# 2250 8:05am Lectures Week 16 S2016

**Last Modified**: January 08, 2016, 12:27 MST. **Today**: March 20, 2018, 23:12 MDT.
### Week 16: Section 9.4, Final Exam Review

**Saturday Apr 26** Special Final Exam Review, 2pm JTB 140

The final exam sample with solutions can also be found at the course web site CALENDAR.

**Text**: Final exam study guide (0.0 K, txt, 31 Dec 1969)

**Pdf**: Final exam sample 2012 with alternate problems, questions only (0.0 K, pdf, 31 Dec 1969)

**Pdf**: Final exam sample 2012 with alternate problems, questions and answers (0.0 K, pdf, 31 Dec 1969)
**Monday**: Nonlinear Mechanical Systems. Section 9.4

**Nonlinear mechanical systems.**
Hard and soft springs.
Nonlinear pendulum.
Undamped pendulum.
Damped pendulum.
Phase diagrams.
Energy conservation laws and separatrices.

**Monday Review**: Stability. Almost Linear systems. Phase Diagram. Sections 9.2, 9.3

**Review of last week's 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.
**Spiral, saddle, center, node.**
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.
**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.

**Tuesday and Wednesday Final Exam Review**

**Final exam review started.**
The final exam sample with solutions can also be found at the course web site CALENDAR.

**Text**: Final exam study guide (0.0 K, txt, 31 Dec 1969)

**Pdf**: Final exam sample 2012 with alternate problems, questions only (0.0 K, pdf, 31 Dec 1969)

**Pdf**: Final exam sample 2012 with alternate problems, questions and answers (0.0 K, pdf, 31 Dec 1969)

**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 (273.7 K, pdf, 14 Mar 2016)

**Slides**: Introduction to dynamical systems (144.9 K, pdf, 14 Mar 2016)

**Slides**: Phase Portraits for dynamical systems (221.2 K, pdf, 14 Mar 2016)

**Slides**: Stability for dynamical systems (158.2 K, pdf, 14 Mar 2016)

**Slides**: Nonlinear classification spiral, node, center, saddle (97.9 K, pdf, 14 Mar 2016)

**Slides**: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016)

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

**Slides**: Algebraic eigenanalysis (173.4 K, pdf, 14 Mar 2016)

**Slides**: What's eigenanalysis 2008 (161.5 K, pdf, 14 Mar 2016)

**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. (137.4 K, pdf, 14 Mar 2016)

**Slides**: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)

**Slides**: Laplace second order systems (273.7 K, pdf, 14 Mar 2016)

**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, 14 Mar 2016)

**Text**: Lawrence Page's pagerank algorithm (0.0 K, txt, 31 Dec 1969)

**Text**: History of telecom companies (0.0 K, txt, 31 Dec 1969)

**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. (188.3 K, pdf, 14 Mar 2016)

**Slides**: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016)

**Slides**: Laplace rules (144.9 K, pdf, 14 Mar 2016)

**Slides**: Laplace table proofs (151.9 K, pdf, 14 Mar 2016)

**Slides**: Laplace examples (133.7 K, pdf, 14 Mar 2016)

**Slides**: Piecewise functions and Laplace theory (98.5 K, pdf, 14 Mar 2016)

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

**Manuscript**: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014)

**Slides**: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016)

**Slides**: Laplace second order systems (273.7 K, pdf, 14 Mar 2016)

**Slides**: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)

**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, 22 Feb 2015)

**Text**: Laplace theory problem notes (0.0 K, txt, 31 Dec 1969)

**Text**: Final exam study guide (0.0 K, txt, 31 Dec 1969)