Consider a dynamical systems over complex numbers $$ z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}},\qquad n=0,1,\ldots $$

where the parameters $\alpha, ~\beta$ are complex numbers, and the initial conditions $% z_{-1}$ and $z_{0}$ are arbitrary complex numbers.

For any initial values $z_{-1}$ and $z_{0}$, does the sequence $\{z_n\}$ bounded for all $n$? If so, what is the proof?

Does this sequence $\{z_n\}$ converge always? If so, how that be ensured?

  • $\begingroup$ What if, for instance, $z_{-1}=\beta$ and $z_0=-\alpha$? $\endgroup$ Sep 7 '14 at 13:20
  • $\begingroup$ @Michael Renary, correctly mentioned. It is an open problem in Difference equation that to find out good set of initial conditions for which the sequence is well defined. So we choose these initial conditions in the forbidden set. So we are interested in good set of initial conditions and for which we need to understand rest. $\endgroup$ Sep 7 '14 at 16:42
  • $\begingroup$ Duplicate of mathoverflow.net/questions/180331/… , which has also been posted math.stackexchange math.stackexchange.com/questions/923353/… $\endgroup$ Sep 8 '14 at 9:26

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