Response of an artificially blown clarinet to different blowing pressure profiles

Using an artificial mouth with an accurate pressure control, the onset of the pressure oscillations inside the mouthpiece of a simplified clarinet is studied experimentally. Two time profiles are used for the blowing pressure: in a first set of experiments the pressure is increased at constant rates, then decreased at the same rate. In a second set of experiments the pressure rises at a constant rate and is then kept constant for an arbitrary period of time. In both cases the experiments are repeated for different increase rates. Numerical simulations using a simplified clarinet model blown with a constantly increasing mouth pressure are compared to the oscillating pressure obtained inside the mouthpiece. Both show that the beginning of the oscillations appears at a higher pressure values than the theoretical static threshold pressure, a manifestation of bifurcation delay. Experiments performed using an interrupted increase in mouth pressure show that the beginning of the oscillation occurs close to the stop in the increase of the pressure. Experimental results also highlight that the speed of the onset transient of the sound is roughly the same, independently of the duration of the increase phase of the blowing pressure.Comment: 14 page

Feynman diagrams versus Fermi-gas Feynman emulator

Precise understanding of strongly interacting fermions, from electrons in modern materials to nuclear matter, presents a major goal in modern physics. However, the theoretical description of interacting Fermi systems is usually plagued by the intricate quantum statistics at play. Here we present a cross-validation between a new theoretical approach, Bold Diagrammatic Monte Carlo (BDMC), and precision experiments on ultra-cold atoms. Specifically, we compute and measure with unprecedented accuracy the normal-state equation of state of the unitary gas, a prototypical example of a strongly correlated fermionic system. Excellent agreement demonstrates that a series of Feynman diagrams can be controllably resummed in a non-perturbative regime using BDMC. This opens the door to the solution of some of the most challenging problems across many areas of physics

Bifurcation delay and difference equations

International audienceWe prove the existence of complex analytic solutions of difference equations of the form $y(x+eps)=f(x,y(x))$, where x and y are complex variables and $eps$ is a small parameter.We also show that differences of two solutions are exponentially small.We apply these results to the problem of delayed bifurcation at a point of period doubling for real discrete dynamical systems. In contrast to previous publications,the results obtained in this article are global

Classification of resonant equations

International audienceWe consider singularly perturbed linear ordinary differential equations of the second order with coefficients analytic near some point, say 0. We assume that the coefficients are real valued on the real axis, i.e. that there is a turning point at the origin. Such equations are called resonant in the sense of Ackerberg-O'Malley, if there is a solution, analytic in some neighborhood of 0, which tends to a non-zero limit as the parameter tends to 0. The article presents a classification of such resonant equations with respect to linear transformations having analytic coefficients. Besides a formal invariant (considered fixed below), we associate three formal series of Gevrey order 1 to any resonant equation which are invariant under analytic transformations. It is shown that this correspondence between equivalence classes of resonant equations and triples of Gevrey 1 series is essentially bijective, and that each equivalence class contains an equation of a particular form

Exponentially Small Splitting of Separatrices for Difference Equations With Small Step Size

Solutions of the vector difference equation y(x + ") \Gamma y(x \Gamma ") = 2"f (x; y(x)), x a complex variable, " ? 0 a small parameter, are constructed that are analytic on x--domains\Omega independent of ". As a first case, horizontally convex bounded domains are considered, i.e. domains having the property that for each x; x 0 2\Omega with same imaginary part, the segment [x; x 0 ] is contained in \Omega\Gamma also considered are unbounded domains such as sectors open to the left or right. Using these results, it is shown that the Hausdorff distance between separatrices of certain systems of difference equations is exponentially small with respect to ". As an application, the so--called ghost solutions of the discretized logistic equation are considered in detail and, in particular, the lengths of the levels are estimated. Other applications, e.g. to the standard mapping, are presented. Abbreviated title: Exponentially small splitting of separatrices 1 Introduction Consid..