# 2250-1 7:30am Lecture Record Week 15 S2014

Last Modified: April 24, 2014, 11:29 MDT.    Today: August 16, 2018, 00:44 MDT.

### Week 15: Sections 9.1, 9.2, 9.3

``` Edwards-Penney, sections 9.1 to 9.4
The textbook topics, definitions and theoremsEdwards-Penney 9.1 to 9.4 (12.1 K, txt, 19 Dec 2013)```

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

```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 calculates 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.
HTML: Extra Credit Sources (4.8 K, html, 17 Dec 2013)
```

#### Monday and 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 ==> asymptotic 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.
```
```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 some of ch10.
Review packet distributed on the web.
Final exam details
Less contact with ch3, ch4, ch5 due their appearance on
exams 1,2,3.
From F2008 to S2013, there are extra chapters 8,9 on the final.
In S2014, chapter 8 is removed and ch1+ch2 added.
A good sample is the S2013 final exam, removing Ch8 and adding Ch1+Ch2.
Chapters 5,6,7,10 will undergo changes and spins. For ch10, more
contact with the second shifting theorem and the Dirac Impulse. For
ch7, there are additional methods for solving DE, especially
Cayley-Hamilton-Ziebur 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.
```

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

```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 (730.9 K, pdf, 10 Apr 2014)   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.
Slides: Algebraic eigenanalysis (187.6 K, pdf, 04 Mar 2012) Slides: What's eigenanalysis 2008 (174.2 K, pdf, 04 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. (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 (730.9 K, pdf, 10 Apr 2014) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 10 Apr 2014) 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 (151.6 K, pdf, 18 Mar 2014) Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014) 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 (99.3 K, pdf, 10 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, 18 Mar 2014) Text: Final exam study guide (8.0 K, txt, 20 Apr 2014)```