Recall that a function gy is oyp if there exists a

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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 2 @f 6 fy := @ y =...
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