Direction Fields and Trajectories and Plots Versus t

The highlighted regions of this tutorial are sensitive to double
clicking with the left mouse button. Use the arrow keys to move up or
down in the tutorial.

The first thing we wish to do is plot direction fields and then
trajectories which represent solutions. For a differential equation
in two variables, the variables are presumed to be y and
x, with the differential being dy/dx. You may plot the
value of y and of x versus t, by either clicking on on the "Plot
Versus t" button on the menu that appears when you move the mouse to
the top left corner of the plot window, or by specifying that this
happen every time by adding "-versus_t 1" to the commands below.

If you wish to plot a second function (eg a solution you have computed
in another manner), in the "X and Y .." window, in that window get the
menu by moving the mouse to the upper left corner, and then click on
Config. Here add a function or functions to plot in the entry
"y=f(x)" for example putting "sin(x)", would plot points (x,sin(x))
where x varies in the allowed horizontal (x) range.

Interesting Equations:

• Duffing's equation:
  plotdf -ode {d(x,t) = y; d(y,t)= -(k*x + c*y + l*x^3)/m} -sliders {k=-2:2,m=-1:1} -parameters {k=-1,m=1.0,c=0,l=1} -bbox {-3 -3 3 3} -tstep .1
• Van der Pol's equation:
  plotdf -ode {d(x,t)=M*x - y - x^3;d(y,t)= x} -sliders {M=-4:4} -parameters {M=2} -bbox {-5 -5 5 5} -tstep .1 -versus_t 1
• Competing Species:
  plotdf -ode {d(x,t)= (1 - x - y)*x;d(y,t)= (A - B*x - C*y)*y} -sliders {A=-2:6,B=-2:4,C=-2:10} -parameters {A=4,C=7,B=2} -tstep .1 -bbox { 0 0 1 1} -versus_t 1
• Predator Prey:
  • x=prey
  • y=predator
  • plotdf -ode {d(x,t)=(A - B*y)*x;d(y,t)=(D*x - C)*y} -parameters {A=.4,C=.3,B=.01,D=.005} -bbox { 0 0 120 80} -versus_t 1
• Pendulum:

  
                               2
                          2   d x       dx
                         L  m --- + C L -- + g L m SIN(x)
                                2       dt
                              dt
  

  • L= length
  • x = angle
  • m=mass
  • c is positive and c*diff(x,t) is the damping force.
  • With Damping:
    plotdf -ode { d(x,t)=y;d(y,t)=-c/(m*L)*y-g/L*sin(x)} -parameters {L=10,g=9.80,c=1.0,m=1} -versus_t 1
  • With Zero Damping:
    plotdf -ode { d(x,t)=y;d(y,t)=-c/(m*L)*y-g/L*sin(x)} -parameters {L=10,g=9.80,c=0.0,m=1} -versus_t 1
• Vibrating String:
  
  • With No Damping:
    plotdf -ode {d(x,t) = y;d(y,t) = -(k*x + d*y)/m} -parameters {k=3,d=0.0,m=1} -versus_t 1 -bbox { -5 -5 5 5}
  • With Damping:
    plotdf -ode {d(x,t) = y;d(y,t) = -(k*x + d*y)/m} -parameters {k=3,d=1.0,m=1} -versus_t 1 -bbox { -5 -5 5 5}