Algebraic entropy for differential-delay equations

We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations

On the algebraic structure of rational discrete dynamical systems

We show how singularities shape the evolution of rational discrete dynamical systems. The stabilisation of the form of the iterates suggests a description providing among other things generalised Hirota form, exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties. We illustrate the phenomenon explicitly with examples covering a wide range of models

Scenarios to explain extreme Be depletion in solar-like stars: accretion or rotation effects ?

Studies of beryllium abundance in large samples of solar-type stars show a small fraction of extremely beryllium-deficient stars, which challenges our current understanding of light element depletion in these stars. We suggest two possible scenarios that may explain this high level of Be depletion: early accretion and rotational mixing. We show that in both cases, the conditions required to reach the observed level of Be depletion are quite extreme, which explains the very small fraction of detected Be outliers. We suggest that substantial Be depletion can be obtained in stars if they were fast rotators in the past, with high initial rotational velocities and short disc lifetimes. Our analysis suggests that rotational mixing may not be efficient enough to deplete Be in less than 10 Myr. Consequently, the detection of strongly Be-deficient stars in clusters younger than $\sim$ 10 Myr may provide a genuine signature of accretion process and the proof that some protostars may undergo many extreme bursts of accretion during their embedded phases of evolution.Comment: 7 pages, 6 figures, accepted for publication in A&

An exercise in experimental mathematics: calculation of the algebraic entropy of a map

We illustrate the use of the notion of derived recurrences introduced earlier to evaluate the algebraic entropy of self-maps of projective spaces. We in particular give an example, where a complete proof is still awaited, but where different approaches are in such perfect agreement that we can trust we get to an exact result. This is an instructive example of experimental mathematics

On the relevance of bubbles and potential flows for stellar convection

Recently Pasetto et al. have proposed a new method to derive a convection theory appropriate for the implementation in stellar evolution codes. Their approach is based on the simple physical picture of spherical bubbles moving within a potential flow in dynamically unstable regions, and a detailed computation of the bubble dynamics. Based on this approach the authors derive a new theory of convection which is claimed to be parameter free, non-local and time-dependent. This is a very strong claim, as such a theory is the holy grail of stellar physics. Unfortunately we have identified several distinct problems in the derivation which ultimately render their theory inapplicable to any physical regime. In addition we show that the framework of spherical bubbles in potential flows is unable to capture the essence of stellar convection, even when equations are derived correctly.Comment: 14 pages, 3 figures. Accepted for publication in Monthly Notices of the Royal Astronomical Society. (Comments and criticism are welcomed

Lithium depletion in solar-like stars: effect of overshooting based on realistic multi-dimensional simulations

We study lithium depletion in low-mass and solar-like stars as a function of time, using a new diffusion coefficient describing extra-mixing taking place at the bottom of a convective envelope. This new form is motivated by multi-dimensional fully compressible, time implicit hydrodynamic simulations performed with the MUSIC code. Intermittent convective mixing at the convective boundary in a star can be modeled using extreme value theory, a statistical analysis frequently used for finance, meteorology, and environmental science. In this letter, we implement this statistical diffusion coefficient in a one-dimensional stellar evolution code, using parameters calibrated from multi-dimensional hydrodynamic simulations of a young low-mass star. We propose a new scenario that can explain observations of the surface abundance of lithium in the Sun and in clusters covering a wide range of ages, from $\sim$ 50 Myr to $\sim$ 4 Gyr. Because it relies on our physical model of convective penetration, this scenario has a limited number of assumptions. It can explain the observed trend between rotation and depletion, based on a single additional assumption, namely that rotation affects the mixing efficiency at the convective boundary. We suggest the existence of a threshold in stellar rotation rate above which rotation strongly prevents the vertical penetration of plumes and below which rotation has small effects. In addition to providing a possible explanation for the long standing problem of lithium depletion in pre-main sequence and main sequence stars, the strength of our scenario is that its basic assumptions can be tested by future hydrodynamic simulations.Comment: 7 pages, 3 figures, Accepted for publication in ApJ Letter

Solvable Chaos

We present classes of discrete reversible systems which are at the same time chaotic and solvable