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- United-kingdom
- University of Bristol
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- Mathematics 20101
- Stephen Wiggins
- Ch6: Stable And Unstable Manifolds Of Equilibria

Harry W.

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Jacobian Matrix

Given

ẋ = f(x,y)

ẏ = g(x,y)

where (x,y)∈R^{2}. The Jacobian matrix is:

( a b )

( c d )

where:

- a = ∂f/∂x (x
_{0}, y_{0}) , - b = ∂f/∂y (x
_{0}, y_{0}) , - c = ∂g/∂x (x
_{0}, y_{0}) , - d = ∂g/∂y (x
_{0}, y_{0}).

Source and Sink

Considering ẋ = f(x,y) , ẏ = g(x,y) and denoting φ_{t}(⋅):

- (x
_{0,}y_{0}) is a**source**is there exists a neighbourhood U of (x_{0},y_{0}) s.t. for any p∈U, φ_{t}(p) leaves U as t increases. - (x
_{0},y_{0}) is a**sink**if there exists a neighbourhood S of (x_{0},y_{0}) s.t for any p∈S, φ_{t}(p) approaches (x_{0},y_{0}) at an exponential rate as t increases, In this case (x_{0},y_{0}) is an example of an attracting set and its basin of attraction is given by: B = ∪_{t≤0}φ_{t}(S).

Stable and Unstable Manifold Theorem for Hyperbolic Equilibria

Considering ẋ = f(x,y), ẏ = g(x,y), we can transform this locally near (x_{0},y_{0}):

(Ṗ,Q̇) = (-α 0, 0 -β)(P,Q) + (u(P,Q), v(P,Q))

where the jacobian at the origin: (-α 0, 0 -β)(P,Q), reflects the hyperbolic nature of the equilibrium. The linearization about the origin is given by:

(Ṗ,Q̇) = (-α 0, 0 -β)(P,Q).

It is easy to see for the linearised system that:

- E
^{s}= {(P,Q) | Q = 0}, - E
^{u}= {(P,Q) | P = 0}

(1) is the invariant *stable* subspace and (2) is the invariant *unstable* subspace.

To state how the saddle point structure is inherited by the nonlinear system, we state the results of the stable and unstable manifold theorem for hyperbolic equilibria:

First, consider two intervals of the coordinate axis containing the following as follows: I_{P} = { -ε < P < ε } and I_{Q} = { -ε < Q < ε } for some small ε>0. A neighbourhood of the origin is constructed by taking the cartesian product of these two intervals:

B_{ε} = { (P,Q) ∈ R^{2} | (P,Q) ∈ I_{P} x I_{Q} }.

The stable and unstable manifold theorem for hyperbolic equilibrium points states the following:

There exists two C^{r} curves given by the graph as a function of the P and Q variables (resp.):

1. Q = S(P), P∈I_{P},

2. P = S(Q), Q∈I_{Q}.

The *first* curve has three important properties: it passes through the origin, i.e. S(0) = 0; It is tangent to E^{S} at the origin; it is locally invariant. The curve satisfying these three properties is unique and is referred to as the **local stable manifold of the origin**, denoted by:

W^{s}_{loc}((0,0)) = {(P,Q) ∈ B_{ε} | Q = S(P) }.

The *second* curve has three important properties: it passes through the origin, i.e. U(0) = 0; it is tangent to E^{U} at the origin; it is locally invariant.. For these reasons it is referred to as the local unstable manifold of the origin, and it is denoted by:

W^{U}_{loc}((0,0)) = {(P,Q) ∈ B_{ε} | P = U(Q) }..

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