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A trichotomy result for non-autonomous rational difference equations

Frank PalladinoMichael Radin — 2011

Open Mathematics

We study non-autonomous rational difference equations. Under the assumption of a periodic non-autonomous parameter, we show that a well known trichotomy result in the autonomous case is preserved in a certain sense which is made precise in the body of the text. In addition we discuss some questions regarding whether periodicity preserves or destroys boundedness.

Some boundedness results for systems of two rational difference equations

Gabriel LugoFrank Palladino — 2010

Open Mathematics

We study k th order systems of two rational difference equations x n = α + i = 1 k β i x n - 1 + i = 1 k γ i y n - 1 A + j = 1 k B j x n - j + j = 1 k C j y n - j , y n = p + i = 1 k δ i x n - i + i = 1 k ε i y n - i q + j = 1 k D j x n - j + j = 1 k E j y n - j n . In particular, we assume non-negative parameters and non-negative initial conditions, such that the denominators are nonzero. We develop several approaches which allow us to extend well known boundedness results on the k th order rational difference equation to the setting of systems in certain cases.

Unboundedness results for systems

Gabriel LugoFrank Palladino — 2009

Open Mathematics

We study k th order systems of two rational difference equations x n = α + i = 1 k β i x n - i + i = 1 k γ i y n - i A + j = 1 k B j x n - j + j = 1 k C j y n - j , n , In particular we assume non-negative parameters and non-negative initial conditions. We develop several approaches which allow us to prove that unbounded solutions exist for certain initial conditions in a range of the parameters.

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