# 2250-1 7:30am Lecture Record Week 14 S2013

Last Modified: April 22, 2013, 06:04 MDT.    Today: August 18, 2018, 17:30 MDT.

### Week 14: Sections 9.1, 9.2, 9.3, 9.4

``` Edwards-Penney, sections 9.1 to 9.4
The textbook topics, definitions and theoremsEdwards-Penney 9.1 to 9.4 (12.1 K, txt, 17 Apr 2013)```

### Week 14: Sections 9.1, 9.2, 9.3, 9.4

#### Monday and Tuesday: Stability. Classifications. Phase Diagram. Sections 9.1, 9.2

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

#### Tuesday: Intro to stability theory for autonomous systems. Section 9.2

```Review of topics
Phase diagram.
Stability and the three pictures: Node, Center, Spiral
Detecting stability and instability for u'=Au at x=y=0:
Main Theorem: Re(lambda)<0 ==> asym. stability.
Stable center picture. Definition of stability.
Stability at t=-infinity classifies Unstable solutions.
Maple Demonstration
Maple phase diagram tools.
Example
x' = x + y,
y' = 1 - x^2
Classification pictures
Set 1: Stable node, stable center, stable spiral
Set 2: Unstable node, unstable saddle, unstable spiral
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 calculates 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.
```
```Spiral, saddle, center, node.
Linearization theory.
Jacobian.

Algebraic Detection of Linear stability for u'=Au:
Rule: det(A) not zero of all classifications!
Re(lambda)<0 ==> asymptotic stability
Re(lambda)=0 and lambda not zero ==> Center picture
Stability at t=-infinity classifies Unstable solutions.
When testing stability, we check t=infinity and t=-infinity.
Nonlinear stability theory u'=f(u)
When the linearized classification and stability transfers to
the nonlinear system.
stability of almost linear [nonlinear] systems,
phase diagrams,
classification of nonlinear systems.
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 S2012 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 order systems. For ch5, deeper problems on the topics of
variation of parameters and undetermined coefficients, resonance,
and beats.
```

#### Wednesday and Friday: Nonlinear Stability. Classification. Predator-Prey. Section 9.3

```Final exam review
Some chapter 8 and chapter 9 problems.

Nonlinear stability
phase diagrams,
classification.
Predator-Prey systems. How to tell which is the predator and which is
the prey.
Calculations for equilibrium points,
linearization,
classification of equilibria,
impact on the phase diagram.
Using DEtools and DEplot in maple to make phase diagrams.
Exercises 9.1, 9.2.
```
```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 (288.1 K, pdf, 04 Mar 2012)   Slides: Introduction to dynamical systems (158.0 K, pdf, 04 Mar 2012)   Slides: Phase Portraits for dynamical systems (239.3 K, pdf, 04 Mar 2012)   Slides: Stability for dynamical systems (170.8 K, pdf, 04 Mar 2012)   Slides: Nonlinear classification spiral, node, center, saddle (75.3 K, pdf, 12 Dec 2009)   Slides: Matrix Exponential, Putzer Formula, Variation Parameters (130.1 K, pdf, 04 Mar 2012)
References for Eigenanalysis and Systems of Differential Equations.
Manuscript: Algebraic eigenanalysis (187.6 K, pdf, 04 Mar 2012) Manuscript: What's eigenanalysis 2008 (174.2 K, pdf, 04 Mar 2012) 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. (152.9 K, pdf, 04 Mar 2012) Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012) Manuscript: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008) Slides: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012) Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008) Text: History of telecom companies (1.9 K, txt, 04 Apr 2013)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. (200.2 K, pdf, 04 Mar 2012) Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012) Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012) Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012) Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012) Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013) MAPLE: Maple Lab 7. Laplace applications (156.0 K, pdf, 04 Dec 2012) Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008) Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012) Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012) Slides: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012) 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 (351.3 K, pdf, 09 Apr 2013) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010) Text: Laplace theory problem notes (17.2 K, txt, 03 Dec 2012) Text: Final exam study guide (8.2 K, txt, 05 Dec 2012)```