Yuanxun  Bill  Bao  Simon  Fraser  University  

July  9th  ,  2009  

7/9/2009     Yuanxun  Bill  Bao   Simon  Fraser  University  

}  System of Ordinary Differential Equations }  Direction Fields }  A 2-D example }  Motivation }  Arrows of the direction field }  3-D problem }  Animate curves }  Animate fields }  More examples

7/9/2009    Overview   2  

}  System of 1st order ODEs }  Autonomous system }  Applications ◦  Dynamical systems ◦  Physical and Biological

models

7/9/2009    System  of  ODEs   3  

}  A graphical representation of the solutions of the 1st order ODE system

}  Qualitatively visualize the solutions

}  Very helpful when an analytical solution is hard to achieve

7/9/2009    DirecDon  Fields   4  

7/9/2009    A  2-­‐D  example   5  

7/9/2009    A  2-­‐D  example   6  

}  3-Dimensional system of 1st order non-linear ODEs

}  In general, hard to solve

analytically

}  Use direction fields to get a qualitative visualization of the solution

7/9/2009    MoDvaDon   7  

7/9/2009    MoDvaDon   8  

}  Let’s try DEplot3d command in Maple

}  For a better visual, needs to be 3D }  Computationally efficient }  Maple PLOT3D data structure ◦  PLOT3D(objects, options) ◦  square:= hfarray([[0,0,0],[1,1,0],[1,1,1],[0,0,1]]): ◦  PLOT3D(POLYGONS(square), COLOR(RGB,1,0,0),

AXESSTYLE(BOX)); ◦  Picture

7/9/2009    Arrows  of  the  DirecDon  Fields   9  

7/9/2009    Arrows  of  the  DirecDon  Fields   10  

7/9/2009    Arrows  of  the  DirecDon  Fields   11  

}  Make a 3D grid over the domain

}  Calculate the direction at

each grid point

}  Compute each arrow centered at its corresponding grid point

7/9/2009    Back  to  3D  case   12  

}  Make a 3D grid over the domain

}  Calculate the direction at

each gird point

}  Compute each arrow centered at its corresponding grid point

7/9/2009    Back  to  3D  case   13  

}  To get a better visual, consider placing the arrows RANDOMLY over the domain

}  Still missing something? ◦  Speed of direction fields

}  How to resolve this? ◦  Scale the magnitude of the

arrows ◦  Apply a color scale on the

arrows

7/9/2009    Back  to  3D  case   14  

}  To get a better visual, consider placing the arrows RANDOMLY over the domain

}  Still missing something? ◦  Speed of direction fields

}  How to resolve this? ◦  Scale the magnitude of the

arrows ◦  Apply a color scale on the

arrows

7/9/2009    Back  to  3D  case   15  

}  Static visualization is not enough to fully exhibit dynamical behaviors

}  By using animation, the dynamics of the I.V.P

are recovered

}  Animate the solution curves

}  3D example

7/9/2009    Animate  Curves   16  

}  Model the direction fields as a “School of Fish”

}  What trajectory should each “fish” follow ?

}  animation

7/9/2009    Animate  Fields   17  

}  Frame-by-frame animation }  In the 1st frame, initialize the

direction fields }  Treat each point where the

arrow lies on as the starting point of I.V.P

}  Solve the I.V.P for each point

numerically }  Compute the arrows for each

frame

7/9/2009    Animate  Fields   18  

}  To have a smooth animation, at least 25 frames are needed

}  If we have a direction field of 100 arrows, run

the animation requires computing at least ◦  25 × 100 = 2,500 arrows OR ◦  2500 × 18 = 45,000 points (square-base arrow) ◦  2500 × 48 = 120,000 points (hexagon–base arrow)

7/9/2009    Animate  Fields   19  

7/9/2009    Rössler  System   20  

}  animation

}  animation

7/9/2009    Lorenz  System   21  

}  animation

7/9/2009    Chemical  Oscillators   22  

}  animation

7/9/2009    Future  Enhancements   23  

 

1.  M.B. Monagan, K.O. Geddes, K.M. Heal, G. Labahn, S.M. Vorkoetter, J. McCarron and P. DeMarco, Maple 10 Advanced Programming Guide. Maplesoft, Waterloo, Ontario, Canada, 2005.

2.  Richard H. Enns, George C. McGuire, Nonlinear Physics with Maple for Scientists and Engineers. Birkhäuser, Boston 1997

3.  C. Henry Edwards, David E. Penney, Differential Equations and Boundary Value Problems, Computing and Modeling, 2008

7/9/2009    References   24  

7/9/2009    Thanks  Note   25