Dynamics in the centre manifold around equilibrium points in periodically perturbed three-body problems

A new application of the parameterization method is pre- sented to compute invariant manifolds about the equilib- rium points of Periodically Perturbed Three-Body Problems ( PPTBP ). These techniques are applied to obtain high-order semi-numerical approximations of the center manifolds abo ut the points L 1 , 2 of the Sun-perturbed Earth-Moon Quasi- Bicicular Problem ( QBCP ), which is a particular case of PPTBP . The quality of these approximations is compared with results obtained using equivalents of previous normal form procedures. Then, the parameterization is used to ini- tialize the computation of Poincaré maps, which allow to get a qualitative description of the periodically-perturb ed dynamics near the equilibrium pointsPostprint (published version

Two-particle irreducible effective actions versus resummation: analytic properties and self-consistency

Approximations based on two-particle irreducible (2PI) effective actions (also known as $\Phi$-derivable, Cornwall-Jackiw-Tomboulis or Luttinger-Ward functionals depending on context) have been widely used in condensed matter and non-equilibrium quantum/statistical field theory because this formalism gives a robust, self-consistent, non-perturbative and systematically improvable approach which avoids problems with secular time evolution. The strengths of 2PI approximations are often described in terms of a selective resummation of Feynman diagrams to infinite order. However, the Feynman diagram series is asymptotic and summation is at best a dangerous procedure. Here we show that, at least in the context of a toy model where exact results are available, the true strength of 2PI approximations derives from their self-consistency rather than any resummation. This self-consistency allows truncated 2PI approximations to capture the branch points of physical amplitudes where adjustments of coupling constants can trigger an instability of the vacuum. This, in effect, turns Dyson's argument for the failure of perturbation theory on its head. As a result we find that 2PI approximations perform better than Pad\'e approximation and are competitive with Borel-Pad\'e resummation. Finally, we introduce a hybrid 2PI-Pad\'e method.Comment: Version accepted for publication in Nuclear Physics B. 31 pages, 16 figures. Uses feynm

Splitting of inviscid fluxes for real gases

Flux-vector and flux-difference splittings for the inviscid terms of the compressible flow equations are derived under the assumption of a general equation of state for a real gas in equilibrium. No necessary assumptions, approximations or auxiliary quantities are introduced. The formulas derived include several particular cases known for ideal gases and readily apply to curvilinear coordinates. Applications of the formulas in a TVD algorithm to one-dimensional shock-tube and nozzle problems show their quality and robustness

First-order approximation of strong vector equilibria with application to nondifferentiable constrained optimization

Vector equilibrium problems are a natural generalization to the context of partially ordered spaces of the Ky Fan inequality, where scalar bifunctions are replaced with vector bifunctions. In the present paper, the local geometry of the strong solution set to these problems is investigated through its inner/outer conical approximations. Formulae for approximating the contingent cone to the set of strong vector equilibria are established, which are expressed via Bouligand derivatives of the bifunctions. These results are subsequently employed for deriving both necessary and sufficient optimality conditions for problems, whose feasible region is the strong solution set to a vector equilibrium problem, so they can be cast in mathematical programming with equilibrium constraints

Multilevel coarse graining and nano--pattern discovery in many particle stochastic systems

In this work we propose a hierarchy of Monte Carlo methods for sampling equilibrium properties of stochastic lattice systems with competing short and long range interactions. Each Monte Carlo step is composed by two or more sub - steps efficiently coupling coarse and microscopic state spaces. The method can be designed to sample the exact or controlled-error approximations of the target distribution, providing information on levels of different resolutions, as well as at the microscopic level. In both strategies the method achieves significant reduction of the computational cost compared to conventional Markov Chain Monte Carlo methods. Applications in phase transition and pattern formation problems confirm the efficiency of the proposed methods.Comment: 37 page

Convex computation of the region of attraction of polynomial control systems

We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description