American Institute of Mathematical Sciences

Advanced
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.
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Table 1.  Chaotic Solutions: The parameters, delay terms and corresponding Lyapunov exponent for about 5000 solutions.
Parameters Delay Terms Estimated Interval of Lyapunov Exponent
$p=83; q=2$ $l=23; k=39$ $(1.2047, 2.6210)$
$p=11; q=2$ $l=5; k=7$ $(1.5959, 2.8415)$
$p=64; q=57$ $l=13; k=29$ $(1.8484, 3.0188)$
$p=9; q=4$ $l=9; k=17$ $(0.782, 1.7173)$
$p=70; q=34$ $l=5; k=9$ $(1.8132, 2.8781)$
$p=61; q=20$ $l=9; k=17$ $(0.2173, 1.4842)$
Table Options

Download as excel

Parameters Delay Terms Estimated Interval of Lyapunov Exponent
$p=83; q=2$ $l=23; k=39$ $(1.2047, 2.6210)$
$p=11; q=2$ $l=5; k=7$ $(1.5959, 2.8415)$
$p=64; q=57$ $l=13; k=29$ $(1.8484, 3.0188)$
$p=9; q=4$ $l=9; k=17$ $(0.782, 1.7173)$
$p=70; q=34$ $l=5; k=9$ $(1.8132, 2.8781)$
$p=61; q=20$ $l=9; k=17$ $(0.2173, 1.4842)$
[1]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140
[2]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[3]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[4]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824
[5]

Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
[6]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094
[7]

D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
[8]

Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151
[9]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[10]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
[11]

Monica Lazzo, Paul G. Schmidt. Convergence versus periodicity in a single-loop positive-feedback system 1. Convergence to equilibrium. Conference Publications, 2011, 2011 (Special) : 931-940. doi: 10.3934/proc.2011.2011.931
[12]

Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69
[13]

Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939
[14]

Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781
[15]

Boling Guo, Zhaohui Huo. The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121
[16]

Rolci Cipolatti, Otared Kavian. On a nonlinear Schrödinger equation modelling ultra-short laser pulses with a large noncompact global attractor. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 121-132. doi: 10.3934/dcds.2007.17.121
[17]

Aslihan Demirkaya. The existence of a global attractor for a Kuramoto-Sivashinsky type equation in 2D. Conference Publications, 2009, 2009 (Special) : 198-207. doi: 10.3934/proc.2009.2009.198
[18]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155
[19]

Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure & Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301
[20]

Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (123)
  • HTML views (274)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences