On second-order rational difference equations, Part 2

Amleh, A. M.; Camouzis, E.; Ladas, G.

doi:10.1080/10236190701761482

Cited by 34 publications

(46 citation statements)

References 42 publications

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“…In a special case when equation (2) has three equilibrium solutions and all solutions are asymptotic to one of these equilibrium solutions only partial description of the basins of attraction of these equilibrium solutions was given in [1,2]. Here we improve these results by obtaining the complete description of the basins of attraction of all equilibrium solutions.…”

Section: Introductionmentioning

confidence: 84%

“…which was considered in [1,2] and some precise results on global dynamics were obtained. In a special case when equation (2) has three equilibrium solutions and all solutions are asymptotic to one of these equilibrium solutions only partial description of the basins of attraction of these equilibrium solutions was given in [1,2].…”

Section: Introductionmentioning

confidence: 98%

“…0. Equation (1), which is reduced form of one of the systems of linear fractional difference equations considered in [4], was considered in great details in [1,2] and all 49 special cases, when one or more parameters were equal to zero, were considered. In number of cases the authors gave the complete or partial description of the global dynamics.…”

Section: Introductionmentioning

confidence: 99%

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Basins of attraction of equilibrium and boundary points of second-order difference equations

Jašarević

Kulenović

2013

Journal of Difference Equations and Applications

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Dedicated to Gerasimos Ladas on occasion of his retirement.We investigate the global behaviour of the difference equation of the formwith non-negative parameters and initial conditions such thatWe give a precise description of the basins of attraction of different equilibrium points, and show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are in fact the global stable manifolds of neighbouring saddle or non-hyperbolic equilibrium points. Different types of bifurcations when one or more parameters b; d; e; f are 0 are explained.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Basins of attraction of equilibrium and boundary points of second-order difference equations

Jašarević

Kulenović

2013

Journal of Difference Equations and Applications

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“…and the result follows by employing Theorem 1.6 in[1]. Clearly, one can see that the component {y n } of the solution is bounded.…”

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confidence: 84%

Patterns of boundedness of a rational system in the plane

Amleh

Camouzis

Ladas

et al. 2010

Journal of Difference Equations and Applications

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We investigate the boundedness character of non-negative solutions of a rational system in the plane. The system contains 10 parameters with non-negative real values and consists of 343 special cases, each with positive parameters. In 342 out of the 343 special cases, we establish easily verifiable necessary and sufficient conditions, explicitly stated in terms of 10 parameters, which determine the boundedness character of solutions of the system. In the remaining special case, we conjecture the boundedness character of solutions. It is interesting to note that this special case can be transformed to the well-known May's Host-Parasitoid model.

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“…Note that even if the system trajectory lies out of region (22) initially, (2) and (15) ensure that the trajectory will enter region (22) and will not leave this region once it enters. In other words, for our present objective of arriving at the global asymptotic stability result, the initial condition may be assumed to be such as to satisfy (22).…”

Section: State-space Criterionmentioning

confidence: 98%