Unformatted text preview: Systems of ﬁrst-order differential equations 177
Figure 4.27. of the cycle, which is 2π/β . The critical point is called the centre. These situations
are illustrated in ﬁgure 4.27. Summary
From the ﬁve cases discussed we arrive at a number of observations.
3. After a sufﬁcient time interval, the trajectory of the system tends towards
three types of behaviour:
(i) the trajectory approaches inﬁnity
(ii) the trajectory approaches the critical point
(iii) the trajectory traverses a closed curve surrounding the critical point.
Through each point (x0 , y0 ) in the phase plane there is only one trajectory.
Considering the set of all trajectories, then three possibilities arise:
(i) All trajectories approach the critical point. This occurs when
(a) tr(A)2 > 4det(A), r < s < 0
(b) tr(A)2 < 4det(A), r = α + β i, s = α − β i and α < 0. ...
View Full Document