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

**Last Modified**: April 25, 2010, 17:55 MDT. **Today**: December 17, 2017, 04:49 MST.
### Week 14, Apr 19 to 23: Sections 8.1, 8.2, 9.1, 9.2

## 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.

**Slides on Dynamical Systems**

**Manuscript**: Systems theory and examples (785.8 K, pdf, 16 Nov 2008)

**Slides**: Laplace second order systems, spring-mass,boxcars, earthquakes (248.9 K, pdf, 01 Nov 2009)

**Slides**: Introduction to dynamical systems (126.2 K, pdf, 30 Nov 2009)

**Slides**: Phase Portraits for dynamical systems (205.5 K, pdf, 11 Dec 2009)

**Slides**: Stability for dynamical systems (125.7 K, pdf, 30 Nov 2009)

**Slides**: Nonlinear classification spiral, node, center, saddle (75.3 K, pdf, 12 Dec 2009)

**Slides**: Matrix Exponential, Putzer Formula, Variation Parameters (85.3 K, pdf, 14 Dec 2009)

**References for Eigenanalysis and Systems of Differential Equations.**

**Manuscript**: Algebraic eigenanalysis (127.8 K, pdf, 23 Nov 2009)

**Manuscript**: What's eigenanalysis 2008 (126.8 K, pdf, 11 Apr 2010)

**Manuscript**: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008)

**Manuscript**: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007)

**Slides**: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (111.4 K, pdf, 30 Nov 2009)

**Slides**: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)

**Slides**: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)

**Manuscript**: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)

**Slides**: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)

**Text**: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)

**Text**: History of telecom companies (1.4 K, txt, 30 Dec 2009)

**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.
**Laplace theory references**

**Slides**: Laplace and Newton calculus. Photos. (145.3 K, pdf, 01 Nov 2009)

**Slides**: Intro to Laplace theory. Calculus assumed. (109.5 K, pdf, 01 Nov 2009)

**Slides**: Laplace rules (112.2 K, pdf, 01 Nov 2009)

**Slides**: Laplace table proofs (130.3 K, pdf, 01 Nov 2009)

**Slides**: Laplace examples (101.2 K, pdf, 07 Nov 2009)

**Slides**: Piecewise functions and Laplace theory (64.7 K, pdf, 01 Nov 2009)

**MAPLE**: Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969)

**Manuscript**: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008)

**Slides**: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)

**Slides**: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)

**Slides**: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)

**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 (186.8 K, pdf, 20 Oct 2009)

**Manuscript**: Laplace theory 2008 (350.5 K, pdf, 06 Mar 2009)

**Transparencies**: Ch10 Laplace solutions 10.1 to 10.4 (1968.3 K, pdf, 13 Nov 2003)

**Text**: Laplace theory problem notes F2008 (8.9 K, txt, 31 Dec 2009)

**Text**: Final exam study guide (7.6 K, txt, 12 Dec 2009)