# 2250-4 12:55pm Lecture Record Week 14 S2010

**Last Modified**: April 25, 2010, 18:03 MDT. **Today**: January 21, 2018, 05:46 MST.
### Week 15, Apr 26 to 29: Sections 9.1, 9.2, 9.3, Final Exam Review

**27 Apr**: Stability. Almost Linear systems. Phase Diagram. Section 9.2

**Review of last week's topics**
**Maple Demonstration**
Maple phase diagram tools.
Example
x' = x + y,
y' = 1 - x^2
**Spiral, saddle, center, node.**
Classification
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.

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.
Exercises 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.
**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.
**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 F2009 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 and second order systems. For ch5, deeper problems on
variation of parameters and undetermined coefficients, resonance,
and beats.

**27 Apr**: 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.

**27 Apr**: Nonlinear Mechanical Systems. Section 9.4

**Final exam review continued**
Some chapter 8 and chapter 9 problems.
**Survey of Nonlinear mechanical systems.**
Hard and soft springs.
Nonlinear pendulum.
Undamped pendulum.
Damped pendulum.
Phase diagrams.
Energy conservation laws and separatrices.

**28 Apr**Final Exam Review

Ben Murphy, WEB 104 12:55 - 3:00pm

**Text**: Final exam study guide (8.1 K, txt, 30 Dec 2009)
**Thu 29 Apr**: Final Exam Review, WEB 104

Gustafson: 1-3pm

**Text**: Final exam study guide (8.1 K, txt, 30 Dec 2009)
**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 (8.1 K, txt, 30 Dec 2009)