Unformatted text preview: ), we see that the stability of an autonomous system is related to
that of the simple linear IVP y= y t>0 0 y(0) = : (1.2.7) Additional details on autonomous and non-autonomous systems appear in Birkho and
Rota 2] or Hairer et al. 5], Section 1.13. The solution of the linear IVP (1.2.7) is y(t) = e t
hence, (1.2.7) is
stable when Re( ) 0,
asymptotically stable when Re( ) < 0, and
unstable when Re( ) > 0.
These conclusions remain true for the original nonlinear autonomous problem when
Re( ) 6= 0 and y is small enough for the O(y2) term to be negligible relative to y. This
cannot happen in the stable case (Re( ) = 0) hence, it requires a more careful analysis.
Remark 4. The analysis of non-autonomous systems is similar 1].
Analyzing the stability of autonomous vector systems y = f (y(t))
0 12 (1.2.8) is more complicated. Once again, assume that y = 0 is an equilibrium point and expand
f (0) in a series y = f (0) + fy (0)y + O(kyk2) = fy (0)y + O(kyk2):
0 We need a brief digression for a few de nitions. De nition 1.2.4. The Jacobian matrix of a vector-valued function f (y) with respect to
y is the matrix
fy := @ y =...
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- Spring '14
- Boundary value problem, lipschitz condition, IVPs, di erential equations