Dynamics of a rational difference equation

We investigate the global behavior of the solutions of rational difference equations. In particular, we study the global stability, the periodic nature, and especially the boundedness character of their solutions. In the first manuscript we study the equation[special characters omitted]and establish that every solution of this equation converges to a (not necessarily prime) period-two solution when γ = β + δ + A and β + A is positive, and that unbounded solutions of this equation exist whenever γ > β + δ + A. In this case, we establish a very large set of initial conditions that produce unbounded solutions. In subsequent manuscripts, we establish conditions under which every solution of the special case of the above equation with α = 0, A = 1, and δ = 1 converges to β + δ. We also present a number of open problems and conjectures related to the general third-order rational difference equation. In all equations, the parameters are nonnegative real numbers and the initial conditions are arbitrary nonnegative real numbers such that the denominators are always positive.

PurposeThe aim in this paper is to investigate the dynamics of difference equation yn+1=(pyn+yn−k)/(qyn+yn−k), n=0,1,2,… where k∈{1,2,3,…}, the initial conditions y−k, … ,y−1,y0 and the parameters p and q are non‐negative.Design/methodology/approachThe paper studies characteristics such as the character of semicycles, periodicity and the global stability of the above mentioned difference equation.FindingsIn particular, the results solve the open problem introduced by Kulenovic and Ladas in their monograph, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures.Originality/valueThe global behaviour of the solutions of equation yn+1=(pyn+yn−k)/(qyn+yn−k), n=0,1,2,… were investigated providing valuable conclusions on practical data.

We show that the second order rational difference equation has several qualitatively different types of positive solutions. Depending on the non-negative parameter values A,B,C,α,β, all solutions may converge to 0, or they may all be unbounded. For some parameter values both cases can occur, or coexist depending on the initial values. We find converging solutions of both monotonic and oscillatory types, as well as periodic solutions with period two. A semiconjugate relation facilitates derivations of these results by providing a link to a rational first order equation.

We consider the higher‐order nonlinear difference equation xn+1 = (α + xn)/(A + Bxn + xn−k), n = 0, 1, …, where parameters are positive real numbers and initial conditions x−k, …, x0 are nonnegative real numbers, k ≥ 2. We investigate the periodic character, the invariant intervals, and the global asymptotic stability of all positive solutions of the abovementioned equation. We show that the unique equilibrium of the equation is globally asymptotically stable under certain conditions.

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

We prove that every positive solution of the following difference equation: <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript n Baseline equals 1 plus StartFraction x Subscript n minus 2 Baseline Over x Subscript n minus 3 Baseline EndFraction comma n equals 0 comma 1 comma ellipsis comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mfrac> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x_n = 1 + \frac {x_{n-2}}{x_{n-3}}, \quad n = 0,1,\ldots ,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> converges to a period two solution.

We investigate the global asymptotic stability and Naimark-Sacker bifurcation of the dierence equation xn+1 = F bxnxn 1 +cx 2 1 +f ; n = 0; 1;:::

We study first-order systems of two rational difference equations, In particular, we assume non-negative parameters and non-negative initial conditions. We develop several approaches which allow us to resolve some conjectures regarding the boundedness of first-order systems of two rational difference equations.

Book Reviews

The aim of this paper is to investigate the global stability and periodic nature of the positive solutions of the difference equation $$x_{n + 1} = \frac{{A + Bx_{n - 2k - 1} }} {{C + D\prod\limits_{i = 1}^k {x_{n - 2i} } }}, n = 0,1,2, \ldots ,$$ where A, B are nonnegative real numbers, C,D &gt; 0 and l, k are nonnegative integers such that l ≤ k.

An investigation is made of the transition from periodic solutions through nearly-periodic solutions to chaotic solutions of the differential equation governing forced coplanar motion of a weakly damped pendulum. The pendulum is driven by horizontal, periodic forcing of the pivot with maximum acceleration Є g and dimensionless frequency ω As the forcing frequency ω is decreased gradually at a sufficiently large forcing amplitude Є, it has been shown previously that the pendulum progresses from symmetric oscillations of period T (= 2 π/ω) into a symmetry-breaking, period-doubling sequence of stable, periodic oscillations. There are two related forms of asymmetric, stable oscillations in the sequence, dependent on the initial conditions. When the frequency is decreased immediately beyond the sequence, the oscillations become unstable but remain in the neighbourhood in (θ,) phase space of one or other of the two forms of periodic oscillations, where θ(t) is the pendulum angle with the downward vertical. As the frequency is decreased further, the oscillations move intermittently between the neighbourhoods in (θ,) phase space of each of the two forms of periodic oscillations, in paired nearly-periodic oscillations. Further decrease of the forcing frequency leads to time intervals in which the motion is strongly unstable, with the pendulum passing intermittently over the pivot, interspersed with time intervals when the motion is nearly-periodic and only weakly unstable. The strongly-unstable intervals dominate in fully chaotic oscillations. Windows of independent, stable, periodic oscillations occur throughout the frequency range investigated. It is shown in an appendix how the Floquet method may be interpreted to describe the linear stability of the periodic and nearly-periodic solutions, and the windows of periodic oscillations in the investigated frequency range are listed in a second appendix.

We study the dynamical behaviour of the following rational difference equation x n + 1 = γ + β x n 2 + α x n − l 2 ρ + A x n + B x n − m , n , m , l ≥ 0 , where the parameters γ , β , α , ρ , A , and B, as well as the initial conditions x − k , … , x − 1 , x 0 are non-negative real numbers. Also, the denominator is assumed to be nonzero. We examine the local stability, existence of invariant intervals, boundedness of solutions, presence of semi-cycles, periodic solutions of prime period two, and global stability of the positive fixed points. To support our theoretical findings, we provide numerical examples with graphical illustrations.

For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures, including as special cases Choquet and Sugeno integrals. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced.

Let T be an edge weighted tree, let dT(u, v) be the sum of the weights of the edges on the path from u to v in T, and let d min and d max be two non-negative real numbers such that d min ≤ d max . Then a pairwise compatibility graph of T for d min and d max is a graph G = (V, E), where each vertex u' ∈ V corresponds to a leaf u of T and there is an edge (u', v') ∈ E if and only if d min ≤ dT(u, v) ≤ d max . A graph G is called a pairwise compatibility graph (PCG) if there exists an edge weighted tree T and two non-negative real numbers d min and d max such that G is a pairwise compatibility graph of T for d min and d max . Kearney et al. conjectured that every graph is a PCG [3]. In this paper, we refute the conjecture by showing that not all graphs are PCG s . Moreover, we recognize several classes of graphs as pairwise compatibility graphs. We identify two restricted classes of bipartite graphs as PCG. We also show that the well known tree power graphs and some of their extensions are PCGs.

A set S equipped with two algebraic operations: addition and multiplication is called a semiring if the following conditions are satisfied: both operations are associative; the addition is commutative; the multiplication is distributive with respect to the addition a.(b+ c) = (a.b) + (a.c) and (a+ b).c = (a.c) + (b.c) ∀a, b, c ∈ S . A semiring S is called an idempotent semiring if a+ a = a for all a ∈ S. Employing additively idempotent semirings semirings as a platform for a cryptographic scheme arose several years ago. In the present work we show how to apply different dual pairs of idempotent semirings for constructing new cryptographic protocol. We are interested in four idempotent semirings: Max-plus semiring: Consider Rmax = R ∪ {−∞}. Given a, b ∈ Rmax, define: a ⊕ b = max{a, b}; a⊙ b = a+ b. If we add the top element ⊤ = +∞ to this set, the resulting semiring is complete and denoted by Rmax. Min-plus semiring: Consider Rmin = R ∪ {+∞}. Given a, b ∈ R+∞, define: a ⊕ b = min{a, b}; a⊙ b = a+ b. If we add the top element ⊥ = −∞ to this set, the resulting semiring is complete and denoted by Rmin. Max-time semiring: Consider Rmax,× = R ∪ {+∞} (non-negative real numbers). Given a, b ∈ Rmax,×, define: a⊕ b = max{a, b}; a⊗ b = a.b. Min-time semiring: Consider Rmin,× = R∪{+∞} (non-negative real numbers). Given a, b ∈ Rmin,×, define: a⊕ b = min{a, b}; a⊗ b = a.b. We give a generalization of the Diffie-Hellman key exchange protocol, to the context of semigroup actions. Our protocol in its most general form consists of the following: two commutative semirings S1, S2 act on a set X i.e. ((S1 × S2)×X) → X. We propose two practical realizations of this scheme based on different dual pairs of idempotent semirings. For the semirings S1 and S2 we suggest commutative semirings generated by two given matrices M and N. These semirings are semirings of polynomials in M and N with coefficient of two dual pairs of idempotent semirings. Set X should be selected carefully with respect to the chosen semirings. In the first protocol: S1 = Mn(Rmax), S2 = Mn(Rmin), M ∈ Mn(Rmax), N ∈ Mn(Rmin),X ∈ Mn(R). In the second protocol: S1 = Mn(Rmax,×), S2 = Mn(Rmin,×), M ∈ Mn(Rmax,×), N ∈ Mn(Rmin,×), X ∈ Mn(R).