The parameter space of the Hénon map is reported to contain a regular structure-parallel-to-structure sequence of shrimp-shaped robust isoperiodic domains. They appear densely concentrated on a neighborhood along a main α direction, extending across both orientation-preserving and -reversing domains. There is also a secondary β direction, roughly perpendicular to a very dense ‘‘foliation of legs’’ emanating from the isoperiodic domains. Familiar bifurcation phenomena observed in unimodal maps correspond to particular cuts along the β direction. The α direction is rich in new phenomena. The topology along α is conjectured to be typical of bimodal maps.

We present molecular dynamics simulations of granular material submitted to vibrations in a two-dimensional system. We find various types of convection cells, due either to the existence of walls or to spatial modulations in the amplitude of the vibration. The direction of the motion relative to the walls depends on shear friction. We measure the strength of the convection velocity and find a characteristic resonance frequency as in experiments. We propose an explanation for the mechanisms that are at the origin of the different motions.

We show the standard two-level continuous-time model of loss-modulated ${\mathrm{CO}}_2$ lasers to display the same regular network of self-similar stability islands known so far to be typically present only in discrete-time models based on mappings. Our results suggest that the two-parameter space of class $B$ laser models and that of a certain class of discrete mappings could be isomorphic.

In this paper general working equations for the Morse ${(r-r_e)}^l$ matrix elements are given. These equations can be used to calculate the diagonal ($m=n$) matrix elements and, for the off-diagonal ($m\ne n$) elements, are simpler to use than the ones currently available in the literature. Also, in this paper a new approach is given which allows one to obtain simple formulas, in closed form, for the off-diagonal matrix elements. Explicit expressions are given for $l=1, 2, \mathrm{and} 3$.