Lecture Notes in Differential Equations - Bruce E. Shapiro

377
Using the cos π/4 = √ 2/2 and angle addition formulas,
y = − cos π 4 cos t + sin π sin t (34.29)
(
4
π
)
= − cos
4 + t (34.30)
x = cos π 4 sin t + sin π cos t (34.31)
(
4
π
)
= sin
4 + t (34.32)
The trajectory is also the the unit circle, because we still have x 2 + y 2 = 1,
but not the solution starts at the point 45 degrees above the x-axis.
The two solutions are different, but they both follow the same trajectory.
Definition 34.2. A Critical Point of the system x ′ = f(x, y), y ′ = g(x, y)
(or fixed point or local equilibrium) is a any point (x ∗ , y ∗ ) such that
both of the following conditions
hold simultaneously.
f(x ∗ , y ∗ ) = 0 (34.33)
g(x ∗ , y ∗ )0 (34.34)
A critical point is a unique kind of trajectory: anything that starts there,
stays there, for all time. They are thus zero-dimensional, isolated trajectories.
Furthermore, no other solution can pass through a critical point,
because once there, it would have to stop. 1
If there is an open neighborhood about a critical point that does not contain
any other critical points, it is called an isolated critical point. Certain
properties (that we will discuss presently) of linear systems at isolated critical
points determine the global geometry of the phase portrait; for nonlinear
systems, these same properties determine the local geometry.
We will classify a critical point P based on the dynamics of a particle
placed in some small neighborhood N of P, and observe what happens as
t increases. We call P a
1. Sink, Attractor, or Stable Node if all such points move toward P ;
1 This does not prevent critical points from being limit points of solutions (ast → ±∞)
and thus appearing to be part of another trajectory but this is an artifact of how we
draw phase portraits.