Volume 10, Number 3
Volume 10, Number 3, 2005
150th anniversary of H.Poincaré
Marchal C.
Abstract
At the end of the nineteenth century the triumphant "scientism" left almost no room to consciousness and claimed that it will soon rule everything. The determinism was considered as the main property of scientific facts while freedom, will, freewill were considered by most scientists as illusions.
The danger of this evolution was pointed by Henri Poincaré (1854–1912) who developed many philosophical considerations on the future of science and its relations with mankind. He was also a major scientist opening the gate to the theory of chaos that was for decades considered as an odd singularity and revealed its fundamental importance in the seventies when chaos was acknowledged in most domains of science and technology. The other main philosophical upheavals of science were of course the intrinsic presence of random, irreducible to determinism, (theory of quanta) and the trouble of scientists confronted with the terrible misuses of science. All this has led to new perspectives on consciousness and freedom while materialism is no more a must. 
Lim C. C., Assad S. M.
Abstract
A low temperature relation $R^2=\Omega\beta/4\pi\mu$ between the radius $R$ of a compactly supported 2D vorticity (plasma density) field, the total circulation $\Omega$ (total electron charge) and the ratio $\mu/\beta$ (Larmor frequency), is rigorously derived from a variational Principle of Minimum Energy for 2D Euler dynamics. This relation and the predicted structure of the global minimizers or ground states are in agreement with the radii of the most probable vorticity distributions for a vortex gas of $N$ point vortices in the unbounded plane for a very wide range of temperatures, including $\beta = O(1)$. In view of the fact that the planar vortex gas is representative of many 2D and 2.5D statistical mechanics models for geophysical flows, the Principle of Minimum Energy is expected to provide a useful method for predicting the statistical properties of these models in a wide range of low to moderate temperatures.

Borisov A. V., Mamaev I. S.
Abstract
We consider various generalizations of the Kepler problem to threedimensional sphere $S^3$, (a compact space of constant curvature). In particular, these generalizations include addition of a spherical analogue of the magnetic monopole (the Poincaré–Appell system) and addition of a more complicated field which is a generalization of the MICZsystem. The mentioned systems are integrable superintegrable, and there exists the vector integral which is analogous to the Laplace–Runge–Lenz vector. We offer a classification of the motions and consider a trajectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space $L^3$.

Fasso F., Giacobbe A., Sansonetto N.
Abstract
It has been recently observed that certain (reduced) nonholonomic systems are Hamiltonian with respect to a ranktwo Poisson structure. We link the existence of these structures to a dynamical property of the (reduced) system: its periodicity, with positive period depending continuously on the initial data. Moreover, we show that there are in fact infinitely many such Poisson structures and we classify them. We illustrate the situation on the sample case of a heavy ball rolling on a surface of revolution.

Albeverio S., Galperin G. A., Nizhnik I. L., Nizhnik L. P.
Abstract
A constructive description of generalized billiards is given, the billiards being inside an infinite strip with a periodic law of reflection off the strip's bottom and top boundaries. Each of the boundaries is equipped with the same periodic lattice, where the number of lattice's nodes between any two successive reflection points may be prescribed arbitrarily. For such billiards, a full description of the structure of the set of billiard trajectories is provided, the existence of spatial chaos is found, and the exact value of the spatial entropy in the class of monotonic billiard trajectories is found.

Blackmore D.
Abstract
New type of strange chaotic "attractor" models for discrete dynamical systems of dimension greater than one are constructed geometrically. These model, unlike most of the standard examples of chaotic attractors, have very complicated dynamics that are not generated by transverse (homoclinic) intersections of the stable and unstable manifolds of fixed points, and may include transverse heteroclinic orbits. Moreover, the dynamics of these model are not generally structurally stable (nor $\Omega$stable) for dimensions greater than two, although the topology and geometry of the nonwandering set $\Omega$ are invariant under small continuously differentiable perturbations. It is shown how these strange chaotic models can be analyzed using symbolic dynamics, and examples of analytically defined diffeomorphisms are adduced that generate the models locally. Possible applications of the exotic dynamical regimes exhibited by these models are also briefly discussed.

Sadetov S. T.
Abstract
It is established that the restricted circular planar threebody problem (RCPTBP) [1], [15], [5] admits a nonconstant algebraic integral on a level of energy only in cases when it can be reduced to the Kepler problem. The Hill problem [1], [7], [5] is the limit case of the RCPTBP if by analogy with the MoonEarthSun system we put the mass of the Sun and the distance between the Sun and the Earth to be infinitely large. It is established that the Hill problem also does not admit a nonconstant algebraic integral on any level of energy. The proof is based on the Husson method [8], [2], improved by the author [21], [22]. At the end of the proof we expand the result of J. Liouville [13] that the integral $\int f(z)e^z$ for $f$ algebraic in $z$ is not generally an algebraic function times the exponent function.
