# 2250-1 7:30am Lecture Record Week 14 S2012

Last Modified: April 23, 2012, 05:01 MDT.    Today: July 18, 2018, 12:33 MDT.

## Mon and Tue, Apr 16,17: Sections 6.2, 7.3

```Diagonalization Theory
In the case of a 2x2 matrix A,
FOURIER'S MODEL is
A(c1 v1 + c2 v2) = c1(lambda1 v1) + c2(lambda2 v2)
where v1,v2 are a basis for the plane
equivalent to DIAGONALIZATION
AP=PD, where D=diag(lamba1,lambda2), P=augment(v1,v2),
where det(P) is not zero
equivalent to EIGENPAIR EQUATIONS
A(v1)=lambda1 v1,
A(v2)=lambda2 v2,
where vectors v1,v2 are independent
Drill Problems
1. Problem: Given P and D, find A in the relation AP=PD.
2. Problem: Given Fourier's model, find A.
3. Problem: Given A, find Fourier's model.
4. Problem: Given A, find all eigenpairs.
5. Problem: Given A, find packages P and D such that AP=PD.
6. Problem: Give an example of a matrix A which has no Fourier's model.
7. Problem: Give an example of a matrix A which is not diagonalizable.
8. Problem: Given 2 eigenpairs, find the 2x2 matrix A.
Cayley-Hamilton topics, Section 6.3.
Power Method
Computing powers of matrices.
Stochastic matrices.
Example of 1984 telecom companies ATT, MCI, SPRINT with discrete
dynamical system u(n+1)=A u(n). Matrix A is stochastic.
EXAMPLE:
[ 6  1  5 ]               [ a(t) ]
10 A = [ 2  7  1 ]        u(t) = [ m(t) ]
[ 2  2  4 ]               [ s(t) ]

Meaning: 60% stay with ATT and 20% switch to MCI, 20% switch to SPRINT.
70% stay with MCI and 20% switch to SPRINT, 10% switch to ATT.
40% stay with SPRINT and 50% switch to ATT, 10% switch to MCI.
Lawrence Page's pagerank algorithm, google web page rankings.
```
```  Methods to solve dynamical systems
Consider the 2x2 system
x'=x-5y, y'=x-y, x(0)=1, y(0)=2.
Cayley-Hamilton-Ziebur method.
Laplace resolvent.
Eigenanalysis method.
Exponential matrix using maple
Putzer's method to compute the exponential matrix
Spectral methods [ch8; not studied in 2250]
```
``` 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). For the algorithm, see the slides.
2. Laplace resolvent L(u)=(s I - A)^(-1) u(0)
See slides for details about the resolvent equation.
3. Eigenanalysis  u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2
See chapter 7 in Edwards-Penney for examples and details.
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
See the slides and manuscript on systems for proofs and details.
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.
```

## Wed Apr 18: Second Order Systems. Section 7.4

```Exam 3 Review
Shortest trial solution in undetermined coefficients.
Example: Sample exam.
Eigenvalues
A 4x4 matrix.
Block determinant theorem.
Eigenvectors for a 4x4.
B:=matrix([[5,0,0,0],[0,5,0,0],[0,0,0,3],[0,0,-3,0]]);
lambda=5,5,3i,-3i
v1=[1,0,0,0], v2=[0,1,0,0], v3=[0,0,i,-1], v4=[0,0,i,1]
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, t2 to 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.
Final exam: Second shifting theorem in Laplace theory.
Second Order Systems
How to convert mx''+cx'+kx=F0 cos (omega t) into a
dynamical system  u'=Au+F(t).
Electrical systems u'=Au+E(t) from LRC circuit equations.
Electrical systems of order two: networks
Mechanical systems of order two: coupled systems
Second order systems u''=Au+F
Examples are railway cars, earthquakes,
vibrations of multi- component systems,
electrical networks.
Second Order Vector-Matrix Differential Equations
The model u'' = Ax + F(t)
Coupled Spring-Mass System. Problem 7.4-6
A:=matrix([[-6,4],[2,-4]]); eigenvals(A);
lambda1= -2, lambda2= -8
Ziebur's Method
roots for Ziebur's theorem are plus or minus sqrt(lambda)
Roots = sqrt(2)i,  sqrt(8)i, -sqrt(2)i, -sqrt(8)i
Atoms = cos (sqrt(2)t), sin(sqrt(2)t), cos(sqrt(8)t), sin(sqrt(8)t)
Vector x(t) = vector linear combination of the above 4 atoms
Maple routines for second order
de1:=diff(x(t),t,t)=-6*x(t)+4*y(t); de2:=diff(y(t),t,t)=2*x(t)-4*y(t);
dsolve({de1,de2},{x(t),y(t)});
x(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)+_C3*sin(2*sqrt(2)*t)+_C4*cos(2*sqrt(2)*t),
y(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)-(1/2)*_C3*sin(2*sqrt(2)*t)-(1/2)*_C4*cos(2*sqrt(2)*t)}
Eigenanalysis method section 7.4
u(t) = (a1 cos(sqrt(2)t) + b1 sin(sqrt(2)t)) v1 + (a2 cos(sqrt(8)t) + b2 sin(sqrt(8)t)) v2
where (-2,v1), (-8,v2) are the eigenpairs of A.  The two vector terms in u(t) are called
the natural modes of oscillation. The natural frequencies are sqrt(2), sqrt(8).
Eigenanalysis of A gives v1=[1,1], v2=[2,-1].
Railway cars. Problem 7.4-24
Cayley-Hamilton-Ziebur method
Laplace Resolvent method for second order
Eigenanalysis method section 7.4
```

## Fri Apr 20: Non-Homogeneous Systems. Sections 8.1, 8.2

```Non-Homogeneous Systems
Direct solution methods with the Laplace Resolvent
Computer Algebra System methods
Variation of Parameters Formula for systems
Exercise solutions: ch7 and ch8.
Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.
Extra Credit Maple Project: Earthquakes. Explore a 5-story or 7-story building
and the resonant frequencies of oscillation of the building which might make it destruct
during an earthquake. See Edwards-Penney, application section in 7.4.
```

## Fri Apr 20: Intro to stability theory for autonomous systems. Section 9.1

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

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

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

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

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