Solutions in the neighborhoods of singular points

Let us first consider the relationship between e and i near the points defined by equations (8)-(ll). In the neighborhood of (i = 0, e = 0) [equations (8) and (9)], we must assume that F (t, (p) — 0 over a short range of i in order to remedy the cold-boundary difficulty (see Section 5.3.2). Hence equation (5-36) implies that e = 0 as i increases in the neighborhood of either of these points (that is, i e here). 

The nature of the singularities (i = 1,6 = 1) [equations (10) and (11)] Can be deduced by expanding equation (5-36) about these points. In this manner, these singularities are found to be saddle points in the (t, e) plane only one integral curve reaches (i = 1, e = 1) from the permissible region (i 1, e 1) all others bend away from the singularities. The slope of the curve that reaches (i = 1, = 1) from (i 1, 1) is 

Let us next investigate the relationship between x and p near the points defined by equations (8)-(l 1). For equations (8) and (9) we may assume (in view of the preceding discussion) that = 0. Expanding equation (5-35) about I = 0, (p = (p+(0) then yields the equidimensional equation 

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