18.03 Class 3
9, May 12, 2010
Dangers of linearization; limit cycles; chaos; next steps in mathematics
1. Stability
2. Limits of linearization
3. Limit cycles
4. Strange attractors
5. Essential skills
6. Next steps
[1] Let's say that an equilibrium is
 "stable" if all nearby trajectories stay near to it and converge to
it as
t > infinity
 "unstable" otherwise
We classified linear phase portraits using the (Tr,Det) plane.
In the (Tr,Det) plane, the stable region is the upper left quadrant.
It is
bounded by the straight halflines where either the eigenvalues
are purely imaginary, or one is zero and the other is negative.
[2] The method we sketched on Friday and Monday works well "generically,"
i.e. almost all the time. It lets you predict what the phase portrait of
an autonomous system looks like near equilibrium, most of the time,
and whether the equilibrium is stable or not.
The facts: (1) If the linearization is not on the borderline of the stable
quadrant, it correctly predicts the stability of the equilibrium.
(2) If the eigenvalues of the Jacobian have nonzero real parts  so
(tr,det)
is neither on the horizontal axis nor on the tr = 0, det > 0 line 
then the phase portrait is a small deformation of the linear phase portrait.
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 Spring '09
 vogan
 Chaos Theory, Derivative, Limits, LTI system theory

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