If for any [member of] > 0, there must exist [delta] > 0 and [t.sub.0] [greater than or equal to] 0, such that for any point [P.sub.1] [member of] U(P, [delta]) intersection N{x, y], we have [rho](f([P.sub.1], t), [GAMMA]) < [epsilon] for t > [t.sub.0]; then we call the order one periodic solution [GAMMA]

orbitally asymptotically stable.

(1) If Re([lambda]'([[sigma].sub.0])) < 0 and [[beta].sub.2] < 0 ([[beta].sub.2] > 0), then the bifurcating periodic solution is

orbitally asymptotically stable (unstable) and the direction of Hopf bifurcation is [sigma] < [[sigma].sub.0] ([sigma] > [[sigma].sub.0]).

In a neighborhood of a transversal Hopf point of codimension 2 with [l.sub.2] [not equal to] 0 the dynamic behavior of the system (8), reduced to the family of parameter-dependent continuations of the center manifold, is

orbitally topologically equivalent to [omega]' = ([eta] + i[[omega].sub.0]) [omega] + [tau][omega][[absolute value of [omega]].sup.2] + [l.sub.1][omega][[absolute value of [omega]].sup.4], where [eta] and [tau] are unfolding parameters.

According to condition (14), [absolute value of [[mu].sub.2]] < 1, so the order-1 periodic solution is

orbitally stable according to the analogue of the Poincare criterion.

If [absolute value of (((1 - (1 - p)[h.sub.2]) - ([[eta].sub.0]/((1 - p)[h.sub.2] + a)))/(1 - [h.sub.2] - (([[eta].sub.0] - [tau])/((1 - q)([h.sub.2] + a)))))] [less than or equal to] 1 and [beta] [greater than or equal to] 1, the periodic solution of system (6) with initial point [C.sub.0] ((1 - p)[h.sub.2], [[eta].sub.0]) is

orbitally asymptotically stable.

Therefore, if [absolute value of [mu]] < 1 holds, the semitrivial periodic solution (0, [eta](t)) is

orbitally asymptotically stable.

Reciprocating-clamp puller system has adjustable pressure features; an automatic flying cutoff saw mounted below the table is

orbitally operated and has a manually adjustable envelope.

These interplanetary particles are

orbitally related to the short-period Comet Encke and to the daytime Taurids the Earth encounters in late June.

The piping system is manufactured in mirror polished tubing 0.6 um Ra,

orbitally welded with sanitary flanges and pharmaceutical grade Teflon gaskets.

630/892-9000 FAX 630/892-2573 7 21 31

Orbitally riveted swivels 32 Olympus Lock Inc., 2720 NE 115th St., Seattle, WA 98125 Tel.

In a neighborhood of a transversal Hopf point with [l.sub.1] [not equal to] 0 the dynamic behavior of the system (12), reduced to the family of parameter-dependent continuations of the center manifold, is

orbitally topologically equivalent to the following complex normal form [omega]' = ([eta] + i[omega])[omega] + [l.sub.1][omega][[absolute value of [omega]].sup.2], where [omega] [member of] C, [eta], [omega] and [l.sub.1] are real functions having derivatives of arbitrary higher order, which are continuations of 0, [[omega].sub.0], and the first Lyapunov coefficient at the Hopf point.

If [sigma] > [bar.[sigma]] and [[gamma].sub.1][m-r[[eta].sub.0]-q[[gamma].sub.0] (K + d[[eta].sub.0])] (1-[beta]) (K + d[[eta].sub.1])/[[gamma].sub.0] (m-r[[eta].sub.1]-q[[gamma].sub.1]) (K + d[[eta].sub.0]) < 1, then the OOPS of system (7) is

orbitally asymptotically stable.