On the asymptotic character of a generalized rational difference equation

Esha Chatterjee; Sk. Sarif Hassan

doi:10.3934/dcds.2018070

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2018, Volume 38, Issue 4: 1707-1718. Doi: 10.3934/dcds.2018070

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On the asymptotic character of a generalized rational difference equation

Esha Chatterjee^1, , and
Sk. Sarif Hassan^2,

1.
Department of Mathematics, Indian Institute of Science, Bangalore, Karnataka, 560012, India
2.
Department of Mathematics, Maligram, Paschim Medinipur, 2421140, India

Corresponding author. First author's work is supported by DST (India) Grant D.O. No SR/FTP/MS-013/2011. Also supported in part by UGC(India).
Corresponding author. First author's work is supported by DST (India) Grant D.O. No SR/FTP/MS-013/2011. Also supported in part by UGC(India).

Received: March 31, 2016

Revised: October 31, 2017

Published: June 2018

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Abstract

We investigate the global asymptotic stability of the solutions of $X_{n+1}=\frac{β X_{n-l} + γ X_{n-k}}{A + X_{n-k}} $ for $n=1,2, ...$, where $l$ and $k$ are positive integers such that $l≠ k$. The parameters are positive real numbers and the initial conditions are arbitrary nonnegative real numbers. We find necessary and sufficient conditions for the global asymptotic stability of the zero equilibrium. We also investigate the positive equilibrium and find the regions of parameters where the positive equilibrium is a global attractor of all positive solutions. Of particular interest for this generalized equation would be the existence of unbounded solutions and the existence of prime period two solutions depending on the combination of delay terms ($l$, $k$) being (odd, odd), (odd, even), (even, odd) or (even, even). In this manuscript we will investigate these aspects of the solutions for all such combinations of delay terms.

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Mathematics Subject Classification: 39A10, 39A11.

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Figure 1. Orbit plots with higher order periodicities.

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Figure 2. Chaotic solutions for different cases as adumbrated in the Table-1.

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Table 1. Chaotic Solutions: The parameters, delay terms and corresponding Lyapunov exponent for about 5000 solutions.

Parameters	Delay Terms	Estimated Interval of Lyapunov Exponent
$p=83; q=2$	$l=23; k=39$	$(1.2047, 2.6210)$
$p=11; q=2$	$l=5; k=7$	$(1.5959, 2.8415)$
$p=64; q=57$	$l=13; k=29$	$(1.8484, 3.0188)$
$p=9; q=4$	$l=9; k=17$	$(0.782, 1.7173)$
$p=70; q=34$	$l=5; k=9$	$(1.8132, 2.8781)$
$p=61; q=20$	$l=9; k=17$	$(0.2173, 1.4842)$

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