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Unformatted text preview: ime by [4], this system defines a complete 4.4 Linearization
Let , where and is an dimensional manifold. Consider . [1] Suppose has an equilibrium point
about : . Let and expand sing Taylor’s theorem .
Use to find
. These expressions use
as notation (“littleoh” notation). In general:
if there is a neighborhood .
There is also notation (“bigoh” notation): of such that 7
Chapter 4A as
.
Example. If if there is a neighborhood of and a such that .■ then Taylor’s theorem wo ld give The linearization of [1] at is
[2] where . Here and . Recall the decomposition of
into unstable, center and stable eigenspaces of :
, where
is the unstable eigenspace, and so forth.
An equilibrium point is hyperbolic if none of the eigenvalues of
have zero real part.
Then
is empty.
Hyperbolic equilibrium points are important because, by the HartmanGrobman theorem
(section 4.8), the flow of [1] in some neighborhood of a hyperbolic equilibrium point is
topologically equivalent to the flow of [2].
Distinguish three classes of hyperbolic equilibrium points : is a sink if all of the eigenvalues of
have negative real parts. Then
.
is a source if all of the eigenvalues of
have positive real parts. Then
.
is a saddle if some of the eigenvalues of
have negative real parts and some
have positive real parts. Then
. Often very useful information about the behavior of
can be obtained by finding the
equilibrium points and analyzing the associated linearized systems using the methods of
chapter 2.
Example. The LotkaVolterra Competition Model.
,
.
The equilibrium points are
sketch , (3,0), (0,2) and (1,1). , Linearized systems have the form , eq. pts.
, where . 8
Chapter 4A The Jacobean matrix is Linearize about . Obtain . Notice that linearizing about the origin is equivalent to dropping the nonlinear terms in the LotkaVolterra equations. See that , . This is a source. Add to sketch. and
Linearize about Obtain . See , . See . This is a sink. Add to sketch.
Linearize about Obtain . See , and find . . This is also a sink. Add to sketch. See
Linearize about Obtain . Find . Find the corresponding eigenvectors and the eigenvalues
and . This is a saddle. add to sketch. ■
Example. The Lorenz model with . ,
.
Linearize about the equilibrium point at
terms. Obtain . This is equivalent to dropping the nonlinear .
This has block diagonal form. The
block has eigenvalues
. For
all of
the eigenvalues of the Lorenz model are negative and the origin is a sink. For
one
eigenvalue is positive and the origin is a saddle. The origin loses stability as increases through
1. A bifurcation occurs at
.■ 9
Chapter 4A 4.5 Stability
An equilibrium of a flow sketch , If , is Lyapunov stable or stable if
. , , trajectory based at stays in such that is not stable then it is unstable. An equilibrium is asymptotically stable if it is Lyapuno...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
 Fall '14
 MarcEvans

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