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April  2018, 38(4): 1707-1718. doi: 10.3934/dcds.2018070

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).

Received  April 2016 Revised  November 2017 Published  January 2018

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.

Keywords: Difference equation, global attractor, invariant interval, convergence, periodicity.
Mathematics Subject Classification: 39A10, 39A11.
Citation: Esha Chatterjee, Sk. Sarif Hassan. On the asymptotic character of a generalized rational difference equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1707-1718. doi: 10.3934/dcds.2018070
References:
[1]

R. M. Abu-Saris and R. DeVault, Global Stability of $ \displaystyle{y_{n+1}=A + \frac{ y_{n} }{ y_{n-k} }}$, Applied Mathematics Letters, 16 (2003), 173-178.   Google Scholar

[2]

E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, 2008.  Google Scholar

[3]

E. Camouzis, E. Chatterjee and G. Ladas, On the dynamics of $ \displaystyle{x_{n+1}=\frac{\delta x_{n-2} + x_{n-3}}{ A+x_{n-3} }}$, Journal of Mathematical Analysis and Applications, 331 (2007), 230-239.   Google Scholar

[4]

E. Chatterjee, R. DeVault and G. Ladas, On the Global Character of $ \displaystyle{x_{n+1}=\frac{\beta x_{n} + \delta x_{n-k}}{ A+x_{n-k} }}$, International Journal of Applied Mathematical Sciences, 2 (2005), 39-46.   Google Scholar

[5]

C. W. Clark, A delayed recruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol., 3 (1976), 381-391.  doi: 10.1007/BF00275067.  Google Scholar

[6]

R. DeVault, G. Ladas and S. W. Schultz, On the recursive sequence $\displaystyle{x_{n+1}=\frac{A}{x_{n}}+\frac{1}{x_{n-2}}}$, Proc. Amer. Math. Soc., 126 (1998), 3257-3261.  doi: 10.1090/S0002-9939-98-04626-7.  Google Scholar

[7]

E. A. Grove, G. Ladas, M. Predescu and M. Radin, On the global charecter of $x_{n+1}=\frac{p x_{n-1}+x_{n-2}}{q+x_{n-2}}$, Math. Sci. Res. Hot-line, 5 (2001), 25-39.   Google Scholar

[8]

E. A. Grove, G. Ladas, M. Predescu and M. Radin, On the global character of the difference equation$ \displaystyle{x_{n+1}=\frac{\alpha + \gamma x_{n-(2k+1)} + \delta x_{n-2l}}{ A+x_{n-2l} }}$, Journal of Difference Equations and Applications, 9 (2003), 171-199.   Google Scholar

[9]

V. L. Kocic and G. Ladas, Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, Holland, 1993.  Google Scholar

[10]

V. L. Kocic, G. Ladas and I. W. Rodrigues, On Rational Recursive Sequences, J. Math. Anal. Appl., 173 (1993), 127-157.  doi: 10.1006/jmaa.1993.1057.  Google Scholar

[11]

M. R. S. Kulenovi$\acute{c}$ and G. Ladas, Dynamics of Second Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, 2002.  Google Scholar

[12]

M. R. S. Kulenović, G. Ladas and N. R. Prokup, On a rational difference equation, Computers and Mathematics with Applications, 41 (2001), 671-678.  doi: 10.1016/S0898-1221(00)00311-4.  Google Scholar

[13]

V. G. Papanicolaou, On the Asymptotic Stability of a Class of Linear Difference Equations, Mathematics Magazine, 69 (1996), 34-43.  doi: 10.2307/2691392.  Google Scholar

[14]

A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9.  Google Scholar

show all references

References:
[1]

R. M. Abu-Saris and R. DeVault, Global Stability of $ \displaystyle{y_{n+1}=A + \frac{ y_{n} }{ y_{n-k} }}$, Applied Mathematics Letters, 16 (2003), 173-178.   Google Scholar

[2]

E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, 2008.  Google Scholar

[3]

E. Camouzis, E. Chatterjee and G. Ladas, On the dynamics of $ \displaystyle{x_{n+1}=\frac{\delta x_{n-2} + x_{n-3}}{ A+x_{n-3} }}$, Journal of Mathematical Analysis and Applications, 331 (2007), 230-239.   Google Scholar

[4]

E. Chatterjee, R. DeVault and G. Ladas, On the Global Character of $ \displaystyle{x_{n+1}=\frac{\beta x_{n} + \delta x_{n-k}}{ A+x_{n-k} }}$, International Journal of Applied Mathematical Sciences, 2 (2005), 39-46.   Google Scholar

[5]

C. W. Clark, A delayed recruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol., 3 (1976), 381-391.  doi: 10.1007/BF00275067.  Google Scholar

[6]

R. DeVault, G. Ladas and S. W. Schultz, On the recursive sequence $\displaystyle{x_{n+1}=\frac{A}{x_{n}}+\frac{1}{x_{n-2}}}$, Proc. Amer. Math. Soc., 126 (1998), 3257-3261.  doi: 10.1090/S0002-9939-98-04626-7.  Google Scholar

[7]

E. A. Grove, G. Ladas, M. Predescu and M. Radin, On the global charecter of $x_{n+1}=\frac{p x_{n-1}+x_{n-2}}{q+x_{n-2}}$, Math. Sci. Res. Hot-line, 5 (2001), 25-39.   Google Scholar

[8]

E. A. Grove, G. Ladas, M. Predescu and M. Radin, On the global character of the difference equation$ \displaystyle{x_{n+1}=\frac{\alpha + \gamma x_{n-(2k+1)} + \delta x_{n-2l}}{ A+x_{n-2l} }}$, Journal of Difference Equations and Applications, 9 (2003), 171-199.   Google Scholar

[9]

V. L. Kocic and G. Ladas, Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, Holland, 1993.  Google Scholar

[10]

V. L. Kocic, G. Ladas and I. W. Rodrigues, On Rational Recursive Sequences, J. Math. Anal. Appl., 173 (1993), 127-157.  doi: 10.1006/jmaa.1993.1057.  Google Scholar

[11]

M. R. S. Kulenovi$\acute{c}$ and G. Ladas, Dynamics of Second Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, 2002.  Google Scholar

[12]

M. R. S. Kulenović, G. Ladas and N. R. Prokup, On a rational difference equation, Computers and Mathematics with Applications, 41 (2001), 671-678.  doi: 10.1016/S0898-1221(00)00311-4.  Google Scholar

[13]

V. G. Papanicolaou, On the Asymptotic Stability of a Class of Linear Difference Equations, Mathematics Magazine, 69 (1996), 34-43.  doi: 10.2307/2691392.  Google Scholar

[14]

A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D, 16 (1985), 285-317.  doi: 10.1016/0167-2789(85)90011-9.  Google Scholar

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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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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