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page 391 —————————————————————————— CHAPTER 7. —— 14 . The roots of the characteristic equation, . Note that the roots are complex when
, the equilibrium point
when , are . For the case when
is a stable spiral. On the other hand, , the equilibrium point is an unstable spiral. For the case , the roots are purely imaginary, so the equilibrium point is a center. When
,
the roots are real and distinct. The equilibrium point becomes a node, with its stability
dependent on the sign of
Finally, the case
marks the transition from spirals
to nodes.
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page 392 —————————————————————————— CHAPTER 7. —— 17. The characteristic equation of the coefficient matrix is
, with
roots given formally as
. The roots are real provided that
First note that the sum of the roots is
and the product of the roots is
. For
negative va...
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 Spring '08
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