Entropy and Poincar\'e recurrence from a geometrical viewpoint

We study Poincar\'e recurrence from a purely geometrical viewpoint. We prove that the metric entropy is given by the exponential growth rate of return times to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss theorem. Moreover, we show that minimal return times to dynamical balls grow linearly with respect to its length. Finally, some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures are given.Comment: 11 pages, revised versio

Analysis of Round Off Errors with Reversibility Test as a Dynamical Indicator

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analysed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator. The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same. Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.Comment: International Journal of Bifurcation and Chaos, 201

Synthesis and reductive cleavage of 3-(2-flurophenyl) and 3-(4-flurophenyl)-cyclopent-5-en[d]isoxazolines by raney nickel in trifluroacetic acid

3-Fluoro arylcycloopent-5-en[d]isoxazolines have been obtained via the 1,3-dipolar cycloaddition of cyclopentadiene to aromatic nitrile oxides. The reductive cleavage of these isoxazolines by Raney nickel in trifluoroacetic acid led to corresponding acylcyclopentenes along with acylcyclopentanes. The synthesized compounds are the precursors of new prostanoids as well as the analogues of cyclic ?-triketones with fluorinated acyl side chain

The catalytic hydrogenation of 3-(2-fluorophenyl)- and 3-(4-fluorophenyl)-4,4-ethylenedioxycyclopenta[d]isoxazolines

The catalytic hydrogenation of 3-(2-fluorophenyl)- and 3-(4-fluorophenyl)-4,4-ethylenedioxycyclo-penta[d ]isoxazolines led with good yields to corresponding fluorinated β -hydroxyketones. The synthesized compounds are precursors of new fluorinated prostanoids and carbocyclic analogs of acetogenins being of great interest as potential bi ologically active substances as well

Synthesis of (2-fluorophenyl) and (4-fluorophenyl)-(2-nitromethyl)-methanones as precursors of fluorinated prostanoids

The synthesis of new fluorinated primary nitrocompounds ?(2-fluorophenyl)and (4-fluorophenyl)-(2-nitromethylcyclopentyl)methanones has been realized by the nitromethane 1,4-addition to 1-acylcyclopentenes, available by the reductive cleavage of corresponding cyclopent-5-en[d]isoxazolines. The obtained nitrocompounds are the precursors of new fluorinated prostaglandin analogues, which could be synthesized via the formation of second prostanoids side chain by nitrile oxides method

A "metric" complexity for weakly chaotic systems

We consider the number of Bowen sets which are necessary to cover a large measure subset of the phase space. This introduce some complexity indicator characterizing different kind of (weakly) chaotic dynamics. Since in many systems its value is given by a sort of local entropy, this indicator is quite simple to be calculated. We give some example of calculation in nontrivial systems (interval exchanges, piecewise isometries e.g.) and a formula similar to the Ruelle-Pesin one, relating the complexity indicator to some initial condition sensitivity indicators playing the role of positive Lyapunov exponents.Comment: 15 pages, no figures. Articl

Smooth and Non-Smooth Dependence of Lyapunov Vectors upon the Angle Variable on a Torus in the Context of Torus-Doubling Transitions in the Quasiperiodically Forced Henon Map

A transition from a smooth torus to a chaotic attractor in quasiperiodically forced dissipative systems may occur after a finite number of torus-doubling bifurcations. In this paper we investigate the underlying bifurcational mechanism which seems to be responsible for the termination of the torus-doubling cascades on the routes to chaos in invertible maps under external quasiperiodic forcing. We consider the structure of a vicinity of a smooth attracting invariant curve (torus) in the quasiperiodically forced Henon map and characterize it in terms of Lyapunov vectors, which determine directions of contraction for an element of phase space in a vicinity of the torus. When the dependence of the Lyapunov vectors upon the angle variable on the torus is smooth, regular torus-doubling bifurcation takes place. On the other hand, the onset of non-smooth dependence leads to a new phenomenon terminating the torus-doubling bifurcation line in the parameter space with the torus transforming directly into a strange nonchaotic attractor. We argue that the new phenomenon plays a key role in mechanisms of transition to chaos in quasiperiodically forced invertible dynamical systems.Comment: 24 pages, 9 figure

Cross sections for geodesic flows and \alpha-continued fractions

We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced \alpha-variants.Comment: 20 pages, 2 figure

A Poincar\'e section for the general heavy rigid body

A general recipe is developed for the study of rigid body dynamics in terms of Poincar\'e surfaces of section. A section condition is chosen which captures every trajectory on a given energy surface. The possible topological types of the corresponding surfaces of section are determined, and their 1:1 projection to a conveniently defined torus is proposed for graphical rendering.Comment: 25 pages, 10 figure