(i)If g([a.sup.2]M + [b.sup.2] g) < 0, then [e.sup.M.sub.1] is an unstable equilibrium point
In this case, system (1) has a stable equilibrium point, [C.sup.+.sub.2] (the solid curve in Figure 5(b)), and an unstable equilibrium point
, [C.sup.-.sub.2] (the dashed curve in Figure 5(b)).
Moreover, for each b < [b.sub.0], but close to [b.sub.0], there is a stable limit cycle close to the unstable equilibrium point
(1) When the initial point [x.sub.0] = ([x.sub.1](0), [x.sub.2](0), [x.sub.3](0)) is closed to the unstable equilibrium point
[p.sup.+] or [q.sup.+], the new system (2) has the same chaotic attractor.
A chaotic attractor appears, which corresponds to a unstable equilibrium point
, indicating the chaotic attractor is a self-excited attractor.
The subactuated biped robot has an unstable equilibrium point
and the control main task is to maintain the robot state near to neighborhood of the equilibrium point in order to prevent falls.
A time delay system will generate a limit cycle if the system has a unique unstable equilibrium point
and bounded solutions.
A linear MPC was designed based on a linearized model around the unstable equilibrium point
with sample time of 0.1 h.
For x = 0, we have an unstable equilibrium point
and two new symmetrical stable equilibrium points.
Condition (A1a) shows that, when the trait dynamic equation in isolation has an unstable equilibrium point
, larger rates of adaptive change (larger v) tend to make the entire system unstable.
Note that with the chosen coordinated system the unstable equilibrium point
is located at ([theta], [??], [phi], [??]) = (0 ,0, 0, 0).
As for [E.sub.2], [J.sub.1] = 1 + [alpha][e.sub.1]a(1 - b[e.sub.2]/(2[c.sub.2] + 2b[e.sub.2])) >1 is one eigenvalue which corresponds to [E.sub.2], so [E.sub.2] is an unstable equilibrium point
. In the same way we can prove that E3 and [E.sub.4] are unstable equilibrium points