Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE RULE OF TRAJECTORY STRUCTURE AND GLOBAL ASYMPTOTIC STABILITY FOR A FOURTH-ORDER RATIONAL DIFFERENCE EQUATION

  • Li, Xianyi (SCHOOL OF MATHEMATICS AND PHYSICS NANHUA UNIVERSITY) ;
  • Agarwal, Ravi P. (DEPARTMENT OF MATHEMATICAL SCIENCES FLORIDA INSTITUTE OF TECHNOLOGY)
  • Published : 2007.07.30
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Abstract

In this paper, the following fourth-order rational difference equation $$x_{n+1}=\frac{{x_n^b}+x_n-2x_{n-3}^b+a}{{x_n^bx_{n-2}+x_{n-3}^b+a}$$, n=0, 1, 2,..., where a, b ${\in}$ [0, ${\infty}$) and the initial values $X_{-3},\;X_{-2},\;X_{-1},\;X_0\;{\in}\;(0,\;{\infty})$, is considered and the rule of its trajectory structure is described clearly out. Mainly, the lengths of positive and negative semicycles of its nontrivial solutions are found to occur periodically with prime period 15. The rule is $1^+,\;1^-,\;1^+,\;4^-,\;3^+,\;1^-,\;2^+,\;2^-$ in a period, by which the positive equilibrium point of the equation is verified to be globally asymptotically stable.

Keywords

References

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