# 2250 Lecture Record Week 15 F2009

**Last Modified**: December 14, 2009, 09:18 MST. **Today**: October 23, 2017, 18:09 MDT.
### Week 15, Dec 7-11: Sections 9.1,9.2,9.3,9.4. Final Exam Review.

**07 Dec**: Intro to stability theory for autonomous systems. Section 9.1

**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.
**Nonlinear stability theory**
When the linearized classification and stability transfers to
the nonlinear system.

**08 Dec**: Stability. Almost Linear systems. Phase Diagram. Section 9.2

More on stability theory.
Linear and almost linear systems,
Jacobian,
stability of linear systems,
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 S2009 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.

**09 Dec**: 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.

**09-10-11 Dec**: Fusi and Richins. Final Exam Review, WEB 103

Richins: 2pm on Dec 09 in WEB 103
Richins-Fusi: 7:30am on 10 Dec in WEB 103
Fusi: 2pm, on Dec 11 in WEB 103
Richins: 4pm, on Dec 11 in WEB 103
Gustafson: 1-3pm Saturday Dec 12 in LCB 219

**11 Dec**: Nonlinear Mechanical Systems. Section 9.4

**Final exam review continued**
Some chapter 8 and chapter 9 problems.
Subspace problems from chapter 4.
**Nonlinear mechanical systems.**
Hard and soft springs.
Nonlinear pendulum.
Undamped pendulum.
Damped pendulum.
Phase diagrams.
Energy conservation laws and separatrices.

**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 (135.8 K, pdf, 07 Apr 2008)

**Manuscript**: What's eigenanalysis 2009 (129.5 K, pdf, 07 Apr 2008)

**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 (1.1 K, txt, 05 Oct 2008)

**Systems of Differential Equations references**

**Manuscript**: Systems of DE examples and 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)