Parameters | Delay Terms | Estimated Interval of Lyapunov Exponent |
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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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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 |
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E. Camouzis and G. Ladas,
Dynamics of Third Order Rational Difference Equations; With Open Problems and Conjectures,
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E. Camouzis
, E. Chatterjee
and G. Ladas
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, 230-239.
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E. Chatterjee
, R. DeVault
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, 39-46.
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C. W. Clark
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, 381-391.
doi: 10.1007/BF00275067.![]() ![]() ![]() |
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, G. Ladas
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, 3257-3261.
doi: 10.1090/S0002-9939-98-04626-7.![]() ![]() ![]() |
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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.
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E. A. Grove
, G. Ladas
, M. Predescu
and M. Radin
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, 171-199.
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V. L. Kocic and G. Ladas,
Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications,
Kluwer Academic Publishers, Dordrecht, Holland, 1993.
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V. L. Kocic
, G. Ladas
and I. W. Rodrigues
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Appl., 173 (1993)
, 127-157.
doi: 10.1006/jmaa.1993.1057.![]() ![]() ![]() |
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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.
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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.![]() ![]() ![]() |
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V. G. Papanicolaou
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, 34-43.
doi: 10.2307/2691392.![]() ![]() ![]() |
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A. Wolf
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, 285-317.
doi: 10.1016/0167-2789(85)90011-9.![]() ![]() ![]() |
Orbit plots with higher order periodicities.
Chaotic solutions for different cases as adumbrated in the Table-1.