\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the asymptotic character of a generalized rational difference equation

  • 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). 
Abstract Full Text(HTML) Figure(2) / Table(1) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 39A10, 39A11.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Orbit plots with higher order periodicities.

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

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

    Parameters Delay Terms Estimated Interval of Lyapunov Exponent
    $p=83; q=2$ $l=23; k=39$ $(1.2047, 2.6210)$
    $p=11; q=2$ $l=5; k=7$ $(1.5959, 2.8415)$
    $p=64; q=57$ $l=13; k=29$ $(1.8484, 3.0188)$
    $p=9; q=4$ $l=9; k=17$ $(0.782, 1.7173)$
    $p=70; q=34$ $l=5; k=9$ $(1.8132, 2.8781)$
    $p=61; q=20$ $l=9; k=17$ $(0.2173, 1.4842)$
     | Show Table
    DownLoad: CSV
  •   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. 
      E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, 2008.
      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. 
      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. 
      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.
      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.
      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. 
      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} }}$, Journal of Difference Equations and Applications, 9 (2003) , 171-199. 
      V. L. Kocic and G. Ladas, Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, Holland, 1993.
      V. L. Kocic , G. Ladas  and  I. W. Rodrigues , On Rational Recursive Sequences, J. Math. Anal. Appl., 173 (1993) , 127-157.  doi: 10.1006/jmaa.1993.1057.
      M. R. S. Kulenovi$\acute{c}$ and G. Ladas, Dynamics of Second Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, 2002.
      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.
      V. G. Papanicolaou , On the Asymptotic Stability of a Class of Linear Difference Equations, Mathematics Magazine, 69 (1996) , 34-43.  doi: 10.2307/2691392.
      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.
  • 加载中

Figures(2)

Tables(1)

SHARE

Article Metrics

HTML views(480) PDF downloads(307) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return