On rational difference equations
We study the global qualitative behavior of solutions of the general rational difference equation with nonnegative parameters and nonnegative initial conditions. We focus on finding general patterns of boundedness, general patterns of trichotomy behavior, and general convergence results for the general rational difference equation. ^ In the first manuscript of this dissertation, we find the general pattern for boundedness by iteration. We show that every solution is bounded by iteration whenever a diophantine equation involving the delays present in the numerator and the denominator is satisfied. We also develop some important comparison tests in the first manuscript of this dissertation which we use in the subsequent manuscript. ^ In the second manuscript of this dissertation, we give a complete characterization of the behavior of solutions to the general rational difference equation whenever A is greater than or equal to the sum of the βi terms in the numerator. This characterization again depends on the integer delays present in the numerator and denominator, particularly on the greatest common divisor of delays in the numerator and whether or not the greatest common divisor of delays in the numerator divides any of the delays in the denominator. ^ In the third manuscript of this dissertation, we expand on the work given in the second manuscript. We show that whenever A is equal to the sum of the βi terms in the numerator, which itself is greater then 0, and there exist periodic solutions of prime period P for the general rational difference equation, then we have a periodic trichotomy result. ^ In the fourth manuscript of this dissertation, we give a general global asymptotic stability result which strengthens the already powerful m-M theorem. ^
Frank J Palladino,
"On rational difference equations"
Dissertations and Master's Theses (Campus Access).