Integrability versus topology of configuration manifolds and domains of possible motions

We establish a generic sufficient condition for a compact $n$-dimensional manifold bearing an integrable geodesic flow to be the $n$-torus. As a complementary result, we show that in the case of domains of possible motions with boundary, the first Betti number of the domain of possible motions may be arbitrarily large

A model for separatrix splitting near multiple resonances

We propose a model for local dynamics of a perturbed convex real-analytic Liouville-integrable Hamiltonian system near a resonance of multiplicity $1+m, m\geq 0$. Physically, the model represents a toroidal pendulum, coupled with a Liouville-integrable system of $n$ non-linear rotators via a small analytic potential. The global bifurcation problem is set-up for the $n$-dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on an $n$-torus, whose $k$th Fourier coefficient satisfies the estimate $$O(e^{- \rho|k\cdot\omega| - |k|\sigma}), k\in\Z^n\setminus\{0\},$$ where $\omega\in\R^n$ is a Diophantine rotation vector of the system of rotators; $\rho\in(0,{\pi\over2})$ and $\sigma>0$ are the analyticity parameters built into the model. The estimate, under suitable assumptions would generalize to a general multiple resonance normal form of a convex analytic Liouville integrable Hamiltonian system, perturbed by O(\eps), in which case \omega_j\sim\omeps, j=1,...,n.Comment: 24 page

Aspects of average response computation by aperiodic stimulation

A mathematical analysis of the variance of the average evoked-response computation as a function of the numberN of stimuli presented is made for the case when the response is disturbed by additive stationary noise. A comparison is made between the variance for purely periodic stimuli and that for stimuli of which the interstimulus durations are Gaussian distributed. In the latter situation, the interval durations may be correlated with each other, e.g. according to a Gaussian Markov process. It is deduced that, in general, the introduction of aperiodic stimulation tends to make the functional relationship between the variance andN behave as though it holds for noise with a very broad frequency spectrum; the variance is proportional to 1/N

Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations

This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier-Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective--diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective--diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259--280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection--diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin (SUPG) and the Galerkin/least-squares method (GLS). The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large- and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method.Comment: Update to postprint versio