Part II: Global Behavior

Next we consider the long term effect of the flow upon a given volume. For purposes of illustration we consider the interval from t=0 to t=1.2. Instead of using 1-step Euler we use Maple's built-in differential equations solver.

> with(DEtools):

> sys:=diff(x(t),t)=F(x(t),y(t))[1], diff(y(t),t)=F(x(t),y(t))[2] ;

> unknowns:= x(t), y(t):

> init1:= x(0)=a, y(0)=b:

> init2:= x(0)=aplus, y(0)=b:

> init3:= x(0)=a, y(0)=bplus:

> init4:= x(0)=aplus, y(0)=bplus:

> collection:= NULL:

> for j from 1 to 6 do

> LLcorner:=dsolve({sys, init1}, {unknowns}, numeric)(0.2):

> LRcorner:=dsolve({sys, init2}, {unknowns}, numeric)(0.2):

> ULcorner:=dsolve({sys, init3}, {unknowns}, numeric)(0.2):

> URcorner:=dsolve({sys, init4}, {unknowns}, numeric)(0.2):

> LL:= eval( [x(t),y(t)], LLcorner ):

> LR:= eval( [x(t),y(t)], LRcorner ):

> UL:= eval( [x(t),y(t)], ULcorner ):

> UR:= eval( [x(t),y(t)], URcorner ):

> collection:= collection, plot([LL, LR, UR, UL, LL ], style=LINE):

> init1:= x(0)=LL[1], y(0)=LL[2]:

> init2:= x(0)=LR[1], y(0)=LR[2]:

> init3:= x(0)=UL[1], y(0)=UL[2]:

> init4:= x(0)=UR[1], y(0)=UR[2]:

> od:

> Fplot2:= fieldplot( [F(x,y)], x = -6.5 .. 2.5 , y = 0 .. 10, arrows=SLIM ):

> collection:= Fplot2, Boxplot, collection :

> display( [collection], insequence=true);

> display( {collection}, title=`Time interval [0, 1.2], step=0.2`);

What is going on here? Does this calculation shed any more light (or perhaps cast a shadow) on the 1-step Euler approximation that we did earlier, especially when we took deltat = 0.9? To explore these concepts further, rework everything (both local and global portions of this worksheet) with a new vector field, say for example, F(x,y) = (y,-x). What do you observe this time around?

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