# 2250-4 12:55pm Lecture Record Week 14 S2010

Last Modified: April 25, 2010, 17:59 MDT.    Today: August 18, 2018, 17:30 MDT.

## 20 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.
```

## 20 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.

```

## 22 Apr: Stability. Almost Linear systems. Phase Diagram. Section 9.2

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

Detecting stability:
Re(lambda)<0 ==> asym. stability.
Stability at t=-infinity classifies Unstable solutions.

<
Final exam review started.
Cover today ch8 and some of ch10.
Review packet distributed on the web.

Final exam details
Less contact with ch3, ch4, ch6 due their appearance on
exams 1,2,3.
Since F2008, there are extra chapters 8,9 on the final.
A good sample is the F2009 final exam.
Chapters 5,6,7,10 will undergo changes and spins. For ch10, more
contact with the second shifting theorem and the Dirac Delta. For
ch7-ch8, there are additional methods for solving DE, especially
Cayley-Hamilton-Ziebur, matrix exp(At) and the Laplace resolvent
for first and second order systems. For ch5, deeper problems on
variation of parameters and undetermined coefficients, resonance,
and beats.
```

## 27 Apr: Nonlinear Stability. Classification. Predator-Prey. Section 9.3

```B>Nonlinear stability theory
When the linearized classification and stability transfers to
the nonlinear system.
stability of almost linear [nonlinear] systems,
phase diagrams,
classification of nonlinear systems.

Nonlinear stability
phase diagrams,
classification.
Using DEtools and DEplot in maple to make phase diagrams.
Jacobian.

```

## Thu 29 Apr: . Final Exam Review, WEB 104

```  Gustafson: 1-3pm
```
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.