Riddled-like Basin in Two-Dimensional Map for Bouncing Motion of an Inelastic Particle on a Vibrating Board

Motivated by bouncing motion of an inelastic particle on a vibrating board, a simple two-dimensional map is constructed and its behavior is studied numerically. In addition to the typical route to chaos through a periodic doubling bifurcation, we found peculiar behavior in the parameter region where two stable periodic attractors coexist. A typical orbit in the region goes through chaotic motion for an extended transient period before it converges into one of the two periodic attractors. The basin structure in this parameter region is almost riddling and the fractal dimension of the basin boundary is close to two, {\it i.e.}, the dimension of the phase space.Comment: 4 pages, 5 figures. to be published in J. Phys. Soc. Jpn. (2002

Stiff knots

We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are: (i) Their equilibrium energy as well as the equilibrium tension for open knots depend on the type of knot as the square of the bridge number; (ii) Braid localization is found to be a general feature of stiff strings entanglements, while angles and knot localization are forbidden. Moreover, we identify a family of knots for which the equilibrium shape is a circular braid. Two other equilibrium shapes are found from Monte Carlo simulations. These three shapes are confirmed by rudimentary experiments. Our approach is also extended to the problem of the minimization of the length of a knotted string with a maximum allowed curvature.Comment: Submitted to Phys. Rev.

Layer dynamics of a freely standing smectic-A film

We study the hydrodynamics of a freely-standing smectic-A film in the isothermal, incompressible limit theoretically by analyzing the linearized hydrodynamic equations of motion with proper boundary conditions. The dynamic properties for the system can be obtained from the response functions for the free surfaces. Permeation is included and its importance near the free surfaces is discussed. The hydrodynamic mode structure for the dynamics of the system is compared with that of bulk systems. We show that to describe the dynamic correlation functions for the system, in general, it is necessary to consider the smectic layer displacement $u$ and the velocity normal to the layers, $v_z$, together. Finally, our analysis also provides a basis for the theoretical study of the off-equilibrium dynamics of freely-standing smectic-A films.Comment: 22 pages, 4 figure

Jumping particle model. Period doubling cascade in an experimental system

An experimental model of a modification of the Fermi acceleration problem is described. Evidence is presented for three consecutive bifurcations on the period doubling route of the system from regular to chaotic behaviour.Nous présentons une réalisation pratique d'une variante du modèle d'accélération de Fermi. Nous mettons en évidence trois bifurcations successives sur le chemin vers le comportement chaotique

Quasi-quantization of writhe in ideal knots

The values of writhe of the tightest conformations, found by the SONO algorithm, of all alternating prime knots with up to 10 crossings are analysed. The distribution of the writhe values is shown to be concentrated around the equally spaced levels. The "writhe quantum" is shown to be close to the rational $4/7$ value. The deviation of the writhe values from the $n (4/7)$ writhe levels scheme is analysed quantitatively

Brownian motion of screw dislocations in the hexagonal Blue Phase

By quickly decreasing the electric field, we observe a transition from the two-dimensional hexagonal Blue Phase (BPH2D) to the three-dimensional hexagonal Blue Phase (BPH3D) and the appearance of screw dislocations in some crystallites. After more or less long time, some of these dislocations may escape from the crystallites. We analyse the phenomena through the Brownian motion and consider the influence of the electric field on the diffusion time.En diminuant rapidement le champ électrique, la transition de la Phase Bleue hexagonale bidimensionnelle (BPH2D) à la Phase Bleue hexagonale tridimensionnelle (BPH3D) fait apparaître des dislocations vis dans certains cristallites. Certaines de ces dislocations s'échappent des cristallites en des temps plus ou moins longs et l'on analyse ce phénomène en terme de mouvement brownien en regardant l'influence du champ électrique sur le temps de diffusion

Facetting of cholesteric phases in electric fields

Electric field effects on crystal shapes of Blue Phases and of the Helicoidal Phase of cholesteric liquid crystals are considered We show that dielectric energy of these phases contributes to the energy βhkl, of steps on (hkl) facets and finally affects global crystal shapes : (1) facets perpendicular to the helix axis appear on droplets of the Helicoidal Phase, (2) (100) facets appear on monocrystals of the Blue Phase II. We point out that shapes and sizes of facets, existing without the field, should be modified by the field.On considère les effets du champ électrique sur les formes cristallines des Phases Bleues et de la Phase Hélicoïdale de cristaux liquides cholestériques. On démontre que l'énergie diélectrique de ces phases contribue à l'énergie β hkl des marches sur les facettes (hkl) et, par conséquent, modifie les formes cristallines globales. On met en évidence : (1) l'apparition de facettes perpendiculaires à l'axe hélicoïdal de la Phase Hélicoïdale, (2) l'apparition de facettes (100) sur les monocristaux de la Phase Bleue 11. On indique que les dimensions et les formes des facettes qui existaient sans champ sont modifiées par celui-ci

Orientation of crystals of blue phases by electric fields

Blue Phases are cubic crystals with large unit cells. In two systems we studied we find experimentally that they orient with their [100] axis parallel to an electric field, E. By analysing the non-linear polarizability, which we describe in terms of the relevant fourth rank tensor, we show that cubic blue phase crystals are absolutely stable with their four-fold or threefold axis directed along E, depending on the sign of this tensor.On trouve expérimentalement que dans deux systèmes étudiés les monocristaux des Phases Bleues soumis au champ électrique E s'orientent avec leurs axes [100] parallèlement à la direction du champ. En analysant la susceptibilité diélectrique non-linéaire, décrite par un tenseur du quatrième ordre, nous démontrons que, suivant son signe, les cristaux des Phases Bleues minimisent leur énergie électrostatique en s'orientant avec leurs axes quaternaires ou ternaires parallèlement au champ