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0 the notion of a well posed system is related to the

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Unformatted text preview: ition. De nition 1.2.3. Consider the di erential equation y = f (t y) and (without loss of 0 generality) let the origin (0 0) be an equilibrium point, i.e., f (0 0) = 0. Then, the origin is: stable if a perturbation of the initial condition jy(0)j < grows no larger than for subsequent times, i.e., if jy(t)j < for t > 0. asymptotically stable if it is stable and jy (0)j < implies that limt !1 jy (t)j = 0. unstable if it is not stable. Remark 3. This de nition could, like De nition 1.2.2, also involve perturbations of f (t y). We have omitted these for simplicity. 11 An autonomous system is one where f (t y) does not explicitly depend on t, i.e., f (t y) = f (y). If (0 0) is an equilibrium point then, in this case, f (0) = 0. Expanding the solution in a Taylor's series in y, we have y (t) = f (y) = f (0) + fy (0)y(t) + O(y2): 0 Since f (0) = 0, y (t) = fy (0)y(t) + O(y2) 0 where fy = @[email protected] Recall that a function g(y) is O(yp) if there exists a constant C > 0 such that jg(y)j C yp as y ! 0. Letting = fy (0...
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