In two and three dimensions, there are only two possible dynamics near a stable equilibrium
. Either the flow is directly downhill, or it spirals down like water near a sink drain--or air and debris near a tornado.
where [[xi].sub.1], [[xi].sub.2] [member of] R, [N.sub.0], [B.sub.0] are parameters of manipulator system, and the system has a globally asymptotically stable equilibrium
at [[xi].sub.*] and [x.sub.*] = [pi]([[xi].sub.*]).
As a closed set, the set Im(EC) has the boundary given by images of some stable equilibrium
points of the system through the energy-Casimir mapping.
The solid curve stands for the stable equilibrium
and the dashed curve stands for the unstable equilibrium.
(1) If 0 < y < ([c.sub.1] + [f.sub.1])/(a[DELTA][pi] + [f.sub.1] - [E.sub.1]), then there is a stable equilibrium
solution for RDE named x = 0 after taking x = 0 and x = 1 into formula (9) and concluding [a.sub.11] < 0 and [a.sub.11] > 0 respectively.
Caption: Figure 2: An asymptotically stable equilibrium
for [tau] = [[tau].sub.1] = [[tau].sub.2] = [[tau].sub.3] = 0.
As the time delay increases to the critical value, it can cause a stable equilibrium
to become unstable and Hope bifurcation can occur.
Coexistence between Stable Equilibrium
Point and Limit Cycle.
Caption: FIGURE 1: (a) Stable equilibrium
Also, as the value of recovery rate increases the infected cells revert to the healthy cells more rapidly and converge to stable equilibrium
. We can see that the infection would always keep stability when the cure rate [gamma] is larger.
Bourguignon (1981) shows that, for a given proportion of rich a.sub.2 satisfying 0<[a.sub.2]<[bar.A], the possible equilibria are pairs ([k.sub.1],[k.sub.2]) each consisting of an unstable equilibrium [k.sub.1] and a stable equilibrium
[k.sub.2] with [k.sub.2] > [k.sub.1].