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WOLFRAM|DEMONSTRATIONS PROJECT

Difference Equation versus Differential Equation

​
step size: γ
0.438
Let
x=(
x
1
,
x
2
)
be a generic point in the plane. The figure illustrates the relation between the difference equation
Δx
n
=
x
n+1
-
x
n
=γ·f(
x
n
)
and the differential equation
·
x
=f(x)
for the particular case
f(x)=(
x
2
,-
x
1
)
. For decreasing values of the step size parameter
γ
and for a chosen initial value
x
0
, you can see how the discrete process
x
n
(in white) tends to follow the trajectory of the differential equation that goes through
x
0
(in black). The background is colored using the norm of the expected motion, rescaled to be in the interval
(0,1)
.
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