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- United-kingdom
- University of Bristol
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- Mathematics 20101
- Stephen Wiggins
- Ch7: Lyapunov's Method And The Lasalle Invariance Principle

Harry W.

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Lyapunov's Theorem on Stability of an Equilibrium Point

Consider the following C^{r} (r≥1) autonomous vector field on R^{n}:

^{}

ẋ = f(x), x∈R^{n}

Let x = x̄ be a fixed point and let V:U→R be a C^{1} function defined in some neighbourhood U of x̄ such that:

- V(x̄) = 0 and V(x)>0 if x≠x̄
- V̇(x) ≤ 0 in U - {x̄}

Then x̄ is Lyapunov stable. Moreover if V̇(x) < 0 in U - {x̄} then x̄ is asymptotically stable.

The function V(x) is referred to as a Lyapunov function.

CLaSalles Invariance Principle

Consider a C^{r} autonomous ODE defined on R^{n}:

^{}

ẋ = f(x), x∈R^{n}

Let φ_{t}(⋅) denote the flow and let M⊂R^{n} denote a positive invariant set that is compact (i.e. closed and bounded in this setting). Suppose we had a scalar valued function:

V:R^{n} → R such that V̇(x) ≤ 0 in M.

Let E = { x∈M | V̇(x) ≤ 0 } and M = { the union of all trajectories that start in E and remain in E for all t ≥ 0 }

Then, **LaSalle's invariance principle** is that for all x∈M, φ_{t}(⋅) → M as t → ∞.

Bendixson's Criterion

Consider a C^{r} (r≥:1) vector field on the plane of the following form:

ẋ = f(x,y)

ẏ = g(x,y)

If on a simply connected region D ⊂ R^{2} the expression

__∂f__ (x,y,) = __∂g__ (x,y,)

∂x ∂y

is not identically zero and does not change sign, then the vector field has no periodic orbits lying entirely in D.

Index Theorem

Inside any periodic orbit there must be at least one fixed point. If there is only one, then it must be a sink, source or centre.

If all fixed points inside the periodic orbit are hyperbolic, then there must be an odd number, 2n+1, of which n are saddles, and n+1 are either sinks or sources.

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