Unformatted text preview: Systems of ﬁrst-order differential equations 179
Figure 4.28. If both roots are opposite in sign, we have found that the det(A) is negative
and the critical point is a saddle. Hence, below the x-axis, marked region III, the
critical point is an unstable saddle point. Notice that this applies whether the trace
is positive or negative.
The complex region is sub-divided into three categories. In region IV the sign of
α in the complex conjugate roots α ± β i is strictly negative and the spiral trajectory
tends towards the critical point in the limit. In region V α is strictly positive and
the critical point is an unstable one with the trajectory spiralling away from it.
Finally in region VI, which is the y-axis above zero, α = 0 and the critical point
has a centre with a closed curve as a trajectory.
It is apparent that the variety of possibilities can be described according to the
tr(A) and det(A) along with the characteristic roots of A. The list with various
nomenclature is given in table 4.1. 4.10 Limit cycles9
A limit cycle is an isolated closed integral curve, which is also called an orbit.
A limit cycle is asymptotically stable if all the nearby cycles tend to the closed
orbit from both sides. It is unstable if the nearby cycles move away from the
closed orbit on either side. It is semi-stable if the nearby cycles move towards
the closed orbit on one side and away from it on the other. Since the limiting
trajectory is a periodic orbit rather than a ﬁxed point, then the stability or instability
is called an orbital stability or instability. There is yet another case, common
9 This section utilises the VisualDSolve package provided by Schwalbe and Wagon (1996). It can
be loaded into Mathematica with the Needs command. This package provides considerable visual
control over the display of phase portraits. ...
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