One way to get some insight into discretized finite difference
equations is with simple toy cases. The simplest model of this
class is one dimensional in time and has a single point in space.
This defines a sequence of integers, 
, with the entire sequence being determined by the
first two elements. The discretized finite difference equation is
, where 
is a constant that determines the
frequency of the solution and 
truncates x towards 0, i.e., 
and 
. The exact solution to the difference equation
for 
without the
truncation operation is 
, where 
and 
.
The discretized equation approximates the continuous one but has some additional interesting properties. The discretization forces the sequence of values to repeat after some finite initial segment provided it does not diverge. There are only a finite number of pairs of integers below any fixed upper limit and thus after some time one of these pairs must be repeated. Since all subsequent values are determined by any two adjacent values in the sequence repetition of such a pair means the sequence is looping. Because the discretized equation preserves the exact time symmetry of the differential equation any repeat loop must include the initial state. We can reverse the sequence by reversing the order of two successive values. This will generate the values in the loop in reverse order. Thus the initial state must be in the loop.
For a given value of 
the equation partitions all pairs of integers into
disjoint sets that are in the solution for some set of initial
values. One interesting result is the combinatorial complexity
introduced by discretization. In trying 186 similar initial
conditions with a fixed value of 
the repeat length ranges from 2,880 to 148,092
samples. Table 
on page gives these lengths. The
program that generated this table is available as TO BE
DETERMINED.
Table: FDE repeat length for similar initial
conditions
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