Solvability and semi‐cycle analysis of a class of nonlinear systems of difference equations

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We have studied recently solvability and semi‐cycles of eight systems of difference equations of the following form: xn=a+pn−1qn−2pn−1+qn−2,yn=a+rn−1sn−2rn−1+sn−2,n∈double-struckN0, where a ∈ [0, + ∞), the sequences pn, qn, rn, sn are some of the sequences xn and yn, with positive initial values x−j,y−j, j = 1,2, in detail. This paper is devoted to the study of the other eight systems of the form. We show that these systems are also solvable in closed form and describe semi‐cycles of their solutions complementing our previous results on such systems of difference equations.

Section: Introductionsupporting

confidence: 85%

“…As in Stević and Tollu, we use a modification of the method in solving the eight systems not considered therein.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Namely, motivated by Berg and Stević and Stević, in Stević and Tollu, we studied eight systems of difference equations in the following form:

x_{n} = \frac{a + p_{n - 1} q_{n - 2}}{p_{n - 1} + q_{n - 2}}, y_{n} = \frac{a + r_{n - 1} s_{n - 2}}{r_{n - 1} + s_{n - 2}}, n \in N_{0},

where a ∈ [0, + ∞ ), and the sequences p n , q n , r n , s n are some of the sequences x n and y n .…”

Section: Introductionmentioning

confidence: 99%

“…If a semi‐cycle is eventually positive or negative, then m + and m − is equal ∞ . By using the notation, semi‐cycle patterns of each component can be written as

m_{1}^{+}, m_{2}^{-}, m_{3}^{+}, ⋯,

m_{1}^{-}, m_{2}^{+}, m_{3}^{-}, ⋯ .

If one of these sequences is eventually periodic, then we use, for example, the following notation

m_{1}^{+}, m_{2}^{-}, m_{3}^{+}, ⋯, m_{2 s - 1}^{+}, {false(m_{2 s}^{-}, ⋯, m_{2 t}^{-} false)}_{p}

, or any of the other seven possible notations (for details, see Stević and Tollu).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Solvability of eight classes of nonlinear systems of difference equations

Stević

Tollu

2019

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Flip bifurcations of two systems of difference equations

Cheng

Deng

2020

This paper investigates the bifurcations of the following difference equations x n+1 = ax n + b n e −x n , n+1 = c n + dx n e − n , x n+1 = a n + bx n e − n , n+1 = cx n + d n e −x n , where a, b, c, and d are positive constants and the initial conditions x 0 and y 0 are positive numbers. Psarros, Papaschinopoulos, and Schinas (Math. Methods Appl. Sci., 2016, 39: 5216-5222) presented the semistability of the fixed point (0, 0) when one eigenvalue is equal to 1 and the other eigenvalue has absolute value less than 1. In this paper, we consider another case: one eigenvalue is equal to −1. With the aid of the center manifold reduction theorem, we rigorously show that these two systems undergo flip bifurcations or generalized flip bifurcations. Moreover, the stability of the fixed point (0, 0) and the existence of period-two cycles are also given.

New classes of hyperbolic‐cotangent–type systems of difference equations solvable in closed form

Stević

2020

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In the case of homogeneous linear difference equations with constant coefficients, only a relatively small part of the equations is practically solvable. This is connected to impossibility to solve all polynomial equations of degree bigger than four. In the case of nonlinear difference equations and systems the situation is even worse. So it is of interest to find new classes of practically solvable difference equations and systems. Recently, there have been presented several classes of practically solvable hyperbolic‐cotangent–type systems of difference equations. Here, we present a new class of solvable systems of this type. It is a bit surprising that all the systems in the class are practically solvable, especially since several special cases are connected to some polynomials of degree eight.