave solution as a basic state. We determine solutions bifurcating from the basic state that describe counterpropagating traveling waves in directions orthogonal to the direction of propagation of the basic state and determine their stability. Specifically, we derive long wave modulation equations for the amplitudes of the counterpropagating traveling waves that are coupled to an equation for a mean field, generated by the translation of the basic state in the direction of its propagation. The modulation equations are then employed to determine stability boundaries to long wave perturbations for both unidirectional and counterpropagating traveling waves. The stability analysis is delicate because the results depend on the order in which transverse and longitudinal perturbation wavenumbers are taken to zero. For the unidirectional wave we demonstrate that it is sufficient to consider the cases of (i) purely transverse perturbations, (ii) purely longitudinal perturbations, and (iii) longitudinal perturbations with a small transverse component. These yield Eckhaus type, zigzag type, and skew type instabilities, respectively. The latter arise as a specific result of interaction with the mean field. We also consider the degenerate case of very small group velocity, as well as other degenerate cases, which yield several additional instability boundaries. The stability analysis is then extended to the case of counterpropagating traveling waves

Characterizing mixed mode oscillations shaped by noise and bifurcation structure

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and reset and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise.Comment: 22 page