Periodic solutions of autonomous systems under discretization

The existence of a sequence of periodic trajectories of a general one-step numerical scheme corresponding to a null sequence of constant time-steps is established under the assumption that the autonomous ordinary differential equation has an isolated periodic solution with non-zero topological index. The convergence of the linearly interpolated numerical curve to the original invariant curve with respect to the Hausdorff metric is also shown

Asymptotically optimal weigthed numerical integration

We study numerical integration of Hölder-type functions with respect to weights on the real line. Our study extends previous work by F. Curbera, [2] and relies on a connection between this problem and the approximation of distribution functions by empirical ones. The analysis is based on a lemma which is important within the theory of optimal designs for approximating stochastic processes. As an application we reproduce a variant of the well known result for weighted integration of Brownian paths, see e.g., [8]

Continuous and inverse shadowing

By the Shadowing Lemma we can shadow any sufficient accurate pseudo-trajectory of a hyperbolic system by a true trajectory of a hyperbolic system. If we are interested in finite trajectories, at least from one side, then a pseudo trajectory usually has many possible shadows. Here we show that we can choose a continuous single-valued selector from the corresponding multi-valued operator "pseudo-trajectory ↦ the totality of possible shadows". We do this in the context of Lipschitz mappings which are semi-hyperbolic on some compact subset, which need not be invariant. We also prove that semi-hyperbolicity implis inverse shadowing with respect to a very broad class of nonsmooth perturbations

Maximal monotonicity and convex programming

We introduce an explicit constraint qualification condition which is necessary and sufficient for the nondegenerate Lagrange multipliers rule to hold. We compare it with metric regularity conditions and we show that it is strictly weaker than the Slater assumption. Under certain weak smoothness hypotheses, our condition, the Slater condition and the existence of nondegenerate Lagrange multipliers are equivalent. The basic ingredient in the proof of the main result is the theory of maximal monotone operators (Minty's theorem). Another approach using a direct exact penalization argument yields a modified nondegenerate Lagrange multipliers rule involving the positive part of the constraint mapping. Examples and applications to abstract optimal control problems are also indicated

Fast periodic oscillations in singularly perturbed relay control systems and sliding modes

As a mathematical model of chattering in the small neighbourhood of switching surface in the sliding mode systems we examine the singularly perturbed relay control systems (SPRCS). The sufficient conditions for existence of fast periodic solutions in such systems are found. Their stability is investigated. It is proved that the slow motions in such SPRCS are approximately described with equations obtained from the equations for the slow variables of SPRCS by averaging along fast periodic motions. It is shown that in the case when the original SPRCS contains the relay control linearly the averaged equations and equations which describe the motions of the reduced system in the sliding mode are coincide. The algorithm is proposed which allows to solve the problem of eigenvalues assignment for averaged equations using the additional dynamics of fast actuator

Reduction of the number of particles in the stochastic weighted particle method for the Boltzman equation

Different ideas for reducing the number of particles in the stochastic weighted particle method for the Boltzmann equation are described and discussed. The corresponding error bounds are obtained and numerical tests for the spatially homogeneous Boltzmann equation presented. It is shown that if an appropriate reduction procedure is used then any effect on the accuracy of the numerical scheme is negligible

Fluctuations of the Phase Boundary in the Ising Ferromagnet

We discuss statistical properties of phase boundary in the 2D low-temperature Ising ferromagnet in a box with the two-component boundary conditions. We prove the weak convergence in C [O, 1] of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them. The limiting Gaussian measure coincides with the conditional distribution of certain Gaussian process obtained by the integral transformation of the white noise

Dobrushin-Kotecký-Shlosman theorem up to the critical temperature

We develop a non-perturbative version of the Dobrushin-Koteck'y-Shlosman theory of phase separation in the canonical 2D Ising ensemble. The results are valid for all temperatures below critical

Asymptotically Optimal Weighted Numerical Integration

We study numerical integration of Hölder-type functions with respect to weights on the real line. Our study extends previous work by F. Curbera, [2] and relies on a connection between this problem and the approximation of distribution functions by empirical ones. The analysis is based on a lemma which is important within the theory of optimal designs for approximating stochastic processes. As an application we reproduce a variant of the well known result for weighted integration of Brownian paths, see e.g., [8]

Asymptotically optimal weighted numerical integration

We study numerical integration of Hoelder-type functions with respect to weights on the real line. Our study extends previous work by F. Curbera and relies on a connection between this problem and the approximation of distribution functions by emperical ones. The analysis based on a lemma which is important within the theory of optimal designs for approximating stochastic processes. As an application we reproduce a variant of the well known result for weighted integration of Brownian paths. (orig.)Available from TIB Hannover: RR 5549(318)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman