Figure 8.1 In the live figure, drag the gold state onto either invariant line. Its
trajectory remains on that line and decays at the corresponding exponential
rate. Away from the lines, trajectories curve toward the origin and
eventually become tangent to the slower direction. The moving point makes
that slow final approach visible.
Figure 8.2 The two panels show the same flow in different coordinates. In standard
coordinates the invariant grid is slanted; in eigenvector coordinates it is
square, and each coordinate decays independently at its own rate. In the
live figure, drag the gold state in either panel and follow its synchronized
copy in the other.
Figure 8.3 For 𝑥″+𝑏𝑥′+2𝑥=0, the state matrix has trace −𝑏 and determinant 2.
In the live figure, drag the gold point along this horizontal path in the
trace--determinant plane. As 𝑏 increases, the phase portrait changes from
a center to a spiral sink, reaches a critically damped improper node on the
parabola, and then becomes an overdamped node. The spiral sinks lie above the parabola.
Figure 8.4 In the live figure, choose a point in the trace--determinant plane and compare
it with the phase portrait beside it. Saddles lie below the horizontal axis,
nodes between that axis and the parabola, and spirals above the parabola. Centers occur
where the trace is zero and the determinant is positive. The live controls
snap to the axes and the parabola so that the boundary cases can be examined
directly.
Figure 8.5 In the live figure, switch among the three flows and follow the same small
region of initial states. A center carries it around without shrinking, a
spiral sink draws it toward the origin, and the trace-zero saddle stretches
it into a thin filament without changing its area. The area readout tracks
det𝑒𝑡𝐴=𝑒𝑡tr𝐴 throughout the motion.
