### 2280 Lectures Week 13 S2016

Last Modified: January 08, 2016, 12:26 MST.    Today: September 23, 2018, 13:32 MDT.
``` Edwards-Penney, sections 6.1, 6.2, 6.3, 6.4
The textbook topics, definitions and theoremsEdwards-Penney 6.1, 6.2, 6.3, 6.4 (11.8 K, txt, 05 Apr 2015)```

#### Monday and Tuesday: Stability. Classifications. Phase Diagram. Sections 6.1, 6.2

```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)   Manuscript: Dynamical systems (1037.6 K, pdf, 05 Apr 2015)   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)```
```Dynamical Systems 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:
Each vector solution u(t) is a vector linear combination of Euler atoms.
An atom has limit zero at t=infinity if and only if real part of the root is negative.
A root with zero real part has Euler atoms x^n or pure sine and cosine times x^n.
Theorem. Re(lambda)<0 ==> asym. stability.
Stability at t=-infinity classifies Unstable solutions.

Maple phase diagram tools. Demonstration for the example
x' = x + y,
y' = 1 - x^2

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.
```
```Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.
Extra Credit Maple Project: Earthquakes. Explore a 5-story or 7-story building
and the resonant frequencies of oscillation of the building which might make it destruct
during an earthquake. See Edwards-Penney, application section in 5.3.
HTML: Extra Credit Sources (4.6 K, html, 09 Jan 2016)
```

#### Monday and Tuesday: Intro to stability theory for autonomous systems. Section 6.2

```Review of 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 ==> asymptotic stability.
Stable center picture. Definition of stability.
Stability at t=-infinity classifies Unstable solutions.
Maple Demonstration
Maple phase diagram tools.
Example
x' = x + y,
y' = 1 - x^2
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.
```
```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.
Nonlinear stability
phase diagrams,
classification.
Using DEtools and DEplot in maple to make phase diagrams.
Jacobian.
```
```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: Nonlinear Stability. Classification. Predator-Prey. Section 6.3

```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 6.1, 6.2.
```

#### Friday: Nonlinear Mechanical Systems. Section 6.4

```Nonlinear mechanical systems.
Hard and soft springs.
Nonlinear pendulum.
Undamped pendulum.
Damped pendulum.
Phase diagrams.
Energy conservation laws and separatrices.
```
```Review of the 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.
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.

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.

```
```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)```