Google Search in:

2250-2 7:30am Lecture Record Week 14 F2010

Last Modified: November 29, 2010, 20:48 MST.    Today: October 17, 2017, 05:35 MDT.

Week 14, Nov 29 to Dec 3: Sections 7.3, 7.4, 8.1, 8.2

Nov 29 and 30: Sections 6.2, 7.3

Diagonalization Theory
   In the case of a 2x2 matrix A,
   FOURIER'S MODEL is
        A(c1 v1 + c2 v2) = c1(lambda1 v1) + c2(lambda2 v2)
          where v1,v2 are a basis for the plane
   equivalent to DIAGONALIZATION
        AP=PD, where D=diag(lamba1,lambda2), P=augment(v1,v2),
          where det(P) is not zero
   equivalent to EIGENPAIR EQUATIONS
        A(v1)=lambda1 v1,
        A(v2)=lambda2 v2,
        where vectors v1,v2 are independent
Drill Problems
   1. Problem: Given P and D, find A in the relation AP=PD.
   2. Problem: Given Fourier's model, find A.
   3. Problem: Given A, find Fourier's model.
   4. Problem: Given A, find all eigenpairs.
   5. Problem: Given A, find packages P and D such that AP=PD.
   6. Problem: Give an example of a matrix A which has no Fourier's model.
   7. Problem: Give an example of a matrix A which is not diagonalizable.
   8. Problem: Given 2 eigenpairs, find the 2x2 matrix A.
Cayley-Hamilton topics, Section 6.3.
   Power Method
    Computing powers of matrices.
    Stochastic matrices.
    Example of 1984 telecom companies ATT, MCI, SPRINT with discrete
    dynamical system u(n+1)=A u(n). Matrix A is stochastic.
     EXAMPLE:
                    [ 6  1  5 ]               [ a(t) ]
             10 A = [ 2  7  1 ]        u(t) = [ m(t) ]
                    [ 2  2  4 ]               [ s(t) ]

     Meaning: 60% stay with ATT and 20% switch to MCI, 20% switch to SPRINT.
              70% stay with MCI and 20% switch to SPRINT, 10% switch to ATT.
              40% stay with SPRINT and 50% switch to ATT, 10% switch to MCI.
   Google Algorithm
        Lawrence Page's pagerank algorithm, google web page rankings.
  Methods to solve dynamical systems
  Consider the 2x2 system
    x'=x-5y, y'=x-y, x(0)=1, y(0)=2.
   Cayley-Hamilton-Ziebur method.
   Laplace resolvent.
   Eigenanalysis method.
   Exponential matrix using maple
   Putzer's method to compute the exponential matrix
   Spectral methods [ch8; not studied in 2250]
 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). For the algorithm, see the slides.
  2. Laplace resolvent L(u)=(s I - A)^(-1) u(0)
     See slides for details about the resolvent equation.
  3. Eigenanalysis  u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2
      See chapter 7 in Edwards-Penney for examples and details.
  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
      See the slides and manuscript on systems for proofs and details.
    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.


30 Nov: Second Order Systems. Section 7.4

Exam 3 Review
   Shortest trial solution in undetermined coefficients.
      Example: Sample exam.
   Eigenvalues
     A 4x4 matrix.
     Block determinant theorem.
   Eigenvectors for a 4x4.
      B:=matrix([[5,0,0,0],[0,5,0,0],[0,0,0,3],[0,0,-3,0]]);
         lambda=5,5,3i,-3i
         v1=[1,0,0,0], v2=[0,1,0,0], v3=[0,0,i,-1], v4=[0,0,i,1]
     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, t2 to 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.
   Final exam: Second shifting theorem in Laplace theory.
Second Order Systems
     How to convert mx''+cx'+kx=F0 cos (omega t) into a
       dynamical system  u'=Au+F(t).
     Electrical systems u'=Au+E(t) from LRC circuit equations.
     Electrical systems of order two: networks
     Mechanical systems of order two: coupled systems
     Second order systems u''=Au+F
       Examples are railway cars, earthquakes,
       vibrations of multi- component systems,
       electrical networks.
 Second Order Vector-Matrix Differential Equations
  The model u'' = Ax + F(t)
  Coupled Spring-Mass System. Problem 7.4-6
    A:=matrix([[-6,4],[2,-4]]); eigenvals(A); 
    lambda1= -2, lambda2= -8
    Ziebur's Method
    roots for Ziebur's theorem are plus or minus sqrt(lambda)
       Roots = sqrt(2)i,  sqrt(8)i, -sqrt(2)i, -sqrt(8)i
       Atoms = cos (sqrt(2)t), sin(sqrt(2)t), cos(sqrt(8)t), sin(sqrt(8)t)
       Vector x(t) = vector linear combination of the above 4 atoms
    Maple routines for second order
         de1:=diff(x(t),t,t)=-6*x(t)+4*y(t); de2:=diff(y(t),t,t)=2*x(t)-4*y(t);
         dsolve({de1,de2},{x(t),y(t)});
           x(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)+_C3*sin(2*sqrt(2)*t)+_C4*cos(2*sqrt(2)*t),
           y(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)-(1/2)*_C3*sin(2*sqrt(2)*t)-(1/2)*_C4*cos(2*sqrt(2)*t)}
     Eigenanalysis method section 7.4
           u(t) = (a1 cos(sqrt(2)t) + b1 sin(sqrt(2)t)) v1 + (a2 cos(sqrt(8)t) + b2 sin(sqrt(8)t)) v2
             where (-2,v1), (-8,v2) are the eigenpairs of A.  The two vector terms in u(t) are called
             the natural modes of oscillation. The natural frequencies are sqrt(2), sqrt(8).
             Eigenanalysis of A gives v1=[1,1], v2=[2,-1].
    Railway cars. Problem 7.4-24
     Cayley-Hamilton-Ziebur method
     Laplace Resolvent method for second order
     Eigenanalysis method section 7.4

1 Dec and 3 Dec: Non-Homogeneous Systems. Sections 8.1, 8.2

Non-Homogeneous Systems
    Direct solution methods with the Laplace Resolvent
    Computer Algebra System methods
    Variation of Parameters Formula for systems
Exercise solutions: ch7 and ch8.
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 7.4.

1 Dec and 3 Dec: Intro to stability theory for autonomous systems. Section 9.1

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

3 Dec: Stability. Classifications. Phase Diagram. Section 9.1, 9.2

Spiral, saddle, center, node.
  Linearization theory.
  Jacobian.

Detecting stability:
   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 calculated the
    sub-classification.
B>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.
    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. (152.6 K, pdf, 23 Nov 2010)
    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 (1068.7 K, pdf, 28 Nov 2010)
    Text: Laplace theory problem notes F2008 (8.9 K, txt, 18 Nov 2010)
    Text: Final exam study guide (0.0 K, txt, 31 Dec 1969)