131F08.classnotes2

131F08.classnotes2 - For any u let g u,t be the solution...

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Class notes #2 for MTH G131: Fall 2008 First order autonomous ODE’s du dt = f ( u ) A curve u = g ( t ) which satisfies this ODE is called a trajectory of the system. We identify trajectories which differ only by a translation of the time coordinate. Points that satisfy f ( u ) = 0 are called equilibrium points or critical points . Assume that f and f 0 are continuous. 1) Through any point u there is at most one trajectory. 2) A trajectory starting at a point that is not a critical point cannot reach a critical point in finite time. 3) A trajectory cannot reverse direction. Stability Consider the ODE du dt = f ( u,t
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Unformatted text preview: ) For any u let g ( u,t ) be the solution satisfying g ( u, 0) = u . (i) A point u * is Lyapunov stable if and only if for all ± > 0 there exists δ > 0 such that if | u *-v | < δ then | g ( u * ,t )-g ( v,t ) | < ± for all t ≥ 0. (ii) A point u * is quasi-asymptotically stable if and only if there exists δ > 0 such that if | u *-v | < δ then | g ( u * ,t )-g ( v,t ) | → 0 as t → ∞ . (iii) A point u * is asymptotically stable if and only if it is both Lyapunov stable and quasi-asymptotically stable....
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