Period-two Solutions Research Articles

Existence and Local Stability of Prime Period-two Solutions of Certain Quadratic Rational Second Order Difference Equation

In this paper we proved the existence and local stability of prime period-two solutions for the equation 𝐱𝐧􀬾𝟏 􀵌 𝛂𝐱𝐧 𝟐 􀬾𝛃𝐱𝐧􀬾𝛄𝐱𝐧􀰷𝟏 𝐀𝐱𝐧 𝟐 􀬾𝐁𝐱𝐧􀬾𝐂𝐱𝐧􀰷𝟏 , for certain values of parameters ,,,A,B,C0, where ++>0 , A+B+C>0, and where the initial conditions x₋₁, x₀>0 are arbitrary real numbers such that at least one is strictly positive. For the obtained periodic solutions, it is possible to be locally asymptotically stable, saddle points or nonhyperbolic points. The existence of repeller points is not possible.

The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation

We present the bifurcation results for the difference equation x n + 1 = x n 2 / a x n 2 + x n − 1 2 + f where a and f are positive numbers and the initial conditions x − 1 and x 0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.

Local and Global Stability of Certain Mixed Monotone Fractional Second Order Difference Equation with Quadratic Terms

This paper investigates the local and global character of the unique positive equilibrium of a mixed monotone fractional second-order difference equation with quadratic terms. The corresponding associated map of the equation decreases in the first variable, and it can be either decreasing or increasing in the second variable depending on the corresponding parametric values. We use the theory of monotone maps to study global dynamics. For local stability, we use the center manifold theory in the case of the non-hyperbolic equilibrium point. We show that the observed equation exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which can be locally stable, non-hyperbolic when there also exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle. Numerical simulations are carried out to better illustrate the results.

Dynamics of a population in two patches with dispersal

A two-dimensional discrete system of a species in two patches proposed by Newman et al. is studied. It is shown that the unique interior steady state is globally asymptotically stable if the active population has a Beverton–Holt type growth rate. If the population is also subject to Allee effects, then the system has two interior steady states whenever the density-independent growth rate is large. In addition, the model has period-two solutions if the symmetric dispersal exceeds a critical threshold. For small dispersal, populations may either go extinct or eventually stabilize. However, populations are oscillating over time if dispersal is beyond the critical value and the initial populations are large.

Basins of attraction of period-two solutions of monotone difference equations

We investigate the global character of the difference equation of the form $$x_{n+1} = f(x_{n}, x_{n-1}),\quad n=0,1, \ldots $$ with several period-two solutions, where f is increasing in all its variables. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period-two solutions. As an application of our results we give the global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types.

Global dynamics of cubic second order difference equation in the first quadrant

We investigate the global behavior of a cubic second order difference equation $x_{n+1}=Ax_{n}^{3}+ Bx_{n}^{2}x_{n-1}+Cx_{n}x_{n-1}^{2}+Dx_{n-1}^{3}+Ex_{n}^{2} +Fx_{n}x_{n-1}+Gx_{n-1}^{2}+Hx_{n}+Ix_{n-1}+J$ , $n=0,1,\ldots$ , with nonnegative parameters and initial conditions. We establish the relations for the local stability of equilibriums and the existence of period-two solutions. We then use this result to give global behavior results for special ranges of the parameters and determine the basins of attraction of all equilibrium points. We give a class of examples of second order difference equations with quadratic terms for which a discrete version of the 16th Hilbert problem does not hold. We also give the class of second order difference equations with quadratic terms for which the Julia set can be found explicitly and represent a planar quadratic curve.

On the relative coexistence of fixed points and period-two solutions near border-collision bifurcations

At a border-collision bifurcation a fixed point of a piecewise-smooth map intersects a surface where the functional form of the map changes. Near a generic border-collision bifurcation there are two fixed points, each of which exists on one side of the bifurcation. A simple eigenvalue condition indicates whether the fixed points exist on different sides of the bifurcation (this case can be interpreted as the persistence of a single fixed point), or on the same side of the bifurcation (in which case the bifurcation is akin to a saddle–node bifurcation). A similar eigenvalue condition indicates whether or not there exists a period-two solution on one side of the bifurcation. Previously these conditions have been combined to obtain five distinct scenarios for the existence and relative coexistence of fixed points and period-two solutions near border-collision bifurcations. In this Letter it is shown that one of these scenarios, namely that two fixed points exist on one side of the bifurcation and a period-two solution exists on the other side of the bifurcation, cannot occur. The remaining four scenarios are feasible. Therefore there are exactly four distinct scenarios for fixed points and period-two solutions near border-collision bifurcations.

Global behaviour of solutions to a class of second-order rational difference equations when prime period-two solutions exist

For non-negative parameters α, β, γ, A, B and C such that A+B+C>0, consider the class of difference equations where either if A = 0, or R = [0,∞)2 if A>0. This class consists of 49 non-trivial second-order rational difference equations. We give a complete qualitative description of the global behaviour of solutions including basins of attraction of equilibria and prime period-two (PP2) solutions for all nonlinear difference equations (E) for which PP2 solutions exist. In particular, we show that every solution of (E) converges to an equilibrium or to a PP2 solution.

Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form x n+1 = x n−1 2/(ax n 2 + bx n x n−1 + cx n−1 2), n = 0,1, 2,…, where the parameters a, b, and c are positive numbers and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

Basins of attraction of an anti-competitive discrete rational system

C Dedicated to Professor Mustafa Kulenovic on the occasion of his 60th birthday Abstract. We investigate the global asymptotic behavior of solutions of the following anti-competitive system of difference equations xn+1 = 1yn A1 + xn ; yn +1 = 2xn + 2yn yn ; n = 0; 1;:::; where the parameters 1; 2; 2;A1 are positive numbers and the initial conditions x0 0;y0 > 0. We find the basins of attraction of all at- tractors of the system, which are the equilibrium point and period-two solutions.

Basins of attraction of equilibrium points of second order difference equations

We investigate the basins of attraction of equilibrium points and period-two solutions of the difference equation of the form x n + 1 = f ( x n , x n − 1 ) , n = 0 , 1 , … , where f is decreasing in the first and increasing in the second variable. We show that the boundaries of the basins of attraction of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points.

Boundedness and some convergence properties of the difference equation

For the difference equation with positive parameters given in the title, we show that the solutions are bounded for all positive initial conditions whenever . Furthermore, if and for a certain class of initial conditions, then the solutions converge to period-two solutions.

Global attractivity of a higher-order nonlinear difference equation

The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation x n + 1 = 1 - x n A + x n - k , n = 0 , 1 , … , where A ∈ ( - ∞ , - 1 ) , k is a positive integer and initial conditions x - k , … , x 0 ∈ ( - ∞ , 0 ] . It is shown that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient.

Complex dynamics in a linear impulsive system

The dynamical behavior of a linear impulsive system is discussed by means of both theoretical and numerical ways. This paper investigates the existence and stability of the equilibrium and period-one solution, the discontinuous jumps of eigenvalues, and the conditions for system possessing infinite period-two, period-three, and period-six solutions. By using discrete maps, the conditions of existence for Neimark–Sacker bifurcation are derived. In particular, chaotic behavior in the sense of Marotto’s definition of chaos is rigorously proven. Moreover, some detailed numerical results of the phase portraits, the periodic solutions, the bifurcation diagram, and the chaotic attractors, which are illustrated by some interesting examples, are in good agreement with the theoretical analysis.

Global asymptotic stability of a second order rational difference equation

The main goal of the paper is to confirm Conjecture 9.5.5 stated by Kulenović and Ladas in Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures (Chapman & Hall/CRC, Boca Raton, FL, 2002). The boundedness, invariant intervals, semicycles and global attractivity of all nonnegative solutions of the equation are studied, where the parameters and the initial conditions are such that . It is shown that if the equation has no prime period-two solutions, then the unique positive equilibrium of the equation is globally asymptotically stable.

Global asymptotical stability of a second order rational difference equation

In this paper, we investigate the boundedness, invariant interval, semicycle and global attractivity of all positive solutions of the equation x n + 1 = α + γ x n − 1 A + B x n + C x n − 1 , n = 0 , 1 , … , where the parameters α , γ , A , B , C ∈ ( 0 , ∞ ) and the initial conditions y − 1 , y 0 are nonnegative real numbers. We show that if the equation has no prime period-two solutions, then the positive equilibrium of the equation is globally asymptotically stable. Our results solve partially the conjecture proposed by Kulenović and Ladas in their monograph [M.R. Kulenović, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman Hall/CRC, Boca Raton, 2001].

Global Asymptotic Behavior of

We investigate the global stability character of the equilibrium points and the period-two solutions of , with positive parameters and nonnegative initial conditions. We show that every solution of the equation in the title converges to either the zero equilibrium, the positive equilibrium, or the period-two solution, for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist, we give a precise description of the basins of attraction of all points. Our results give an affirmative answer to Conjecture 9.5.6 and the complete answer to Open Problem 9.5.7 of Kulenović and Ladas, 2002.

The dynamics of a Prey-Predator model with impulsive state feedback control

The dynamics of a prey-predator model with impulsive state feedbackcontrol is studied by an autonomous system with impulses. Thedynamical behavior of this system is discussed by means of boththeoretical and numerical ways. The sufficient conditions ofexistence and stability of the semi-trivial periodic solution,positive period-one, and positive period-two solutions are obtainedby using Lambert W function and the analogue of thePoincaré criterion. The bifurcation analysis shows thatsolutions appear via a cascade of period-doubling in some intervalof parameters. The bifurcation diagrams, the Lyapunovexponents, and the phase portraits are given in two examples. Thediscussion of prey (pest) control strategy shows that the impulsivestate feedback control is effective.

Spurious behavior of a symplectic integrator

In this article, we study the existence and behavior of spurious solutions of symplectic Euler method for some Hamiltonian systems. It is shown that the symplectic integrator applied to Hamiltonian systems, in general, doesn't avoid spurious behavior, even spurious period-two solutions. The numerical results are presented.

DYNAMICS OF LINEAR MULTISTEP METHODS FOR DELAY DIFFERENTIAL EQUATIONS

In this paper we study the relationship between the asymptotic behavior of a numerical simulation by linear multistep method and that of the true solution itself for fixed step sizes. The numerical method is viewed as a dynamical system in which the step size acts as a parameter. Numerical stability of linear multistep method for nonlinear delay differential equation is investigated and we prove that A-stable linear multistep methods are NP-stable. It is shown that a consistent zero-stable linear multistep method does not admit spurious fixed points. The existence of spurious period-two solutions in the time-step is also studied. Finally we give a simple example to illustrate instability of the spurious period-two solutions.