trajectory based at stays in such that is not stable

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Unformatted text preview: v stable and, in addition, there is a neighborhood of such that . sketch ,, , trajectory based at converges to Remarks Meiss’ example, eq. 4.14 and Fig. 4.6, shows why it is necessary to use the two closed balls and in the definition of Lyapunov stability. It shows a system for which initial conditions must be confined to a with in order to guarantee that trajectories stay within . Meiss’ example, eq. 4.16 and Fig. 4.7, shows why it is necessary to require Lyapunov stability explicitly in the definition of asymptotic stability. It shows a system for which there is a neighborhood of such that and yet the system is not stable. Instead of reiterating these examples, let’s see how the definitions apply to systems we are already familiar with. Stability of 1D systems: asymptotically stable. -, -, -, -, , . decreasing through decreasing through with negative slope, flow on axis. with zero slope, flow on axis. unstable -, -, increasing through with positive slope, flow on semi-stable (a type of unstable) -, -, concave up with a minimum at If asymptotically stable unstable axis. 10 Chapter 4A need more information to make a conclusion on stability Example. The Logistic Equation. -, -, Equilibrium points: unstable, . asymptotically stable. ■ Stability of 2D systems: . , Example. The Harmonic Oscillator. or where Let obtaining and . or or Observe , , . This is a center. -, -, circular trajectories surrounding the origin. is stable but not asymptotically stable. ■ Example. Stable Degenerate Node , , , , Find . , find and -, -, , . . is a line of degenerate equilibria, exponential decay along The equilibrium points in . are stable but not asymptotically stable. ■ Example. Lotka-Volterra Competition Model. -, these The source The sinks -, equilibria , putative trajectories connecting and saddle (1,1) are unstable. and are asymptotically stable. ■ Example. The Lorenz Model with . , , . The equilibrium point has three eigenvalues: , , . It is asymptotically stable for and unstable for .■ 11 Chapter 4A Recall theorem 2.10 (Asymptotic Linear Stability) Let eigenvalues of have negative real parts. . all Theorem 4.6 (Asymptotic Linear Stability Implies Asymptotic Stability) Let be and have an equilibrium such that all of the eigenvalues of have negative real parts. Then is asymptotically stable. Do not give proof. (The statement is plausible and the proof depends on detailed estimates. Re ommend for reading; it is an exer ise in sing Grönwall’s ineq ality. This is also implied by the Hartman-Grobman theorem, although the text does not prove that in detail. The proof of theorem 4.21 is a discrete analog of this.) Remark. For nonlinear systems, the condition that have eigenvalues with negative real parts is sufficient but not necessary for asymptotic stabili...
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