Open problems and conjectures on rational systems in three dimensions

A note on an open conjecture in rational dynamical system

“…It is shown in a paper by Lugo and Palladino [5] that there exist unbounded solutions of (2)in the case that 0 ≤ < 1 and 0 < < 1 3 .Ying Sue Huang and Peter M. Knopf showed in [3] for ′ ≥ 0, > 0 and if ≠ 1 there exist positive initial conditions such that the solutions are unbounded except for the case ′ = 0 and > 1, Question related to Boundedness of solutions of (2) in the case = is the folllowing conjecture which it is proposed as eight open conjecture in this paper [1] by G. LADAS, G. LUGO AND F. J. PALLADINO, In our present paper we disprove in general the only if part of the conjecture 8 in [1] using sub-energy function and some properties of Todd's difference equation [6] and [7] * not university de Batna.2 A -J 21, 2021…”

“…with nonnegative parameters , , , , , , and , one wishes to show that either the solutions remain bounded for all positive initial conditions, or there exist positive initial conditions so that the solutions are unbounded .Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures [4] treats the large class of difference equations described by Equation (1),Some open problems related to (1) in which the boundedness properties were not known was recently solved in [2], By the following assumption = = = = 0 with the variable change → , with ≥ 0, > 0, > 0, > 0 equation (1) reduces to the following form :…”

Global Dynamics and Bifurcations of Certain Second Order Rational Difference Equation with Quadratic Terms

“…), that is the sequences {x 2n } ∞ n=0 and {x 2n+1 } ∞ n=0 are decreasing, so they are convergent with x 2n+1 → l 1 and x 2n → l 2 when n → ∞. Now, from (20) we have that l 1 = l 1 al 2 2 + 1 , l 2 = l 2 al 2 1 + 1 , from which l 1 = l 2 = 0.…”

“…(3) have been considered in the series of papers [3,4,13,14,22,24]. Some special second order quadratic fractional difference equations have appeared in analysis of competitive and anti-competitive systems of linear fractional difference equations in the plane, see [5,[7][8][9]12,20,21]. Describing the global dynamics of Eq.…”