2250-2 7:30am Lecture Record Week 15 F2010

Last Modified: December 07, 2010, 06:04 MST. Today: April 22, 2018, 23:37 MDT.

Week 15, Dec 6 to 10: Sections 9.1, 9.2, 9.3, Final Exam Review

Dec 6, 8 Final Exam Review

Michal, JWB 335 at 12:55pm
Text: Final exam study guide (8.3 K, txt, 09 Dec 2010)

Dec 9 Final Exam Review

Laura, WEB 103 7:30am
Text: Final exam study guide (8.3 K, txt, 09 Dec 2010)

Saturday Dec 11: Final Exam Review

Gustafson: 1-3pm in JWB 335
Text: Final exam study guide (8.3 K, txt, 09 Dec 2010)

Dec 6 and 7: Stability. Almost Linear systems. Phase Diagram. Section 9.2

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.
  Maple Demonstration
  Maple phase diagram tools.
  Example
    x' = x + y,
    y' = 1 - x^2
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.
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 S2010 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 order systems. For ch5, deeper problems on the topics of
   variation of parameters and undetermined coefficients, resonance,
   and beats.

Dec 8: 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.

Dec 8: 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.

Slides on Dynamical Systems

Manuscript: Systems theory and examples (785.8 K, pdf, 16 Nov 2008)

Slides: Laplace second order systems, spring-mass,boxcars, earthquakes (248.9 K, pdf, 01 Nov 2009)

Slides: Introduction to dynamical systems (126.2 K, pdf, 30 Nov 2009)

Slides: Phase Portraits for dynamical systems (205.5 K, pdf, 11 Dec 2009)

Slides: Stability for dynamical systems (125.7 K, pdf, 30 Nov 2009)

Slides: Nonlinear classification spiral, node, center, saddle (75.3 K, pdf, 12 Dec 2009)

Slides: Matrix Exponential, Putzer Formula, Variation Parameters (85.3 K, pdf, 14 Dec 2009)

References for Eigenanalysis and Systems of Differential Equations.

Manuscript: Algebraic eigenanalysis (127.8 K, pdf, 23 Nov 2009)

Manuscript: What's eigenanalysis 2008 (126.8 K, pdf, 11 Apr 2010)

Manuscript: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008)