Loop Numbers for the Stability of Homoclinic Loops of Planar Vector Fields

This paper is devoted to the study of stability and bifurcations of homoclinic loops for planar vector fields. For a given homoclinic loop, a sequence of loop numbers can be defined such that the stability and bifurcations of the loop are determined by the first nonzero term of the sequence. Formulas for the first several loop numbers were established in the past. In this paper, we will introduce general formulas for the loop numbers for both the single and double homoclinic loops.

Wave Heights and Damping Coefficients for Half-Submerged Ellipsoids in Low-Frequency Sway and Yaw

The authors consider the problem of an ellipsoid performing simple-harmonic swaying or yawing oscillations of small amplitude in the free surface of an ideal incompressible fluid, with gravity. The problem is examined in the limit of low frequency. The first nonzero term in the radiation pattern, which is a dipole term, and the damping coefficient, are found for sway. In the case of yaw the first nonzero term in the radiation pattern is a quadrupole term, and involves a higher power of frequency. Hence, the damping coefficient for yaw is of smaller order than the damping coefficient for sway, at low frequency. The strip-theory prediction of the sway damping coefficient is compared with the exact value and is shown to be too large by a factor of order K.