# 2250-1 7:30am Lecture Record Week 14 S2010

Last Modified: April 25, 2010, 17:55 MDT.    Today: July 18, 2018, 12:34 MDT.

## 19 Apr: Sections 8.1, 8.2

``` Survey of Methods for solving a 2x2 dynamical system
1. Cayley-Hamilton-Ziebur method for u'=Au
Solution: u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n)
Atoms: They are constructed by Euler's theorem from roots of det(A-rI)=0
Vectors: Symbols vec(c_1), ..., vec(c_n) are not arbitrary. They are
determined from A and u(0). Algorithm outlined above for 2x2.
2. Laplace resolvent L(u)=(s I - A)^(-1) u(0)
3. Eigenanalysis  u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2
4. Putzer's method for the 2x2 matrix exponential.
Solution of u'=Au is: u(t) = exp(A t)u(0)
THEOREM: exp(A t) = r1(t) I + r2(t) (A-lambda_1 I),
Lambda Symbols: lambda_1 and lambda_2 are the roots of det(A-lambda I)=0.
The DE System:
r1'(t) = lambda_1 r1(t),         r1(0)=0,
r2'(t) = lambda_2 r2(t) + r1(t), r2(0)=0
THEOREM. The formula can be used as
e^{r1 t} - e^{r2 t}
e^{At} = e^{r1 t} I  +  ------------------- (A-r1 I)
r1 - r2
where r1=lambda_1, r2=lambda_2 are the eigenavalues of A.

EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]),
A=matrix([[2,3],[0,4]]) using the matrix exponential,
Zeibur's method, Laplace resolvent and eigenanalysis.
EXAMPLE. Solve a non-homogeneous system u'=Au+F(t), u(0)=vector([0,0]),
A=matrix([[2,3],[0,4]]), F(t)=vector([3,1]) using variation
of parameters.
```

## 21 Apr: Intro to stability theory for autonomous systems. Section 9.1

```Exam 3 Review
Eigenvalues
A 4x4 matrix.
Block determinant theorem.
Eigenvectors for a 4x4.
lambda=5,5,3i,-3i
One panel for lambda=5
First frame is A-5I with 0 appended
Find rref
Apply last frame algorithm
Scalar general solution
Take partials on t1, t2to find v1,v2
Eigenpairs are (5,v1), (5,v2)
One panel for lambda=3i
Same outline as lambda=5
Get one eigenpair (3i,v3)
Other eigenpair=(-3i,v4) where v4 is the conjugate of v3.
Shortest trial solution in undetermined coefficients.
Second shifting theorem in Laplace theory.

Dynamical Systems Topics
Equilibria.
Stability.
Instability.
Asymptotic stability.
Classification of equilibria for u'=Au when
det(A) is not zero, for the 2x2 case.

```

## 23 Apr: Stability. Classifications. Phase Diagram. Section 9.1, 9.2

```Spiral, saddle, center, node.
Linearization theory.
Jacobian.

Maple phase diagram tools. Demonstration for the example
x' = x + y,
y' = 1 - x^2

How to detect saddle, spiral, node, center in the linear case
using Zeibur's method and examples.

Limitations:
In the case of a node, we cannot sub-classify as improper
or proper using the Zeibur method and examples. The finer
sub-classifications require the exponential matrix e^{At}
or else a synthetic eigenvalue theorem which calculated the
sub-classification.

```
Systems of Differential Equations references
Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)

Extra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.