Isochronous centers of polynomial Hamiltonian systems and a conjecture of Jarque and Villadelprat

We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture has already been proved for quadratic and quartic systems. Using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields, we are able to describe a very large class of nonisochronous Hamiltonian systems of even arbitrarily large degree.     

Effet stereodirecteur d'un groupe phenyle par coordination π arene metal dans l'addition d'organometalliques sur la (±) methyl-2 phenyl-2 indanone-1

The steric course of the 1–2 addition of organometallics to the (±) 2-methyl, 2-phenyl, 1-indanone mainly depends on the nature of the metal. Most strikingly opposite stereoselectivities are observed with organolithium and organomagnesium reagents. These results point out the directive effect of the adjacent phenyl group via metal-arene coordination in the transition state.     

A continuous/discrete fractional Noether’s theorem

We prove a fractional Noether’s theorem for fractional Lagrangian systems invariant under a symmetry group both in the continuous and discrete cases. This provides an explicit conservation law (first integral) given by a closed formula which can be algorithmically implemented. In the discrete case, the conservation law is moreover computable in a finite number of steps.     

A note on a derivation method for SDE models: Applications in biology and viability criteria

We discuss a method, which was popularized by E. J. Allen and that is frequently used in applications to construct SDE models. The derivation procedure is based on information about the elementary processes involved in the dynamics and their corresponding probabilities. We formulate criteria for the viability of the resulting models. In particular, explicit necessary and sufficient conditions are deduced for the non-negativity and/or boundedness of solutions. Moreover, we show that the class of deterministic models for which the construction leads to an admissible SDE extension is strongly limited. Several examples are presented to illustrate the implications of our results.     

Variational formulations of differential equations and asymmetric fractional embedding

Variational formulations for classical dissipative equations, namely friction and diffusion equations, are given by means of fractional derivatives. In this way, the solutions of those equations are exactly the extremal of some fractional Lagrangian actions. The formalism used is a generalization of the fractional embedding developed by Cresson [Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys. 48 (2007) 033504], where the functional space has been split in two in order to take into account the asymmetry between left and right fractional derivatives. Moreover, this asymmetric fractional embedding is compatible with the least action principle and respects the physical causality principle.     

Hyperbolicity, transversality and analytic first integrals

Let A be a (normally) hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold M. We assume that the stable and unstable manifold of A intersect transversally (in an admissible way), the dynamics on A is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold, respectively, satisfy a non-resonance condition. In the case where A is a point or a torus, we prove that the discrete dynamical system associated to f does not admit an analytic first integral. The proof is based on a triviality lemma, which is of combinatorial nature, and a geometrical lemma. The same techniques, allow us to prove analytic non-integrability of Hamiltonian systems having Arnold diffusion. In particular, using results of Xia, we prove analytic non-integrability of the elliptic restricted three-body problem, as well as the planar three-body problem.     

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.     

A λ-Lemma for Partially Hyperbolic Tori and the Obstruction Property

Using a scheme given by Marco, we prove that partially hyperbolic tori along resonant surfaces of near-integrable Hamiltonian systems possess the obstruction property in Arnold's terminology. The proof is based on a specific lambda lemma for these tori.