Periodic orbits for three and four co-orbital bodies

We investigate the natural families of periodic orbits associated with the equilibrium configurations of the planar-restricted 1 + n-body problem for the case 2 ≤ n ≤ 4 equal-mass satellites. Such periodic orbits can be used to model both trojan exoplanetary systems and parking orbits for captured asteroids within the Solar system. For n = 2, there are two families of periodic orbits associated with the equilibria of the system: the well-known horseshoe and tadpole orbits. For n = 3, there are three families that emanate from the equilibrium configurations of the satellites, while for n = 4, there are six such families as well as numerous additional connecting families. The families of periodic orbits are all of the horseshoe or tadpole type, and several have regions of neutral linear stability

Phototropism: Mechanism and Outcomes

Plants have evolved a wide variety of responses that allow them to adapt to the variable environmental conditions in which they find themselves growing. One such response is the phototropic response - the bending of a plant organ toward (stems and leaves) or away from (roots) a directional blue light source. Phototropism is one of several photoresponses of plants that afford mechanisms to alter their growth and development to changes in light intensity, quality and direction. Over recent decades much has been learned about the genetic, molecular and cell biological components involved in sensing and responding to phototropic stimuli. Many of these advances have been made through the utilization of Arabidopsis as a model for phototropic studies. Here we discuss such advances, as well as studies in other plant species where appropriate to the discussion of work in Arabidopsis