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2280 Lectures Week 12 S2017

Last Modified: October 19, 2016, 03:30 MDT.    Today: November 24, 2017, 22:48 MST.

Week 12: Systems, Sections 5.1 to 5.6

 Edwards-Penney, sections 4.2, 5.1 to 5.6, 6.1, 6.2
  The textbook topics, definitions and theorems
Edwards-Penney 5.1 to 5.6 (24.0 K, txt, 06 Jan 2015)
Edwards-Penney 6.1 to 6.4 (11.8 K, txt, 05 Apr 2015)
Maple Example to find roots of the characteristic equation
 Consider the recirculating brine tank example, section 5.2:
    20 x' = -6x + y,
    20 y' = 6x - 3y
 The maple code to solve the char eq:
   A:=(1/20)*Matrix([[-6,1],[6,-3]]);
   linalg[charpoly](A,r);
   solve(%,r);
   # Answer: -9/40+(1/40)*sqrt(33), -9/40-(1/40)*sqrt(33)
EIGENANALYSIS WARNING
  Reading Edwards-Penney Chapter 5 may deliver the wrong ideas
  about how to solve for eigenpairs. The examples emphasize a
  clever shortcut, which does not apply in general to solve for
  eigenpairs.

Solving DE System u' = Au by Eigenanalysis
  Example: Solving a 2x2 dynamical system
     Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,3],[0,4]]).
     Dynamical system scalar form is
         x' = 2x + 1y,
         y' = 3y,
         x(0)=1, y(0)=2.
     Find the eigenpairs (2, v1), (3,v2) where v1=vector([1,0])
     and v2=vector([1,1]).
     THEOREM. The solution of u' = Au in the 2x2 case is
         u(t) = c1 exp(lambda1 t) v1 + c2 exp(lambda2 t) v2
     APPLICATION:
       u(t) = c1 exp(2t) v1 + c2 exp(4t) v2
                        [ 1 ]            [ 1 ]
       u(t) = c1 e^{2t} [   ] + c2 e^4t} [   ]
                        [ 0 ]            [ 1 ]
       which means
          x(t) = c1 exp(2t) + 3 c2 exp(4t),
          y(t) = 2 c2 exp(4t).

Section 5.1: Topics from linear systems:

Systems of two differential equations
   Solving a system from Chapter 1 methods
   The Laplace resolvent method for systems.
    Cramer's Rule,
    Matrix inversion methods.
   EXAMPLE: Solving a 2x2 dynamical system using Laplace's resolvent method.
     Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,1],[0,3]]).
   EXAMPLE: Problem 10.2-16,
     This problem is a 3x3 system for x(t), y(t), z(t) solved
     by Laplace theory methods. The resolvent formula
                     (sI - A) L(u(t)) = u(0)
     with u(t) the fixed 3-vector with components x(t), y(t), z(t),
     amounts to a shortcut to obtain the equations for L(x(t)),
     L(y(t)), L(z(t)). After the shortcut is applied, in which
     Cramer's Rule is the method of choice, to find the formulas,
     there is no further shortcut: we have to find x(t), for example,
     by partial fractions and the backward table, followed by Lerch's
     theorem.

 EXAMPLE. Recirculating brine tanks
    20 x' = -6x + y,
    20 y' = 6x - 3y
    x(t)=pounds of salt in tank 1 (100 gal)
    y(t)=pounds of salt in tank 2 (200 gal)
    x(0), y(0) = initial salt amounts in each tank
    t=minutes
    20=inflow rate=outflow rate
     0=inflow salt concentration
 EXAMPLE. Solve x'=-2y, y'=x/2.
   ANSWER: x(t)=A cos(t) + B sin(t), y(t) = (-A/2) cos(t) +
           (B/2) sin(t).
Conversion Methods to Create a First Order System
    The position-velocity substitution.
    How to convert second order systems.
    EXAMPLE. Transform to a first order system
             2x'' = -6x + 2y,
              y'' = 2x - 2y + 40 sin(3t)
           ANSWER: u1=x,u2=x',u3=y,u4=y' ==>
              u1' = u2,
              u2' = -3u1 + u3, [a division by 2 needed]
              u3' = u4,
              u4' = 2u1 - 2u3 + 40 sin(3t)
    How to convert nth order scalar differential equations.
    EXAMPLE. x''' + 2x'' + x = 0
      Use u1=x(t), u2=x'(t), u3=x''(t)
    Non-homogeneous terms and the vector matrix system
         u' = Au + F(t)
    Non-linear systems and the vector-matrix system
         u' = F(t,u)
    Answer checks for u'=Au
      Example: The system u'=Au, A=matrix([[2,1],[0,3]]);

Systems of two differential equations
   The Laplace resolvent method for systems.
        Solving the resolvent equation for L(x), L(y).
          Cramer's Rule
          Matrix inversion
          Elimination
   Example: Solving a 2x2 dynamical system
     Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,1],[0,3]]).
     Dynamical system scalar form is
         x' = 2x + y,
         y' = 3y,
         x(0)=1, y(0)=2.
        The equations for L(x), L(y)
             (s-2)L(x)  +  (-1)L(y)=1,
                (0)L(x)  + (s-3)L(y)=2
        REMARK: Laplace resolvent method shortcut.
        How to solve the [resolvent] equations for L(x), L(y).
          Cramer's Rule
          Matrix inversion
          Elimination
          Answers: L(x) = delta1/delta, L(y)=delta2/delta
                       delta=(s-2)(s-3), delta1=s-1, delta2=2(s-2)
                       L(x) = -1/(s-2)+2/(s-3), L(y)=2/(s-3)
        Backward table and Lerch's theorem
        Answers: x(t) = - e^{2t} + 2 e^{3t},
                     y(t) = 2 e^{3t}.
      Edwards-Penney Shortcut Method in Example  1, 4.2. Uses Chapter 1+3 methods.
      This is the Cayley-Hamilton-Ziebur method. See below.
        Solve w'+p(t)w=0 as w = constant / integrating factor.
        Then  y' -2y=0 ==> y(t) = 2 exp(3t)
        Stuff y(t) into the first DE to get the linear DE
           x' - 2x = 2 exp(3t)
        Superposition: x(t)=x_h(t)+x_p(t),
           x_h(t)=c exp(2t),
           x_p(t) = d1 exp(t) = 2 exp(3t) by undetermined coeff.
        Then x(t)= - exp(2t) + 2 exp(3t).

Cayley-Hamilton Theorem
   A matrix satisfies its own characteristic equation.
   ILLUSTRATION: det(A-r I)=0 for the previous example
     is (2-r)(3-r)=0 or r^2 -5r + 6=0. Then C-H says
        A^2 - 5A + 6I = 0.
Cayley-Hamilton-Ziebur Method
  ZIEBUR'S LEMMA.
        The components of u in u'=Au are linear combinations of
        the atoms created by Euler's theorem applied to the
        roots of the characteristic equation det(A-rI)=0.
  THEOREM. Solve u'=Au without complex numbers or eigenanalysis.
        The solution of u'=Au is a linear combination of atoms
        times certain constant vectors [not arbitrary vectors].
             u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n)
Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (137.4 K, pdf, 14 Mar 2016) PROBLEM: Solve by Cayley-Hamilton-Ziebur the 2x2 dynamical system x' = 2x + y, y' = 3y, x(0)=1, y(0)=2. The characteristic equation is (2-lambda)(3-lambda)=0 with roots lambda = 2,3 Euler's theorem implies the atoms are exp(2t), exp(3t). Ziebur's Theorem says that u(t) = exp(2t) vec(v_1) + exp(3t) vec(v_2) where vectors v_1, uv_2 are to be determined from the matrix A = matrix([[2,1],[0,3]]) and initial conditions x(0)=1, y(0)=2. ZIEBUR ALGORITHM. To solve for v_1, v_2 in the example, differentiate the equation u(t) = exp(2t) v_1 + exp(3t) v_2 and set t=0 in both relations. Then u'=Au implies u_0 = v_1 + v_2, Au_0 = 2 v_1 + 3 v_2. These equations can be solved by elimination. The answer: v_1 = (3 u_0 -Au_0), v_2 = (Au_0 - 2 u_0) = vector([-1,0]) = vector([2,2]) Vectors v_1, v_2 are recognized as eigenvectors of A for lambda=2 and lambda=3. ZIEBUR SHORTCUT [Edwards-Penney textbook method, Example 1 in 4.2] Start with Ziebur's theorem, which implies that x(t) = k1 exp(2t) + k2 exp(3t). Use the first DE to solve for y(t): y(t) = x'(t) - 2x(t) = 2 k1 exp(2t) + 3 k2 exp(3t) - 2 k1 exp(2t) - 2 k2 exp(3t)) = k2 exp(3t) For example, x(0)=1, y(0)=2 implies k1 and k2 are defined by k1 + k2 = 1, k2 = 2, which implies k1 = -1, k2 = 2, agreeing with a previous solution formula.


Tuesday and Wednesday: Second Order Systems. Section 5.3

Exam 2 Review
      Sample exam 2
   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.

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

Slides: Cayley-Hamilton-Ziebur for second order systems (130.4 K, pdf, 22 Mar 2017) The model u'' = Ax + F(t) Coupled Spring-Mass System. Problem 5.3-6 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 5.3 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 5.3-24 Cayley-Hamilton-Ziebur method Laplace Resolvent method for second order Eigenanalysis method section 5.2
Some Essential Topics
  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 eigenvalues 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.

Wednesday and Monday: Dynamical Systems. Sections 6.1, 6.2

Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.

MAPLE: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969) 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.
Dynamical Systems Topics
  Equilibria.
  Stability.
  Instability.
  Asymptotic stability.
  Classification of equilibria for u'=Au when
    det(A) is not zero, for the 2x2 case.
    Impact of Cayley-Hamilton-Ziebur on classification
Slides on Dynamical Systems
   
Manuscript: Systems theory and examples (730.9 K, pdf, 09 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)

Systems of Differential Equations references
 
Manuscript: Systems of DE examples and theory (730.9 K, pdf, 09 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, 20 Aug 2008)
References for Eigenanalysis and Systems of Differential Equations.
 
Sildes: Algebraic eigenanalysis (173.4 K, pdf, 14 Mar 2016)
Slides: What's eigenanalysis 2008 (161.5 K, pdf, 14 Mar 2016)
Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (137.4 K, pdf, 14 Mar 2016)
Manuscript: Systems of DE examples and theory (730.9 K, pdf, 09 Apr 2014)
Slides: Cayley-Hamilton-Ziebur for second order systems (130.4 K, pdf, 22 Mar 2017)
Text: Final exam study guide (7.6 K, txt, 20 Apr 2017)