# 2250 Lectures Week 14 S2015

Last Modified: April 13, 2015, 05:15 MDT.    Today: September 24, 2018, 11:21 MDT.

### Week 14: Sections 7.3, 7.4, 9.1, 9.2

``` Edwards-Penney, sections 6.2, 6.3, 7.3, 7.4, 9.1, 9.2
The textbook topics, definitions and theoremsEdwards-Penney 6.1, 6.2 (7.6 K, txt, 18 Dec 2013)Edwards-Penney 7.1, 7.2, 7.3, 7.4 (25.6 K, txt, 18 Dec 2013)Edwards-Penney 9.1,9.2,9.3,9.4 (12.1 K, txt, 18 Dec 2013)```

## Monday: Sections 6.2, 6.3, 7.3

```Exam 2 Review
Sample exam 2, Problem types
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 [slides, not studied in 2250]
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(d_1)+ (atom_2)vec(d_2)
Atoms: They are constructed by Euler's theorem from roots of det(A-rI)=0
Vectors vec(d_1),vec(d_2) are found from the equation
[d1 | d2]=[u(0) | Au(0)](W(0)^T)^(-1)
where W(t) is the Wronskian matrix of the two atoms.
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.
This method fails when matrix A is not diagonalizable.
EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]),
A=matrix([[2,3],[0,4]]) using
Zeibur's method, Laplace resolvent and eigenanalysis.
```

## Monday and Wednesday: Second Order Systems. Section 7.4

```Exam 2 Review
Sample exam 2, Chapter 6 problems
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.
Second and 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)
Characteristic Equation for a Second Order System
Euler's Substitution: u = exp(lambda t) v
u = solution vector in u'' = Au
v = unknown vector, to be determined
lambda = scale factor, a number real or complex
Substitute Euler's expression into u'' = Au, solve for lambda and v.
Then the pair (lambda,v) determines a solution u of u'' = Au
Coupled Spring-Mass System. Problem 7.4-6
A:=matrix([[-6,4],[2,-4]]); eigenvals(A);
lambda1= -2, lambda2= -8

Ziebur's Method and Euler's Characteristic Equation
Why Ziebur's method applies and the First Order System w'=Bw
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
```
```Some Deprecated Topics
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 eigenvalues 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.
```

## Monday: Dynamical Systems. Sections 9.1, 9.2

```Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.
MAPLE: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)
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.
MAPLE: Maple Lab 8. Earthquakes (0.0 K, pdf, 31 Dec 1969)```
```Dynamical Systems Topics
Equilibria.
Stability.
Instability.
Asymptotic stability.
Classification of equilibria for u'=Au when
det(A) is not zero, for the 2x2 case.
Impact of Cayley-Hamilton-Ziebur on classification
```
```Slides on Dynamical Systems
Manuscript: Systems theory and examples (730.9 K, pdf, 10 Apr 2014)   Slides: Laplace second order systems, spring-mass,boxcars, earthquakes (288.1 K, pdf, 03 Mar 2012)   Slides: Introduction to dynamical systems (145.4 K, pdf, 05 Apr 2015)   Slides: Phase Portraits for dynamical systems (221.3 K, pdf, 05 Apr 2015)   Slides: Stability for dynamical systems (154.4 K, pdf, 05 Apr 2015)   Slides: Nonlinear classification spiral, node, center, saddle (98.0 K, pdf, 05 Apr 2015)   Slides: Matrix Exponential, Putzer Formula, Variation Parameters (130.1 K, pdf, 03 Mar 2012)
References for Eigenanalysis and Systems of Differential Equations.
Slides: Algebraic eigenanalysis (187.6 K, pdf, 03 Mar 2012) Slides: What's eigenanalysis 2008 (174.2 K, pdf, 03 Mar 2012) Slides: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008) Slides: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007) Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (137.7 K, pdf, 28 Mar 2015) Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012) Manuscript: Systems of DE examples and theory (730.9 K, pdf, 10 Apr 2014) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014) Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008) Text: History of telecom companies (1.9 K, txt, 03 Apr 2013)
Systems of Differential Equations applications
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.
MAPLE: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)
Laplace theory references
Slides: Laplace and Newton calculus. Photos. (200.2 K, pdf, 03 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (144.8 K, pdf, 25 Mar 2015) Slides: Laplace rules (160.3 K, pdf, 03 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 03 Mar 2012) Slides: Laplace examples (133.2 K, pdf, 27 Mar 2015) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969) Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014) Slides: Laplace resolvent method (88.1 K, pdf, 03 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 03 Mar 2012) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 09 Apr 2014) Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008) Manuscript: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes (17.7 K, txt, 17 Mar 2014) Text: Final exam study guide (8.3 K, txt, 06 Jan 2015)```