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A simple toy discrete model



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 gif on page gif 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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