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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
in the definition of Lyapunov stability. It shows a system for which
initial conditions must be confined to a
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
unstable axis. 10
Chapter 4A need more information to make a conclusion on stability
Example. The Logistic Equation. -,
unstable, . asymptotically stable. ■ Stability of 2D systems: . , Example. The Harmonic Oscillator.
and . or
. 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. -,
The sinks -, equilibria , putative trajectories connecting and saddle (1,1) are unstable.
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
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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- Fall '14