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Volume 10, Number 3

Volume 10, Number 3, 2005
150th anniversary of H.Poincaré

Marchal C.
Determinism, random, chaos, freedom. Henri Poincaré and the revolution of scientific ideas in the twentieth century
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, free-will 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.
Keywords: determinism, random, freedom, philosophy of science
Citation: Marchal C., Determinism, random, chaos, freedom. Henri Poincaré and the revolution of scientific ideas in the twentieth century , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 227-237
DOI:10.1070/RD2005v010n03ABEH000312
Lim C. C.,  Assad S. M.
Self containment radius for rotating planar flows, single-signed vortex gas and electron plasma
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.
Keywords: rotating planar flows, vortex gas, equilibrium statistical mechanics, ground state
Citation: Lim C. C.,  Assad S. M., Self containment radius for rotating planar flows, single-signed vortex gas and electron plasma , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 239-255
DOI:10.1070/RD2005v010n03ABEH000313
Borisov A. V.,  Mamaev I. S.
Superintegrable systems on a sphere
Abstract
We consider various generalizations of the Kepler problem to three-dimensional 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 MICZ-system. 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 $L^3$.
Keywords: spaces of constant curvature, Kepler problem, integrability
Citation: Borisov A. V.,  Mamaev I. S., Superintegrable systems on a sphere , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 257-266
DOI: 10.1070/RD2005v010n03ABEH000314
Fasso F.,  Giacobbe A.,  Sansonetto N.
Periodic flows, rank-two Poisson structures, and nonholonomic mechanics
Abstract
It has been recently observed that certain (reduced) nonholonomic systems are Hamiltonian with respect to a rank-two 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.
Keywords: Poisson structures, non-holonomic systems, periodic flows
Citation: Fasso F.,  Giacobbe A.,  Sansonetto N., Periodic flows, rank-two Poisson structures, and nonholonomic mechanics , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 267-284
DOI:10.1070/RD2005v010n03ABEH000315
Albeverio S.,  Galperin G. A.,  Nizhnik I. L.,  Nizhnik L. P.
Generalized billiards inside an infinite strip with periodic laws of reflection along the strip's boundaries
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.
Keywords: billiards, dynamical systems, spatial chaos, entropy
Citation: Albeverio S.,  Galperin G. A.,  Nizhnik I. L.,  Nizhnik L. P., Generalized billiards inside an infinite strip with periodic laws of reflection along the strip's boundaries , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 285-306
DOI: 10.1070/RD2005v010n03ABEH000316
Blackmore D.
New models for chaotic dynamics
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.
Keywords: Axiom A, heteroclinic and homoclinic points, hyberbolic, strange attractor, structural stability, subshift, transversality
Citation: Blackmore D., New models for chaotic dynamics , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 307-321
DOI:10.1070/RD2005v010n03ABEH000317
Sadetov S. T.
On algebraic integrals of the Hill problem and restricted circular planar three-body problem on a level of energy
Abstract
It is established that the restricted circular planar three-body 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 Moon-Earth-Sun 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 non-constant 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.
Keywords: Hill problem, restricted circular planar three-body problem, algebraic integrals, improved Husson method, Hamiltonian perturbation
Citation: Sadetov S. T., On algebraic integrals of the Hill problem and restricted circular planar three-body problem on a level of energy , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 323-332
DOI: 10.1070/RD2005v010n03ABEH000318

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