By Choquet-Bruhat Y., Dewitt-Morette C., M.Dillard-Bleick

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A solid torus is the homeomorphic image of the cross product GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS 43 of a circle and a disk. Suppose the region is that depicted in Fig. 5 and the coordinate system (r, (), 4» is as shown with 0 ~ r ~ 2 and 0 ~ (), 4> ~ 21T. The value r = 0 is a circle C which lies in a plane P and the surface r = constant, 0 ~ 4>, () ~ 21T is a torus, which has C at its center. For the differential. equations, choose ¢=w, f =r(I-r), where w is a constant. The torii r = 0 and r = 1 are invariant in the sense that any solution with initial value in these surfaces remain in them for all t in ( - 00, (0).

Differential Inequalities Let Dr denote the right hand derivative of a function. 1). 1. If x(t) is a continuously differentiable n-vector function on a ~ t ~ b, then Dr Ix(t)1 exists on a ~ t < b and IDr(lx(t)1>1 ~ li(t)1 ' a ~ t0, we haye Ix+ Ohul-IOx+ Ohul ~ (1 - 0) Ixl. Therefore, Ix + Ohul-Ixl Oh ~ Ix + hul -Ixl h ; GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS 31 that is, the difference quotient Ix+hul-Ixl h is a nondecreasing function of h.

OnCAnt. ervals of time. J In the remainder of this sec1;ion we assume f smooth enough to ensure existence, uniqueness and continuous dependence on parameters. x Definitions. Suppose f: [0, 00) X Rn --+ Rn. Consider = f (t, x), f(t, 0) ==0, t E [0, 00). The solution x = 0 is called UG;punov 8tahZe if for any 8> 0 and any to ~ 0, there is a 8 = 8(8, to) such that Ixol < 8 implies Ix(t, to, xo) I < 8 for t E [to, 00). ~lJm'TJtoticall1J stahle if it is stable and there exists a b = b(to) such that Ixol < b implies· Ix(t, to, xo) 1--+0 as t --+00.