By Banesh Ed. Vaclav Hlavaty; Hoffman
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Additional info for Perspectives in Geometry and Relativity: Essays in Honor of Václav Hlavatý
However, when a family of second species orbits is followed, the stability index occasionally jumps from +00 to -00, or vice versa. For M > 0, the stability index jumps then from large positive values to large negative values (or vice versa). There exists a-short interval of stable orbits. Such jump happens when, for one of the arcs, the following condition "perturbations in the direction of the departure velocity have no effect on the impact parameter at the next encounter" (Hitzl and H6non 1977b, p, 1029).
We recover Fig. 2 = = the results of Sect. 1'. e. if the complement is not symmetric, quantitative approach is unable to establish the junctions for K < 1. 1 is not then the It seems 13. e. in the complement, through higher-order perturbations exerted at the two ends of the bifurcating arc. Stability and Jacobian. A critical bifurcating are corresponds 0. 48) . 50) . only if K < 1; it corresponds to the point of interfamily. The second solution exists only if 1 < K < I +3 < K; it corresponds to the extremum in K along the family.
IV). For the bifurcating arc itself, however, the above condition can be realized. 31) for a given value of W, with the last equation deleted, We have then a system of 2n 1 equations for 2n variables. Starting from a given value X, and using the equations one by one, we obtain successively Y,,. in the same way, starting from a variation dX1, we can Y1, X2, compute successively dY1, dX2, dYn. The direction of the initial velocity is a function of X1, and the final impact parameter is proportional to Yn.
Perspectives in Geometry and Relativity: Essays in Honor of Václav Hlavatý by Banesh Ed. Vaclav Hlavaty; Hoffman