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2250 8:05am Lectures Week 16 S2016

Last Modified: January 08, 2016, 12:27 MST.    Today: November 17, 2017, 16:11 MST.

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)