2250-1 7:30am Lecture Record Week 14 S2010

Last Modified: April 25, 2010, 17:55 MDT.    Today: October 20, 2017, 19:25 MDT.

Week 14, Apr 19 to 23: Sections 8.1, 8.2, 9.1, 9.2

19 Apr: Sections 8.1, 8.2

 Survey of Methods for solving a 2x2 dynamical system
  1. Cayley-Hamilton-Ziebur method for u'=Au
    Solution: u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n)
    Atoms: They are constructed by Euler's theorem from roots of det(A-rI)=0
    Vectors: Symbols vec(c_1), ..., vec(c_n) are not arbitrary. They are
         determined from A and u(0). Algorithm outlined above for 2x2.
  2. Laplace resolvent L(u)=(s I - A)^(-1) u(0)
  3. Eigenanalysis  u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2
  4. Putzer's method for the 2x2 matrix exponential.
    Solution of u'=Au is: u(t) = exp(A t)u(0)
    THEOREM: exp(A t) = r1(t) I + r2(t) (A-lambda_1 I),
      Lambda Symbols: lambda_1 and lambda_2 are the roots of det(A-lambda I)=0.
      The DE System:
         r1'(t) = lambda_1 r1(t),         r1(0)=0,
         r2'(t) = lambda_2 r2(t) + r1(t), r2(0)=0
    THEOREM. The formula can be used as
                                 e^{r1 t} - e^{r2 t}
         e^{At} = e^{r1 t} I  +  ------------------- (A-r1 I)
                                       r1 - r2
         where r1=lambda_1, r2=lambda_2 are the eigenavalues of A.

    EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]),
             A=matrix([[2,3],[0,4]]) using the matrix exponential,
             Zeibur's method, Laplace resolvent and eigenanalysis.
    EXAMPLE. Solve a non-homogeneous system u'=Au+F(t), u(0)=vector([0,0]),
             A=matrix([[2,3],[0,4]]), F(t)=vector([3,1]) using variation
             of parameters.

21 Apr: Intro to stability theory for autonomous systems. Section 9.1

Exam 3 Review
   Eigenvalues
     A 4x4 matrix.
     Block determinant theorem.
   Eigenvectors for a 4x4.
     lambda=5,5,3i,-3i
     One panel for lambda=5
       First frame is A-5I with 0 appended
       Find rref
       Apply last frame algorithm
       Scalar general solution
       Take partials on t1, t2to find v1,v2
       Eigenpairs are (5,v1), (5,v2)
     One panel for lambda=3i
       Same outline as lambda=5
       Get one eigenpair (3i,v3)
       Other eigenpair=(-3i,v4) where v4 is the conjugate of v3.
   Shortest trial solution in undetermined coefficients.
   Second shifting theorem in Laplace theory.

Dynamical Systems Topics
  Equilibria.
  Stability.
  Instability.
  Asymptotic stability.
  Classification of equilibria for u'=Au when
    det(A) is not zero, for the 2x2 case.


23 Apr: Stability. Classifications. Phase Diagram. Section 9.1, 9.2

Spiral, saddle, center, node.
  Linearization theory.
  Jacobian.
  
  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 calculated the 
    sub-classification.
  
    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)
    Manuscript: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007)
    Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (111.4 K, pdf, 30 Nov 2009)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
    Manuscript: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)
    Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
    Text: History of telecom companies (1.4 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.
    Laplace theory references
    Slides: Laplace and Newton calculus. Photos. (145.3 K, pdf, 01 Nov 2009)
    Slides: Intro to Laplace theory. Calculus assumed. (109.5 K, pdf, 01 Nov 2009)
    Slides: Laplace rules (112.2 K, pdf, 01 Nov 2009)
    Slides: Laplace table proofs (130.3 K, pdf, 01 Nov 2009)
    Slides: Laplace examples (101.2 K, pdf, 07 Nov 2009)
    Slides: Piecewise functions and Laplace theory (64.7 K, pdf, 01 Nov 2009)
    MAPLE: Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969)
    Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)
    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 (186.8 K, pdf, 20 Oct 2009)
    Manuscript: Laplace theory 2008 (350.5 K, pdf, 06 Mar 2009)
    Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1968.3 K, pdf, 13 Nov 2003)
    Text: Laplace theory problem notes F2008 (8.9 K, txt, 31 Dec 2009)
    Text: Final exam study guide (7.6 K, txt, 12 Dec 2009)