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# 2250 8:05am Lectures Week 16 S2015

Last Modified: December 12, 2014, 13:08 MST.    Today: September 23, 2018, 13:49 MDT.

## 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)
```
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.