Math 5490 11/3/2014 Dynamical Systems Math 5490 Summary So Far - PDF document

Math 5490 11/3/2014 Dynamical Systems Math 5490 Summary So Far November 3, 2014 Topics in Applied Mathematics: dx   a x a y   x  a a  d dt 11 12 x Introduction to the Mathematics of Climate    dt  A A 11 12 x x    dy   y  a a   a x  a y 21 22 21 22 Mondays and Wednesdays 2:30 – 3:45 dt http://www.math.umn.edu/~mcgehee/teaching/Math5490-2014-2Fall/ Eigenvalues and eigenvectors      Streaming video is available at Av v v 0 http://www.ima.umn.edu/videos/ v u If and are linearly independent eigenvectors Click on the link: "Live Streaming from 305 Lind Hall".   with corresponding eigenvalues and , Participation: then the general solution is https://umconnect.umn.edu/mathclimate    t  t t c e v c e u x ( ) 1 2 c c where and are arbitrary constants. 1 2 Linear independence : one is not a multiple of the other. Math 5490 11/3/2014 Dynamical Systems Dynamical Systems Changing Coordinates Changing Coordinates   x   x d   A       x             d x x d   dt y y  A S  S 1 AS                  y y y dt   dt              v   u  v  1 u  1 A Suppose that   and   are linearly independent eigenvectors of v u          S  v u  Av v Au u   2 2     with corresponding eigenvalues and . Introduce new variables and :    0   x   x  v   u    v u          AS  A v u     Av Au    v u    v u    S              v   u  1 1  1 1  S          , i.e.     0  v   u  v u     y   y         2 2 2 2  v u  S  1 1  v u    v  u  where   =   . v u     v  u   1 1      a a   v u  2 2  v  u    11 12 1 1   A v u       2 2 a a v u        d   d   d   x   x   21 22 2 2  v u     0       S S A AS 1 1 Then                 a v a v a u a u     dt   dt   dt   y   y   v u    11 1 12 2 11 1 12 2    0   2 2    a v a v a u a u         d 21 1 22 2 21 1 22 2   0   S 1 AS         v u        Av Au dt          0  Math 5490 11/3/2014 Math 5490 11/3/2014 Dynamical Systems Dynamical Systems Changing Coordinates Changing Coordinates     x           x         d   x   x d d   x   x d      S  1  S  1 A     S AS A     S AS                 y    y    dt   y   y     dt     dt   y   y     dt     S   v u  Av   v Au   u S   v u  Av   v Au   u           0 0 AS  A v u     Av Au     v  u    v u    S  AS  A v u     Av Au     v  u    v u    S                        0  0     S 1 AS    d         1   S AS          dt        dx d     a x a y dt 11 12 dt  dy d  a x  a y   dt 21 22 dt Math 5490 11/3/2014 Math 5490 11/3/2014 Richard McGehee, University of Minnesota 1

Math 5490 11/3/2014 Dynamical Systems Dynamical Systems Changing Coordinates Changing Coordinates Example Example     i  i  2 1 dx dx S   S    x  y     y i  i  eigenvalues: 2 and 1 2 eigenvalues:    1 1      1 1  dt dt 1 2 0 1       A     A   i  i   2 1  dy dy     2 0    1 0 eigenvectors:   and   .  1 0 eigenvectors:      i  x    x 0             1 1 1 1 dt  dt   0 1    i  0     x   z  i i    z  iz iw  x  iz  iw     x    2 S                           x 2 1 2  y w w z w  S        1 1      y  z  w               y        y         1 1 dx dx dz  d  x  y   y  iz 2   2 dt     dt   dt x x iz iw 2 dt dy dy dw        y d y z w     x x iw    dt dt dt dt Math 5490 11/3/2014 Math 5490 11/3/2014 Dynamical Systems Dynamical Systems Changing Coordinates Changing Coordinates Example dz Example  iz dx     y x iz iw dt dx  ax   y dt   x  a      x   x  a    d dw   dt y z w   A A              iw dy dt y  a y y  a            x dt dy   x  ay dt dt Note that one of these equations is redundant. z  y  ix 2   A  a  a  a  w  z trace( ) 2 w  y  ix 2      A a 2 2 det( ) dr dx    y 0 A   I   2       2  a  a 2   2 dz det( ) 2 dt dt dt  iz d  dy   1 x dt      a  a  a   2 4 2 4 2 4 2 4 2 dt       a i The eigenvalues are 2 2 complex Cartesian polar     a i a i and Math 5490 11/3/2014 Math 5490 11/3/2014 Dynamical Systems Dynamical Systems Changing Coordinates Complex Eigenvalues dx dz dx dy    ax y   i Then dt   dt dt dt z x iy Let . A If is a matrix with real elements and dy  ax   y  i x   iay    x ay  dt A v if is an eigenvalue of with corrresponding eigenvector ,  a  i  x  iy ( )( )  A v then is an eigenvalue of with corresponding eigenvector . dz        a i   t i a i   t at i t a i z z t  z e ( )  r e e ( )  r e e ( ) ( ) Solution: ( ) 0 0 dt 0 0 0         Av v Av v Av v re  z  i Let . dr  dz dr d  ar   i  i        i  r t  r e at e rie a i z a i re Then ( ) ( ) dt ( ) 0 dt dt dt Solution:   t     t d ( )  dr d   0  ri  a  i  r ( ) dt dt dt Math 5490 11/3/2014 Math 5490 11/3/2014 Richard McGehee, University of Minnesota 2