# 2250-1 7:30am Lecture Record Week 15 S2010

Last Modified: April 25, 2010, 18:03 MDT.    Today: July 15, 2018, 13:31 MDT.

## 26 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

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.

```
```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.
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.
```

## 28 AprFinal Exam Review

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

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

## 28 AprFinal Exam Review

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)
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.