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In this work, we consider an adaptive control method, which is simpler and generalized to obtain some conditions on the parameters for hyperchaotic models determined by using a Lyapunov direct method. Further, an adaptive controller for synchronization is designed by using Lyapunov functions by which the deriving system and the response system can realize adaptive modified function projective synchronization up to scaling matrix. Numerical simulation of each system is discussed in detail with graphical results. The graphical results are presented in detail in order to validate the theoretical results. These results in this article generalize and improve the corresponding results of the recent works. [ABSTRACT FROM AUTHOR]

This work present the adaptive modified function projective synchronization of two systems with different order, which is a further extension of many existing synchronization schemes, such as function projection synchronization, modified projective synchronization and so on. Based on Lyapunov direct method of stability, an adaptive control is proposed to realize the modified function projective synchronization. Finally, numerical results are provided to illustrate the effectiveness of the obtained result. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the form of the solutions and the periodicity character of the following systems of nonlinear difference equations of order two Zn+1 = Zntn-1/±tn ± tn-1, tn+1 = tnZn-1/±Zn ± Zn-1, where the initial conditions Z-1, Z0, t-1 and t0 are nonzero real numbers. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear di¤erence equation... where the parameters α,β,γ, a; b; c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l,k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

This paper purports to study the optical solitons solutions regarding the nonlinear schr¨odinger equation with dual power law of nonlinearity along with self steeping and repercussion. The optical solutions mostly comprise of singular solitons with nonlinear chirping. The adopted solitary wave ansatz and the importance of the obtained chirped soliton pulses are also discussed. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study the global stability character and the periodic nature of the solutions of the difference equation ... where the initial conditions x-r, x-r+1, x-r+2, ..., x0 are arbitrary positive real numbers, r = max{l, k, s, t} is nonnegative integer and a, b, c, d, e are positive constants. Finally, some numerical examples are presented and graphed by Matlab. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear difference equation xn+1 = αxn+βxn-l+γxn-k + axnxn-k/bxn + cxn-l + dxn-k, n = 0, 1, ..., where the parameters α, β, γ, a, b, c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l, k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

In this paper, a Noval chaotic dynamical system is proposed and the basic properties of the system are investigated. Linear feedback control technique is used to suppress chaos. The controlled system is stable under some conditions on the parameters of the system determined by Lyapunov direct method. In addition, a function projective synchronization of two identical Noval system is presented. Lyapunov method of stability is used to prove the asymptotic stability of solutions for the error dynamical system. Numerical simulations results are included to show the effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

The main goal of this paper, is to study the existence of solutions for a class of nonlinear systems of difference equations of order four, in four-dimensional with the initial conditions are real numbers. Moreover, we study some behavior such as the periodicity and boundedness of solutions for such systems. Finally, some numerical examples are presented and graphed by Matlab. [ABSTRACT FROM AUTHOR]

The paper studies the solitons of the generalized resonanat dispersive nonlinear schödinger equation(GRD-NLSE) with power law nonlinearity. Subject to the enhanced and protracted chirped solitons, the model is studied with dispersion of self phase and self steepening coefficients.The results thus obtained shows that bright,dark and singular solitons depends on the intensity of the propagating pulse. [ABSTRACT FROM AUTHOR]

This work describes a new study to achieve a combination of modified function projective synchronization between three different chaotic systems through adaptive control. Using the Lyapunov function theory, the asymptotic stability of the error dynamics is obtained and discussed. Further, we set some appropriate initial conditions for the state variables and assigning specific values to the parameters and obtain the graphical results, which shows the efficiencies of the new method. Finally, we summarized our work with conclusion and references. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear difference equationxn+1 = βxn+βxn-l+γxn-k + axnxn-k/bxn + cxn-l + dxn-k, n = 0, 1, ..., where the parameters α, β, γ, a, b, c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l, k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions, the boundedness and the periodic character of the difference equation xn+1 = Axn + Bxn ¡ l + Cxn ¡ k + γxn ¡ k/Dxn ¡ s + Exn ¡ t, n=0, 1, ..., where the parameters A, B, C, D, E,γ ∊ (0,∞) and the initial conditions x¡ σ, x¡ σ+1,..., x¡1, x0 are positive real numbers where σ= max{l, k, s, t}. Numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = αxn + βxn-l + γxn-k + axn-l + bxn-k/cxn-l + dxn-k, n + 0, 1, ..., where the parameters α,β,γ,a,b,c,d ∊(0,∞) and the initial conditions x-s, x-s+1 ..., x-l and x0 are positive real numbers where s = max{l,k}. Examples to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the local and the global stability of the solutions, the periodic character and the boundedness of the difference equation xn+1 = axn + bxn-1 + α + cxn-2/β + dxn-2 where the parameters β, α, a, b, c and d are positive real numbers and the initial conditions x-2, x-1 and x0 are positive real numbers. Some numerical examples will be given to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the form of the solutions and the periodicity of the following third order systems of rational difference equations xn+1 = yn-2/1 - yn-2xn-1yn, yn+1 = xn-2/±1 ±xn-2yn-1xn, with initial conditions x-2, x-1, x0, y-2, y-1 and y0 are real numbers. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to find the form of solutions for the following systems of rational difference equations:... where the initial conditions have non-zero real numbers. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = A kΣi=0 xn-i/B+C kπi=0xn-i, n=0, 1, ..., where the parameters A, B and C are positive real numbers and the initial conditions x-k, x-k+1, ..., x-, x0 are nonnegative real numbers. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = axn + . . ., n=0,1, ..., where the parameters α, β, γ and a are positive real numbers and the initial conditions x-t, x-t+1 ..., x-1 and x0 are positive real numbers where t = max{l, k}. Numerical examples to the difference equation are given to explain our results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global stability of the positive solutions and the periodic character of the difference equation xn+1 = axn + αxnxn-l/β+γn-k,n = 0, 1, ..., where the parameters αβγ and a are positive real numbers and the initial conditions x-t,x-t+1 ..., x-1 and x0 are positive real numbers where t = {l,k}. Numerical examples to the difference equation are given to explain our results. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the periodic character and the global stability of the positive solutions of the difference equation xn+1 = axn + bxn-k + cxn-l - dxn-s/exn-s - αxn-t, n = 0, 1, ..., where the parameters a, b, c, d, e and α are positive real numbers and the initial conditions x-σ, x-σ+1, ..., x-1, x0 are positive real numbers where σ = max{s, t, k,}. Some numerical examples were given to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the periodic nature of the solution of the following max-type difference equation: [ABSTRACT FROM AUTHOR]

In this paper, we investigate the form of the solution of the following systems of difference equations xn+1 = xn-2/±1+xn-2zn-1yn, yn+1 = yn-2/±1+yn-2xn-1zn zn+1 = zn-2/±1+zn 2yn-1xn with initial conditions are nonzero real numbers. [ABSTRACT FROM AUTHOR]

This paper is devoted to obtain the form of the solution and the qualitative properties of the following systems of a rational difference equations of order two xn+1 = ynyn-1/xn (± 1 ± ynyn-1), yn+1 = xnxn-1/yn (± 1 ± xnxn-1), with positive initial conditions x-1, x0, y-1 and y0 are nonzero real numbers. If we let un = xnxn-1 and vn = ynyn-1, then these systems can be viewed as special cases of the system of the form un+1 = f(vn), vn+1 = g(un). This system has applications in modeling population growth with age structure or the dynamics of plant-herbivore interaction. Let wn = u2n, we have wn+1 = f(g(wn)) ≡ h(wn). At a nonzero steady state w+ of the last difference equation, we have |h'*)| = |f'(g(w*))g'(w*)| =1, indicating that the system is degenerate at this steady state. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of solutions and the periodicity character of the following systems of rational difference equations with order two xn+1 = xnyn-1/y-1 ± yn, yn+1 = ynxn-1/xn-1 ± xn, where the initial conditions x-1, y-1 and y0 0are nonzero real numbers. [ABSTRACT FROM AUTHOR]