I would suggest searching for **round-off error**, **floating-point representation** and **floating-point precision** and **numerical simulation of dynamical systems**.

This is equivalent to doing numerical simulation computations using integers for all of the variables. Integers, short and long, signed and unsigned, are represented using a fixed number of bits in computational systems. Floating point numbers are also represented with a fixed number of bits, with a certain number allocation to the *mantissa*, and the remainder of bits allocated to the *exponent* in base $2$, representing what the *mantissa* is multiplied by

This is also equivalent to what must happen in fixed-digit or limited precision representation of floating point digits in any computerized simulation of any dynamical systems. The numerical simulation is only correct up to a certain number of digits, and the imprecision can build up rather quickly depending upon the variance of the elements of the matrix. Rounding occurs in every floating-point representation using a fixed number of bits; it just occurs at smaller magnitudes with larger number of bits used for the mantissa.

There are two points to consider:

fixed point errors in the numerical simulation (or round-off error), due to limited precision representation of floating point numbers or fixed-number-of-bits representation of integers

the dynamics of the system itself. The dynamics may be such that the sequence is diverging, and that the series will quicly go beyond the limit of what the numerical representation system is capable of dealing with: effectively hitting infinity (for the representation system) and not being able to go further. Even if the dynamics are such that the system is oscillatory or chaotic around an attractor, the round-off error in the numerical simulation may begin to dominate after a set number of iterations.

You will find a lot of literature about this from the 1950's and 1960's about matrix representations and matrix manipulations with computers, and a lot about this if you research floating-point representation standards.

iterative rounding, used in multiobjective optimization problems to obtain approximations. – Joseph O'Rourke Oct 29 '10 at 0:57